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Compression, correlation and detection for energy efficient wireless sensor networks
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Compression, correlation and detection for energy efficient wireless sensor networks
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COMPRESSION, CORRELATION AND DETECTION FOR ENERGY EFFICIENT WIRELESS SENSOR NETWORKS by Caimu Tang A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (COMPUTER SCIENCE) August 2005 Copyright 2005 Caimu Tang Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3196901 Copyright 2005 by Tang, Caimu All rights reserved. INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3196901 Copyright 2006 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To my parents Dedication Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements First, I am deeply grateful for the advice I received from my advisor Dr. Cauligi S. Raghavendra. His patient guidance and his insightful discussions are invaluable to me. I also would like to express my sincerest thanks to my other guidance committee members for their generous help. Especially, Prof. Prasanna (co-advisor) and Prof. Ortega have also spent countless hours on helping on some of technical details of this dissertation and my doctorate study as a whole. I have always felt blessed for these great professors I can work with and receive advice from. Through their advice, I have acquired methodologies on how to conduct research and discourse research results. It is their inspiration and encouragement that have driven me to go this far on my academic pursuit. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents Dedication ii Acknowledgements iii List of Tables vi List of Figures vii Abstract x 1 Introduction 1 1.1 Contributions and Applications of Our Techniques .............................. 7 1.2 Thesis Organization................................................................................... 11 2 Related Work and Background 14 2.1 Distributed Compression and Delay E stim ation.................................... 15 2.2 Energy Conservation in Sensor N e tw o rk s............................................. 20 3 Wavelet Slepian-Wolf Compression 25 3.1 Introduction................................................................................................ 26 3.1.1 Problem and Our C ontributions................................................. 26 3.2 Our Models for Distributed Compression .............................................. 28 3.3 Code Design of B itp la n e .......................................................................... 34 3.3.1 Bitplane Generation and Encoding.............................................. 36 3.3.2 Bitplane D ecoding....................................................................... 38 3.4 Experiments and Simulation S tu d y .......................................................... 44 3.5 Conclusions................................................................................................ 49 4 Wavelet Source Broadcast for Sensor Array Data 50 4.1 Introduction................................................................................................ 50 4.1.1 Problem and Our C ontributions................................................. 55 4.2 Source Broadcast Model and Codec D e s ig n ......................................... 55 4.3 Our Proposed C o d e c ................................................................................. 62 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4 Experimental R esu lts............................................................................... 65 4.5 Conclusions............................................................................................... 68 5 Power Aware Compression of Sensor Data 69 5.1 Introduction............................................................................................... 69 5.1.1 Problem Formulation and Our Contributions .......................... 71 5.2 Energy Efficiency of C o d in g .................................................................. 72 5.2.1 Energy Savings on C od in g......................................................... 75 5.2.2 SNR Loss Upper Bound of Coding .......................................... 82 5.3 Proposed Power Aware Coding S c h e m e ................................................ 85 5.3.1 Rate Selection Based on Energy Dissipation............................. 88 5.4 Experimental Evaluations......................................................................... 93 5.5 Conclusions................................................................................................ 98 6 Correlation Analysis in Sensor Networks 99 6.1 Introduction................................................................................................ 99 6.1.1 Problem Formulation and Our Contributions .......................... 105 6.2 Correlation Analysis and Tracking Scheme .......................................... 106 6.2.1 Initial Coefficient Estimation....................................................... 107 6.2.2 Selection of LP Order ................................................................ 109 6.2.3 Tracking U p d a te .......................................................................... 112 6.3 Experiments and A pplications................................................................ 116 6.4 Conclusions................................................................................................ 125 7 Cueing and Detection in Wireless Sensor Networks 126 7.1 Introduction................................................................................................ 127 7.1.1 The Cueing and Detection Problems and Our Contributions . 129 7.2 Proposed Multi-hop Cueing Protocol....................................................... 130 7.2.1 Radio Models and Wake-up Channel Emulation....................... 131 7.2.2 State-Machine M odel.................................................................... 133 7.2.3 Beacon Communication and Tracker H a n d -o ff........................ 135 7.3 Iterative Detection through Sensor Collaboration ................................. 138 7.3.1 Models and Signal D e-noising.................................................... 139 7.3.2 Alarm Threshold Detection.......................................................... 140 7.3.3 False Alarm Detections................................................................ 141 7.4 Simulation and Experimental Results....................................................... 148 7.5 Conclusions................................................................................................ 158 8 Conclusions, Future Directions and Open Problems 159 8.1 Future Research Directions and Open Problem s........................................ 160 Bibliography 165 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables 2.1 Slepian-Wolf code d e s ig n s ..................................................................... 18 3.1 Type IEG-LDPC base code param eters............................................... 46 3.2 Encoding and decoding tim in g .............................................................. 49 5.1 IPOP parameters and results of the exam ple........................................ 90 5.2 Simulation parameter settings.................................................................. 95 5.3 Energy dissipation com parison.............................................................. 98 6.1 Experiment s e ttin g .................................................................................. 122 6.2 Performance and cost comparisons (per frame) ...................................... 122 7.1 Characteristics of tripwire and tracker radios......................................... 131 7.2 Characteristics of common commercial rad io s...................................... 131 7.3 State machine of tripwires ..................................................................... 133 7.4 State-beacon tab le..................................................................................... 134 7.5 Linear block codes for beacons with 4 content b its............................... 137 7.6 Power profiles of tracker and tripwire .................................................. 150 7.7 Field simulation parameters..................................................................... 152 7.8 Key scheme assum ptions........................................................................ 152 7.9 Cueing simulation parameters ............................................................... 157 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures 1.1 Automatic target recognition using sensor netw orks............................ 3 1.2 Network of sensor a rra y s ........................................... 5 1.3 Two targets in a two-tier wireless sensor n e tw o rk ............................... 7 2.1 Sensor array with seven n o d e s .............................................................. 15 2.2 Rate region of doubly BSS for the Wyner-Ziv coding problem .... 19 2.3 Models for generalized likelihood ratio t e s t ........................................ 23 3.1 Encoding and decoding wavelet coefficient trees.................................. 29 3.2 Bit proabilities in bitplanes with varying capsule s i z e ......................... 31 3.3 Capsules to form wavelet coefficient tre e ............................................... 36 3.4 Iterative identification of significant capsules ..................................... 37 3.5 SNR gain vs. capsule size........................................................................ 47 3.6 Bitplanes coding performance using EG-LDPC c o d es......................... 48 3.7 Codec rate-distortion c u r v e s .................................................................. 49 4.1 Source broadcast in wireless sensor n etw o rk s...................................... 52 4.2 Rate-Distortion performance comparison............................ 66 4.3 Rate comparison of SW-WISP with different block len g th s................ 67 5.1 System m o d el............................................................................................ 74 5.2 Relationships of three factors.................................................................. 74 5.3 Search of significant coefficients............................................................ 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.4 Simulation topology................................................................................. 94 5.5 Energy dissipation comparisons ........................................................... 96 5.6 SNR com parison.................................................................................... 96 5.7 Power consumption com parison........................................................... 98 6.1 Frame correlation.................................................................................... 101 6.2 Node calibration over t i m e ..................................................................... 102 6.3 Selection of S-Backward and S-Forward prediction la g s...................... 110 6.4 Fundamental index and auxiliary Indices............................................... I l l 6.5 Optimal coefficient v ecto rs..................................................................... 112 6.6 Illustration of interleaving of estimation and tra c k in g ......................... 119 6.7 Bitrate comparisons of codecs with/out correlation analysis................ 120 6.8 Tracking performance.............................................................................. 123 6.9 Kalman filter input controls..................................................................... 124 6.10 Fidelity effects on analysis s te p s ............................................................ 124 7.1 EAP in a two-tier sensor network............................................................ 128 7.2 Tracker in different wake-up radius......................................................... 132 7.3 State diagram of trip w ire......................................................................... 135 7.4 Beacon propagation boundary ............................................................... 136 7.5 Beacon error rate for Hamming (8,4,4) c o d e ..........................................138 7.6 Quantile plots of subbands with a — 1 .5 ............................................... 142 7.7 Subband used in detection p ro c e ss......................................................... 143 7.8 Decision statistics from detail su b b an d s................................................ 147 7.9 Vehicle signal in time domain................................................................... 148 7.10 Energy comparison of detection and tracking ...................................... 149 7.11 Power consumption of duty-cycling d etectio n...................................... 149 7.12 A two-target scenario............................................................................... 151 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.13 Field power and energy co m p ariso n ...................................................... 151 7.14 The one-target c a s e .................................................................................. 153 7.15 The two-target c a s e .................................................................................. 154 7.16 Average energy dissipation with varying number of trackers................ 154 7.17 Energy in sampled variance fo rm ............................................................ 155 7.18 Detection predicates.................................................................................. 156 7.19 Alarm notification latency ...................................................................... 157 8.1 Feasible region of in-network processing................................................ 161 8.2 Do encoding communications h e l p ? ...................................................... 163 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract Wireless sensor networks are enabling embedded sense-and-respond applications that were previously unimaginable. A high degree of spatial and temporal correlation ex ists in sensor readings. Data communication and storage costs can be reduced by effectively exploiting these correlations which can achieve high compression. Another source of energy savings is duty-cycling, which aims to extend the system lifetime by reducing the fraction of time a sensor node is on. In this research, we devised algorithms and protocols to improve energy efficiency of wireless sensor networks. The focus was on three related areas: (i) compression of sensor readings, (ii) correlation of sensor readings, (iii) detection and cueing of events. Our techniques made it possible to keep the average power consumption for a typical target tracking application in the 10-mW regime. The results can be summarized as follows. Our compression scheme for exploiting the spatio-temporal correlations uses set-partitioning on wavelet transformed data to extract correlated bitplanes. The spatial correlation is further exploited by the low- density parity-check based Slepian-Wolf codes at the bitplane levels. This scheme has also been extended to support the source broadcast problem where one sensor needs to send its readings to multiple receivers. A power aware joint coding scheme for corre lated sensor readings has been developed that takes into account transmission power, channel coding rate and packet retransmission to minimize energy per transmitted bit. It is crucial for distributed compression to be able to correlate sensor readings. We x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. have developed an energy efficient scheme to quantify and track spatial correlation of sensor readings. This scheme uses linear prediction to establish initial correlation and tracks the correlation using a Kalman-filter based approach. Finally, a tripwire cueing scheme has been developed, which uses detection predicates to wake up signal- processing nodes only when necessary. This multi-hop wake-up protocol enables a two-tier heterogeneous sensor network for greater energy savings, and coupled with a distributed two-stage detection algorithm, it can solve both the false alarm problem and the exposed alarm problem. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction Inexpensive sensors capable of significant computations and wireless communication are increasingly becoming available [65, 75, 16, 2], Sensor networks are becoming important in a number of areas, including target detection, biomedical engineering, environmental awareness and security surveillance. They are generally employed in a field for collection of signal data to collaboratively make decisions. Lightweight wireless enabled sensors are becoming more attractive due to their low cost, ease of deployment and high efficiency. These types of sensors have limited CPU processing power and memory resources. Sensors in this context are mostly powered by batteries; therefore, it is also impor tant to efficiently use this limited energy supply of a node and a network. As sensors have been built with more intelligence and smaller size, the power issue becames a key obstacle to exploit the full potential of wireless sensor networks. Energy conservation by sensor nodes is therefore critical for many applications of sensor networks. One typical application of wireless sensor networks is the automatic target tracking and recognition (ATR) which uses a number of sensor arrays to classify or track targets in a sensor field using acoustic, seismic or imagery sensors. There are a few processing steps in ATR including: 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (1) issuing an alarm of a potential target (2) confirming the alarm, and (3) classifying the target and/or tracking the target At all these steps, carefully designed protocols and algorithms can help applications to conserve energy so that a network can continuously work in a field with a longer lifetime. Figure 1.1(a) shows that three sensor arrays are tracking one target and Fig ure 1.1(b) shows these processing tasks inside a particular array. There are three types of beamforming algorithms dependent on how the beamforming is performed and what data it needs. They are depicted as follows: 1. FFT transforms are computed in each sensor and FFT coefficients are aggregated to the cluster head node. Beamforming is performed in the frequency domain. 2. Sensor readings are aggregated to the cluster head and FFT transform on these data is performed at the cluster head all together. Beamforming is also performed in the frequency domain. 3. Sensor readings are aggregated to the cluster head. Beamforming is performed directly in the time domain. In all these cases, a large volume data needs to be exchanged between a sensor and its cluster head. As it is a well-known fact that communication energy per bit is signif icantly higher than that of a single CPU arithmetic operation (we shall elaborate this later), it is therefore important to reduce this communication cost. Since not all components of a sensor node are used by an application at all times, one way to save energy for a sensor node is to turn off these components which are not needed, and switch them back on when a task execution needs them. Even when the 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LOB3 LOB1 Array! FFT |A/r VI FFI’ (a) Target tracking (b) Processing inside an array Figure 1.1: Automatic target recognition using sensor networks cost of switching components is taken into account, it still can save energy as long as a component can be put in low power mode for a sufficiently long period of time. This idea has been well exploited in hardware design of sensor nodes and has been proven successful, for example, a deep-sleep mode is introduced in LUTONIUM [55], and a low power tripwire module is introduced to the stack of PASTA architecture [38]. The same idea can be extended to a sensor network. For a sensor network, a portion of a network can be switched off when events are far off and switched on when an event is imminent (via event cueing) or is actually happening. To enable this idea in a sensor network for energy conservation, there is a critical problem which has to be addressed, namely, a network has to have a mechanism for identifying events and disseminating the detection predicate, and most importantly, this mechanism should not incur high energy cost on itself. Since radio communications are deemed to consume relatively high power [77,93] than task processing, it is equally important to reduce communication energy dissipa tion in a hop-by-hop fashion. Unlike that in cellular communications, sensor nodes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. are normally deployed on the ground with an antenna close to the ground, and they are also equipped with low power radios. Due to these factors, ground scatter and reflector can seriously affect the radio signal, hence the per-bit communication cost in wireless sensor networks is normally very high where a high rate error correction codes may be needed or an ARQ protocol is needed for reception with some erroneous bits. In sensor network applications, there are in general two types of communications required: local communications and global communications. In-network processing is one of principle useful for communication energy reduction; however, local communi cations are also required by many algorithms for achieving robustness and exploiting field diversity for more accurate results. Compression therefore is useful to reduce communication cost among local sensors. Sensor readings collected by sensor nodes in a sensor network are correlated. There are three types of correlations. First, since nodes are deployed in close proximity, these readings are spatially correlated. Second, since sensor readings are sampled in a short sampling interval, these readings are also highly temporally correlated. Third, these spatial and temporal correlations also extend to the multi-modality sensor readings, i.e., readings from different sensor modalities are correlated. Better compression gain is expected when these correlations are fully exploited. In many sensor network applications, sensor nodes collect acoustic, seismic or imagery data, and sensors collaboratively process signal data to detect/classify/track targets. In one scenario, a network consists of a number of sensor arrays, with each array having several homogeneous sensors. Sensors in an array are geometrically close to each other in the range of 10 to 200 meters while inter-array distances could be much farther. As an example, Figure 1.2 shows a field test network, which has 4 sensor arrays and each has 3 sensors, based on ground truth data. One sensor in each array is elected 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 1.2: Network of sensor arrays as the head which aggregates the data from the members, e.g, nodes with a line-of- bearing (LOB) line shown in Figure 1.2, and the head sensor node also observes the same event. The head performs LOB beamforming on the aggregated data. There are a number of advantages to have signal processing performed inside a sensor array. Some of these advantages are low communication cost and low latency. Further information fusion at the inter-array level can proceed only when the local processing results, which can be represented by fewer number of bits, are received from neighboring sensor arrays. Referring to Figure 1.2, these four sensor arrays track a target over one period of its movement, where, the lower right sensor array started its processing first on the target. The other two sensor arrays highlighted in white color started their processing on the target upon results from the first sensor array. Finally, the upper right sensor array starts the tracking once the target moves close to it. In this scenario in Figure 1.2, tracking is done by successive arrays based on the earlier tracking results. This example shows one common application characteristics of sensor network on target tracking, that is, the in-network processing method can save resources, e.g., energy savings due to less communications or less memory activities. This leads to energy conservation. 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. There is much work done on the minimization of the energy cost for the above scenario from both the network and the hardware standpoints [112,107, 38,16]. These approaches may not provide sufficient energy savings on above scenarios especially when the number of events is small and events are happening infrequently. These previous works are focused on the case where a sensor node is kept awake all the time or with duty-cycling. Even though they can be put in duty-cycle mode when no event happens in a long period time, the energy dissipated may still be substantial due to various hardware constraints [39] and state switch cost. And the delay is increased on event processing due to the delay introduced by state switch. For example, a PXA255 CPU consumes around 10 mW in sleep mode in order to keep its oscillators running and needs 10 ms to switch from sleep mode to active mode, and the long-haul data radio consumes more than 100 mW on idle. A two-tier sensor network can help conserve energy for wireless sensor networks, and this approach uses a two-tier sensor architecture which does not incur extra delay. In this type of networks, there are two types of sensors deployed, namely, trackers and tripwires. Trackers are signal processing sensor nodes which may equip with spe cific processor (e.g. digital signal processor) and need a high power level to operate. Tripwire consumes much less power than a tracker even at a low duty-cycle mode, for instance, picoRadio acting as a tripwire can operate at a power less than 1 mW [65]. Tripwires are deployed along with trackers, and form a tripwire network to monitor a sensor field so that trackers can be kept in sleep mode to conserve energy when no events are happening nearby. Tripwires wake up periodically to check field status and they wake up trackers for various signal processing tasks only when necessary. Since there is at least one order of magnitude difference on power consumption between a tripwire and a tracker, by using tripwires, energy can be saved since trackers con sume energy only for these periods when events are happening. Figure 1.3 shows a 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 1.3: Two targets in a two-tier wireless sensor network field example with two targets whose trajectories are marked by white curves. In Fig ure 1.3, the star shaped objects and the ‘L’ shaped objects are tripwires and trackers, respectively, and different colors on tripwires and trackers are used to indicate their different power states (i.e. lighter color for a higher power). This network configura tion is feasible and it has been demonstrated in a field test. For example, PASTA nodes, which cannot communicate with each not directly, are connected by a UDP tunnel via a MICA2 Mote network for the notification of detection predicates from the sensors of Motes. 1.1 Contributions and Applications of Our Techniques Surrounding the energy efficiency issue in wireless sensor networks, this thesis presents a portfolio of algorithms and protocols on compression of spatio-temporally correlated sensor data, correlation of sensor data and event cueing and detection. To explain how our techniques can be applied for energy efficiency, taking the ATR application, we shall illustrate how our techniques can be used to reduce the energy dissipation for an ATR application. Next, we first summarize our contributions in this thesis. 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The first two schemes are on distributed source coding and source broadcast of correlated sensor array data. First, a compression scheme was developed to exploit the spatio-temporal correlation. It uses set-partitioning on wavelet transformed data to ex tract correlated bitplanes and the spatial correlation is further exploited at the level of significant coefficients. Low-density parity-check based Slepian-Wolf codes are used to compress bitplanes extracted from the wavelet coefficients. Since wavelet trans form and the set-partitioning process enable that more significant bits in significant coefficients are coded earlier than less significant bits, our scheme achieves excellent rate-distortion performance. For a given rate at SNR 20 dB, the proposed codec uses a rate about a third of that by the standard wavelet set-partitioning codec based on the set-partitioning in hierarchical trees (SPIHT) [73]. Second, to facilitate the in-network processing, we proposed a scheme to compress sensor array data in which one sensor needs to send its readings to multiple neighboring sensors. This broadcast problem arises in many practical sensor network applications especially for the in-network processing purpose. For this broadcast source coding problem, we extend the above scheme to incrementally broadcast sensor’s readings and different receivers can receive the data at their respective correlated levels. When all receivers have decoded the data successfully, the sender stops its broadcast. Moreover, a priori correlation statistics are not required before encoding, and the reception rate of each receiver can achieve close to conditional entropy with regard to the sender. Communication energy dissipation can further be reduced when the transmission power, bits fidelity importance and error correction coding rates are jointly considered, we present a joint design methodology to make the compressed data generated by our codec more resilient to channel errors while the overall transmission cost in terms of per-bit energy dissipation is kept at the minimum level. We take into account the chan nel coding rate, retransmission and the transmission power to minimize the overall 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. joules-per-bit metric. Lower bound of energy savings and upper bound of signal dis tortion have been derived for this scheme. Using this scheme, we are able to reduce the communication cost by more than 65% compared to unprocessed data and 35% compared to compressed and non-jointly coded data. Distributed source coding techniques have been shown effective only when the inter-sensor correlation of readings is high. We also present a scheme to correlate sensor readings and quantify the spatial correlation of sensor readings. Our scheme uses linear prediction to establish initial correlation and tracks the correlation using a Kalman filter once the initial correlation is established. This tracking approach is an extension to an existing scheme based on the optimal filtering technique. This scheme requires much less bits to be sent for inter-sensor frame alignment. This technique can lead to better compression results due to the fact that better correlated sensors’ readings can be compressed better. Our technique boosts the distributed source coding codec performance by about 1.5 to 3 dB on reconstructed signal SNR. Tripwire cueing in sensor networks, which uses detection predicates to cue signal processing tracker nodes, can allow that a signal processing tracker be kept in sleep mode and woken up only when necessary. A multi-hop wake-up protocol is developed which enables a two-tier network configuration for a greater energy savings. Coupled with our distributed two-stage detection scheme, this protocol can solve both the false alarm problem and the exposed alarm problem. For the first time, we proposed to use wavelet to perform detection on resource constrained devices. Its detection rate is around 90% which is comparable to that of the FFT-based detection algorithm, and yet processing is significantly reduced which makes the real-time detection processing possible on a microcontroller. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To apply our developed techniques, recall the ATR application, tripwires can be deployed to monitor the field and signal processing trackers are in sleep-mode initially. Referring to Figure 1.3, the chain of events could happen in practice is as follows: (11) - Initiation: Tripwires monitor the sensor field and an event starts (i.e. target enters the field). (12) - Alarm: Tripwires alarm the close-by sensor arrays upon signal energy thresh old detection. (13) - Detection: Signal processing trackers first verify the event when they are wo ken up, if it is a false alarm, they go back to sleep and alarm the field again by activating the tripwires. (14) - Aggregation: When an alarm is confirmed, the head tracker aggregates sen sors’ readings. (15) - Beamforming: The head tracker computes one LOB using a beamforming algorithm. (16) - Fusion: One elected tracker node fuses at least two LOBs from neighboring head trackers and reports the target location. (17) - Leave: The event ends (i.e., the target moves off the monitored field). For these seven steps, we have focused on (II) - (14) and (17) where our developed algorithms and protocols can be directly applied. Our techniques can also be applied to reduce energy consumption of (15) when distributed beamforming is used and multiple LOBs are to be computed in a single sensor array. For (II) and (17), our event cueing protocol presented in Section 7.2 provides an energy efficient mechanism to monitor 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the field for (II) and alarm the field for (17). Our detection scheme presented in Sec tion 7.3 is applied to (12) and (13). To aggregate sensor readings on trackers, sensor readings are correlated using the Kalman filter based approach presented in Chapter 6, and our distributed source coding algorithms presented in Chapter 3 is then used to compress the data. To further reduce the communication cost, our power-coding technique presented in Chapter 5 is used to select proper radio transmission power and error correcting rate to combat channel errors. To enable in-network processing in (15), our source broadcasting algorithm presented in Chapter 4 is applied for efficient data gathering for multiple LOB computations in a single sensor array. 1.2 Thesis Organization This thesis is organized as follows: In Chapter 2, we shall first review previous work on compression and correlation, and previous work on energy conservation in wire less sensor networks in general. In particular, we will discuss in details the distributed source coding problem using Slepian-Wolf theorem and Wyner-Ziv rate-distortion the orem for compression of sensor readings, and the node scheduling and duty-cycling for sensor networks. In Chapter 3, we present the wavelet based Slepian-Wolf coding scheme. This scheme applies the Slepian-Wolf distributed correlated coding theorem to the bitplanes extracted from wavelet coefficients. We begin by identifying the problem and showing our model under which the proposed scheme is based. The bitplane generation, bit- plane encoding and decoding algorithms are presented next. Experimental results in terms of rate-distortion performance of the proposed codec are then presented based on field data sets. 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In Chapter 4, we investigate another distributed source coding problem where one sensor’s readings broadcast to multiple sensors. We begin this chapter by identifying the problem followed by the model on which our solution is based. Codec design based on the bin superposition idea is presented next. Experimental results in terms of rate-distortion performance on the reception rates of receivers at different correlation levels are presented. In Chapter 5, since compressed data are more sensitive to bit error, we studied the energy optimization of communications of a compressed bitstream. We first give the problem formulation, then we present the analytical results on joint source coding and error correcting coding from the perspective SNR gain and the energy savings. The power-aware coding scheme is then presented followed by the detailed experimental study on the SNR performance and the energy savings of our proposed codec from Chapter 3 and SPIHT, the well-known wavelet set-partitioning codec. Since both the Slepian-Wolf coding problem and Wyner-Ziv coding problem re quire that the correlation statistics are known a priori to both the sender and the re ceiver, the sender needs this information to determine the coding rate and codes, and the decoder needs this information to initialize the decoding algorithm, e.g. sum- product algorithm for low-density parity-check codes. Chapter 6 starts with the prob lem formulation on correlating sensor data. It then presents the sensor data correlation scheme on how to establish the initial correlation and how to track the correlation without re-establishing the correlation. Experimental results on the performance of the scheme in terms of processing efficiency and reduction on data communication are presented next with discussions on how other problems including distributed source coding, data abstraction and beamforming, can benefit from this scheme. Chapter 7 starts with the identification of two problems in wireless sensor net works, namely, the false alarm problem and the exposed alarm problem. It then 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. presents a simple and effective multi-hop cueing protocol using only three beacons. This protocol can enable a two-tier sensor network architecture which is now under consideration by many research groups. It next proposes a general multi-party detec tion algorithms in wavelet domain. Coupled with detection and cueing, the proposed protocol can solve the false alarm problem and exposed alarm problem at the same time. Simulation results on cueing protocol and experimental results on detection al gorithms are also presented. We conclude this thesis in Chapter 8. We give future research directions on energy efficiency issues in sensor networks, and we also present three open problems in this chapter on distributed source coding, distributed detection on the energy efficiency aspects of wireless sensor networks. 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Related Work and Background Wireless sensor networks have attracted much attention from researchers in different fields. It has been shown that sensor networks deployed in an ad-hoc fashion can attack problems previously thought not possible. Typically, sensor nodes are severely resource constrained by energy, memory, processing power. As communication energy costs are high, compression techniques can help reduce communication cost when sen sor readings are to be transmitted and reduce memory when sensor readings have to be stored. An important sensor network application is the target detection as shown in Figure 2.1 where there are seven sensor nodes deployed in a close proximity. Nodes in an array need to send data to a data processing node which uses these data to compute one line-of-bearing (LOB). In this application and many others, data from individual nodes are temporally correlated, and data from neighboring nodes are spatially corre lated after some correlation processing [10, 96, 89]. In this chapter, we will review work on distributed source coding, delay estimation and correlation of sensor readings. It has been proven that it is a promising energy saving technique to control sensor nodes or duty cycle nodes to fit in field activities. We will also review work on energy conservation in this direction in this chapter. 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.1: Sensor array with seven nodes 2.1 Distributed Compression and Delay Estimation Compression is one of the important approaches for communication energy reduction. Distributed source coding has been studied in [80,63,25,15]. In [25], a two-stage iter ative approach is devised followed by an index-reuse which is aimed to exploit spatial correlation. In [63], a data compression scheme, called DISCUS that exploits spatial correlation, is presented. The idea is to use trellis codes to partition the code space into a number of cosets, and it sends only few coded bits in the form of syndrome bits representing a coset. The recipient in DISCUS can infer the correct codewords through transmitted syndromes with the help of side-information. A trellis based Viterbi alike algorithm is employed in DISCUS at the decoder to achieve efficient decoding. DIS CUS has been shown effective when the sensor is densely deployed and it suits for data collection type of applications since encoding complexity is low while decoding is performed in a node of sufficient power supply; however this technique only exploits spatial correlation. The correlated coding and transmission problem is studied in [15] where the rates at sensing nodes can be considered as another degree of freedom on the optimization of the cost function on the data gathering problem. Additional gains can 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. be achieved by exploiting both spatial and temporal correlation, and it is especially use ful when temporal correlation in sensor networks is also high as in many target track ing and classification applications. We have proposed a spatio-temporal compression scheme in [87, 91]. An interesting observation by our experimental study reveals that spatio-temporal, spatio-only and temporal-only schemes can apply to different types of sensor network applications and our proposed spatio-temporal compression can fit well to all correlation scenarios via some simple codec parameter adjustments [87], As pointed out in [62, 87], a high correlation exists in sensor readings. Correlation analysis therefore is important for applications, which are aimed to exploit these cor relations in wireless sensor networks, e.g. compression [96, 87, 10], beamforming [7], and storage systems in sensor networks [27] [28], and routing with compression [60]. If the correlation can be correctly tracked without calibration on a continuous basis, the additional benefit is that there is no need to send calibration parameters or to calibrate each frame before compression; furthermore, synchronization can also be relaxed to a certain degree which could benefit many applications. Since spatial correlation in these applications is high, the linear prediction [54] method is normally effective and it can also be made efficient. In [10], an algorithm based on linear prediction and gra dient method has been applied to the compression of sensor data and a zero-pole model of linear prediction is used. An implicit assumption in [10] is that sensor readings are already calibrated so that the single-zero prediction model could work to exploit the spatial correlation (the temporal correlation is exploited via an all-pole model in [10]). In [96], a time-domain design of application specific compression with detection called CP-MaxMI is proposed. The maximum a posteriori detector under CP-MaxMI can obtain the time delay estimates across sensors which is shown close to the time 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. delay based on the original signal, and the optimality performance of estimates is per sistent with varying levels of signal-noise distortions. In contrast to previous compres sion work, this work uses a realistic model assuming that the sensor readings are not pre-aligned. This algorithm is able to aggregate data with the delay information from different sensors computed at the fusion center. This delay information is very use ful for many distributed source coding codecs and beamforming algorithms for sensor networks. Slepian-Wolf coding theorem provides another way for compression of correlated sensor readings. The Slepian-Wolf problem refers to the lossless correlated source coding problem, and the sum of the achievable rates is bounded below by the joint en tropy. Encoding under Slepian-Wolf coding is performed independently while decod ing is done jointly. For two i.i.d. sources X, Y, the achievable rate region which can describe X and Y with asymptotically arbitrary accuracy is determined by Rx + Ry > H(X]Y), Rx > H(X\Y) and Ry > H(Y\X), where Rx is the rate for X and RY is the rate for Y. In this region, two comer points correspond to the cases where one source is described by its entropy H(X) (H (Y )) and the other source is described by the conditional entropy H(Y\X) (H(X\Y)). Practical codes whose performances are close to these comer points have been designed as shown in Table 2.1. There are two general design methodologies - syndrome partitioning [102,111, 63] and random binning [80, 14, 30]. Syndrome partitioning is a deterministic binning approach since a codeword’s bin is pre-determined by its syndrome, not changeable during encod ing/decoding process. Different error correcting codes are employed by these designs including Trellis codes, Turbo codes and low density parity-check (LDPC) codes. In [74], a general design method is proposed to approach any point on the lower boundary of the achievable region and it uses LDPC [26] codes and partitions the parity-check matrix for different points on the boundary. 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 2.1: Slepian-Wolf code designs Design Codes Used Binning Method [63] Trellis Code Syndrome Partitioning [1] Turbo Code Random Binning [11] Turbo Code Syndrome Partitioning [29] Turbo Code Random Binning [49] LDPC Code Syndrome Partitioning The Wyner-Ziv coding problem refers to the lossy version of the Slepian-Wolf problem in which the achievable rate of the source to be coded is specified by the Wyner-Ziv rate-distortion function. For the case of doubly binary symmetric sources (BSS) where the source and side information can be considered to form a virtual binary symmetric channel, at encoding with an error rate at d, a rate between Rx \y (d) and h(p( 1 — d) + (1 — p)d) — h(d) is achievable, where h(.) is the entropy function; p is the crossover probability, and Rx\y{d) is the minimum rate of X when it is encoded with knowledge of Y . More specifically, the achievable rate is determined by a lower convex envelope of the fol lowing set: {(d, h(p( 1 - d) + (1 - p)d) - /i(d))|0 < d < p} U {(p, 0)} Figure 2.2 shows a few curves where the solid curve below each Wyner-Ziv rate dis tortion curve is the corresponding rate distortion curve when encoding is done in an intra-mode (i.e. without knowledge of the side-information at the encoder). From Fig ure 2.2, these tangent points, as shown in the legend, could arbitrarily close to both 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.49 (0.48, 0.0011) Joint (0.49) 0.4 (0.3128, 0.1) Joint (0.4) 0.25 (0.1,0.4149) Joint (0.25) 0.1 (0.008, 0.421) Joint (0.1) 0.05 (0.0015, 0.59) Joint (0.05) 0.01 (0.00004, 0.08) Joint (0.01) Distortion (error prob.) Figure 2.2: Rate region of doubly BSS for the Wyner-Ziv coding problem extreme points, i.e. (0,0) and (0.5,0). This tells that the achievable rate at distortion d can be estimated by h(p( 1 — d) + (1 — p)d) — h(d) when p is close to 0.5, and that is almost always alone a straight line of the rate- distortion curve as r(d) _ ( r - d ) W ) when p is close to 0. One possible approach to the Wyner-Ziv coding problem can use the methodologies in Table 2.1 with bin dilution for a smaller rate with a possible error probability bounded away from 0 even when the codeword length asymptoti cally approaches to infinity. Another more practical approach to the Wyner-Ziv coding problem is to use a high performance quantizer followed by the lossless Slepian-Wolf coding methods [47, 63]. By this way, the rate loss of the Wyner-Ziv problem in gen eral is overcame and a small loss is only introduced by the quantizer. 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In Figure 2.2, we have shown that a rate loss (cf. [110]) occurs for the doubly BSS. In general, only when the following condition holds, is there no rate loss for the Wyner-Ziv coding problem: I(X-Z\X,Y) = 0 (2.1) where Z is the random variable derived from X under the Markov constraint: Y i — ► X i -> Z forms a Markov chain, and X in (2.1) is the estimate of X under a decoding function from (Y, Z). A special case found in [104] where X and Y are both Gaussian from an infinite alphabet, and the distortion function is mean squared error, it does not result in any rate loss for the Wyner-Ziv problem at any distortion level. A general ization to the above result which only requires that the correlated noise is Gaussian is given in [61]. Most recently, another such case for finite symmetric sources has been found and proven using the well-known random binning method [30]. These re sults are crucial for practical code design for the distributed correlated compression problem. 2.2 Energy Conservation in Sensor Networks Previous endeavors have been on how to schedule trackers to go to sleep or to wake up trackers based on detection outcomes. On one hand, nodes wake up according to a predefined schedule so that only involved nodes are kept active for the duration of a task execution. Recently S-MAC [109] has been proposed, and it enables nodes to sleep not only for a scheduled period, but also for other periods for which it can infer by observing neighbor’s behavior. This node scheduling approach only fits well for certain applications and can save significant energy when the network topology and field configuration are known and a schedule which matches task execution behaviors 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. can be set; however, it can incur high initiation and coordination cost when these conditions do not meet. On the other hand, if tripwires are used to monitor a field, trackers are blindly woken up whenever there are some high energy activities in a field; even when some of these activities may not be of interest. In this thesis, we extend this tripwire idea and propose a framework for effectively utilizing these tripwires in a two-tier sensor network setting. An alternative approach to a two-tier network is to directly use sensor nodes to monitor field and wake up other components whenever necessary, e.g. radio mod ule for wireless data communication. Compared to the alternative, one problem in a two-tier network is an increased latency of notification. The energy-latency trade-off has been studied in [46] for the data gathering applications in a homogeneous sensor network setting. This is also an interesting problem in the context of two-tier sensor networks. A tracker (i.e. a signal processing node) wake-up scheme has been presented in [82], where the focus was on how to optimally select a set of trackers from the appli cation point of view to perform the tracking task. Although that work is orthogonal to tracker cueing scheme to be studied in this thesis, an alternative cueing scheme (denoted by INIT for notational convenience) can be constructed for that algorithm by using the following steps: (1) any tripwire initially constructs a local database of track ers which are connected to itself; (2) online optimization algorithm proposed in [82] selects a set of trackers to execute an application. This INIT scheme however does not solve the so-called exposed alarm problem and false alarm problem. The false alarm problem refers to the wake-up of inappropriate trackers which is caused by a simple energy threshold detection, and the exposed alarm problem refers to the wake-up of 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. trackers which is not in an optimal position to process the event. The algorithm pro posed in [82] can be executed by a set of woken trackers to select an optimal set of trackers for the task in question. Efficient and effective detection algorithms are critical for energy efficient sensor networks. Only when events are detected accurately, should a processing on it start. Receiver operating characteristics (ROC) based approach for narrowband signal detec tion has been well studied in sonar, acoustic and imagery applications [83]. There are limited works on detection on far-field wideband signal. In [45], an adaptive thresh old detection algorithm has been proposed and it uses field noise to adjust the energy threshold so that a constant false alarm rate (CFAR) can be achieved. For narrowband far-field signals, the amplitude envelope with Gaussian noise can be modeled by the Ricean distribution as shown in (2.2). and a is the real noise variance or imaginary noise variance and S is the narrowband signal amplitude. The noise amplitude follows the Rayleigh distribution as shown in (2.3) which is for a random variable as a square root of sum squares of two Gaussian random variables. (2.2) where, i r h(y) = ^ I exp(y cos(t))dt (2.3) The false alarm rate based on an amplitude threshold T is given by (2.4) (2.4) 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 2 4 6 8 10 12 Chi-square quantiles 0.000 0.004 0.008 0.012 Q uantile of Exponential Distribution (a) M-distance of DFT coefficients (b) Quantile plot of PSD Figure 2.3: Models for generalized likelihood ratio test Likewise, the false positive rate with the threshold T can be obtained by (2.5) However, when event signals are wideband as in many acoustic target tracking appli cations, above closed-form formulae for false positive and false alarm rates cannot be easily obtained. The generalized maximum likelihood ratio test (GLRT) detector has been studied intensively for many applications in frequency domain [98, 41] and in time domain [96, 71]. The frequency domain maximum likelihood ratio detector has shown to be effective for narrowband signals. The underlying models are either i.i.d. complex Gaussian on discrete Fourier coefficients [41] or i.i.d. exponential on two-sided power spectrum density [98]. For vehicle acoustic signals (wheeled or tracked), these models are not appropriate. As shown in Figure 2.3 these models are not proper where the quantile lines do not fit the points well. If the complex Gaussian model had been ap propriate, the Mahalanobis distances (or M-distance) from the real and complex parts (2.5) 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of Fourier coefficients should have Chi-squared distribution with two degrees of free dom, and the corresponding quantile line would fit in the quantile plot of Mahalanobis distances. Figure 2.3(a) shows quite the contrary. If the exponential distribution had been appropriate, the quantile transformed plot would form a line. Figure 2.3(b) shows that a line fitting is not possible. When the number of partitions is a priori, the exhaustive search of partitions, where the underlying signal occupys, takes exponential time on the number of partitions. The dynamic programming algorithm in [41] reduces the cost to quadratic complexity by introducing to the cost function an adjusting factor - minimum description length [71]. Even under that improvement, the computation cost of frequency domain GLRT is almost prohibitive due to the non linear Fourier transform cost and the quadratic search cost on possible partitions where the underlying signal occupies. Therefore, alternative approach is needed to detect events for sensor networks. In [86], a distributed false alarm detection algorithm called DFAD in wavelet domain is proposed. It uses a group of tripwires to perform a false alarm detection once an alarm is received or generated. Decision fusion is used in the end to generate a predicate to determine if to wake up trackers. In this thesis, we will further expand this idea for efficient event detection for resource constrained sensors with low false negative and false positive detection rates. 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Wavelet Slepian-Wolf Compression In this chapter, we propose a compression scheme called spatial set-partitioning in hierarchical trees which exploits the spatial and temporal correlations present in sensor data. This scheme allows progressive transmission and provides scalability in adapting to the underlying correlation structure of sensed data. It uses flexible Slepian-Wolf coding based on low density parity-check codes. Two different decoding schemes are proposed for different types of resource constrained sensor nodes. This scheme outperforms known codecs by a large margin of decibel in terms of the signal-to-noise ratio. This scheme has 0(n) complexity for encoding and 0(n log(n)) complexity for decoding using message passing, where n is the codeword length. Experiments and simulation results with field data sets demonstrate the viability of our proposed scheme to sensor data. 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1 Introduction Wireless sensor networks are being proposed for many applications to collect useful data from physical environments. Examples include habitant monitoring, object track ing, etc. We refer to [66] for details. Wireless sensor networks allow ease of deploy ment without sacrificing functionality and flexibility. Due to node size and the need for communication of sensor readings in many of these applications, practical constraints are imposed on sensor networks, namely, limited data storage space and limited energy resource. Since radio communication dissipates high energy as compared to compu tation in sensor nodes, it helps to reduce communication cost by compressing sensed data before transmission; moreover, it reduces storage space if readings have to be stored. 3.1.1 Problem and Our Contributions Sensors are normally densely deployed in a field and many nodes observe the same events. Readings from different sensors are correlated in space; furthermore, field signals also tend to be temporally correlated in nature. Practical compression schemes, e.g. DISCUS in [63], has been shown effective for compression of correlated sensor readings. In [63], a design with low cost encoding and decoding scheme is proposed to exploit the spatial correlations. In these spatio-temporally correlated sensor readings, a better compression performance is expected by exploiting both types of correlations which is the focus of this chapter. For two independent identically distributed sequences X, Y, obtained from read ings of two close-by sensors, when the readings of Y act as side information during decoding of X, the achievable rate is H(X\Y) by the Slepian-Wolf theorem [80]. Code designs with practical encoding and decoding complexities have been proposed 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for sensor networks. The general approach is to use the binning principle [111] to par tition code space into cosets. One approach uses deterministic partitioning [63,49, 11] where the elements that form a coset are determined based on the input sequence, and another approach is to use random binning as shown in [29, 1], which employs a long interleaver for bit sequence scrambling to achieve randomness. These approaches can exploit the spatial correlation in sensor readings. To do so, they are required to randomize the data so that the cosets constructed contain code words whose intra coset pairwise codeword distances (in the code space with a given norm) are as large as possible. The other drawback of these approaches is that the signal temporal correlation are destroyed by the introduced randomness. On the other hand, taking a look at the wavelet based signal compression approach aiming at ex ploitation of the temporal correlation of sensor readings, after set-partitioning at two neighboring sensors, the refinement bits are no longer spatially correlated even if only one significant coefficient is shifted a position as compared to that coefficient in a neighboring band. The spatial correlation is destroyed after wavelet set-partitioning. In light of these approaches, in this chapter, we are focused on how to exploit the temporal correlation while keep the spatial correlation intact (or with minimum effect) after set-partitioning which allows the spatial correlation further to be exploited by the Slepian-Wolf coding theorem. Our contributions in this chapter include a new scheme, called spatial set-partitioning in hierarchical trees (S-SPIHT), with practical bitplane coding for high compression ratio. The advantages include significantly better compression with little process ing overhead as compared to other type of compression schemes. Due to the use of bitplanes, our scheme is progressive. In S-SPIHT, bitplanes extracted during set- partitioning from wavelet coefficients are encoded using low density parity-check code 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (LDPC) based Slepian-Wolf codes; two efficient decoding algorithms are also pro posed - one for sensor nodes which can only perform fixed-point arithmetic and the other with better performance for general-purpose sensor nodes. In S-SPIHT, spa tial correlation of sensor data are fully exploited via flexible finite geometry based LDPC codes from the bitplanes with low crossover probability to bitplanes with high crossover probability. Furthermore, based on our design, the encoding process takes 0(n) time and the decoding process takes 0{n log(n)) time where n is the codeword length. 3.2 Our Models for Distributed Compression In this chapter, we assume that two frames - Frame U and Frame V (a frame is a sequence of samples with a fixed length) are captured by Sensor U and Sensor V, respectively. We assume that Frame U is going to be coded in an inter-mode and Frame V is coded in an intra-mode. A spatial capsule (or capsule for short) is defined as a block of consecutive wavelet coefficients of a fixed length, and we will explain the use of capsule later. With the use of capsules, the wavelet coefficient trees consist of nodes of capsules instead of individual coefficients. The set-partitioning as that in SPIHT [73] proceeds iteratively to identify significant coefficients, and three types of bits are generated in an iteration, namely, significance bits, refinement bits and sign bits. Significance bits are simply coded in an intra-mode, and the other two types of bits are coded in an inter-mode in S-SPIHT. In what follows, bitplanes are coded from sign position to the most significant position to other positions which are less significant, and there is a bitplane correspondent of each of these bit positions. Two terms are used in this chapter for the exposition of S-SPIHT, namely, bit- plane and refplane. The i-th bitplane under S-SPIHT denoted by Bi is defined as the 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Virtual Root Significance bits (001) __ ii .« ■ in . H uficant n S V h x 'V Sth / ' ,'V ■ ■ ■ ■ I I | | J l “ ■ ■ " v . / / A / \b I \ B i / / N , Figure 3.1: Encoding and decoding wavelet coefficient trees set of all the i-th bits of all significant coefficients extracted from Frame V during set-partitioning. Sign bits also form a separate bitplane and it is denoted by the 0-th bitplane; further sign bits during successive iterations are embedded into correspond ing refinement bitplanes. Left tree of Figure 3.1 shows one iteration which identifies 6 significant capsules as shown by the nodes of two coefficients, and there are totally 14 coefficients identified in this case and used to form bitplanes. In this figure, dark gray nodes or subtrees are insignificant and white nodes are the roots of insignificant subtrees. Refplane is used for decoding, and the receiver, which has received the sig nificance bits of Frame V, applies these significance bits to Frame U and extracts these coefficients information corresponding to significant coefficients of Frame V. By doing so, the receiver has just generated the side information in terms of bitplanes at each bit position. Sign bits also form the 0-th bitplane and they are treated exactly the same as other bitplanes. These bitplanes generated at the decoder are called refplanes, and we denote the i-th refplane from S-SPIHT by Hi. Since the wavelet coefficient trees have the same shape, the spatial correlation is preserved between Bi and Hi. Right tree of Figure 3.1 shows the decoding wavelet coefficient tree corresponding to the encoding tree, and in this figure, it also shows one particular search of a capsule highlighted by a white node. Assume that bits in a bitplane or a refplane are independent of each other. De note Bi(k) and Hi(k) as the k-th bit of the i-th bitplane and refplane, respectively. 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The correlation model of bitplane and refplane is defined as follows: 1) a sender and a receiver observe the sensor field at the same time, and the receiver’s readings are used as side information during decoding. 2) compression is only applied to bits in bitplanes; therefore, the sources are binary. Furthermore, these bits can be considered to be independently drawn according to a joint distribution as two binary sources hav ing an identical distribution. 3) crossover probabilities of Bi(k) w.r.t IZi(k) for all k, are fixed for Bitplane £ > * . 4) crossover probabilities differ among bitplanes; moreover, these probabilities decrease from bitplanes corresponding to more significant position to those corresponding to less significant position. This means that correlation is not independent of bitplanes. The left plot of Figure 3.2 shows the crossover probabilities between Bi(k) and TZi(k) for 1 < k < 13 with four different capsule sizes at 8, 4, 16, 128. The data set comes from field tests with the vehicle acoustic sampled at 16 kHz. The crossover probability increases along bitplane bit positions. In the higher bit position, the cor relation is close to random, and these bitplanes are normally not encoded unless high fidelity is required by applications. In this case, these bitplanes have to be coded in an intra-mode (e.g. entropy coding). The right plot of Figure 3.2 shows these curves of the bit distribution probability within each bitplane. In this chapter, a code space is defined as the set of binary sequences of a fixed length n; and a bitplane of width n is in the code space. With binning (i.e. coset partitioning) as used by Slepian-Wolf cod ing, the bin width (i.e. the number of codewords in a bin) is set close to 2nI(x,Y\ and a generic decoding algorithm consists of two parts: indexing part - decoding for the bin index i, and estimating part - decoding of the codeword x from the desired bin k* based on the conditional probability with Py\x (v\x) is the largest among Py\x {v\x G «:*). Definition 3.1 and Proposition 3.1 establish a relationship between the channel coding (via Shannon second theorem, i.e. pp. 198 in [14]) and our bitplane coding 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - e - Size 8 -*-S iz e 4 - 9- Size 16 -— Size 12E Bitplane Index 0.9 - 9- Size 8 — Size 4 — Size 16 — Size12£ S - 0 .8 = 0.6 . a .O 2 0 .5 Bitplane Index Figure 3.2: Bit proabilities in bitplanes with varying capsule size problem with varying source distributions. Since in S-SPIHT, we consider the case that bitplanes from source V are available during decoding, and such a connection does exist. However, for the general distributed source coding problem where the bitplanes of Y and X are coded jointly, this connection may not be obvious. In this general case, using an auxiliary random variable U satisfying that U i-+ X y forms a Markov chain, the rate region [103] is as follows: {(Rx ,R y )\Rx > W X ) , R y > H(Y\U)} Definition 3.1: Given independent identically distributed random variables X and Y with joint probability pXy (x, y ), compute the conditional probability pY\x(y\x). For nc > 1, a discrete memoryless channel C for this distributed source coding problem is defined as follows: i) Input alphabet X = {0, l} n°. ii) Output alphabet y — X. iii) The transition probability matrix is given by p(y\x) for the output y with the input x to C. □ 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Note that ^JPY\x(y\x) = 1 y and px(x) is immaterial to the definition once the conditional probabilities are ob tained. When nc = 1, channel C degrades to the binary channel. For bitplanes of length knc, the transition probabilities of the k-th extension channel Ck is as follows: k pYn\xn{yk\xk) = n PY\x(Vi\Xi) i= 1 Notation 3.1: |_ .J denotes integer floor function and [.] denotes integer ceiling function, and H(.) denotes the entropy function while Hx denotes a matrix H mul tiplied by a vector x. Exponential and logarithm functions in this chapter use base 2 unless specified otherwise. The crossover probability between and IZi is p;. First the capacity of channel C is given as follows: C o= sup I(Px(x)tC)(X,Y) px(x) where I(Px(x),c)(X, Y) is the mutual information with X under distribution px(x), and Y depends on X by the channel Ck. For a given source distribution Px{x) under binning and 0 < e, we only need to consider typical sequence of X (refer to pp. 410 - 415 in [14]), and there are totally (1 - e) exp(k(HPx{x)(X) - e)) such typical sequences when k is sufficiently large, where HPx^ (X) is the entropy of random variable X under distribution px(x) (refer to Theorem 3.1.2 of [14]). 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. By noticing that each bin can have the maximum number of codewords upto the capacity the channel can handle which is upper bounded by kCo for the lowest source coding rate, and M, the number of bins, needed to represent these typical sequences is lower bounded as follows: M > exp (k(HPxix)(X) - Co)) Therefore, we have the following proposition. Proposition 3.1: Let X be the random variable from population X, and Y be the corresponding random variable from population y, under Slepian-Wolf binning, the limiting source coding rate R of bitplanes under distribution px(%) is lower bounded by the following formula: R > HVx(x){X) - C0 □ Since the binning approach of Slepian-Wolf coding associates a physical channel C with the distributed source coding problem, this channel capacity is then directly linked to the level of correlation between X and Y. More specific, the channel capac ity determines the binning width of Slepian-Wolf coding scheme and the rate for the channel C is the average number of bits needed to represent a bin. Putting in other words, the higher the capacity of C is, the less number of bits are needed to represent these bins. Remark 3.1: 1) error probability Pe under Slepian-Wolf decays exponentially as codeword length increases. Moreover, larger rate beyond H(X\Y) yields a faster de cay. There is a trade-off between the Slepian-Wolf rate and codeword length. For a 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. larger rate, the exponent is larger, and smaller n may result in a same error proba bility. 2) for the Wyner-Ziv problem (WZP) of doubly symmetric binary source 1 , a binning scheme may also be constructed by diluting the bin (i.e. expanding the bin) and the encoder still does not require to know the side information. 3) from the prac tical standpoint, under Slepian-Wolf, codeword can not be too large. It is known that the probability of two correlated source is jointly typical, increases as the codeword length increases (see the proof of Theorem 14.4.1 in [14]). In practical system design in sensor networks, a codec can simply ignore atypical sequences; the error introduced by this rareness is not of direct interest to a codec design since a codec is not working alone in sensor networks, if an atypical pair occurs, others means from application layer can easily correct this by noticing the fact that it has negligible probability that more than one atypical sequence pairs happen at the same time. 4) the estimating part of a decoding algorithm is a procedure which introduces most of distortion and could be of high complexity. 3.3 Code Design of Bitplane Based on the connection between channel coding and Slepian-Wolf coding studied in Section 3.2, we present the following design of bitplane coding for a rate R where H(X\Y) < R < H(X). 'Wyner-Ziv Problem of doubly symmetric binary source: for given distortion d, find a codec at a rate between Rx\Y(d) and h(p( 1 — d) + (1 — p)d) — h(d), where h is the entropy function; p is the crossover probability, and Rx \y(d) is the achievable rate at distortion d when the side information is also available to the encoder. The minimum achievable rate for our problem has to be strictly greater than (4) at distortion d unless encoder has knowledge of the side information. 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Design 1: Rate R is given, first to design a low density parity-check code (LDPC) with rate H(X) — R and codeword length n. Assume the designed code has a parity- check matrix H whose dimension is N x n, where N = \n R ]. Proceed to encode the source and decode for the source as follows: 1. encoding: for a given bitplane x, generate an N-tuple binary sequence by Hx. The encoded sequence for bitplane x is Hx. 2. decoding: given an iV-tuple binary sequence z and side information y, denote g as the function which maps z to its coset, then decode x to x using the following maximum likelihood decoding rule: PY\x{y\x) > PY\x(y\u) Vu < = g(z) where, ties are broken randomly. □ There are a few observations on Design 1.1) only when R > H(X\Y), should the error probability be possibly made arbitrarily small. 2) when R < H(X\Y), an error probability is bounded away from 0 when n is sufficiently large. This follows from the proof of Shannon second theorem (see p.206 - 207, [14]). In fact, for regular LDPC code whose check set size is k, this bound can be estimated as follows: P >k'( l ____ 2_________ I____ e - V H(X) - R n(H(X) - R) where, k' = [k(H(X) — i?)J. Note that each bit error in a message sequence of length [n(H(X) — R)J could cause k check equations unsatisfied simultaneously, in (3.1), and P ^ < l w h « * < W ) 35 (3.1) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. OOOOOOOOOOOOiOO oo C apsule w ilh size 4 i-------------1 ----------- O O O OiO O O OiO O o o;o o o o Figure 3.3: Capsules to form wavelet coefficient tree 3) from a practical standpoint, syndrome computation and maximum likelihood de coding have to be efficient. Fortunately, with LDPC, an algorithm in 0(n log(n)) is possible using message-passing with a straightforward modification to account for the case of any syndrome note that the standard message-passing LDPC decoding algo rithm searches codeword in the bin corresponding to the all-zero syndrome. 3.3.1 Bitplane Generation and Encoding In order to better exploit the spatial correlation, a new technique is introduced to strike a balance between exploitation of spatial correlation and temporal correlation. To better understand S-SPIHT in this context, the set-partitioning process on wavelet co efficients can be taken as a method to reorganize the coefficients so that Slepian-Wolf coding scheme can better exploit the spatial correlation. Figure 3.3 shows the use of capsule to form a tree with a frame of length 32, where the capsules have size 4 (the selection of capsule size in power of two is for process ing efficiency). A wavelet coefficient tree can be constructed as shown in Figure 3.4 where each node corresponds to a capsule of size 4. In this figure, the top subband is the coarsest and the solid vertical lines separate the subbands. A capsule is called significant at a quantization level if at least one of coefficients inside this capsule is sig nificant. Once capsules are defined, the set-partitioning process proceeds in a similar way as SPIHT. 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. am Low Pass I # § f § I High Pass Figure 3.4: Iterative identification of significant capsules S-SPIHT encoding process consists of three stages: 1. identifying significant cap sules and output significance bits into bitstream, 2. generating bitplanes and 3. encod ing bitplanes to bitstream. The first stage is the same as that in the SPIHT except that capsule is used instead of coefficients. Stage (2.) and (3.) are done in the refinement phase which also includes encoding of sign and refinement bitplanes. At a refinement phase, a bitplane of signs is first generated, then a refinement bitplane corresponding to a quantization threshold is generated. At each bitplane, each significant capsule has at least one nonzero bit plus the rest zero bit. Unlike in SPIHT, where most significant bit (MSB) of coefficients in a bitplane can be inferred and they are not encoded into bitstream, S-SPIHT also encodes these MSB bits which are included in the MSB bit- plane. In Figure 3.4, after the first iteration, significant capsules are encoded. For non significant coefficients (if any) in a significant capsule, a bit zero is outputted to the current bitplane. We shall explain with the help of Proposition 3.1 that this does not re sult in a higher bitrate, and in fact, this method can achieve a much higher compression ratio for correlated sensor data when a proper capsule size is selected. 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Lemma 3.1: Let two sequences X k and Y k of length n be associated by channel C, let X 2k be a random variable of sequences of 2k, and Y 2 k is obtained by passing X 2k to channel C symbol by symbol, the following identities hold: Proposition 3.1 can be easily verified. It tells that Slepian-Wolf rate increases at a slower pace than that of bitplane width. Compared to SPIHT, there are possible bits added due to usage of capsule instead of coefficients; however, the rate increases can be excessively compensated by reduction on significance bits on some proper se lected capsule size. The conditional entropy actually decreases since it is reasonable to assume that the extra bits are more correlated since they are at a higher bit posi tion. Although a theoretic analysis is hard to obtain on the change of extra entropy, experimental data have strongly demonstrated this fact for acoustic sensor signals. Another salient feature using capsule in a wavelet set-partitioning algorithm is a great reduction on processing overhead. In fact, the set partitioning cost is reduced by a factor equal to the capsule size. This is very appealing to sensor nodes with strict resource constraints since set-partitioning cost dominates both encoding and decoding overheads. 3.3.2 Bitplane Decoding There are three points on decoding of bitplanes encoded using Design 1 at the begin ning of Section 3.3: 1. based on correlation, bitplane x must be in the close neigh borhood of side information bitplane y, 2. the decoded codeword must yield the same H (X 2k) < 2H (X k) H{X2k\Y2k) = 2H(Xk\Yk) I(X 2k,Y 2k) < 21(Xk,Y k) (3.2) (3.3) (3.4) 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. syndrome; otherwise an error can be correctly declared, and 3. any partial result (i.e. it yields a different syndrome) is not an approximation to x, this is due to the fact that there are many valid codewords in the neighborhood of y which could be significantly different from x. However, if the decoder decodes a codeword other than x whose syndrome is the same as that of x, this also causes a decoding error, but this type of errors is not detectable. We give two decoding algorithms. The first one, so called bit-flipping (BF), is simple and suitable for sensor nodes that are not capable of floating-point arithmetic operations. It may generate some undetectable errors. The second is a sum-product algorithm (SPA), and it yields a lower error probability than BP at a higher processing overhead. Notation 3.2: 1) the error-free syndrome is sx = H x = (sx(l), sx(2), • • • , sx(iV)) 2) side information is y and Sy = H y — (Sy(l), Sy(2), ' ' ' , S y ( N ) ) 3) rii is the number of check equations checking on the i-th bit, i.e. the number of 1 ’s in the i-th column of H. 4) the set of parity equation indices containing the i-th bit is denoted by Q i = Qi2 i ' ' ' > Qini} 5) © is the addition operation in GF(2) - the Galois field of order 1. 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Algorithm 3.1: Let T be the threshold on the number of equations needed for cor recting a bit. 1) Initialization: Compute sy. 2) Iteration: For each bit, perform steps: 2-1) For the i-th bit, compute the following number: T = 'y ^ (sj/{(jij) © sx(qij)) 1 < j< r ii 2-2) If T' > T, y' = y © exp(z — 1); otherwise, y' = y. 2-3) If y' y, compute Hy'. 2-4) If Hy' = Hx, set x = y' and return; 3) Repeat (2) or maximum number of trials reached and set x — y (the side infor mation). □ Note that y is probably the best approximation if the iteration diverses. In this BP algorithm, we force the codeword search in the designated bin in Step (2-4) while the bin in standard BF algorithm corresponds to the zero syndrome. Our second decoding algorithm is a message-passing algorithm. Standard LDPC sum-product algorithm [50] does not work directly in our case since syndrome s cor responding to the message at encoder can be any k dimension vector in vector space GF(2fc ) - extension Galois field of order k, while the syndrome in standard LDPC is assumed to be in the neighborhood of the zero vector. The changes we made comprise of two parts: 1. the initialization of posteriori probabilities and 2. modification of local 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. kernel to deal with any syndrome. This second algorithm can directly decode a mes sage with multiple refplanes. When multiple refplanes (e.g. Y and Z) are available, since H(X\Y,Z) < mm{H(X\Y),H{X\Z)} (i.e., conditioning does not increase entropy), Bitplane X can be compressed at a pos sible lower rate than the previous case. We have also tested on the two-refplane case, based on our test from our data set with an array of size 7 sensors, the additional rate reduction on the bitplane is quite substantial as shown in Section 3.4. Assume that a bipartite graph is created for the parity-check matrix of a LDPC code. Let vi be a variable node corresponding to the 1-th bit, and wm be the check node corresponding to m-th check. Define a predicate from a binary sequence 0 as below which is needed for our local kernel during decoding. 1 if £ ® = r v & 4 > 0 otherwise where the summation uses the addition in GF(2). To support multiple refplanes, denote 71 = (yl5 y 2, • • • , yn), and each element of 7 ?, is a vector of dimension q if there are q > 2 refplanes and they degrade to one bit (0 or 1) for the single refplane case as in Algorithm 3.1. A message in SPA is a pair of two conditional probabilities corresponding to bit assignment 0 and 1, and actually only one probability is needed to store since all pairs of probabilities in SPA are normalized to 1. Notation 3.3: 1) n(r) denotes the set of neighbors of node r (check node or vari able node). 2) n(r, s) denotes the set of neighbors of node r with one of its neighbor s removed. 3) qf ml denotes the probability (i.e. posteriori probability) of the 1-th bit of 2 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. being t (0 or 1) given the information via checks in n(uj, wm). 4) denotes that the bits corresponding to variable nodes in n(wm) are independent. 5) f) is the index set of variable nodes of wm defined as follows: 6) $ is the sequence set of binary assignments with each item in the sequence corre sponding to a unique variable node in n(wm, v{). It is given as follows: where, symbol xy in < f > is used for an assignment (0 or 1) to vi> . The a priori probability in each iteration can be updated based on the posteriori probability of the previous iteration as follows: where, xv = (0, or, 1) is a binary assignment of I'-th bit in fl, and x(0, S[) and si © 1) are the local kernels corresponding to m-th syndrome bit being 0 or 1, respectively. Addition of 1 to syndrome bit sm in (3.6) is to make sure the syndrome is kept as sm after this assignment, i.e. xi = 1, since sm ® 1 © xi = sm. For (3.5), the syndrome is also kept as sm since sm © xi = sm in this case. This purpose is to limit the codeword search in a given bin specified by the syndrome. probability of check m is satisfied if the 1-th bit of y is decoded to t (0 or 1) given that = {V\vv € n(wm,vi)} $ = {(••• xv = (0, or, 1) • • • )\l' e fi} (3.5) (3.6) 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The initial a priori probabilities of variable nodes should utilize the side informa tion TZ as shown in the initialization step of Algorithm 3.2. Assume all codewords are equally likely, the decoding algorithm SPA then follows. Algorithm 3.2: - SPA 1) Initialization: (Qml, Q m l) = (P°> 1 - P°), l< l< n where p° = Pr(0|yj), Vm G n(v{). (Note that for the two-node case, Pr((%j) equals to the crossover probability pi if yi ■ = £ ■ 0. ) 2) Iteration: iterate these steps as follows: 2-1) Check node: For the message from wrn to vt, use (3.5) to update and set r 1 ^ = 1 - r ° m l. 2-2) Variable node: for the message from vi to wm, R °= [I m 'en(vi,w m) R1= II t1 ~ m 'en(vi,w m) ( R ? R} \ message:(qQ ml, q ^) = , R o + R i ) ^ and rll,l and r^,, are the messages sent from check nodes in n(vi, (Vm ). 2-3) Decoding: on the l-th bit, use (3.7) on n(v{) instead of n(vi, wm) to com pute the posteriori probability, qf, then set £i to 1 if q] > 0.5 or 0 otherwise. 3) Repeat to step 2) until H x = H x return x, or maximum number of iterations reached. □ 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The incorporation of syndrome in Algorithm 3.2 is done through the local kernel in Step (2-1). This completely captures the essence of binning and it forces the search of a; in a designated bin only. Messages initiate from variable nodes; all check nodes are updated for the first time. After the initial step, the flooding scheduling scheme is used, i.e. messages are passed at each step over all edges. Algorithm 3.2 eliminates loga rithm function all together, since these n(wm, vi)s’ have to be rather small for possible implementation in a sensor processor, the precision loss due to multiplication of small probabilities is manageable. However, when n(wm, vi) is large (although this is rarely the case in practice), LLR version has to be used. The conversion of Algorithm 3.2 to an LLR version is rather straightforward as shown in [44]. Further modification, for a so-called hybrid decoding algorithm to achieve processing efficiency, can be done via sequentially running Algorithm 3.2 with less number of iterations and generate a partial result as the input to Algorithm 3.1. 3.4 Experiments and Simulation Study We design codes for coding of bitplanes based on the type-I Euclidean Geometry based LDPC (EG-LDPC) [43] in this experimental study. To fit for different correlation level of bitplanes, codes can be constructed by splitting columns of a base type I EG- LDPC code for a higher compression ratio when the crossover probability is low, or by puncturing the base code for bitplanes with a higher crossover probability. This method is flexible for coding bitplanes with different spatial correlations. Type I EG-LDPC codes and their extended codes (via column splitting and puncturing) can be kept to be cyclic or made to be quasi-cyclic. This allows an efficient syndrome computation, and this feature is particularly appealing for sensor nodes which have limited computation capability. This guarantees that the syndrome generation takes 0{n) time where n is 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the width of bitplane; moreover, codes constructed in this way has a girth 6, and this allows a close to optimal performance of SPA. Denote H as the parity-check matrix of the base code, let r be the rate of the base LDPC code. For a given compression ratio R < 1, the problem boils down to find an extended LDPC code with rate r' = \n — nR\ n There are three cases: 1. r' > r, 2. r' = r and 3. r' < r. For Case (2.), the base code is used, and when H has a full rank, the relationship between R and r is: R = 1 — r. For Case (3.), column splitting is needed and column puncturing is needed for Case (1.). When exact length of codeword is not possible to match a bitplane, it can be sent either in several transmissions, or several bitplanes at the same bit position from consecutive frames can be combined to send. To explain how to extend a base code for bitplane with different level of spatial correlations, assume H is reduced to be a full rank parity-check matrix by Gaussian elimination. For Case (1.), splitting of columns is needed to extend the number of columns of H to n. Let q be the splitting factor, then q = R The new LDPC code has a parity-check matrix with dimension of x nR T - r 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.1: Ty D e I EG-LDPC base code parameters Code Len. Synd. Len. Min. Dist. Density 65535 6560 257 0.0039 16383 2186 129 0.0078 For Case (3.), column puncturing on the incidence matrix is needed. Let q be the number of columns needs to be deleted from H, then q = Lf~J — n, write q in radix exp(s) form as follows: q = < 5 o + 5\2S + ■ • • + 5k 2ks where, s is a positive number and GF(2S ) is the ground field used in the construction of the base code, then the code with desired rate can be obtained by iteratively delet ing exp((i + j).s) columns for each non-zero j-th binary bit in 0 < 5i < 2s, where 0 < j < s. In these iterations, all-zero rows are removed from the punctured parity- checked matrix. The new LDPC code has columns close to n and has a corresponding number of rows which can explicitly computed, and the compression ratio is close to the desired ratio R. This construction of extended type-I EG-LDPC codes guarantees that these codes have a good structure for efficient syndrome generation. In the de sign of our LDPC codes, the PARI/GP package [59] is used for operations of finite geometry and generation of incidence matrices. Table 3.1 shows the code generation parameters, and these two codes are used as the base codes in this experimental study, the selection of (0, s)-th order type I EG-LDPC code is due to its ease of implementa tion and the fact that an explicit formula for syndrome sequence length is available as exp3(s) — 1 [81, 52]. Optimal capsule size generally depends on signal characteristic and the number of transform level. For vehicle acoustics, Figure 3.5 shows that the compression per formance with different capsule sizes with varying bitrates. The SNR gains are com puted using the standard mean square error logarithm formula. As a comparison, two 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. — 6—0.5 bps —— 1.0 bps - a - 1.5 bps - ^ - 2.0 bps L U 0 T3 3 : $ 3C Z 25 1 2 4 8 16 32 64128 DISC. SPIHT Capsule Size Figure 3.5: SNR gain vs. capsule size other schemes’ performance curves are plotted. One is for DISCUS which compresses wavelet coefficients in a subband by subband fashion, and the other is for SPIHT with out Slepian-Wolf coding on bitplanes. From Figure 3.5, both DISCUS and SPIHT perform worse than S-SPIHT with capsule size of 1, and S-SPIHT with capsule size 8 yields a good performance for our field acoustic data set. Experimental data that follow are based on capsule size 8. Figure 3.6 shows the bit error probabilities at four most significant bitplanes, where the bitplanes are denoted by the bit position and the corresponding conditional en tropies (i.e. H (X\Y)) at these bitplanes are shown by a vertical line close to it, and only the first point in MSB uses the base code and all other codes are punctured ver sions of the base code. From Figure 3.6, as one would expect that the compression ratios increase from bitplanes at a more significant position to less significant posi tions. Furthermore, the bit error probabilities in a bitplane seem not depending on the bit positions of bitplanes, and they all have a sharp waterfall characteristic. We also compared EG-LDPC codes with some irregular LDPC code [51] in S-SPIHT as that one example for coding the MSB bitplane is shown in Figure 3.6. Although the ir regular gallager code (IGC) performs well at codes with long lengths, its performance 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. MSB 9 (Base) 3rd 4th Pune.) » (Pune.) .-3 to Q ) C L ID C O (Pune.) > v-6 0.5 Figure 3.6: Bitplanes coding performance using EG-LDPC codes degrades to close to that of EG-LDPC when shorter codes are used. For some appli cations like acoustic tracking, bitplanes have an average length only in the order of 10000. We obtained compression rates within 0.05 bits of Slepian-Wolf limits. Based on our experiments using 7 acoustic sensors for vehicle detection, the pro posed codec outperforms both the subband approach, where each subband is Slepian- Wolf coded without set-partitioning, and the intra-bitplane approach, where the bit planes are intra-coded after the set-partitioning process. Figure 3.7 shows three curves from our codec with 1,3 and 6 refplanes used during decoding, and two curves for the subband approach and intra-bitplane approach. Table 3.2 shows timing of the encoding and decoding in two sensor processors where the hybrid decoding method uses SPA 3 followed by BF 4. C8051 F020 MCU is used for BF decoding algorithm and PXA 255 of PASTA node [38] for SPA. To trade-off the decoding performance with the processing efficiency, the hybrid approach uses SPA with fewer iterations followed by BF and it is shown in the last column in Table 3.2. These times makes our scheme amenable to resource constraints. 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 Q . Q . ® — 1 Refplane - * - 3 Refplanes -® -6 Refplanes -^ B itp la n e — Subband 1 25 30 35 40 SNR Figure 3.7: Codec rate-distortion curves Table 3.2: Encoding and decoding timing Time (ms) Enc. BF 4 BF 8 SPA 3 SPA 6 Hyb. 8051 580 200 260 X X X PXA 120 35 50 120 180 160 3.5 Conclusions In this chapter, a new approach for exploiting correlations in sensor readings is pro posed, and it uses a wavelet set partitioning algorithm to reorganize coefficients so that the conditional entropy of bitplanes extracted from the source given the bitplanes from the side information is decreased for better compression performance. This scheme can be made progressive, and an application therefore can control the reconstructed signal SNR. Since applications in sensor networks normally require that sensor readings of one node being made available to near-by sensors for collaborative processing, we next study the source broadcasting problem where one sensor independently compresses its readings which is to send to multiple receivers. We will also study the limit rates of this source broadcasting problem and the associated issues. 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 Wavelet Source Broadcast for Sensor Array Data In-network processing is an appealing approach for resource conservation in sensor network applications. To collaboratively process a signal, sensor readings have to be compressed, transmitted and processed locally within a sensor network. We propose a scheme to compress sensor array data in which one sensor has to send its readings to multiple neighbor sensors. This scheme uses wavelet transform to compact signal energy into a few large coefficients, and bitplanes extracted from these coefficients are correlated to bitplanes of neighbor sensors. Then the proposed scheme uses low- density parity-check code based Slepian-Wolf codes to compress these bitplanes with flexible rates and each receiver uses the sum-product algorithm to decode the progres sive transmission. Furthermore, a priori correlation statistics are not required, and the reception rate of each receiver can achieve close to conditional entropy with regard to the sender. 4.1 Introduction By the nature of sensor networks and its applications, processing should be performed locally. By this way, communications can be reduced to the minimal possible level 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. since only processed results instead of raw data are sent to distant nodes. To enable this in-network processing, local compression is important. In the target tracking ap plication, in order to locate one target, at least two line-of-bearings (LOB) are needed, i.e. the intersection of two LOBs is the target position, and one sensor data needs to be sent to multiple LOB fusion nodes. These data are spatially and temporally correlated. Wavelet based compression with distributed source coding approach can exploit these correlations [91, 92], Figure 4.1(a) shows that two LOB fusion nodes (Y and Z) simultaneously track a target. In this chapter, we use “a sensor node” or “a random variable” interchange ably. These fusion nodes are also equipped with an acoustic sensor and each fusion node needs two sensor data (for one target tracking) or more sensor data (for better accuracy or multiple target tracking) to compute one LOB. Sensor X in 4.1(a) needs to send its readings to both fusion node Y and fusion node Z. Figure 4.1(b) shows the coding setup studied in this chapter. There is an encoder function f n{xn) which upon observing xn from random variable X n generates the compressed version of xn. There are two decoding functions, gn for Y and hn for Z. gn which have yn available and received f n(xn) generates the decoded version of xn, and hn which have zn available and received f n(xn) generates the decoded version of xn. By Slepian-Wolf theorem [80], the achievable rate to transmit X to Y is bounded below by H(X\Y) provided that the correlation statistic is known to both X and Y, and the achievable rate to transmit X to Z is bounded below by H(X\Z) when X and Z know the correlation statistics between them. The problem on how to efficiently broadcast one source to multiple receivers have not received much attention despite the renewed interests on the distributed source coding for sensor networks [62, 106], distributed video coding [31] and correlated image compression [85], 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Target LOB (X:Y) Sensor Y X:Z) Sensor X Sensor Z (a) LOB Computation Decoder (Y) X — Encoder (X) Broadcast — Decoder(Z) (b) Source broadcast Coding Figure 4.1: Source broadcast in wireless sensor networks In general, if receivers do not directly have the side-information, the problem to aggregate multiple sources, say X, Y and Z, has been studied in [103] and later a generalization in [17]. To aggregate random variables X, Y, Z, the approach in [103] is to broadcast Y in an intra mode to X, Z and the fusion center at a rate bounded below by H(Y) and send source X to the center at a rate bounded below by H(X\Y) given the broadcasted Y and send source Z to the center at a rate bounded below by H(Z\Y, X ) given the broadcasted Y and received X . The achievable total rate is then bounded below by total rate > H(Y) + H(X\Y) + H (Z\Y , X ) which equals to the joint entropy H(X, Y, Z). When transmissions have to be done inside the network as necessary for In-network processing, the source broadcast is needed in which for the three-node case, source X can broadcast its readings to both Y and Z at rate lower bounded by the maximum conditional entropy of H(X\Y) and H(X\Z). 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For this broadcast source coding problem with one transmitter X and two receivers Y and Z, there are three possible alternative solutions as follows: 1) Use the techniques from [62,106] for each receiver, and two transmissions are required and the combined rate is lower bounded by H(X\Y) + H(X\Z) at the transmitter and a reception rate of H(X\Y) for Receiver Y and a reception rate of H(X\Z) for Receiver Z. 2) Use excessive broadcast rate with only one transmission, the achievable rate is max{H(X\Y),H(X\Z)} (cf. Corollary 1 of [76]) and the reception rate for both receivers is max{H(X\Y), H(X\Z)}. The advantage of 2) over 1) is one less transmission. There is possible a third solution which is our interest in this chapter. 3) Use source broadcast with one transmission with rates between mm{H(X\Y),H(X\Z)} and max{H(X\Y),H(X\Z)} for different receivers and the achievable reception rate for Y is H(X\Y) and that for Zis H(X\Z). For one transmitter (X)/two receiver (Y and Z) setting, on one hand, the statistic for computing H(X\Y) or H(X\Z) can not or too costly to estimate a priori. On the other hand, different receivers generally have different levels of correlations to the transmitter (i.e. H(X\Y) ^ H{X\Z)). In one extreme, if the transmitter selects a rate of min{Tf(X|Y), H(X\Z)}, the receiver with less correlation to the source may not be able to decode the message which renders its effective rate being 0 (an effec tive rate is a rate that the receiver is actually achieved, and it is not the rate that the transmitter sends). In the other extreme, if the transmitter selects an excessive rate of 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. max{H(X\Y), H(X\Z)}, the rate to the receiver with the higher level of correlation is artificially increased. In [24,78], a scheme using sequential random bits is proposed, and it has been shown to be optimal. In [22], a similar idea as that in [24, 78] has been studied. In [101], the broadcast version of the Slepian-Wolf coding problem has been stud ied and the linear block code is used for this problem. The work in this chapter is inspired by these ideas. We propose a low-density parity-check (LDPC) code based broadcast Slepian-Wolf scheme with wavelet compression, called Slepian-Wolf Wavelet Based Iterative Set Partitioning (SW-WISP) scheme to compress sensor data. SW- WISP uses a set-partitioning algorithm to extract large wavelet coefficients after the time-domain samples having been transformed by a wavelet transform. Significance bits generated during the set-partitioning process are encoded in an intra-mode, and they are used by receivers to identify the position information of these large coeffi cients. Bitplanes extracted from these large coefficients are encoded in an inter-mode in terms of syndrome bits using low-density parity-check codes using an embedded source broadcast algorithm. Since large coefficients contain most of the signal en ergy, bit rates are allocated to significant bits of large coefficients; therefore, excellent reconstructed signal-noise ratio of our codec has been obtained. There are two places which require a priori the correlation statistic: 1) encoder needs this information to derive the compression rate and 2) decoders need this in formation to initialize the prior probabilities for the sum-product decoding algorithm. WS-WISP does not require a priori the correlation statistics. For a bitplane of width n, the encoder selects a LDPC code so that it has a parity-check matrix which has a size of [Yn] x n and ma,x{H(X\Y), H(X\Z)} < r < H{X) where [.] is the integer ceiling function. The embedded source broadcast approach allows both receivers to decode a respective version of X upto a given error probability. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To run the sum-product algorithm, the decoder needs the crossover probability to initialize the prior probabilities of variable nodes of the bipartite graph corresponding to the underlying parity-check matrix. It is important to properly initialize these prior probabilities which can affect 1) convergence time of the sum-product algorithm if it does converge and 2) the decoded error probability. WS-WISP utilizes the number of received syndrome bits to set a proper prior probability for a variable node. 4.1.1 Problem and Our Contributions As explained above, this prior correlation estimation in distributed source coding of sensors readings can be costly, and it could diminish the benefit of “independent en coding and joint decoding” provided by the Slepian-Wolf theorem. This problem is even more pronounced when correlations of multiple sensors have to be estimated for the in-network processing energy saving principle. In this chapter, for the first time since this source broadcast problem has been iden tified [24, 78], we have developed a practical codec which can compress the readings in a broadcast setting. This proposed codec can allow receivers correlated at different levels to receive the data at their respective rates which on average are off the limit rates only by about 0.05 bits. Since this codec uses wavelet set-partitioning and the spatial correlation is exploited at the level of bitplanes from significant coefficients, both spatial and temporal are effectively exploited by the codec for this source broad cast problem. 4.2 Source Broadcast Model and Codec Design In this chapter, we use one transmitter/two receiver model. However, our scheme can be easily extended to the model for one transmitter/multiple receivers. We assume that 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X is correlated to both Y and Z, and this correlation will be defined quantitatively shortly. However, neither X nor any of its receivers knows the correlation statistic. We assume that a receiver stops its reception once X is correctly decoded or the decoded error probability is sufficiently small. Since we use low-density parity-check codes with a sum-product decoding algorithm, the receiver knows this error probability by checking the number of simultaneous satisfied check equations. For a more corre lated receiver (to the source), it is normally able to receive the same information in less time as compared to a coarser correlated receiver. Therefore, the better correlated receiver stops its reception earlier. After set-partitioning on wavelet coefficients, bitplanes are generated along with significance bits. Significance bits are used by the receivers to generate the correlated side-information bitplanes based on its own readings (details are presented in Sec tion 4.3). Therefore, our model has binary symmetric sources, and the same binary alphabet is assumed for both the transmitter and these two receivers. Let X, Y and Z denote these correlated bitplanes, without of loss generality, we further assume that the crossover probability py between X and Y is smaller than the crossover probability pz between X and Z, i.e., X is more correlated to Y than Z. In this chapter, the binning approach is taken for our source broadcast problem. This problem is modeled as follows. There are two sets of bins, one for receiver Y and one for receiver Z. Assume that a source message has n bits, and the number of bins for Y is My and that for Z is Mz. A code C is defined in (4.1) for the source broadcast problem. C = (y, My, Z, Mz, n) (4.1) 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the definition given in (4.1), the code C consists of an encoding function: x:m £{ 0, l}r e ^ (s„, sz) € {0,1}M " x {0, l} M z and two decoding functions: f y ■ ({0,1 }My,y) ■ -> {o, l}n f y ( s y) = Xy fz ■ ({o, 1}M % z) !-» • {0, l} n f z (sz ) = xz A broadcast channel [12] can be formulated for this source broadcast coding prob lem. In this chapter, the common information to the two receivers is the common information they actually received, not the common information the transmitter sends. For the Slepian-Wolf binning approach, the capacity of this channel is directly related to how the best code can distinguish a codeword from a designated bin. Furthermore, classic results on the channel capacity region of broadcast channels, i.e. Section 14.6 of [14], dictates the compression rates of our problem as a similar connection on one transmitter/one receiver model has been established in [91]. In particular, our cor related bitplane broadcast problem follows the degraded binary symmetric broadcast channel model. Roughly speaking, the connection between Cover’s broadcast channel [12] and our source broadcast model lies in the following two points: 1) the difference between H (X ) and the compression rate R is the rate of a code over this source broad cast channel note that H(X) — R > 0. The intuition is that the reduction on bitrate of transmission of source X comes from this source broadcast channel. 2) the capacity of a broadcast channel corresponds to the mutual information between the source and the side information. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We assume that the virtual channel between X and Z has a crossover probability P»=Py*Q-= Py( 1 - g) + (1 - Py)q for some q G [0,0.5], i.e., the channel between X and Z is a cascade of a new chan nel with crossover probability q and the channel connecting X and Y. We have the following proposition. Proposition 4.1: Let U be an auxiliary random variable which has a crossover probability (3 G [0,0.5] with X, the average bin width (Cy, Cz) where Cy for receiver Y and Cz for receiver Z for this one transmitter/two receiver model must satisfy the following conditions: Since the capacity is directly related to the bin width from Slepian-Wolf coding with either random binning or deterministic binning, this proposition is a direct result from the theorem in [42] or the discussion in (14.6.2) of [14] for our equivalent source broadcast channel. Proposition 4.1 can help determine the compression rates for the source broadcast problem. We have the following theorem on compression rates for our one transmitter/two receiver model. Theorem 4.1: In the degraded binary source broadcast problem, the following com pression rate (Ry, Rz) for receiver Y and Z is simultaneously achievable for any e > 0. Cz < I(U, Z) — H(U) — H(pz * P) Cy - C z < I(X, Y\U) = H(J3 * py) - H(py) C y < l(X ,Y ) (4.2) (4.3) (4.4) Ry > H(X\Y) + e Rz > H{X\Z) + e 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Proof: In this proof, the superposition binning coding is used. In this super position binning method, the codebook for Z is generated by random binning with \nH{X\Z)\ bins, where, |_ .J is the floor function, and the codebook for Y is generated by random merging these bins of Z so that the codebook has a total of \nH{X\Y)\ bins. For a given message of length n, X sends the index of the bin from codebook of Z to which the message belongs. Both Y and Z can determine the bin index based on their respective codebook, and the common information both receivers can obtain is the decoded version xz although the better receiver Y does not actually decode for it. The private information Y can obtain a better estimate xy. Since the codebook of Y has less bins, it can achieve a better compression rate. To show that (H(X\Z), H(X\Y)) is achievable, consider a special case (3 = 0 in Proposition 4.1, i.e. U = X. To transmit a codeword to Z, we need to send the bin index corresponding to this codeword. Since only typical sequence of X is sufficient to evaluate for the error probability, the number of bits required to represent a bin is as follows: n{H(X) - Cx) > nH(X\Z) Following the standard error probability estimation technique (i.e. the proof of Theo rem 14.6.2 of [14]), when n is large enough, the decoding error probability approaches zero provided that a maximum likelihood decoding algorithm (i.e. satisfying condition (14) of [80]) is used as in the proof of Theorem 3 of [80]. By (4.4) of Proposition 4.1, for the above U, the corresponding number of bits required to represent a bin for Y is as follows: n(H(X) - Cy) > n(H(X) - I(X,Y )) = nH(X\Y) 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. What left to complete this proof is to show that (4.3) in Proposition 4.1 is satisfied. For the above U, the additional information Y (due to that X is more correlated to Y than Z) can receive subjects to the following constraint: I(X, Y ) - I(X, Z ) < I(X, Y\U) (4.5) Equation 4.5 holds since I(X,Y\U) = I(X,Y) This is due to the fact that X = U. ■ Theorem 4.1 shows that the compression rate pair (H(X\Y), H(X\Z )) is achiev able under the superposition coding scheme. We next show that a practical code using embedded irregular Gallagar code (IGC) [51] allows both receivers to simultaneously achieve their respective minimum compression rate. Algorithm 4.1 shows encoding and decoding of this embedded coding scheme. Algorithm 4.1: For a given random parity-check matrix H of an IGC with m parity-check equations and n columns, and a message vector x. Assume that H is known to both transmitter and receivers, so is the random seed for the generation of H used by the encoder. 1. encoding: compute syndrome s — Hx, and it periodically picks and transmits a block of k bits from syndrome vector s. 2. decoding: upon receiving a new block, update its decoding bipartite graph based on received blocks and run the Sum-Product LDPC decoding algorithm using its side information. 3. encoding exit: either all bits of s are transmitted or it receives a confirmation from both Y and Z to indicate that encoding is finished. 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4. decoding exit: if the number of equations which are satisfied after a given num ber of iterations is greater than a threshold, return the decoded x and exit, other wise, wait for more blocks. □ Algorithm 4.1 is called embedded source broadcast since the codewords recognizable by the better receiver Y are actually embedded into the codeword used by X from the codebook of the coarser correlated receiver Z. To see this, consider the partial parity-check matrix from H with the syndrome bits corresponding to the bits that X transmits, since Y is a better receiver and it needs less syndrome bits, it must exit this transmission earlier than Z. Therefore, a codeword for Y is one segment of some codeword for Z. The number of parity-check equations is set sufficiently large, and it should be greater than nmax{H(X\Y), H(X\Z)} and this number is also upper bounded by nH(X). There is no need to know the correlation statistics in advance in Algo rithm 4.1. The parity-check matrix H can be selected offline with proper girth for better decoding performance, and the receivers use a derived parity-check matrix (i.e. a partial parity-check matrix of H ) for decoding. This derived parity-check matrix has a girth at least as large as that of H since removing rows of H to derive the decoding parity-check matrix could remove the shortest cycles, and this process never creates a new cycle. The rates given by Theorem 4.1 are asymptotic, and the actual rates which can be achieved under Algorithm 4.1 also depend on the transmission block length. That is the rate pair (H(X\Y), H(X\Z )) is achievable when codeword size goes to infinite and the transmission block length of syndrome bits goes to one bit. Experimental results show that fairly close rates to (H(X\Y), H(X\Z)) can be achieved in practice. 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 Our Proposed Codec We first briefly describe the general set-partitioning method on wavelet coefficients. This set-partitioning approach uses a significance test on wavelet coefficients to parti tion them into the significant set and the insignificant set where the insignificant coef ficients form zero-trees. Bitplanes can be extracted from significant coefficients. The significance bits, which are used by the decoder to recover positions of significant co efficients in the wavelet coefficient tree, are directly encoded into a bitstream. Signs of coefficients form separate bitplanes. The process of SW-WISP is as follows. SW-WISP performs set-partitioning on wavelet coefficients, and the significance bits of X are intra-coded and broadcast (with out compression) to both receivers. Upon reception of the significance bits of X, the two receivers Y and Z apply these bits to their wavelet coefficients to perform set- partitioning to extract bitplanes. By Proposition 4.2 (below), we know these bitplanes between X and Y and these between X and Z are correlated. Notation 4.1: 1) denote bx(i), by(i) and bz(i) as the i-th bitplanes (i.e. bits gener ated during the i-th iteration) of X, Y and Z, respectively. 2) denote bx(i,j), by(i,j ) and bz(i,j) as the j-th bit of bx(i), by(i) and bz{i), respectively. 3) denote sy(i) and sz(i) as the syndrome bits computed from the i-th bitplanes bx(i) for Y and Z, re spectively. 4) denote by(i) and bz(i) as the reconstructed bx(i) by receiver Y and Z, respectively. 5) denote by(i,j) and bz(i, j) as the j-th bit of by(i, j) and bz(i,j), respec tively. Proposition 4.2: Let N be the bit position of the sign bitplane (highest bitplane), bitplane bx(i) and by(i) (1 < i < N) have the same length; furthermore, by(i) is extracted from these coefficients whose position information in F ’s tree is the same as that of corresponding coefficients in X ’s tree. The same holds true between X and Z. 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Proof: The proof uses mathematical induction on zero-trees of set-partitioning. Denote that Cx(i) = {cj|SIG, (c,&( 1 « (i - 1))) = bx(i,j)} where, SIG means that Cj is a significant coefficient during the i-th set-partitioning iteration on coefficients of X, and and ‘<C’ are the binary AND and SHIFT (to left) bit operators, respectively. Let f(cj,i) be a function mapping c, identified as significant at the i-th iteration to its position in a tree. We can construct a similar set on Y ’s coefficients as follows: C y(i) = {c'jlfic'j) = f ( cj ) } Since both set-partitioning on X and Y start from a single root node and the paths from the root node are the same, these coefficients identified must have the same position information. Therefore, after the first iteration, both X and Y have the same set of zero trees. Assume that all first k iterations (including the k-th iteration) generate the same set of zero-trees for both X and Y. At the k-th bit position, there have been N — k + 1 iterations performed. Since the set-partitioning on Y and coefficient identifying using significance bits from X start from the same set of zero-trees at the (k+l)-th iteration, for each root of a zero-tree, they should yield the same set of zero-trees by following the same argument above but on each root of zero-tree instead of the tree root at the first iteration. Therefore, after the (k+l)-th iteration, the same set of zero-trees are formed at both X and Y . By induction, we have shown that Cy(i) = Cx(i) (1 < i < N) which is exactly the set constructed by applying significance bits of X to the wavelet coefficient tree of Y. For the second statement on X and Z, the same induction proof applies on Z ’s tree. ■ 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. By Proposition 4.2, the bitplanes between X and Y or those between X and Z are correlated if the time-domain signal is correlated. Our codec is given by Algorithm 4.2 which exploits both spatial and temporal correlations of sensor data. Let K be the number of iterations SW-WISP performs. Algorithm 4.2: - SW-WISP(K) 1) initialization: transform time-domain signals to obtain Y and Z for the purpose of generating correlated side-information. 2) iteration Vi (N — K + 1 < i < N) 2a) for each bit position i, Y runs the sum-product decoding algorithm with 2b) Y construct set Cy(i) using significance bits of X\ Z construct set Cz{i) using significance bits of X. 3) reconstruction Y sets all coefficients not in Cy(N — K + 1 ) to zero and perform inverse transform to recover time domain signal of X, and Z sets all coefficients syndrome bits sy(i) and side information bitplane by(i) to obtain by(i), and Z does the same to obtain bz{i). 2c) Y updates coefficients in Cy(i) using newly decoded by(i) by the following formula: cj + (1 « (* - 1)) if by(i,j) ^ 0 Cj o th erw ise V c,- < E Cy(i) Z updates coefficients in Cz(i) using newly decoded bz(i)) by the following formula: Cj + (1 « (i - 1)) if bz(i,j) ^ 0 Cj otherwise \/cj € Cz(i) 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. not in CZ(N — K + 1) to zero and perform inverse transform to recover time domain signal of X. □ 4.4 Experimental Results In this experimental study, we will first examine rate-distortion performance of SW- WISP compared to two other schemes. The first scheme is the subband coding scheme where subband coefficients are directly compressed by LDPC based Slepian-Wolf codes. The second scheme is an intra-code scheme where set-partitioning is performed on wavelet coefficients and bitplanes extracted during set-partitioning are sent in an intra-mode directly. For the subband coding scheme, since the correlations to two re ceivers are different, we use individual correlation statistic when selecting the rates to code subbands. In this scheme, bitplanes from subband coefficients are extracted with out using set-partitioning. We denote the first scheme by “Subband”, and denote the second scheme - the bitplane intra-coding set-partitioning scheme by “Bitplane”. In all cases, we kept the decoding error probability close to zero if the distributed source coding is used, i.e. a rate greater than the Slepian-Wolf limit rate is used. For the comparison, the significance bits from set-partitioning of SW-WISP can also take into account when determining the bitrates. We have a data set which contains vehicle acoustic data from a sensor array of size 7. In our experiments, we selected three sensors using GPS ground truth with which two of them are physically close and one of them is the better correlated receiver. The third which is the coarsely correlated receiver is further from these two. Figure 4.2(a) has three rate-distortion curves for each of these three schemes for the better correlated receiver and Figure 4.2(b) has three curves rate-distortion curves for each of these three schemes for the other receiver. The SNR gain of SW-WISP is close to 5 dB compared 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ° 30 -*-W S -W IS P . -»-Subband — Bitplane Bitrate (bits-per-sample) o 30 -o-W S-W ISR. — —Subband — Bitplane Bitrate (bits-per-sample) (a) Better correlated receiver (b) Coarser correlated receiver Figure 4.2: Rate-Distortion performance comparison to the “Subband” curve, when the bitrate is close to 3 bits-per-sample (bps). Also from these figures, inter-coding schemes (i.e. WS-WISP and “Subband”) using Slepian- Wolf coding perform better than the intra-coding scheme, and only when correlation is high and bitrate is low, does “subband” perform close to SW-WISP. All other cases, SW-WISP performs much better. In particular, for moderate bitrates which are used in practice by many applications, SW-WISP performs significantly better as compared to the other two schemes. We next examine the source broadcast of SW-WISP with respect the limit rates given in Theorem 4.1. There is one more complication involved as compared to dis tributed source coding schemes where side information is known. That is the trans mission block length. So we compare three different block lengths and one of them is the bitwise SW-WISP which uses one bit per transmission. This bitwise version is not practical but it can serve as a baseline for comparison. Figure 4.3 shows four curves and the outer envelope curve is for the theoretic limit rates. This figure tells that SW-WISP with different transmission block lengths are 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -e-B Ik Length 1 — — Blk Length 256 — Blk Length 512 — Limit Rates c 0.8 < d 0.6 o 0 .4 a .0 .2 6000 Codeword Length (bits) 8000 2000 4000 Codeword Figure 4.3: Rate comparison of SW-WISP with different block lengths fairly close this limit-rate curve with practical codeword length. Figure 4.3 also tells that the transmission block length does not affect much the compression rate as both curves with large transmission block lengths are close to the bitwise curve. Remark 4.1: There are three remarks on the transmission block length from our experimental study: 1) unless the required syndrome length from the true correlation is multiples of transmission block length, extra bits are transmitted which degrades the compression performance; however, these extra bits yield a quicker convergence speed of the SPA decoding algorithm. This leads to savings on processing. 2) in general, small transmission block length yields better approximation to the limit rates. How ever, it may cause high decoding processing overhead since the decoding algorithm may have to perform many partial decoding. 3) practical transmission block lengths (i.e. 256 bits) used in our study give good enough performance and the decoding cost is also kept low. 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5 Conclusions In this chapter, we have proposed a compression scheme for source broadcast of sensor data in wireless sensor networks. It does not require knowledge of correlation statis tics of the sources and achieves high compression performance by exploiting both the spatial and temporal correlation present in sensor data. Compressed data is sensitive to channel errors. In order to prevent communication channels from degrading signal-noise ratio of sensor data. We next examine how to transmit distributed source coded sensor data in an energy efficient way. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Power Aware Compression of Sensor Data In this chapter, a novel coding scheme is proposed for applications over wireless mi crosensor networks to meet both wireless link bandwidth and node energy constraints. First, an analysis is performed on energy efficiency of coding schemes in wireless mi crosensor networks. Based on this analysis, we devise a power aware coding scheme, called EESPIHT, which exploits the spatio-temporal correlation of multiple sensor readings. It is also made resilient to channel errors: it selects an error correcting code based on source coding information and transmission power so that energy dissipation is minimized. Experimental results on LINUX implementation and simulation results based on OPNET using field data sets show that the proposed scheme can reduce en ergy on communication by more than 60% and maintain a Signal-to-Noise ratio (SNR) gain of 24 dB or better compared to non-coded case. 5.1 Introduction Radios on sensor nodes typically operate at low power with short ranges, and usually have limited bandwidth. Therefore, data compression is useful in sensor network ap plications. Since sensors are powered by batteries, compression also reduces energy 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. dissipation, as energy for communicating a bit is at least an order of magnitude higher than that for one computation operation [77]. However, compressed data are more sensitive to bit errors. There are three ways to protect bits, namely, channel coding, adaptation of radio power, and automatic repeat request (ARQ). We here propose to protect bits based on their loss sensitivity. In order to provide levels of protection on compressed bits, we make use of these three methods and select a proper combination which minimizes the energy dissipation. For practical systems, it has been shown that joint source-channel coding may re duce distortion, improve efficiency as well as reduce delay [32]. In this chapter, the targeted scheme is for applications over wireless microsensor networks. We combine channel and source coding, ARQ with dynamic power allocation to reduce energy dissipation while maintaining a required SNR gain of reconstructed signals to meet application fidelity requirements. It has been shown that for many sensor network ap plications that sensor readings among neighbor nodes are both spatially and temporally correlated. When spatial correlation is strong, distributed source coding schemes [63] alone can yield high compression ratio. When temporal correlation is strong, SPIHT [73] based scheme (for 1-D data) can also yield high compression ratio. S-SPIHT [91, 87] works well when both spatial and temporal correlations are present. In this chapter, SPIHT and S-SPIHT are used as source codecs, and a power aware error protection scheme is introduced to these codecs. In [6 ], the importance of sym bols is discriminated by designing a modulation scheme with different signal distances on its constellation. However, in this chapter, we use rate compatible punctured convo lutional codes (RCPC) for symbols with different levels of importance. ARQ adopted in this work is Type-I hybrid ARQ protocol [6 8 ], and it is based on ITOH algorithm [108]. We have also derived a lower bound on energy savings and an upper bound on the SNR loss of the proposed scheme. 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.1.1 Problem Formulation and Our Contributions Error correcting problem and source coding problem are two problems which seem to have two different consequences on communication energy consumption. Source cod ing is to compress the data which reduces the data volume to be sent, hence it reduces the energy consumption. Error correcting coding is to introduce structured redundancy so that channel errors occurred during transmission can be correctly during decoding process, hence it increases the energy consumption for a given amount of information. Instead of studying the general problem, in this chapter, we are concentrated on the transmission energy conservation problem of spatio-temporally correlated sensor data from the error correcting and the source coding perspectives. Instead of examining error correcting problem and source coding problem sep arately, we study a more general energy optimization problem. We introduce three factors, namely, compression rate denoted by r, error correcting coding parameter de noted by A and transmission power denoted by r. The problem is to study on what levels these parameters should be set so that the overall communication energy dissi pation is kept at a minimum level. For n number of bits of the representation of the signal, we need to minimize the functional given in (5.1). E(n, SNR) = ([(A — l) 7i] + k)r (5.1) where, SNR is the signal-to-noise ratio of the target reconstructed signal. Our contributions in this chapter include a new coding scheme, which is aimed to save energy, for correlated sensor readings in sensor networks. Furthermore, to make these compressed bits error resilent, different level of protections are employed via channel coding, power allocation and ARQ in an energy optimized manner. Under this scheme, analytic results on SNR loss upper bound and energy savings lower bound are 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. given. Further experimental results show that energy savings of the proposed scheme can be as high as 75%. 5.2 Energy Efficiency of Coding In this section, we study energy efficiency issues of coding schemes in wireless mi crosensor networks. Results obtained here will guide us on devising a power aware coding scheme. Notation 5.1: Notations for this chapter are as follows: (1) radio is power con trolled and has a number of discrete power levels, and it may change its transmission power on different packets; however, the power is fixed at a level during a packet trans mission period; (2) on unrecoverable errors during channel decoding (found by IOTH algorithm) or missed erroneous bits (found by a physical layer checksum mechanism), ARQ protocol is activated by receiver so that a given link reliability level is obtained; (3) on reception, a signal power must be greater than a threshold 7 , and the probability of a signal to be captured is denoted by P( 7 , R), where R is the Tx-Rx separation distance; (4) binary modulation is used; (5) there is no packet loss in transmission, and only bit flips could occur. Note that these assumptions are based on practical systems, for example, the CC1000 radio on MICA2 Motes node supports different power states (i.e. the operating pow ers are different in different duty cycle rates), and they are programmable as in ( 1) in above. Furthermore, ARQ in (2) has been adopted by many standards, e.g. IEEE 802.16, and (3) is more reasonable assumption for a cluttered environment. Capture threshold 7 can be represented as follows: 7 = z(I + N ), where z is the minimum 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. carrier to interference ratio needed for successful capture, and, I and N are the inter ference and noise of the channel, respectively. For a Rayleigh fading channel, capture probability can be approximated as follows: where a'(R) is root mean square (RMS) of received signal amplitudes at a Tx-Rx separation distance of R. In general, capture probability depends on received signal strength and it is indirectly related to transmission power which is inversely propor tional to some power of Tx-Rx distance. This signal strength is normally available in terms of sampled received signal strength indication (RSSI) provided by many radios [9]. Capture probability also depends on the modulation scheme, interference from other co-channel users and noise [35, 113]. In this analysis, we are focused on physical layer energy issues and consider three factors that affect communication energy, namely, antenna emission power, channel coding and ARQ. The model used in this chapter is shown in Figure 5.1. Note that the reference Y is available to the joint decoding process which can be exploited by the channel coding process and/or source coding process; however, its use for spatio- temporal decoding process is treated in this chapter. Putting them together, we have a picture as shown in Figure 5.2 in which there are three axes corresponding to these three factors, and reciprocal relationships between any two of these three factors are approximately inverse. As more redundancy of channel coding is introduced (i.e. lower coding rate), lower radio transmission power is required to maintain a given reception quality, and less ARQ requests are issued; likewise, as radio power increases, the number of ARQ requests decreases, and lower redundancy of channel coding is necessary. 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Joint Encoding Source Channel Encoding Output Spatio-Temporal Encoding Power Allocation Noise Interference Joint Decoding Channel Decoding Spatio-Temporal Decoding Side Information Y Figure 5.1: System model Channel Coding mW Repetition Count W ARQ E R Figure 5.2: Relationships of three factors Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.2.1 Energy Sayings on Coding Before this analysis, we give the following notations used in this chapter. Notation 5.2: (1) Packet: Vb for an uncoded packet, V for a coded packet. (2) Length of a packet: L bits. (3) Radio Tx power: Zb dBm for channel without coding and Z dBm for channel with coding, where Zb = otbF C + /? and Z = a R r + 0 and r is propagation power factor between 2 to 7 based on a communication environment. (4) Channel code rate: j, where A is a rational number and greater than 1, V is expanded to XL symbols. (5) Channel packet reliability probability: Pn , this is targeted probability at which a packet successfully arrives at a receiver. (6 ) Channel symbol rate: B symbols per second, transceiver can clock in B symbols per second into the channel at power Z dBm. (7) Compression ratio of source coding: n, V is reduced to nL bits, where n is rational and less than 1 so that nL is an integer. (8 ) Nominal channel bit error rate (BER): Eb for a channel without coding and E for a channel with coding. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (9) Channel BER: eb for channel without coding and e for channel with coding, where eb = ( l - P ( % R)) + P(1 ,R )E b and e = ( l - P ( y , R ) ) + P{y,R )E (10) Coding scheme: < 3 > , with < f > , Eb i — ► E; L i — > XL, we also use $ to denote the channel with coding scheme $ when the context is clear, when it is necessary we may also use to indicate a channel with channel code of rate j. A baseline is taken as the case where no channel or source coding has been ap plied, in order for a channel to achieve the given PN, ARQ protocol has to be acti vated. Denote the probability of an unsuccessful delivery of Vb as eb and the number of retransmission under baseline as Nb. We have the following lemma. Lemma 5.1: Assume bit flips in Vb follows Bernoulli distribution, the following holds for packet Vb. eb = 1 - (1 - eb)L and Nb = loge6(l - Pn ) Proof of Lemma 1 is straightforward under assumption that a bit flip in Vb is indepen dent and by noticing the fact that P N = 1 - ebN b 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. From Lemma 1, energy consumed under baseline model is as follows: Energy consumed under scheme $ is as follows: n® = /A nL iogel1 ~ Pn) where e = 1 — ( 1 — e)nXL. Definition 5.1: Coding Energy Ratio of < f > w.r.t. baseline: = n$/Ilft. Note, when channel model is clear in a context, we use n instead of /c$ or K\. We use this notation 0(f(x)), the set 9(x) lino7TT = C ° > ° zi-> 0 J[x) Lemma 5.2: With regard to the coding energy ratio, when ARQ is used to obtain packet reliability probability Pn , the following holds for < E > : K — n \ Z In (Leb) + A Zf, ln(nALe) € 0(61 + 6) (5.2) where, 5b = (1 - eb)L - 1 + Lef e S — ( 1 — e)nXL — 1 + nXLe Sb A = Let (Left) 2 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Proof: First show the following claim is correct: < 5 and 6b are positive and are in 0 (e 2) and 0 (eb2), respectively. Let f ( x ) = (1 — x)n — 1 + nx, for a; G (0 ,1) and its first order derivative is as follows: /'(*) = n( 1 - (1 - x)"-1) The above claim holds by noticing that f'{x) is positive for x G (0 ,1) and / ( 0) = 0. In what follows, we use Talyor expansion on function f(x ) = ln(x) in neighbor hoods of Leb and nXLe, we have nXZ In{Leb) + A + 0(61) K ~ Zb ln(nALe) + 0 ( 6 ) Relation ( 5.2) holds by noticing the following fact: a + x a b + x b G 0(x) when a > 0 and b > 0 as these in our case. ■ By Lemma 5.2, we can see that k is affected by the following factors: (1) source coding ratio n, (2) channel coding rate j, (3) channel coding gain which affects the radio transmission power and channel BER. In Lemma 5.2, the Taylor approximation truncation on ln(Leb) does not really affect our analysis as that can be seen later in this section. It is normally difficult to measure the energy gain of channel coding; however, 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. from Lemma 5.2, channel coding can save energy when the coding gain 1 is greater than A . Property 5.1: For joint-coding scheme <hy, without using ARQ, suppose that a channel code can achieve packet reliability probability Py with rate jy, then the fol lowing relation about Ky holds: X'nZ In (Leb) + A K y Zb ln(l — PN) e 0(53) (5.3) where 5b and A are defined as those in Lemma 5.2. The proof is similar to that of Lemma 5.2 and Relation ( 5.3) holds by noticing the following fact: a + x a . € 0{x) b b where b > 0 as in our case. In the remainder of this subsection, we investigate how the proposed scheme saves communication energy in wireless microsensor networks. This method can be applied to both SPIHT and S-SPIHT (We use EESPIHT to denote the proposed scheme which is derived from S-SPIHT with error protection at both bit level and packet level, we may also use jointly coded SPIHT to refer to a scheme derived from SPIHT with the same error protection as that in EESPIHT). The techniques used are: (1) use of unequal error protection (UEP) where the relative importance of the bits in an input bitstream is made available by the source codec, (2 ) use of bit puncturing in convolutional code to adjust RCPC code rate to achieve UEP, (3) rate selection based on an energy criterion, (4) use of ITOH algorithm [108] by a receiver for sending ARQ request. In (4), ARQ is adaptive based on packet importance rather than unacknowledged packets as that is 1 Coding gain is defined as the ratio between SNR (uncoded) and SNR (coded) for equal symbol error probability after decoding under a common channel condition. 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in regular ARQ. In other words, “important” packets trigger receivers to enable ARQ while a receiver accepts other packets which may contain erroneous bits. This can be simply implemented due to the fact that the decoder can precisely infer the bit importance when bits in each segment start to arrive [87]. Since a whole frame with a fixed number of samples, where the same number of wavelet coefficients is generated, is considered at a time, packet sizes can be selected so that code granularity can be controlled for efficient implementation. Before continuing our analysis, some comments might be helpful. First, the effec tiveness of our scheme relies on jointly consideration of coding and dynamic power allocation; it requires that the source codec can provide needed information; certain fidelity property has to be met for the source codec. Second, as examples of source coding schemes, we use either SPIHT or S-SPIHT, both schemes can partition the bitstream into three blocks: path-bits (high loss sensitive), fidelity-bits (median loss sensitive), and refinement-bits (low loss sensitive). It is natural to assign different channel rates to packets of different blocks, one noticeable feature of these source coding schemes is that a path-bit block also contains significant signal fidelity in formation in which the most significant bit (MSB) of significant coefficients can be inferred. For applications of wireless microsensor networks where multiple tests are performed, fidelity-bits and refinement-bits are used to boost performance and they are less loss-sensitive. In what follows, we compare three schemes, namely, a scheme based on base line model (denoted as scheme B), a source codec without error protection (denoted as scheme ^4) and EESPIHT (denoted as scheme £). The source codec selected for scheme A is S-SPIHT. For EESPIHT, these three blocks of encoded bitstream are denoted as 14 for path-bit block, Um for fidelity-bit block, and Ui for refinement-bit block. ARQ is only applied to Uh and different bit puncturing patterns are applied to 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. channel codes for Um and Ut. For a given frame (i.e. a fixed number of samples), assume there are m packets of length L bits. Percentage of £4 packets is ph, these of lA m packets and U\ packets are /im and pi, respectively. Normalization condition is Ph + Pm + Pi = Channel coding rate for £4 is these for Um and Ui are and j-, respectively. Property 5.2: Denote the energy costs for scheme B, scheme A and scheme 8 as Ilg, r u and Ilf, respectively, the following equations hold: r u = tukxB-b (5.4) Ilf = { p h m n x h + p m m n x m + p i m n x ^ B (5.5) Proof: Equation (5.4) holds by noticing that each packet of the frame in question has K\ as its energy ratio compared to the baseline model. We also know energy cost for £4 is p > h m KxhB-B, and these for Urn and Ui are pmm ^ \r J^B and pim nx^B, respectively. So, (5.5) holds. ■ Using Property 2, we can estimate how much energy gain can be obtained by using scheme 8 instead of scheme A. To do so, starting with the following ratio: 9 = \ — ( 5 ‘ 6 ) *hKxh + AmK\m + , Let Ah = A , by Lemma 5.2 and Property 1, Aln(l — PN) Xp,h ln(l - Pn ) + (Xmpm + Xm) ln(AnLe) 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. So the bigger the 4/ is (greater than 1 at least), the more energy savings the scheme £ returns. Energy savings are mainly determined by the following fraction: J n (An£e) ln ( l- P j v ) ' ’ Using (5.7), we rewrite (5.6) as follows: A $ = A/i/i ( A P A Although E can be made as small as 1CT10 via channel coding, e is much larger, for example, 10-5. With A nL — 400, PN can be 0.999, Let A h — A , A m = |A, A * = |A, and fih — 0.3; fxm = 0.5; H i = 0.2, with these values, we have T = 0.8 and 4/ = 1.6, it saves about 39% in energy for this example. A coding scheme yields better energy savings when the corresponding T is made smaller, and different rate assignments for different types of packets also play a role on energy savings. From Lemma 5.2, compression ratio is a direct factor on energy savings; compared to baseline, EESPIHT can reduce energy dissipation by about 65%. 5.2.2 SNR Loss Upper Bound of Coding In this subsection, we study SNR performance of our proposed scheme using recon structed signals. This signal SNR is defined as follows: SNR = 1 0 log10 1 * E i=i where Sj is the i-th source signal amplitude; s / is the corresponding reconstructed sig nal amplitude; J is the frame length. Since a signal is first transformed by a wavelet 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. transform, we approximate SNR using wavelet coefficients instead of source signal amplitudes (this approximation is close by Parseval’s theorem on signal energy invari ant between time domain and wavelet domain [20]). In S-SPIHT, the sign bit and the most significant bit of each coefficient are kept by the same way, and they are guar anteed to be received correctly under EESPIHT. What we need to do is to estimate how bit flips in residuals (i.e. coefficients with their sign bits and most significant bits removed) affect SNR performance. In this analysis, we assume: (1) residuals follow a Gaussian distribution as N (m 0, <r); (2) total number of residuals is M and the num ber of residuals from fidelity-bit block is Mf, and that of these from the refinement-bit block is Mr where M f+ M r — 1; (3) bit error probability for residuals from fidelity-bit block is e/ and that for these from the refinement-bit block is er. In order to quantify how EESPIHT affects SNR performance, we use following notations: (1) Co for SNR of scheme A, (2) Ci for SNR of scheme £, (3) A and B for signal power and noise power, respectively, as those in the SNR formula above, (4) as residual probability density function (pdf), (5) D for distortion in RMS introduced by EESPIHT. With the above notations, we have 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ci can also be written as follows: Ci = C0 - W + O (5.8) The above is obtained by Taylor expansion on function ln(l + x) in the neighborhood of x = 0. Therefore, we can approximate this distortion (in dB) by W and this ap- bit length of g bits up to g + q bits, the following theorem holds. Theorem 5.1: SNR loss due to jointly coded SPIHT is upper bounded by W, and D in the estimation of W is also upper bounded by (erMr + efM f)D ' and D' is as follows: we know W is an upper bound of SNR loss. In order to show an upper bound of D, we partition interval [ 2 9- 1, 29+q — 1 ] into q + 1 sub-intervals with the z-th sub-interval being [29+t~1,29+t — 1]. The number of bits of any number in the z-th sub-interval is g + i. The number of residuals in the z-th sub-interval is To give an upper bound on D, we use the most significant bit of a coefficient which contributes 22 s+ 2 *“ 2 per bit for signal reconstruction. Since any bit flip on MSB gives the worst distortion, an upper bound follows. ■ Theorem 5.1 also holds for S-SPIHT if spatial correlations of sensor readings in S-SPIHT are exploited in such a way that it does not result in any distortion (this can be done via a code design). A change of variable is useful, and it allows us to calculate proximation is in O ( ( ^ ) 4 J when D < B. Suppose SPIHT generates residuals with Proof: By noticing that C i—Cq+ W is negative and it is in 0 ( ( ^ ) 4) as shown by (5.8), 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the SNR loss via looking up the standard normal distribution table. The formula for change of variable on the intergal of D' is as follows: , v - m 0 v = --------- a so D' can be rewritten as follows: D ' = i ( C w M ^ ) dv) 2* +2,- 2 {3 + i) < 5 -9 ) where and 2» « - l - m 0 “l(!) = 2a------- 2 i + M - m , ^ = 2 ? ------ In (5.9), the integral can be obtained via a table look-up of normal distribution. The number of items in the summation above is small since the raw values are quantized based on application fidelity requirement; therefore, D' can be easily calculated. As an example, we have M f = 192; Mr — 89; e/ = 10~5; er = 10~4; g = 3; q = 8 ; C0 = 35dB, by Theorem 1, we have D = 1993.7; B = 4384.2; A = 13864028.3; Ci = 33.4dB. SNR loss due to EESPIHT in this example is at most 1.6 dB. In general, as symbol error probability decreases, so does SNR loss. 5.3 Proposed Power Aware Coding Scheme In this section, following the analysis in Section 5.2, we show how a practical energy efficient scheme can be devised to transmit sensor data. 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Next we will highlight our modifications to the general SPIHT first used in S- SPIHT. The general SPIHT is used for image coding based on 2-D wavelet transform, we have developed a 1-D version of SPIHT which is suitable for compression of sen sor readings (we use SPIHT to refer to our implementation for sensor applications hereafter). Under set partitioning rule of SPIHT, wavelet coefficients are partitioned in each it eration into three types. They are denoted as ISO, LIP and LSP, and defined as follows: LIP type of coefficients are coefficients which are not significant in an iteration; fur thermore, none of their descendants are significant; ISO type of coefficients are these coefficients which are isolated from further processing (i.e. they are parents of some LIP nodes) and are insignificant at an iteration, but they could become significant in further iterations; LSP type of coefficients are these coefficients which are significant in an iteration. At a refinement phase, a bitplane deduced from an LSP queue is output. Changes made to regular SPIHT are as follows: (1) use of stack structure instead of queue structure to reduce memory requirement, (2 ) directly outputting of residuals to reduce the number of bit operations in the refinement phase, (3) use of dynamic pro gramming approach with deepest-first tree traversal to identify significant coefficients, (4) use of two mark bits to facilitate the search in (3) with one bit for coefficient sig nificance and the other for subtree significance which is rooted at the visiting node (a binary subtree is significant if there is one or more significant coefficients in it). After computing the wavelet transform, wavelet coefficients are arranged into wavelet decomposition trees (WDT) with roots in the coarsest subbands following a specific zigzag scanning order of roots in the coarsest subband. SPIHT iterates on bit positions of these coefficients. Assume there is q + 1 iterations, and it will stop at p-th bit po sition, then the largest coefficient has g + q bits. The iteration starts at the (g + q)-th bit position. During the z-th iteration, SPIHT traverses further down the tree to search 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5.3: Search of significant coefficients for any significant coefficient which is greater than or equal to 2*. For each coeffi cient in LIP stack (LIP stack is a stack data structure to hold LIP type of coefficients), the codec records path bits and mark bits if a significant coefficient is found (decoder needs this information to identify the position in the wavelet tree) and pushs any ISO coefficient into an ISO stack. Figure 5.3 shows one example wavelet decomposition tree. The dark nodes in Figure 5.3 are significant coefficients after the current iteration; gray ones are left to successive iterations, and the circle nodes are in LIS after current iteration. Note that there is also an ISO node in Figure 5.3 as shown. Instead of immediately outputting identified significant coefficients, SPIHT pushes them into an LSP stack (LSP stack is a stack to hold any newly identified significant coefficients, and the ISO stack is defined likewise). Coefficients in ISO stack are all checked at the end of current iteration, any significant coefficient will also be pushed into LSP stack. After all significant coefficients are found, these coefficients in LSP stack are popped up with the most significant bit and sign bit removed. Rest bits are then encoded into a bitstream using a non-binary distributed source coding algorithm. There is also a stack called LIS which is defined as the placeholder for subtrees which need to be checked for significance for the next iteration. SPIHT outputs three types of bits: path and mark bits, LSP bits from LIP stack and LSP bits from ISO stack. These types of bits are named as path bits, fidelity bits and refinement bits. Path bits can not tolerate any error, and fidelity and refinement bits may introduce distortion at different levels. The sign bitplane and most significant bitplane from each iteration form the 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fidelity-bit block, all other refinement bitplanes form the refinement-bit block, and the significance bits form the path-bit block. In what follows, we take S-SPIHT or SPIHT as a procedure which iteratively gen erates three types of bits, denoted as Uh, Um and U\ for path bits, fidelity bits and refinement bits, respectively. We also use these three symbols to denote three types of packets consisting of corresponding bits whenever the context is clear. 5.3.1 Rate Selection Based on Energy Dissipation Rates used for bits should adapt to their different importances. As we have shown in Sec. 5.2, unnecessary low rate (high bit redundancy) on channel coding will lose energy without providing extra SNR gain or boosting accuracy of application results. In this subsection, we shall first work on a rate selection of Uh packets, then derive rates for Urn and Ui packets using RCPC codes. Assume events of any packet delivery (repeated packet or original packet) are in dependent, and also assume one ARQ request is sent by a receiver if one packet has erroneous bit(s) upon channel decoding using ITOH algorithm; furthermore, there is no loss of ARQ packets. Let e(A) denote the probability for a packet to be requested indicated by an ARQ with code rate \ , n be the number of packet, and n + k be the number of packet transmission trials for these n packets. Probability mass function (pmf) p(k, A) for k with code rate j- is as follows: p(k, A) = < k < n (5iQ) E L 1 (*)/?e(A )‘ ( l- < W ) * k > n 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where P f satisfies the following recursive formula: fc-i p ? = T , F?-i (5.11) j= i- 1 where Pj = 1, and P( = 1 for Wj > 0 Let Psig is the minimal packet delivery probability, r be an antenna power factor, k0 be the maximal number of repeated requests of packets, and po be the required packet delivery probability. Rate selection based on an energy criterion is formulated as the following integer programming optimization problem (IPOP): where, Psig is usually close to 1 and r is between r0 and 1. Note that r can take one of a fixed number of discrete values, and the number of antenna power level is fixed and the power on each level is also fixed. From the application standpoint, the delivery request is fixed to meet the latency requirement. Constraint (5.12c) of IPOP is derived from an application latency requirement. The objective function in IPOP consists of (A, f) = argmin([(A — 1 )n\ + k)r A ,r subjects to: A > 1 and is rational (implementable); k (5.12a) (5.12b) i = 0 fco (5.12c) fc=0 To < T < 1 (5.12d) of n packets has to be finished in a timely fashion, so the maximal number of repeat 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 5.1: IPOP parameters and results of the example A e e k P' M ) T Cost 2 1.1046e-5 0.00564 4 0.9997 0.73 75.92 V 4 1.601 le-5 0.00816 5 0.9998 0.78 62.4 3 4.601e-5 0.0276 8 0.999 0 . 8 46.4 5 3.7895e-4 0.1764 39 0.9991 0.85 50.15 two parts, namely, [(A — 1 )n\r and kr. The first part is redundancy cost introduced by channel coding and the second part is repetition cost of some packets introduced by ARQ protocol. Analytical optimal solution for the above problem is not always obtainable since the channel is not stationary. Due to constraints on hardware imple mentation, certain coding rate may be too costly and end up with too much circuitry power dissipation. One solution could be to solve IPOP with a feasible solution space with limited pre-selected set of rates (the space in question is presumably smaller). In order to help understand the above formulation, we give an example as follows. Let n = 100; k0 = 40; Psig = 0.99999; p0 = 0.999; P(7 , R) = 0.99999. A common uncoded wireless channel with BPSK modulation (baseline) has BER around 0.01 per channel symbol on arrival signal with strength above -91 dBm. Assume there are 512 symbols in a packet, the baseline is not in the feasible space since e = 0.994176, and the error rate is too high to meet ( 5.12c) of IPOP within 40 repetitions. Table 5.1 gives the parameters and corresponding objective function values for above example. In Table 5.1, Pq = Yli=oP(h A) is a successful probability of packet delivery upto k repetition, P'sig = J2iLoP(h A) is the actual packet successful delivery probability corresponding to k0 repetitions; and Cost is the objective function value corresponding to a given parameter setting. BER in this example is calculated based on P(7 , R) and BER upper bound formula of convolutional code (for an AWGN BPSK 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. modulated channel). We set ^ = 1.7r (= 2.3 + lOlog(r) dB, w is received signal power, and u;0 is noise power). BER formula used in this example is as follows: where d is the free distance which is equal to the lowest exponent of D in T(N, I, D ), and T(N, I, D), the transfer function, is as what in [97] ( refer to VII-B of [97], and [34, 72] for details on BER upper bound estimate). From this simple example, we can see that there is some optimal operational point on channel codes with an appropriate power level for a given parameter setting. In order to see how transmission power affects the objective function value with coding rate fixed at j, denote Ak = k2 — ki, where ki is the number of retransmissions after power adjustment; k2 is the number of retransmissions before power adjustment; r is the power factor before power adjustment. Suppose the power after adjustment is r + A r, in order to save energy, the following must hold: Note that k\ < k2 in general if A r > 0 To select an appropriate power level, the following formula [8 8 ] is used: Pi = ad% + (3 E < Q 2dw \ ((iw \ <9T(N, I, D) Ak > ([(A — l)n\ + k\) — T 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For example, for Orinoco WaveLAN radio in a free space at noise floor of -119dBm, n = 2, a — 14.8mW , and j3 = 11 m W . The path loss estimate uses the following formula (see Chapter 3 and Appendix B of [67]): where PL is average path loss in dB at distance d away from transmitter in a free space; d0 is a close-in distance, and X a is a random variable with zero-mean and variance o. If the close-in distance is selected so that PL(d0) = 0, signal strength at d can be computed in dB as follows: Thus, using the above formula, a received signal SNR can be derived from its associ ated noise figure. We select code for path-bit packets via solving IPOP with a list of feasible channel code rates. This is done offline using exhaustive search based on training data sets. Once a code is determined for path bits, we are able to determine codes for refinement bits and fidelity bits based on an application SNR requirement with the help of Theo rem 5.1 in Sec. 5.2. In practice, an exhaustive search is possible since both the number of RCPC codes and the number of power level are small (We set both number to 10 in the experimental study and the search space is only 100). However, if the granularity is too fine, IPOP turns out to be intractable. 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.4 Experimental Evaluations In this experimental study, we shall first evaluate the proposed scheme in terms of energy savings and SNR performance. We take 802.1 IB (2 Mbps) wireless LAN to verify the data flow with a source codec, this LAN comprises 3 iPAQs so that decoding of distributed source coding can be done. The use of iPAQ is simply because of imple mentation convenience; the implementation in done in fixed point arithmetics with less than 4K memory requirement, there should be no problem to port to microcontroller in most sensor nodes [16]. OPNET is then used to simulate how the proposed scheme with power allocation can save energy for a wireless sensor network. By this way, performances of both coding part and network part of our scheme can be evaluated. In this study, we use ESPIHT and SPIHT (their LINUX implementations are pre sented in [87]). In these experiments, we only measure power dissipation due to a RF component of transceiver, other circuitry power dissipation of radio is excluded. In simulation based on OPNET, bit error thresholds of convolutional codes with differ ent puncturing patterns have been obtained using the estimation method of previous section. The default error correction code (ECC) model in the OPNET radio pipeline is replaced by our ECC procedure based on RCPC code with corresponding ECC bit error thresholds. In order to output energy dissipation data on each frame from OP NET, we have inserted a new pipeline stage procedure in the radio simulation pipeline; therefore, radio standby energy is not included when radios are not active. Input data to OPNET simulator is output from S-SPIHT or SPIHT, and there are three data chunks while each chunk corresponds to one of UhMmMi- A new process module is devel oped to read an input file and feed the source size information to a radio transmitter. The first OPNET simulation has one mobile receiver and one stationary transmitter. 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 300 450 900 750 600 150 300 450 rx 600 750 900 Figure 5.4: Simulation topology There are three segments on the path trajectory of the mobile node. Simulation param eters are shown in Table 5.2; simulation duration is determined by the data set of length 148 seconds. The network topology is shown in Figure 5.4. Although we present the simulation results based on one type of data set, we have tested other two types of data sets (SensIT SITEX00 and ARL APG), the results are similar to what to be presented. In these experiments, convolutional code with rate \ and constraint length of 6 is used for both Uh packets and Um packets, and RCPC code with rate | is used for Ui packets. Figure 5.5(a) shows the energy dissipation using EESPIHT, jointly coded SPIHT vs. baseline. Compared to baseline energy dissipation, average energy savings are about 80% and 65% by EESPIHT and jointly coded SPIHT, respectively. Fig ure 5.5(b) shows the energy savings using EESPIHT and S-SPIHT, while S-SPIHT is uniformly coded by either |, |, or | RCPC codes. Average energy savings due to 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 5.2: Simulation parameter settings Channel Parameter Parameter Value network dimension 1km x 1km x 1 0 m simulation duration 148 seconds modulation scheme bpsk radio rx sensitivity -111 dBm data rate 19.2 kbps base frequency 900 mHz bandwidth 7.2 mHz maximum Tx power 10 dBm number of RCPC codes 1 0 total number of power levels 1 0 Tx/Rx antenna gain OdBi radio closure ray-tracing propagation model free space noise accumulating power aware coding are at least 20%. Large variation on energy savings is due to sig nal amplitude fluctuation among frames. In our implementation, we select a Coiflet basis with 3 vanishing moments; however, other type of basis could be used. Figure 5.6(a) shows SNR loss of EESPIHT as compared to S-SPIHT. In this plot, SNR differences among frames are shown at two bitrates: 4 iterations and 10 iterations. The 4-iteration case gives about 25 dB SNR gain and 10-iteration case gives about 50 dB SNR gain on average among these frames (the corresponding SNR gain of S-SPIHT used in this comparison is shown in Figure 5.6(b)). The i-th iteration corresponds to a tree traversal of MSB with k — i — 1 bits where k is the number of bits of the largest coefficient in the binary tree (it is 13 for the data set used in the experimental study). In worst case, EESPIHT of 4-iteration case has 0.58 dB SNR loss than that of S-SPIHT over all frames. EESPIHT of 10-iteration case has 2.4 dB SNR extra loss than that of S-SPIHT over all frames (This loss is within the bound as that shown in Theorem 5.1). 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 200 150 o > V . C D C L L I 50 0. -Baseline -Jointly Coded SPIHT •EESPIHT 0 50 100 150 200 Time (s) (a) EESPIHT, SPIHT, Baseline >. E ? < D C LU 40 •— EESPIHT S-SPIHT (1/2) ---S -S P IH T (3/4) S-SPIHT (2/3) 30 10 0, 0 50 100 150 200 Time (s) (b) EESPIHT vs. S-SPIHT Figure 5.5: Energy dissipation comparisons 2.5 — 4 Iterations — 10 Iterations c < / ) O ) o — I 0C z CO 0.5 Frame Sequence (a) SNR loss of EESPIHT — 4 Iterations —-1 0 Iterations S * s/’-vv, / v * - O-n / ' vyHj r 1 V+K Frame Sequence (b) S-SPIHT SNR used in comparison Figure 5.6: SNR comparison Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The SNR loss of the jointly coded SPIHT vs. SPIHT with the same experimental configuration as those in Figure 5.6(a) has also been obtained. The worst case SNR loss of jointly coded SPIHT is 0.34 dB and 1.9 dB for the cases of 4-iteration and 10-iteration, respectively. For the next set of simulations, we uses OPNET to simulate the network and verify how EESPIHT based design can save energy using rate-adaptable power allocation, refer to [90] for radio power profile used in these simulations. The simulation is based on pairs of nodes with different distances. The parameters are also shown in Table 5.2. Assume that the distances are all inside the propagation limit and also assume that every node has 2Kx8 bits of information to send in the baseline in every second. The network consists of a number of uniform nodes randomly distributed in field of size lkm x 1km while five of them are mobile nodes. Figure 5.7 shows the normalized powers in EESPIHT compared to the best fixed power case and a scheme using dis tance with given SNR constraint with varying number of nodes. From Figure 5.7, when the fixed or distance/SNR based power allocation is used, the power increases as the number of nodes increase, while the power is almost constant when more nodes are added to the network under EESPIHT. This is simply due to that the power under EESPIHT is the minimal power required under the optimization and there are much less bits transmitted by EESPIHT as compared to other schemes. When less bits are transmitted, interference from neighboring nodes is also much less. The energy dissipation comparison with the 10-node case for a simulation duration of 60 seconds is shown in Table 5.3. From Table 5.3, it can be seen that on average EESPIHT saves about 20 mJ per second network-wide. Note that the whole network (radio RF) without using EESPIHT dissipates at least 25 mJ per second (excluding en ergy consumed by individual nodes) under the distance-SNR based allocation scheme. EESPIHT has the least energy per bit cost among these three schemes. 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.8 -o 0.6 0.2 Number of Nodes Distance Based - Fixed Power EESPIHT Figure 5.7: Power consumption comparison Table 5.3: Energy dissipation comparison Ave. Power E/Bit Tot. Eng. EESPIHT 2.3 mW 0.12/J 370 mJ Distance (SNR) 3.4 mW 0.18 mJ 1.53 J Fixed Power 3.9 mW 0.21 fjJ 1.95 J 5.5 Conclusions In this chapter, a novel power aware coding scheme that exploits spatio-temporal cor relation in sensor readings is presented. This scheme combining optimal power allo cation with ARQ to select proper coding parameters. Lower bound on energy savings and upper bound on SNR loss of the scheme are obtained. Experimental and simula tion results show that more than 60% energy savings can be obtained as compared to the uncoded case and more than 35% energy savings as compared to temporal coding or spatial coding schemes. One important precondition for distributed source coding is that the readings from neighboring sensors have to be “aligned” before extracting side-information for de coding. Next we will propose algorithms how to efficiently align sensor readings for a better rate-distortion performance. 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 Correlation Analysis in Sensor Networks Sensor readings in a wireless microsensor network are correlated both spatially and temporally. Various coding and storage schemes and also other applications have been developed to exploit these correlations; therefore it is crucial to efficiently track the correlations. In this chapter, a linear prediction algorithm is developed to initially es tablish the correlations, and the order of linear prediction has been derived from the prediction error power distribution. A tracking algorithm uses discrete Kalman filter to track the correlation once it is initially obtained. This Kalman filter based algorithm uses the gradient computed at each step as the input control vector. This approach is suitable for quantifying geographical spatial correlation and multi-modality correla tion. Experimental results using various data sets have shown that the proposed scheme can accurately obtain the correlation and consumes much less energy as compared to known schemes. 6.1 Introduction One useful type of sensor networks consists of a number of sensor arrays, where each array has a few sensor nodes. Sensors in an array are generally in close proximity and 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. their readings are both spatially and temporally correlated. When a sensor is capable of read multiple modality of signals, these correlations also extend to a multi-modality domain. Correlation information is very useful for some applications in wireless sen sor networks. Since energy cost for wireless communication is high, techniques to reduce data communication without loss quality of application is valuable. Correla tion analysis among sensor data is the focus of this chapter which can be exploited for reducing communication energy cost for target tracking type of sensor network appli cations. Other applications that correlation analysis is valuable for beamforming used in target tracking [7] and storage systems in sensor networks [27] [28]. Additional benefits with accurate correlation information in these applications include that there is no need to calibrate sensors’ readings, and synchronization can also be relaxed to a certain degree which can help in sensor deployment. The difficulty of tracking correlations in wireless sensor networks lies in the syn chronization of sensor nodes and the need for calibration of the sensor readings. Cal ibration is generally needed even in a perfectly synchronized network. This is due to the non-stationarity of signals in the sensor field and field dynamics. Figure 6.1 shows the cross correlation coefficients of two frames (a fixed number of samples is called a frame) gathered at the same time from two nearby sensors. Denote these two nodes as node A and node B, for simplicity, they are synchronized to within one sampling pe riod. In order for node A to correlate its readings with those of node B, one approach is simply to let node B send a number of readings for each frame to node A. Node A then has both its own readings and the corresponding readings from node B. Node A then can use a matched filter to correlate its readings with those of node B. This correlation can be measured in terms of number of samples that these two nodes’ readings can be aligned. From Fig. 6.1, the correct lag should be 3 which can be computed with at least 72 samples every second. However, if only 56 samples are transmitted by node B, the 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Sensor Readings Calibration 0.6 2 Q - ■ s s ° - 0.6 - 9 i oo -20 -10 Lags ■ *— 296 Samples -------- 72 Samples o- - 56 Samples 48 Samples Figure 6.1: Frame correlation ambiguity is apparent by noticing the two peaks at -8 and 3 for this case. No correct correlating number of lags is possible when only 48 samples are available by noticing that there are at least 4 confusing peaks. Once a correlating lag is found, it may only be valid for a short period of time, sometimes, it may even be less than one frame duration when either signals experience deep fading or other interferences occur. Figure 6.2 shows the correlated number of lags over a period of 200 seconds of node A and node B. From Fig. 6.2, it is known that sensor correlation changes over time, and node calibration has to be done periodically. This incurs serious overhead for many applications; therefore, its use in sensor network application is rather limited. Furthermore, this correlation with calibration approach does not exploit the temporal correlation of sensor readings. Since the spatial correlation in many sensor network applications is high, linear prediction (LP) [54] approach is normally effective and it can also be made efficient. In [10], a simple and effective algorithm based on linear prediction and gradient method has been applied to the compression of sensor data and a zero-pole model of linear 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Calibration in Number of Samples (Sensor 3, 4) 6 4 2 O ) 0 3 -2 -4 -6 -8 80 160 200 0 40 120 Frame Seq. Figure 6.2: Node calibration over time prediction is used. An implicit assumption in [10] is that sensor readings are already calibrated so that the single-zero prediction model could work to exploit the spatial correlation (the temporal correlation is done via an all-pole model in [10]). Before we proceed, we briefly review the linear prediction model and its computa tion complexity. Let T C be the vector space of random variables (one random variable is a vector), and define the following inner product and the norm (H is therefore a Hilbert space): < X , Y >= £[X,Y] ||X|| = V < In other words, the inner product of X and Y is their covariance and the norm of X is the standard deviation of X . For a given index i, the prediction error to Xi (unknown random variable) based on m + I + 1 samples from random variable Y can be represented as | < Xi, Y ' > \ where Y' denotes the random variable derived from the m + 1 + 1 samples where m + 1 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is the order (lag) of the forward prediction and I + 1 is the order (lag) of the backward prediction. Under the linear prediction model, we have m I y = y ' a j V i + j + y "jPjUi-j 3=0 j = 1 and m I < Xi, Y ' > = ^ 2 a 3 < x ii Vi+3 > + ^ 2 & < Xi> > ( 6 - 1 ) 3=0 j = 1 In order to minimize the prediction error, we need an orthogonal basis of the subspace spanned by vectors y^i, • • • ,yi, ■ ■ ■ , iji+m- By the orthogonality principle which states that the error vector is uncorrelated to its input vector [40], if replacing the vectors in (6.1) by the orthogonal basis, the unique decomposition of Xi corresponding to the ba sis has a set of coefficients which are exactly the coefficients which minimize the error of the linear predictor. Fortunately, under the stationary assumption (we shall explain why this assumption is viable and the trade-off between efficiency and accuracy for this particular application), the order recursive Levinson-Durbin algorithm [23] upto order m + l + 1 can efficiently find these coefficients in time in 0((l + m)2) and space in 0(1 + m). Linear prediction model is briefly reviewed above. Linear prediction model is ef fective for certain applications and it is based on the observation that prediction co efficients change in a slower manner than the time series samples. These coefficients can be estimated periodically; however, the computation is heavy for low power sensor nodes. Therefore, linear prediction is used in our initial estimate. The Levinson-Durbin algorithm described above can be used to solve (6.4) in the proposed model. It takes time on the order of 0 ((m + I)3) to directly compute the 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. inverse of P of (6.4). P is an estimate of a standard covariance matrix which is sym metric and positive definite Toeplitz matrix in practice. When L is large enough, this estimate approaches the covariance matrix of Y and we can simply use the Levinson- Durbin algorithm to solve (6.4) for the LP coefficients. Note that in this case, the constant vector of the m + / + 1 simultaneous linear equations is different from that in the regular case which requires it is identical to the first column of P shifted by one element and with opposite signs. The Levinson-Durbin algorithm used here is a modified algorithm based directly on the orthogonality principle shown in the previous section. There is also “superfast” algorithm [3] for the modified system of equations of (6.4) taking time on the order of 0 (n log2(n)), the overhead can be manageable in a microsensor node especially when the algorithm is only executed periodically. A general linear prediction model assumes that signals are in an infinite interval and the samples on both the sender and receiver are available. In practice, instead of ensemble sampling, only a finite number of samples are available. These finite number of sample are used to derive an initial set of LP coefficients, and then Kalman filter [40, 99] based algorithm is used to update these coefficients to track the correlations. The initial estimate of coefficients can be done offline with statistically sufficient number of samples. On one hand, the initial LP coefficients are used at the starting point, the stationary assumption is a viable assumption when the number of samples is large and the sensors are stationary. On the other hand, when the samples are communicated during the initiation period of tracking, and further tracking processing on the initial LP coefficients has to be done online. It may need to sacrifice the accuracy for efficiency by assuming the signal being stationary in the initial estimate. By our experiments, the accuracy loss is rather limited especially when the sample frequency is relatively high (e.g. greater than 1 kHz). When the relative positions of sensors or signal sources 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in question have a noticeable change, a new round of estimation of LP coefficients is generally needed. 6.1.1 Problem Formulation and Our Contributions It is important to correlate sensor readings before processing the sensor readings. Such correlation task is required by distributed source coding programming and many beam- forming algorithms. By noticing the fact that sensor signal readings are time series, correlation has to be adaptable to field dynamics. Since sensors are resource con strained, simple, yet efficient correlation is critical for sensor network applications. Given two time variant signal representations in vector form (for 1-D signals), si and s2, where si is the predictor and s2 is the source, for the linear prediction, we can build a model as s2 = 0 F ( s i) where, /3 is the vector of model coefficients and matrix function F(.) is a transforma tion of the predictor. Note that in the linear first-order predictor, (3 is a vector of degree 2 and F(si) is matrix whose its first column is the all 1 vector and the second column is the vector si. The problem to study is to find an efficient algorithm which can give the matrix function F and the vector (3 so that the mean squares error residuals between the predicted version of s2 s2 and s2 is minimized over a given period and with a pos sible smallest overall energy consumption including processing and communication costs. In this chapter, we propose a scheme which does not require calibration on sensor readings and is able to predict the correlation of spatial correlations in sensor data. It applies discrete Kalman filter (DKF) to further track the correlation over time. The proposed DKF approach is an extension to the steepest descent method used in [10] 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. since our scheme degrades to the gradient method when the state transition matrix in the state-space model is set to the identity matrix. 6.2 Correlation Analysis and Tracking Scheme We assume the sensors’ clocks are partially synchronized, by that we mean that the time difference between two samples recorded at the same time by two different sen sors is within one sampling period. For the case of 1 KHz sampling rate, the time difference for the above is within 1 ms. Note that the proposed algorithm will work with a coarser clock synchronization; however, the processing overhead will increase. There are some efficient synchronization schemes developed for sensor networks, e.g. [33]. We note that the synchronization requirement is not necessary at a network scale, but rather in a direct vicinity of a node to correlate with. This requirement is much eas ier to fulfill than a global synchronization. The proposed tracking algorithm is shown by the following steps: (51) Each sensor samples the signal independently. (52) A sender sends a few samples which are only temporally coded in order to es tablish the initial coefficients. (53) The correlating node estimates the LP coefficients based on the received samples with the aid of offline computed parameters. (54) The correlating node applies DKF to track the correlation by performing the following steps: (S4-A) Compute gradient as input control; (S4-B) Compute Kalman gain matrix and steady state error covariance; 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (S4-C) Compute the innovation and update state vector as the correlation coeffi cients; (S4-D) Repeat from S4-A for a given number of steps. (55) To reconstruct approximate frames, the correlating node uses steepest descent method based on the coefficients from the estimated coefficients of the DKF. (56) The correlating node loops back on S4 after each timeout period. (57) Based on application fidelity requirements, a new round of analysis can start by looping back to S2. The initial estimation requires some fixed number of samples to be transmitted over to the correlating node, and this number of samples is determined by the LP orders as discussed in Sec. 6.2.2 where an example with 64 samples shows very good esti mate. In general, when a fixed set of signals are present, this initial estimate can be done offline and incurs only once. For some acoustic signal in our experiments, this needs to be done periodically and the period of estimate depends on underline signal characteristics. 6.2.1 Initial Coefficient Estimation Let X denote the random variable associated with readings of a sending node and Y denote the random variable associated with that of the correlating node (i.e. receiver). Samples over a fixed sampling period of the sender is made available to the receiver, The task is to infer samples after a fixed period at the receiving node. Assume that samples available to the receiver are T x = {^i, • • • , xL} from a sender and T y = {y~i, • • • ,yo ,-" , UL+m} of its own. To help infer the i-th (i > L ) sample of node X , the receiver starts with T x and Ty. A set of linear coefficients is 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to be derived and it is denoted as {aj | 0 < j < m} (called S-backward prediction coefficients to distinguish with prediction in the same node) and {,6j \ 1 < j < 1} (called S-forward prediction coefficients) for the LP of order m + I + 1 so that the following objective function denoted by Fn(X, Y) is minimized: L / m I \ 2 Fn(X, Y ) = E °W<+i + E + x‘ (6'2) i= 1 \ j = 0 j = 1 / An alternative cost function could have the following form (i.e. MMSE); ( 6.2) (i.e. minimum variance) is adopted in this chapter. Y ) — E ^ | 'y ^ O ijU i+j + 'y ] PjUi-j + 0=0 j=1 2 X i Note that the signal exists over an infinite interval, but only a finite number of samples are available. To minimize Fn(X, Y), the partial derivatives of Fn(X, Y) with respect to the m + l + 1 coefficient variables should be set to zero, that is, d - W l = 0 (6.3) oz where z = (/?!,-•• , Pi, cxo, a i, • • • , am)T & nd t is for the transpose. By (6.3), the following holds: P z + b = 0 (6.4) where P is the coefficient matrix. P is defined as follows: kxj 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for — I < k, j < m. The b in (6.4) is defined as follows: / L L L \ T b = ( " ' ’ X I XiVi' ' ‘ ' ’ ) \i= l i=1 i=1 / Note that PY sequence starts from y^i, where I is the lag on S-forward prediction. Matrix P is symmetric and almost always invertible. The solution to (6.4) is T = p-'b. Using the same notation as before, the cost function can be rewritten as: Fn(X, Y ) - zTP z + bTz + c (6.5) where P and b are defined as above and c = xi2> then (6.5) is in the quadratic form. Matrix P is positive definite and a point can be found in p m+l+1 (a point in TZm+i+l is also called a coefficient vector) and it minimizes the cost function. There are a number of methods for this, including the deepest descent method or the conjugate gradient method. These methods are alternative methods for solving (6.4); however, instead of solving for an optimal solution at each time step, by following the iterative steps of these algorithms, the correlation can be actually tracked in a similar fashion. In our case, the matrix P is a function of time. It is also prohibitive at runtime for a sensor node to solve for the optimal point in 7Zrn+ l+ 1 at each step. A DKF based approach is used to track the optimal points for the best coefficient vectors. 6.2.2 Selection of LP Order Before we introduce our tracking update model, we need to estimate the LP order, i.e. the numbers m and I. There is a trade-off issue in the selection of the LP order. A larger order could give higher accuracy in the prediction results; however, it incurs a 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 6.3: Selection of S-Backward and S-Forward prediction lags higher processing overhead. Denote Q = 71° x 7Z°, where 1Z° is the set of non-negative real numbers, there is a one-to-one correspondence between a point in Q and the LP order. The first coordinate of a point in Q is the lag on S-forward prediction and the second is the lag on S-backward prediction. For a given point in Q, the LP innovations (i.e. estimates residuals) can be obtained and the power of innovations is a function in Q. An operational point in Q is picked when the innovation power is small enough as shown in Fig. 6.3 with L = 64. From Fig. 6.3, let m = 10 and I = 15 and (10,15) is a good starting point as the corresponding error in root mean square (RMS) is less than 20.0 and the corresponding estimation gain is better than 15 dB. One important issue on the selection of LP order is the so-called fundamental coef ficient which is defined as the largest element in a coefficient vector (the corresponding index is called fundamental index). This coefficient is the deciding factor of the accu racy of a linear predictor. Based on our experimental study on various data sets, the coefficient vector consists of two types of elements, i.e. one fundamental coefficient and a number of auxiliary coefficients. In most cases, a 3-D plot of the coefficient 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (seconds) Figure 6.4: Fundamental index and auxiliary Indices vectors can clearly show these coefficients as one example is shown in Fig. 6.4, where the fundamental index is 5. The fundamental coefficient and major auxiliary coeffi cients determine the estimation accuracy; these indices also change over time. It is therefore crucial to have an order greater than any of these corresponding indices over the planned correlating period. Selection of (m, I) is done in two steps: (1) offline estimation with a training data set and (2) online adjustment at the initiation stage of the tracking and the cost is in curred only once in each round as indicated in step S7. In step (2), based on the 2L samples from a sender and its receiver, a local search around point (m0, lo) is con ducted and the one which gives the best trade-off between the LP order and the predic tion accuracy is picked. We shall not delve into this, our heuristic method on the order search works well in experiments. Assume that sources are partially synchronized, this search is limited to a small region in the neighborhood points of (m0, lo) in Q and the point which gives the lowest innovation power is picked. In this way, the initial estimate is done with a low computational overhead. I l l Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C O > c Q ) O o 0 .5 O ■ O a > 4 — » n s D ) O Z -0. Figure 6.5: Optimal coefficient vectors 6.2.3 Tracking Update The tracking update model is based on the discrete Kalman filter and its goal is to adap tively find the LP coefficients which minimize the objective function on new samples. Figure 6.5 shows the changes of coefficient vectors which minimize the corresponding cost functions for a course of 40 seconds (we negate the coefficient vectors for a better 3-D view). Here a combined approach with steepest descent and Kalman filter model is taken, where the Kalman filter model is used to track the coefficient vectors to match the optimal vectors in %m+l+1 in each update step. The basic steps are to use steepest descent method in each step and Kalman filter to fine-tune and track the coefficient vectors. The state-space model is formed by the following coupled state dynamic equation and the discrete measurement equation: zk+i = A kzk + Bkuk + wk (6.6) sk = Ckzk + Dkuk + vk (6.7) 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We use four matrices to represent the state space model denoted as (Ak, B k) Ck, Dk), where Ak is the state transition matrix, B k is the input control matrix, Ck is the state measurement matrix, which relates a state to the measurement, and Dk is the input measurement matrix which is set to zero in this application. AK and Bk are (m + I + 1) x (m+Z+1) matrices, and Ck is an (M) x (rn+l+l) matrix. Parameter M is smaller than L in general and is selected so that the computation cost of the Kalman gain matrix is small. zk is the state vector in 1Zm+l+l\ uk is the input vector in TZm+l+1; sk is the measurement vector in TZM from the sender; wk is the state noise with covariance (m + I + 1) x (m + I + 1) matrix Q; vk is the measurement noise with covariance (M ) x (M) matrix R. We model the state and measurement noises as the independent identically distributed random variables with normal distribution. Denote and r* as the standard deviation of the random variable from i-th element of the state vector and the measurement vector, respectively, then Q = diag{q~i, qi-u • • ■ , Q o, • • ■ , qm} and R — diagjri, r 2, • • • ,rM}. The input control vector in qz m+l+1 uses the gradient on TY > = {y'-n •• • > V L+ - By (6.5) and the fact that P is symmetric, the gradient can be computed as: Note that matrix P depends on T Y’- Putting it in an iterative form, the following formula holds: VF k{TY') = 2{P(TY,)zk + b) V F k(PY> ) = 2 (7 • • • , 7o, • • • , 7m)r (6.8) and for r = — I, • • • , 0, • • • , m, 7r = ' ; a ty + j + pH - o+ xi (6.9) 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where, x\, for % = 1, • • • , L, are the estimated sender’s samples, and The second term in the sum of (6.9) is invariant to the gradient index and they can be computed once and used for the computing of all the elements of the gradient. When the context is clear, VFk is used to denote the gradient at step k. The state transition matrix A k in (6.6) has the form: A k = (dij)iX j where is defined as follows: Matrix Bk = y kI where nk relates the state vector to the changing speed of the gradi ent, and explanation on how to compute it is given below. Matrix Ck is based on the vector of length M + m + l of samples of the receiver. For the given sample vector of the receiver - ■ ■ ■ , y'Q , • • • , y'M_1+m), matrix Ck is defined as: A * j = i 0 otherwise Ck = ( y ' - i y ' i - i y ' l - i y'2-1 \ y r M - l - l y'M -l ' ■ ' y'M -l+ m J Rewrite the state-space model of (6.6 - 6.7) as: Zk+i A kzk -|- y,k^ F k -|- wk (6.10) (6.11) 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. respectively, model equations ( 6.10) and ( 6.11) use two set of parameters, A j and pk. Based on the orthogonality principle, the selection of parameter p,k at zk uses the following formula: zkT zk ^ k ~ ZkTP(FY')zk where the minus sign is for taking the opposite direction of the gradient at zk. Param eters A j change slowly and get updated less frequently than parameters p,k do. The stability of the tracking update model depends on the eigenvalues of the matrix Ak which further depends on parameters Aj. Matrix A k in our model has eigenvalues within the unit circle when these parameters are less than or equal to 0; therefore, the model is marginally stable. In practice, it is always stable since it degrades to the gradient method when the adjustment factors approach 1. Denote G(k \ k) and G(k + 1 | k) as the steady state a prior and a posteriori error covariance for the state estimates at step k, respectively. The Kalman gain matrix K k at step k is obtained by the following equation: K k = G(k | k)Cl(CkG(k | k)Cr k + R r 1 In step k, the a prior error covariance matrix is computed as: G{k | k) = A kG(k | k - 1 )AT + Q The a posteriori error covariance matrix at step k is updated using the a prior error covariance matrix to eliminate the step to solve a difference Riccati equation as it is otherwise required. The update formula is shown as: G{k + 1 | k) = G(k | k) - K kCkG(k | k) 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Then, the estimated state at step k follows: % k -A -kZ k— i “ I - //fcV -F fc 4" Kfc{^Sk CfcZk— i) where zk-i is the previously estimated state vector. To start the iteration, the initial state vector is computed by the linear prediction model presented in the previous sub section, and the initial error covariance matrix G( 1 | 0) is set as the identity matrix I. The parameters Aj’s get updated periodically but less frequently while the input control parameter fj,k is updated in each step. To update these parameters, one way is to couple the trajectory or field characteristics with the model, and one pair of param eters corresponds to one segment of the field or trajectory targeted. This database is precomputed based on the targeted field and the update of parameters is determined dynamically with aid of application level information or via communicating required information from the sender online. In the experiment, the ground truth of the trajec tory is available and the parameter database is created offline. 6.3 Experiments and Applications In sensor networks, sensors readings are spatio-temporally correlated [87] and also correlated among different types of modality. Each sensor observes a time-series rep resentation of a signal in a particular modality, and nearby sensors observe similar representations in the same modality or a different modality due to the close proximity and the physical law of signals propagation. If these correlations can be exploited ef fectively, a high compression ratio can be achieved to reduce the communication cost in the network. This can significantly reduce the energy cost since the communication 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. cost in general is a dominant factor in most applications of wireless sensor networks. Therefore, compression schemes exploiting these correlations can significantly reduce communication cost. For two correlated sources X and Y, let each sample is drawn from an independent, identical distribution, for the lossless compression case, the achievability of Slepian- Wolf theorem [80] gives the lower bound of transmission rate. The achievability of Slepian-Wolf theorem states that for two iid sources, a rate close to the joint entropy H(X, Y ) can encode the two sources and the decoding error can be made arbitrary small when the numer of samples, n, is large enough, the achievable rate provided by the Slepian-Wolf theorem is simply nH(Y\X) by the following identities (the chain rule of joint entropy): H(X,Y) = H(X) + H(Y\X) = H(Y) + H(X\Y) where H{Y\X) and H(X\Y) are the conditional entropy, and H(X) and H(Y) are the entropies of source X and source Y, respectively. Methods on approaching Slepian- Wolf limit with practical compression complexity have been proposed [25,63]. For the lossy compression case, Wyner-Ziv rate-distortion function [103] tells that the mini mum achievable rate R*(d) at distortion level d with side information available at decoder is as follows: R'(d)= mmI(X,Z\Y), p€M(d) where, M(d) is the set of codecs which meet distortion level d\ Z is a transition random variable derived from a codec and /(.,.) is the mutual information function. Wyner-Ziv rate-distortion function requires that the side information be accurate. In 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. applications of sensor networks, side information facilitating compression may be pre sented as an approximation to the codec, and good correlation analysis scheme is cer tainly necessary for a codec to achieve lower transmission rate for a given distortion level. Based on the above discussion, one important issue in the sensor storage and dis tributed source coding of correlated sources in wireless sensor networks is to estimate the encoder’s values from the data available at the decoder. This is critical for many resource constrained sensor network applications. Correlation tracking is therefore important in these types of applications, and it directly affects a codec and storage systems performance. With a high performance correlation analysis scheme, spatial correlation can be effectively exploited to its maximum. On one hand, instead of us ing a decoder’s frame for reference during decoding, the decoder uses a reconstructed frame. A good correlation analysis scheme can reconstruct the encoder’s frame with a very low RMS error; this can greatly improve the performance of a codec. On the other hand, a high performance codec can further improve the analysis results due to that more accurate estimates of source frames are available to adjust estimation param eters. This is the reason why the proposed analysis scheme can significantly improve the compression ratio of a codec. Assuming a dynamic partition structure for a codec and any codeword can be cor rectly decoded as long as there is a “correct” reference at the decoder within the desired partition. To apply the proposed model to a codec, a coarser partitioning (i.e. larger average coset size) is used during Kalman filter tracking, and a measurement vector from encoder can be confidently used in the estimation so we can weigh more on the measurement. After the Kalman update steps, the normal partition structure is restored and the steepest descent method is used for continuous tracking update. With this inter leaving between update based on DKF and update based on steepest descent method, 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acoustic Readings P X O T p ' , ' 0 5 10 15 20 25 30 35 Time (s) Seismic Readings “ 0.5 25 26 27 28 29 Time (s) (a) Acoustic signal case (b) Seismic signal case Figure 6.6: Illustration of interleaving of estimation and tracking we are able to have a comparable SNR performance over other schemes and better compression ratio can be obtained. Figure 6.6 shows two plots of amplitude readings of one acoustic sensor and one seismic sensor over a period of 37 seconds where seismic readings (see Fig. 6.6(b)) are zoomed in. The data of these plots come from sitexOO [19] data set. Here only a segment of a single vehicle (AAV-O) case is shown to illustrate how estimation and tracking can be interleaved for the trade-off of accuracy and efficiency. In Fig. 6.6, correlation is estimated periodically using steepest descent method on frames marked by dotted rectangle note that the reference frame is not shown in this plot. Frames not highlighted in Fig. 6.6 are tracked directly by the proposed DKF method. Each sample is decoded using corresponding tracked samples other than the sample in the reference frame. Figure 6.7 shows the decoding performance with and without cor relation analysis. We select ESPIHT [87] and DISCUS [63] for this comparison, and these schemes using correlation tracking are denoted as ESPIHT-T and DISCUS-T, respectively. When SNR is set at a practical value of 20 dB, based on three different data sets, the bit rate can be reduced in average by at least 0.5 bits per sample because 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Bit Rate Comparison 3.5 ■ 15 0.5 - ARL SITEXOO APG DISCUS-T ■ DISCUS 0ESPIHT-T SESPIHT Figure 6.7: Bitrate comparisons of codecs with/out correlation analysis of correlation analysis. This extra gain is due to that a large coset size can be set so that less bits are required to code coset indices during spatial coding. Storage system has been critical for many sensor network applications because of resource constraint in sensor networks. High performance storage system not only reduces system memory requirement, but also facilitates network operations and con serves energy. Promising approaches have been reported, e.g. [27, 28]. These ap proaches are also to exploit the temporal, spatial and multi-modality correlations for efficiency and storage space reduction. The proposed scheme can be used to facilitate online execution of these storage systems. Summarization is one of ways to help application level queries. With data sum mary, coarser information can be extracted for a query, and only if necessary, should further data communication continue. With the proposed scheme, a data collecting node only needs to store Z 0 + ra0 + 1 LP coefficients for a sequence of frames of an other node. Compared to DIMENSIONS [27] where wavelet transform is done on a node closer to source where source data can be collected easily, the proposed scheme allows much less information to store and very limited source data transmission in lo cal region and no data transmission in a large scale. The main feature can be recovered by online tracking using the proposed tracking algorithm. If further detail is needed, 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. more data can be first compressed using spatio-temporal compression algorithm and sent over from another data collecting node which is presumably closer to the source. The data can also be organized in a hierarchical fashion with multiple layers, and different layers have different granularity of information. As we know, the coefficient corresponding to principal index conveys the most information about correspondings frames of another node, and further refinement information is contained in a number of auxiliary coefficients. Data can be organized in such a way that nodes closer to the source have the principal coefficient and more auxiliary coefficients, query can be done in an iterative fashion with finer detail. In what follows, we shall compare the performances of three schemes, namely, the proposed estimation and tracking scheme denoted by LP/DKF, the scheme used in [10] denoted by LP/SD, and the least squared estimation (using cross correlation coefficients) denoted by LSE. For the comparison, we take into account three aspects: (1) tracking accuracy; (2) required communication; (3) processing overhead of the algorithms; (4) total energy cost which combines both processing and communication costs. The experiment setup is shown in Table 6.1. In this experiment, communication between two nodes is on average one packet every five seconds, and node interference can be totally eliminated via proper scheduling. Therefore, for simplicity, the com munication energy cost estimate does not consider the affects of the network density and communication interfence. Our experiment is directly based on C8051 microcon troller with its integrated development environment, it does not need any operating system support. The program is directly loaded to a designated memory area and it starts to execute once the node is turned on. Table 6.2 shows these comparisons with an experimental run of 235 seconds from one of a few different data sets. In Table 6.2, the memory is based on the worst- 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 6.1: Experiment setting Processing unit [79] CYGNAL C8051 F020 On-Chip Memory 4096 + 256 Bytes Clock Rate 16 MHz Radio type [8] Chipcon CC1000 Communication energy cost 3.2 /r J per bit Table 6.2: Performance and cost comparisons (per frame) Mem. (Bytes) Error (dB) Comm, (bits) Proc. Time (ms) Eng. ( jjJ) LP/SD 128 -10.4 0.9 K 120 470 LP/DKF 128 -11.9 0.3 K 230 330 LSE 144 -11.5 3.2 K 150 830 LSE/DKF 144 -11.8 0.8 K 250 520 case RAM requirement. In the cases of LP/SD and LP/DKF, the worst-case RAM requirements are dominated by linear prediction, and these of LSE and LSE/DKF are dominated by LSE. The error in Table 6.2 is taken as the average estimate error in RMS divided by the average signal RMS (as the signal power), and it is computed in logarithm form (dB). The error in RMS is computed using the innovations with reference to real coefficients over 235 seconds. The low estimation error of LSE, compared to LP/SD, is not due to its algorithmic superiority, but the high volume raw data transmitted. The processing overhead is measured in the average worst-case time over all frames. Table 6.2 shows that the proposed approach when combined with SD or LSE, can reduce communication significantly without loss of accuracy. Figure 6.8 shows the average prediction error in RMS and the signal power of sender, where the estimation and tracking update is done in 1/5 uniform form, in which analysis step size has length of 6 frames with one frame length of samples estimated by SD and 5 frames length of samples updated using tracking, i.e. roughly 17% samples are needed to sent. The other approaches use different analysis step size with varying 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2800 w ^2300 D C ■§1800 3 §1300 < cd 800 O ) c o S > 300 < -200 Signal Power Innovations - 1 _ U - "I 1 I" I I " " 51 101 151 Frame Seq. 201 Figure 6.8: Tracking performance number of tracking frames so that a close estimation error can be obtained for compar ison purpose. Figure 6.9 shows the corresponding gradients as the input controls to the proposed tracking algorithm, and these vectors show some slight temporal correlation. The per-frame energy cost of the compared schemes is shown in Table 6.2, and the proposed scheme has a lower energy consumption which includes both communication energy cost and processing energy cost. Figure 6.10 shows the worst case estimation error vs. different analysis step size for different data sets of two types of modality with an LSE based baseline. Two dif ferent data sets are used in this comparison. One is the far-field sitexOO acoustic and seismic, and the other is near-field ARL acoustic. There are clearly a trade-off between the frequency of analysis and fidelity (low residual power means high fidelity). In Fig ure 6.10, the tracking update interval is determined based on signal energy variation in the adaptive method, and a relatively short interval is used when signal fluctuates and a longer interval is used when signal energy stablizes around a level. The performance of the near-field acoustic data set has the best performance. Without this analysis, it 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4000k 300a 2000. °-100Q -2000 250 200 150 100_ ™ Q U Q j', Time (seconds) Figure 6.9: Kalman filter input controls Multi-Modalilty Analysis Residual FAR-FIELD FAR-FIELD NEAR-FIELD ACOUSTIC SEISMIC ACOUSTIC ■ 1/5 H1/3 B Adaptive 0 Baseline Figure 6.10: Fidelity effects on analysis steps 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. would have been almost impossible to exploit the spatial correlation of sensors read ings in a far-held for any good use. 6.4 Conclusions In this chapter, we have proposed a scheme to track correlations in wireless microsen sor networks and it uses a linear prediction model to find the initial correlation co efficients and then uses a discrete Kalman filter model to track the correlations. We have applied this algorithm to source coding and eliminated the calibration of sensor readings. Experiments show that the algorithm gives an accurate tracking results of the correlations with manageable processing overhead and low communication cost. In the next chapter, we will propose energy efficient protocols and detection algo rithms to enable a two-tier hierarchical sensor architecture. 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 7 Cueing and Detection in Wireless Sensor Networks One principle on energy conservation for wireless sensor network is the JIP notion, “JIP: Just In time Power” [18]. There are two types of sensor nodes - low power tripwire nodes (tripwire in short) and signal processing sensor node (tracker in short) which can be deployed together; tripwires monitor a field and wake up a tracker to process if an event happens. There are two major problems associated with a two- tier wireless sensor network using tripwires, namely, false alarm problem and exposed alarm problem. In this chapter, we propose a multi-hop tripwire detection and cueing scheme for energy conservation in wireless sensor networks. Our proposed scheme enables this JIP notion, and it overcomes the false alarm problem and the exposed alarm problem. For a typical application scenario of sensor networks, our simulation results show that a two-tier sensor network employing the proposed scheme can reduce energy consumption on average by 55% when compared to a homogeneous sensor network. The proposed scheme takes into account the resource constraints of trackers and tripwires, and it fits well to a wireless sensor network. 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.1 Introduction Wireless sensor networks have made many new applications possible [6 6 ]. Sensor nodes are constrained by various resources including energy. Algorithms and proto cols which can relax these constraints and conserve energy are very useful for sensor network applications. One approach to accomplish this is to use two kinds of nodes - signal processing sensor nodes (trackers for short) and tripwire nodes (tripwires for short). The operating power for a tracker is around an order of magnitude higher than that for a tripwire. Tripwire [65, 2, 16] is one type of lightweight microsensor node which can perform some simple processing tasks, and it also has some limited com munication capability. They can be deployed along with trackers, and form a network to monitor a sensor field so that trackers can be kept in sleep mode to conserve energy. Tripwires wake up periodically to check field status and they wake up trackers for var ious signal processing tasks only when necessary. Since there is at least one order of magnitude difference on power dissipation between a tripwire and a tracker, by using tripwires, energy can be saved since trackers consume energy only for these periods when events are happening. To detect events, tripwires use threshold detection - as long as the signal strength is high enough, trackers will be woken up. Tripwire threshold detection suffers from False Alarm Problem (FAP). FAP refers to a situation in which noise strength is large enough and triggers an alarm when no event is actually happening. There are many sources of false alarms in sensor networks and applications: (1) ambient noise; (2) measurement thermal noise; (3) truncation error since a low precision analog-digit converter (ADC) is usually used; (4) scatter and/or reflection to sensing signals; (5) multi-path and multi-source interference. (4) and (5) usually cause false positives (events are identified as false alarms). These factors are not negligible in sensor network applications. 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 7.1: EAP in a two-tier sensor network Figure 7.1 shows an application scenario using acoustic signals where a number of sensors are woken up unnecessarily as the object is moving toward one direction. In Fig. 7.1, star shaped objects are tripwires; light color “L” shaped objects are active sen sor nodes; black color “L” shaped objects are asleep sensor nodes; target is marked by a moving white rectangle with its trajectory highlighted in white. Apparently, results obtained from these nodes in the black circle are discarded and energy consumed by these nodes are wasted. In some case, many more nodes are woken up unnecessarily when an event is in progress. We refer to this problem as the Exposed Alarm Problem (EAP). Note that false alarm detection can not solve this problem since there is actu ally events happening in a field. To overcome both FAP and EAP, we propose a novel multi-hop tripwire cueing scheme with iterative tripwire detection. Previous endeavors have been on scheduling trackers as in S-MAC [109] or use threshold detection to wake up trackers on demand. In [45], a windowed energy thresh old detection is used to detect a potential event, and that scheme is directly applicable to a tripwire node for event detection. Tripwires give great flexibility to an application and conserve energy; however, it suffers from EAP and FAP. 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.1.1 The Cueing and Detection Problems and Our Contributions For the ATR applications when targets are infrequent, the number of false alarms could be high. Without proper detection algorithms to address this problem, energy can be drawn quickly. When events are happening, many nodes are woken up simultaneously whether or not they are really needed. Since signal processing is a computation inten sive process. This exposed alarm problem may also cause substantial energy waste. With the false alarm problem and the exposed alarm problem in sensor networks, the questions we have been interested in are how to enable an energy efficient sen sor network with the aid of tripwires via the means of cueing protocol and detection algorithms. The objective of event cueing in wireless sensor network is to wake up an optimal set of trackers for the energy efficient process of the event. The event detection problem is to detect potential events with low false positive (or false alarm) rate and low false negative rate. False alarms will waste energy on sensor nodes while false negatives will cause the loss of tracking. Our contribution includes a novel scheme which enables a two-tier sensor network while overcoming both problems. It solves FAP using an iterative detection algorithm and solves EAP via implementing a radio wake-up channel protocol using multi-hop tripwire cueing. Our results provide a framework for building an energy efficient het erogeneous sensor networks. We have demonstrated via analysis and experiments that this framework is effective. A tracker wake-up scheme has been presented in [82] where the focus was on how to select a set of optimizing trackers from application point of view. Although that work is orthogonal to tracker cueing scheme to be studied in this chapter, an alternative cueing scheme (denoted by INIT) can be constructed for that algorithm 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. by the following steps: ( 1) any tripwire initially constructs a local database of trackers; (2 ) online optimization algorithm proposed in [82] selects a set of trackers to execute an application. This INIT scheme however does not solve the above two problems. After trackers are woken up upon an occurrence of an event, the algorithm proposed in [82] then can be executed by a set of woken trackers to select an optimal set of trackers for the task in question. 7.2 Proposed Multi-hop Cueing Protocol We assume that any tracker can be reached by at least one tripwire; furthermore, any tripwire can reach at least a tracker. Note that one tripwire hop is usually much smaller than one tracker hop, and it is necessary to distinguish them (When the context is clear, one hop is used instead of one tripwire hop or tracker hop). We also assume a tripwire is equipped with a microcontroller unit (MCU) which enables some level of intelligence to a tripwire. In this chapter, optimization of tripwire power consumption is not the focus, and maximization of tracker sleep time, therefore, energy saving on trackers, is the main goal. Before delving into the details, we first give a high-level description of the pro posed multi-hop cueing (MHC) protocol. It uses three different types of beacons, namely, T-WAKEUP, T-QUENCH and T-ALARM. T-WAKEUP is sent by a tripwire to wake up a tracker; T-QUENCH is sent by trackers to stop tripwires sending T- WAKEUP and notify tripwires that at least one tracker has been woken up; T-ALARM is sent by trackers to notify near-by tripwires to resume its monitoring task. 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 7.1: Characteristics of tripwire and tracker radios bitrate typical freq. power range Beacon Radio < 1 kbps 900 MHz < 1 mW ~ 1 0 m Data Radio > 5 kbps 2.4 GHz > 100 mW > 1 0 0 m Table 7.2: Characteristics of common commercial radios Tx energy Rx energy range bitrate freq. ORiNOCO 2/iJpb 1/zJpb 1 0 0 m 1 mbps ~2.4Ghz CC1000 2 /i,/pb 1 .2 //Jpb > 1 0 m 19.2 kbps ~900Mhz CC2400 0.5/iJpb 0 . 2 fx Jpb > 1 0 m 250 kbps ~2.4 Ghz TR1000 2 /iJpb 1 n Jpb ~ 1 0 m 30 kbps ~900Mhz PicoRadio 5nJpb 5 nJpb ~ 1 0 m < 1 kbps ~2.4Ghz 7.2.1 Radio Models and Wake-up Channel Emulation A tracker can be woken up by a wake-up beacon. A tracker also has a tripwire unit built-in, and it turns on when its associated tracker is asleep. There are two radios in a tracker where, one is a long-haul radio, called data radio, is for data communication of task execution, and the other is a tripwire radio, called beacon radio for beacon exchange. The tripwire radio in a tracker uses one channel and its long-haul radio uses another channel for data communication. The tripwire network forms a so called “wake-up” channel for trackers. The proposed protocol uses this wake-up channel. In this chapter, all tripwire radios and long-haul radios have fixed transmission power; however, this assumption is only for the simplicity of explanation, and the proposed scheme works as well when these radios are power controlled. Table 7.1 summarizes the characteristics of tripwire and tracker radios. Table 7.2 compares some typical commercial radios and their uses for tripwire and tracker radios. Tracker wake-up is performed as follows: a near-by tripwire wakes up the tripwire unit of a tracker, and this tripwire unit then wakes up other components of a tracker it resides. However, the initiating tripwire of waking-up processing may be some hops 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Tripwire Object Tracker Figure 7.2: Tracker in different wake-up radius away from a closest tracker. Figure 7.2 shows trackers in different wake-up radius, and the numbers on each tracker show how many hops the tracker is from the event. As shown in Figure 7.2, it is proper to let trackers close to targets to process events. However, trackers in more than one hop away may also be woken up. This is the so- called EAP. To overcome this, T-QUENCH beacon is used to put tripwires close to targets into TL state and T-ALARM will wake up these tripwires when the target is off the monitoring area. With multi-hop tripwire cueing, at least one tracker is guaranteed to be woken up. There are two undesirable cases: (1) more than required trackers for a task execution are woken up by tripwires and (2) less than required trackers are woken. Since T- WAKEUP floods, Case (2) cannot happen unless some deployment problem causes an insufficient number of trackers in certain region of a field which is not considered in this chapter. Trackers which are awake will collaboratively decide how to put excessive trackers back to sleep in Case (1). It is up to application level protocols [82] to handle 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 7.3: State machine of tripwires mW MCU RADIO SENSOR ADC TM rs j r v on r s j TE on off on off TL off on off off how to select an optimal set of trackers for a task and what optimization criterion should be used; however, this is out of scope of this chapter. 7.2.2 State-Machine Model A tripwire has three operational states which corresponds to three machine states, namely, monitoring state (TM), listening state (TL) and execution state (TE). In a TM state, tripwire radio and MCU are active periodically and its sensor is always on; in a TL state, tripwire’s MCU and its sensor including analog-to-digital converter (ADC) are turned off while its radio is kept on and only radio receiver is actually used; in a TE state, tripwire’s MCU and radio are kept on while its sensor and ADC are turned off. TE is a transit state. Table 7.3 shows component states in different tripwire states. In Table 7.3, symbol ~ denotes a component that is turned on periodically. Once an alarm is confirmed by a detection scheme, a tripwire issues a wake-up beacon called T-WAKEUP and it goes to a TL state. A multi-hop cueing algorithm is used to wake up trackers. The tripwires switch states in the proposed protocol as specified in the following cases. 1. In TM state, on receiving a T-WAKEUP beacon or having detected an event (alarm), this tripwire sends or forwards a T-WAKEUP beacon and goes into TE state. 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 7.4: State-beacon table T-WAKEUP T-QUENCH T-ALARM TM FORWARD X X TE X FORWARD X TL T-QUENCH X FORWARD 2. In TE state, on receiving a T-QUENCH beacon, it forwards this beacon and switches to TL state; on receiving a T-WAKEUP beacon, the tripwire forwards it. 3. In TL state, on receiving a T-ALARM beacon, it switches to TM state. 4. In TL state, on receiving a T-WAKEUP beacon, it replies by a T-QUENCH beacon. 5. On timeout in TE state, a tripwire goes to TM state. 6 . On timeout in TL state, a tripwire goes to TM state. In the above, no action is needed by a tripwire if the combinations of beacons and states other than the cases listed are encountered. Note that the timeout value of a TE state is shorter than that of a TL state. A new try can be initiated when no T-QUENCH beacon is received in the TE timeout period; the TL state timeout is to restore a tripwire to a TM state when a T-ALARM cannot reach certain tripwires. Item 4 above is used to put into a TL state of these tripwires which newly detected a target in the same monitoring region. Note that only tripwires in a TE state response to T-QUENCH beacons. T-QUENCH is used to put tripwires into a TL state when an event is detected and T-ALARM is used to resume tripwires for monitoring when an event is out of the detection region of a tracker. Table 7.4 summarizes the tripwire responses to these three beacons where x represents no response to a beacon at that state. Figure 7.3 summarizes the state transitions of a tripwire. 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 7.3: State diagram of tripwire 7.2.3 Beacon Communication and Tracker Hand-off Latency is one of the important aspects to consider when to devise efficient tripwire beacon communication protocols. A tripwire radio has to be ready to receive beacon virtually any time; optimization of beacon transmission is not really needed since a low power radio consumes the similar power on reception and transmission. Due to various reasons, tripwire radio channel has high bit error rate. In a multi-hop tripwire environment, without proper coding of beacon bits, many retransmission could sig nificantly delay the alarm notification to trackers. In this proposed multi-hop cueing protocol, beacons are designed so that it is recognizable and retransmission is rarely needed in spite of low quality radio. Protocols devised for data communication may be too wasteful to be adapted by tripwires since only identification of beacons is required, unlike in data communication where various mechanisms are employed to ensure that data can be received correctly. We use a system comprising of 16 beacons (i.e. 4 beacon content bits) with 7 bea cons used by the proposed cueing protocol to illustrate the idea (this beacon design can 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Monitoring Region T-WAKEUP. Boundary T-QUENCH ^ T-ALARM Boundaries T r i p w i r e O b j e c t T r a c k e r Figure 7.4: Beacon propagation boundary support up to 12 hop tripwire cueing). Beacon flooding is controlled by a time-to-live (TTL) based mechanism. To implement TTL on T-WAKEUP, each hop uses a different T-WAKEUP beacon. To address the collision problem in a TTL controlled flooding of T-WAKEUP beacon, each tripwire uses a random timer to control the forwarding of received T-WAKEUP beacons. However, there is no need to use TTL to control T-QUENCH and T-ALARM beacons’ flooding, they are implicitly controlled by the state-machine in each tripwire - the flooding boundary of T-QUENCH and T-ALARM is defined by the T-WAKEUP flooding TTL. Figure 7.4 shows the reachable bound aries of different beacons in a multi-hop cueing system. Tripwires out of T-WAKEUP boundary are in a TM state and these inside are in a TL state after the execution of the protocol. Beacon is designed using linear block code. The basic idea is to devise a set of beacons with maximal inter-beacon Hamming distances. The length of the block code is selected so that a desired codeword error probability is minimized. The decoding 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 7.5: Linear block codes for beacons with 4 content bits Beacon Length 6 7 8 9 dmin 2 3 4 4 Pw(x 10-4) 300 2 0 2 0 28 Pu(x 10-6) 300 7 0 . 0 0 1 ~ 0 of the beacon uses the simple standard array decoding (SAD) method [95]. This is efficient since the number of codewords is small. In what follows, linear block codes of dimension 4 for a multi-hop cueing beacons are discussed where codeword error probability is denoted by Pw and undetected codeword probability is denoted by Pu. Hamming code (7,16,3) is a good candidate for the case of 4 beacon bits with Pw of 2.0310-3 and Pu of 6.810-6 at bit error rate at 10~2. Based on the minimum distance table in [4], Table 7.5 can be derived where dmin is the code minimum distance and table item values are rounded where the number of codewords are set to 16 (although for codeword length of 9 with Hamming distance 4, the maximum number of code words is 20, e.g. Hadamard code [95]). Based on channel conditions, a code with an appropriate length can be selected so that a T-WAKEUP beacon can be received with high probability within the first TE timeout by all trackers which are upto 3 hops away. Longer length of codes increase the probability of collision while they also increase the probability of correct reception of a beacon in a noisy channel. The optimization cri terion is to minimize delay. Among the codes listed above. Code (8 ,4,4) (also called extended Hamming code) gives the shortest delay for radio reception BER between 1 0 - 2 to 1 0 “3, and longer length codes are excluded in the code search since the stan dard array in decoding could be too big for a tripwire to store or look up. Figure 7.5 shows the beacon error rate vs. bit error rate using the extended Hamming code. Tracker hand-off can be handled easily in our scheme. Tracker hand-off is for a situation when one tracker has to yield to other trackers when a target is out of its tracking area or signal-to-noise ratio is too low to obtain accurate results. When a 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. _ 2.0E-03 § 1.8E-03 § 1.6E-03 -9 1.4E-03 ~ 1.2E-03 | 1.0E-03 3 8.0E-04 w 6.0E -04 o 4.0E -04 | 2.0E-04 “ O.OE+OO 0 0.002 0.004 0.006 0.008 0.01 0.012 Bit Error R ate (% bits) Figure 7.5: Beacon error rate for Hamming (8 ,4,4) code tracker is about to yield, T-ALARM will first alarm tripwires around the target and a T-WAKEUP flooding will wake up appropriate trackers. When a new set of trackers are woken up, tracker hand-off is finished. 7.3 Iterative Detection through Sensor Collaboration False alarm detection is important for sensor networks since field noise and measure ment noise are present in sensor readings. In essence, false alarm detection is to con firm that an alarm issued by a tripwire is indeed an event. False alarm detection is critical to energy saving since event process is deemed to be high power while detec tion process on tripwire requires much lower power. In a two-tier sensor network, a tracker should be woken up based on detection predicates. In alarm threshold detection, it is difficult to determine an appropriate threshold and avoid false alarms. High threshold could result in loss of events of low power; however, it would cause a high rate of false detections if a threshold is 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. set too low. In order to solve this dilemma, we introduce a collaborative two-stage detection scheme which consists of an energy threshold detection followed by a false alarm detection. There are different false alarm detection algorithms, e.g. detectors using receiver operating characteristics (ROC) curve [83], DFAD [8 6 , 58] in wavelet domain, Power-Law Detector [98, 41, 36] in Fourier domain. In these algorithms, a decision threshold is commonly used to distinguish a false alarm from an event. Assume that the signal is sampled by a Nyquist frequency f z and appropriate ana log low-pass filtering has been done before sampling so that energy aliasing of signal frequency beyond of ^ is negligible. Signal is processed using frames of fixed length. Further assuming that the frame length is long enough so that a frame can include a full cycle of the lowest frequency component of a signal. 7.3.1 Models and Signal De-noising It is assumed that there are p (unknown) signal sources observable by a sensor array of M sensors. These signals are measured at discrete time intervals U, i e {1,2, • • • , N } and N is the frame length. These sources at a sensor are denoted by x which are assumed having a multivariate normal distribution with mean zero vector as x ~ .A/^(0, £ p), where £ 6 TZNxN. Noise z is assumed additive. The measured readings y by these M sensors are then given in (7.1). y (*, j) = x(i, j) + z (i, j ) (7.1) for i e {1,2, • • • , N } and j G {1,2,--* , M }. We further assume that the noise term z (i,j) consists of two additive components, namely, intra-sensor spatially correlated noise which is local to a specific sensor, and inter-sensor independent noise which is present to surrounding sensors. 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Let j3 be the inter-sensor noise level and e be a coefficient. We then use wavelet de-noising procedure to remove this inter-sensor independent noise before false alarm detection process. In this thesis, we use soft-thresholding de-noising on wavelet coef ficients to remove the inter-sensor noise as shown in (7.2). and 7 is the subband energy normalization factor. Inter-sensor noise can be more accurately modeled by a Gaussian distribution than intra-sensor noise due to long propagation degradation. Wavelet soft-thresholding de- noising procedure has shown effective on this Gaussian noise [21]. These de-noised wavelet coefficients are used during the false alarm detection process. 7.3.2 Alarm Threshold Detection The following sample variance formula in dB is used to compute signal energy of a frame of fixed length in time domain for alarm detection: where, A ; is a frame index; n is the frame length and is the i-th sample amplitude of k-th frame. Once a frame is completely received, its energy £ is computed using (7.3). If it is greater than an energy threshold, an alarm is issued. When a necessary number r]{e) = 0 if |e| < Z fc (7.2) sig n (e)(e-t) if |e| > Th where, the threshold Th is given as follows: (7.3) 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of trackers are woken up by our MHC protocol, false alarm detection process can start by sensor arrays. 7.3.3 False Alarm Detections In this section, we present a detection algorithm on de-noised wavelet coefficients. It is a maximum likelihood ratio detector which is performed on wavelet subband data. This wavelet maximum likelihood ratio (WALO) detector gives low false detection rate, i.e., false negative rate and false positive rate. The distribution of denoised coefficients in a subband can be modeled by the gen eralized Gaussian distribution [84] whose probability density function has the form as (7.4). (7.4) p(x) where, 2T(J) m i1 Lr 0 J and (3 is the standard deviation and T(.) is the Gamma function. In general, the shape parameter a falls in (0 , 2 ]. We have found using the maximum likelihood estimator that this generalized Gaus sian distribution with a = 1.5 gives a satisfying model to the ATR data set. We have performed experiments with a number of acoustic signal data sets after dyadic wavelet decomposition. Figure 7.6 shows that the QQ-plots of quantile transformed subbands of a vehicle acoustic signal using the statistics package R [64], and the quantile lines fit well to these plots. Since the spatially correlated noise is more concentrated in the coarsest approxi mation subband, in this thesis we examine denoised coefficients of subbands excluding 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Magnitudes o f Coefficients o o CVI o o o o o T o o CM I -2 1 0 1 2 C O c 0 ] o 0 O O 0 T 3 3 C o > T O 2 O o CM O o o o o o o CM I ■ 3 -2 1 3 0 1 2 Quantiles of Gen. G aussian Quantiles of Gen. Gaussian (a) Detail (level 3) (b) Detail (Level 2) w c 0 ' o % o O L O O CO 0 T 3 3 O C O c O ) (0 2 3 ■ 2 1 0 1 2 3 Quantiles of Gen. Gaussian (c) Detail (level 1) Figure 7.6: Quantile plots of subbands with a = 1.5 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 7.7: Subband used in detection process the coarsest approximation subband. We assume that coefficients in these subbands are independent and drawn from a Generalized Gaussian distribution with a = 1.5. This assumption is reasonable since the coarsest approximation subband contains most of the dependent noise energy which is excluded in the detection process. Due to the processing capability constraint on sensors, we limit the process only to the coarsest detail subband at each iteration level. Figure 7.7 shows these subbands on which the detection process is operated. Assume that there are n coefficients in Subband Vi, i.e. x = (xi, x2, • • • , xn). The log likelihood function is given by (7.5). f(x \n ,/3 ,V i) = nlog 2 r 0 n ^ { k Q.« |x ,|l“} (7.5) i= 1 where a = 1.5 and (3 is the maximum likelihood estimate of /3 given in (7.6). I 0 1 r0J i—1 (7.6) Let the null hypothesis be H0 : (3 < (3 0 where f3 0 is the variance with respect to the field without target. Denote the unconstrained parameter space by © = {/3|/3 > 0} 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in the generalized likelihood ratio test. By (7.5), the log maximum likelihood function with respect to 0 is given in (7.7) as follows: sup f(x \n , (3, Vi) = n l o g ( - ^ r - \ - - lo g ( — ^ |x < |a ) (7.7) / 3 e 0 X ( a ) / a \ n iZ\ ) For the null hypothesis H0 : (3 < (3 0, the log maximum likelihood function value with respect to 0 O := {(3 < f3 0} is simply the function (7.5) evaluated at /3 0. Therefore, we have the generalized likelihood ratio A(x, f3 0, n, Vf) can be represented by (7.8). \f a /(x K A ) ,A ) /n o ^ A(x, Po, n, Vi) = -----— — — — (7.8) sup/(x|n, (3, Vi) p&e where, the denominator is computed by (7.7). The following Lemma 7.1 gives the sufficient statistic for j3. Lemma 7.1: Function T given by (7.9) is a sufficient statistic for f3. n r ( x ) = y > , | “ (7.9) i = l Proof: The likelihood function is given as (7.10): /(x,/3) = [ a r ( i ) J e x p { - r 7 ( a , f3 )a^2 ( l * < r ) } (7 -1 0 ) i=i The statistic { x \,x 2 ) ■ ■ ■ , xn} can affect (3 only through T(x), i.e. /(x , (3) can be factored into two functions as shown in (7.11). /(x,/?) = g{T{x),(3)h(x) (7.11) where h(x) = 1. Therefore, T(x) is a sufficient statistic for (3. ■ 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. By the Rao-Blackwell theorem and the Lehman-Scheffe theorem, the estimate of variance should use (7.9). Then, we have the decision statistic V for testing H0 as given by Proposition 7.1. Proposition 7.1: The decision statistic V for testing H0 is given as follows: n V (x ,(3 o , A ) = ( 7 - 1 2 ) i= 1 Note that for the special case when the exponent of generalized Gaussian distribution is 2, (7.12) is the standard Chi-Squared test with n degrees of freedom. The computation of the decision statistic can be carried out once a reliable estimate of /?o is obtained. Let 7) be the threshold on decision statistic. The algorithm is given in Algorithm 7.1. Algorithm 7.1: WALO (1) Estimate A) using the maximum likelihood estimator of (3 for field noise calibra tion. (2) Iteration (i-th level): (2-i) Perform dyadic decomposition on the current approximation subband. (2-ii) De-noise the detail subband XV (2 -iii) Compute V(x, A)> Vi) (2-iv) If V (x, /30, T > i) > Tj, return an event predicate. (3) When a given number of levels have been examined, return a false alarm predi cate. ■ Remark 7.1: There are a few remarks on WALO: 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (1) Since the noise estimate can be reliably obtained, when an event is happening in the field, the energy distribution among subbands are altered, and WALO utilizes such changes in subbands to infer false alarms. (2) Unlike Fourier transform, wavelet transform has local frequency resolution. There fore, the model has to be adjusted when filed noise is changed, and field calibra tion is necessary. (3) Type I and Type II error probabilities of V can be computed based on individ ual subbands based on the power functions. Hence the false positive and false negative error rates can be roughly estimated. (4) WALO uses only linear operations including the dyadic wavelet transform. For the event predicate, the transformation may stop at an intermediate level which saves both transformation and detection process. (5) WALO can be collaboratively processed by a group of sensors and each sensor can be assigned a subtree of the wavelet decomposition tree and only decision predicates need to be exchanged. By this way, the transformation cost at each sensor is significantly reduced. The communication cost is also kept very low since one predicate needs only one bit. Although WALO is simple, it can handle non narrowband signal to a fairly sat isfactory level while FFT based generalized maximum likelihood ratio detector [98] can not handle this type of signals well as we have tested. Figure 7.8 shows the de cision statistics on vehicle acoustic signal on detail subbands at different levels. To yield a constant false alarm (CFAR) detector, it can be extended via controlling of the detection threshold of each detail subband. 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Decision Statistic (Magnitude) 12 — Detail Subband (Level 1) 10 8 6 4 2 0, 0 200 400 600 800 1000 Time (seconds) (a) Detail subband (level 1) — Detail Subband (Level 2) 200 400 600 800 1000 Time (seconds) (b) Detail subband (level 2) — Detail Subband (Level 3) O ) 200 400 600 800 1000 Time (seconds) (c) Detail subband (level 3) Figure 7.8: Decision statistics from detail subbands Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5000 a > ■ § 4000 1°08 200 400 600 800 940 Time (Seconds) Figure 7.9: Vehicle signal in time domain Figure 7.9 shows the signal used in this experiment in time domain. By comparing Figure 7.9 with the decision statistic plots in Figure 7.8, the detection is fairly robust due to the fact that all three detail subbands can correctly identify the event for the first time, and the decision statistic has persistently significant large magnitudes when the event is happening in the field. A decision fusion can be simply added for generation of the final predicate based on individual detection predicate from one sensor. We used a simple boolean decision fusion rule as follows: If all tripwires give an event predicate, the final predicate is an event predicate, otherwise, the final predicate is a false alarm predicate. 7.4 Simulation and Experimental Results The performance of MHC protocol has been compared with two other schemes, namely, a baseline scheme where all trackers are active all the time, and a homogeneous sen sor network with duty cycling trackers denoted by “DUTY/C”. The proposed scheme using tripwire detection and cueing is denoted by MHC. In this energy/power comparison study, PASTA [38] tracker and PASTA tripwire are used, and their power profiles are shown in Table 7.6, where power consumption 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. p > UJ 10' — Tripwire — Sensor Node 500 1000 1500 Time (ms) Figure 7.10: Energy comparison of detection and tracking Figure 7.11: Power consumption of duty-cycling detection 149 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 7.6: Power profiles of tracker and tripwire mW CPU RADIO MCU/RADIO SENSOR IDLE 185 126 18 n/a ACTIVE 635 171 30 0.9 SLEEP 67 38 0 .1 n/a data of the PXA255 CPU and PASTA radio are itemized since they are not active at the same time. Tripwires are dissected into three parts - MCU, tripwire beacon radio, and ADC where MCU and radio is not always active. ADC on a tripwire or a tracker is active as long as this tracker or tripwire is on. The application used in this experiment is a time-domain line-of-bearing (LOB) beamforming. The energy per LOB computation including transmission is 245 mJ and the duty-cycle energy per detection is 1.5 mJ and tripwires are turned on periodically, where a tripwire radio is on during the first 1 0 0 ms of every second and off during the remaining period of a second; however, tripwire synchronization cost is omitted since it is required very infrequently. Energy dissipation comparison is shown in Figure 7.10. Figure 7.11 shows the power consumptions of a tripwire and tracker. From Figure 7.11, with cueing protocol and detection at 80% false alarm rate, the duty-cycle power consumption can be brought down within 10 mW regime. We shall further quantify energy savings of the proposed protocol. One scenario studied in this section is shown as Figure 7.12 where two targets are present in a field. The positions of trackers and target trajectory combine field measurement (using GPS ground truth data) and off-field simulation with a fixed field dimension, which can not be changed due to GPS real position. Simulation parame ters are defined in Table 7.7. Figure 7.13(a) shows the overall power consumption of tripwires and total power consumption of both tripwires and trackers for a period of 90 seconds. The corresponding energy dissipations are shown in Figure 7.13(b) where 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 7.12: A two-target scenario 4.0E+03 3.5E+03 > 3.0E+03 1 2.5E+03 2.0E+03 5.0E+02 O.OE+OO ■Total -Tripwires I 2E+04 4E+04 6E+04 8E+04 1E+05 Time (ms) 1E+05 1E+04 1E+03 ® 1E+02 = 1E+01 C D i° 1E+00 1E-01 Total Tripwires ?E +04 4E+04 6E+04 8E+04 1 E+05 Time (ms) (a) Power comparison (b) Energy comparison Figure 7.13: Field power and energy comparison 151 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 7.7: Field simulation parameters Number of Tripwires 25 Number of Trackers 18 Field Dimension 8012x4015m2 Tripwire Sensor Radius 10 m Tripwire Radio Radius 10 m Tracker Sensor Radius 15 m Tracker Radio Radius 100 m Tracker Radio Bitrate 1 kbps TL Timeout 1 second Fusion Timeout 100 milliseconds Predication Timeout 50 milliseconds Beacon Length 8 bits Wake-up Channel Bit Error Rate 0.01 Beacon Error Probability 2 x 10~3 Tracker Duty Cycle Rate 50% Table 7.8: Key scheme assumptions TRACKER SENSOR L/H RADIO TRIPWIRE BASELINE ON ON ON n/a Duty-Cycle W(S) ON n/a MHC W(T) W(T) W(T) ON the energy ordinate is measured and plotted in logarithmic scale. From these figures, it can be seen that overall tripwire energy dissipation is comparably negligible, and the savings in trackers using tripwires could be significant as shown next. Assumptions of these three schemes to be compared are defined in Table 7.8, where “L/H radio” for long haul radio and “SENSOR” represents tracker’s sensor(s). “W(S)” is for waking-up by tracker detection, and “W(T)” is for waking-up by tripwire detec tion. In the duty-cycle scheme, tracker’s sensors have to be always kept on for moni toring events and so does its radio for possible wake-up commands from other trackers. To compute the energy dissipation, we use timer to check the power of every sensor (tracker and tripwires) every millisecond and add the joules to the total energy. This should be a very close approximation to the real energy dissipation. 152 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Baseline 3E+4 6E+4 Time (ms) Duty/C ■Duty/C •MHC *-0 3E+4 6E+4 Time (ms) 9E+4 (a) Power comparison (b) Energy comparison Figure 7.14: The one-target case Figure 7.14(a) shows the power consumption for a 90-second simulation with one target. There are three curves corresponding to these three schemes. The power con sumption by MHC is significantly less than those of baseline scheme and duty-cycle scheme. Figure 7.14(b) shows the energy dissipation for a 90-second simulation, The energy saving of MHC is more than 50% when compared to duty-cycle scheme. MHC saves energy because it keeps trackers awake only for the period of task tracking. Figure 7.15(a) shows the power consumption for a 90-second simulation with two simultaneous targets. The power dissipation of MHC is still significantly smaller than these of baseline and duty-cycle schemes. The energy dissipation is shown in Fig ure 7.15(b). When compared to the single target case, the savings are reduced some what. In general, when many targets are in a field, the savings may not sustain since most of the trackers may be awake most of the time. However, in practical appli cations, events happen infrequently; therefore, the proposed scheme gives significant energy savings. 153 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 4E+3 Baseline — On/Off MHC 3E+4 6E+4 Time (ms) 9E+4 1 E + 5 i -f I ! ? ,1 5,-1 ,-, ,-,-| , , , , , , , , ,1 , , | 1 E + 4 - “ 3 E 1 E + 3 >. p 1 E + 2 | ; j | i ! | j j | l i l l | l l l j | i | i l j l | | l a> c L U 1E +1 - i i i i i i i i i i i i i i i i s i i i s i i i i i i u 1 E + 0 • h r On/Off MHC i-O 3 E + 4 6 E + 4 9 E + 4 Time (ms) (a) Power comparison (b) Energy comparison Figure 7.15: The two-target case Figure 7.16: Average energy dissipation with varying number of trackers 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99999999 n X 1(f 4.03 4l04 £35 4J5B £o7 £o8 Seconds x 1 (f (a) Tripwire 2 Figure 7.17: Energy in } x 10' O ) 1 ( Q ^0.8 0.6 0.4 0.2 4.03 Seconds (b) Tripwire 3 sampled variance form Since different applications require different granularities on sensor data accuracy and processing results, e.g. detection or tracking, tracker density may be different to meet this requirement. In general, it is expected that MHC can obtain a better energy efficiency with a higher density of trackers while average tripwires per tracker is kept constant and the assumption of connectivity holds. Figure 7.16 shows the average field energy dissipation measured in terms of one fixed measurement area with varying number of trackers. The curves in Figure 7.16 conforms the above claim on tracker density when the curve for 27 trackers has the least average energy dissipation. We have conducted a set of field tests using acoustic signals on evaluating our two-stage detection detection scheme using field data collected from Fort AP HILL army base. The first stage result is either an alarm predicate or a null-alarm predicate. The second stage result is either a false-alarm predicate or an event predicate. One configuration in these tests uses three tripwires, and results from two of these tripwires are shown below. Figure 7.17 shows energy plots from two of these tripwires. The 155 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 403 £ 0 4 4.05 4.06 J 4.07 4 M 37)9 4.03 4.04 4.05 „4.06_, 4.07 £US 47)9 Seconds x ig* Seconds x ^ (a) Tripwire 2 (b) Tripwire 3 Figure 7.18: Detection predicates final detection predicates from these two tripwires are shown in Figure 7.18 where value 0 is for no alarm, 1 is for false alarm and 2 is for event. From both plots, the detection rate is more than 90% and false positive rate is less than 1%. Since notification latency may be a concern when MHC scheme is used instead of direct detection as that in baseline scheme. A new simulation on cueing has been conducted using NS-2 simulator [57]. The parameters used in the simulation are shown in Table 7.9. We selected ground two-ray module for our short-path loss modeling of radio communication which is practical since tripwires’ radio is close to ground and ground effects are non-negligible. Figure 7.19(a) shows the average alarm notification for 100 random events to four trackers for different tripwire densities and bit error rates where tripwires are opti mally deployed for the 25-tripwire case. As more tripwires are deployed, robustness increases; however, in general, latency increases due to collision and beacon error. Figure 7.19(b) shows the maximum latency of notification to these four trackers. Due 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Average D elay (s) Table 7.9: Cueing simulation parameters Number of Trackers 4 Field Dimension 1000x1000m2 Radio Radius 200 m Radio Pathloss Model Two-Ray Ground Radio Bitrate 1 kbps Beacon Delay 10 ms Beacon Length 8 bits Wake-up Channel BER 0.01 Beacon Code (8 ,4 ,4)block code 9.5. (optimal] 8.5 7.5 1 2 3 4 6 o . 7 8 9V 1 f[3 Bit Error Rate x 1 o (a) Average notification latency Figure 7.19: Alarm 1 2 3 4 ^ t Error Rate 8 % 10~3 (b) Maximum notification latency notification latency 157 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to the beacon design with error resilience, the effects of channel BER on notification latency is rather limited by noticing that the curves are rather flat along the BER axis, and the maximal latency can be controlled under 200 ms. This latency can be fur ther controlled based on application requirements by adjusting TE timer, increasing minimum distance of codes and/or decreasing tripwires per tracker ratio. 7.5 Conclusions In this chapter, we presented a novel multi-hop tripwire cueing and detection scheme for a two-tier wireless sensor network to conserve energy. It solved both false alarm problem and exposed alarm problem using detection algorithm and cueing protocol. The scheme is simple and only primitive communication is required for tripwires; therefore, it fits in well a two-tier wireless sensor network. Simulation and field ex perimental results show that the proposed scheme saves significant amount of energy when compared to other types of network configurations and schemes. 158 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 8 Conclusions, Future Directions and Open Problems In this thesis, we have studied three problems in wireless sensor networks, namely, compression, correlation and detection. We have devised algorithms for correlation analysis, detection and present two compression codecs for different application sce narios in wireless sensor networks. In order to save energy, we apply our detection algorithms to our proposed cueing protocol so that not only are we able to duty cy cle nodes, but also nodes can be kept in sleep mode for the longest possible duration without loss of target tracking. We have verified our algorithms, protocols and codecs using field experiments, simulation and through theoretical analysis. Our results confirm the objectives of these algorithms, protocols and codecs. These techniques readily lead to energy efficient wireless sensor network operations. To make energy efficient wireless sensor networks possible in practice, besides power aware hardware, we should also pay attention to these three aspects of wire less sensor networks, namely, communication including energy efficient protocol de sign, computation including energy aware algorithm design, and sensor data including when and where they get processed, stored or transmitted. In-network processing as a 159 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. general principle for energy efficiency in sensor networks has its merit. However, in- network processing has its limitations. Since one advantage of wireless sensor network is on the utilization of diversity and collaborative processing, many in-processing algo rithms may filter potentially useful information which degrades performance of fusion algorithms. This could waste energy due to incorrect fusion results, e.g. high false alarm rates. One example could be the increased false detection rates from a detection scheme. Therefore, distributed source coding is important in sensor networks and it comple ments this in-network processing principle and helps to overcome these disadvantages of these in-network processing based energy saving approaches. Practical distributed source coding scheme allows excellent trade-off on rate and distortion. To support a high performance fusion algorithm where an in-network processing based approach can not effectively process data, a low distortion codec can be used to compress un processed data for transmission in a local scale or storage. Figure 8.1 shows that dis tributed source coding (DSC) can allow the feasible region of in-network processing becoming more energy efficient by limiting the whole in-network processing region to the shaded area. 8.1 Future Research Directions and Open Problems In this thesis, we have attempted to achieve an energy efficient wireless sensor net works via protocol design and distributed compression while keeping an eye on the in-network processing aspect of wireless sensor networks. Much work still needs to be done including how to efficiently obtain the correlation statistics to facilitate dis tributed compression, and how application performance changes due to these different 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. s I G N A L E N E R G Y Energy Reduction Figure 8.1: Feasible region of in-network processing levels of distortion introduced by the codec. In the latter case, application specific met rics should be used to evaluate the codec instead of conventional rate-distortion metric or signal-to-noise ratio (SNR), since this universal SNR metric in some cases gives little insight on the application performance. We next shall give a few open problems which we believe are crucial for wireless sensor networks. We have shown that highly correlated bitplanes can be effectively compressed by the Slepian-Wolf coding theorem; however, correlation of bitplanes at low bit positions, which are coarsely correlated, can not be effectively exploited by the LDPC Slepian-Wolf codes. The crossover probabilities are high for these low bitplanes, and in general these bitplane widths are large. This causes that in the high SNR regime, the rate-distortion performance of S-SPIHT degrades. Our first open problem is then “P I: how to exploit the correlation of coarsely correlated bitplanes?”. We believe that this problem has its intrinsic complexity derived from the channel 161 Data Fusion DSC with Data F asion Decision Fus Algorithm Protocol Improvement ” —;— Decision Fusion Decision Fusion Algorithm/Protocol Improvement Region Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. coding theory, where more insights might be obtained from latest development on channel coding theory. The other open problem along the same lines is on the significance bits of dis tributed source coding of sensor data. Besides significance tree, these bits also contain information on signal energy distribution. However, these bits have to be sent in intra mode to help the decoder to extract side-information from neighboring sensors’ read ings. These intra-coded significance bits degrade the rate-distortion performance since the information from the most significant bits of significant coefficients is also spatially correlated to the side-information. There are possible methods to mitigate the rate loss caused by the intra-coding of these significance bits. One method could be to make use of the significance bits for compression readings from more than one sensors. This way we can amortize the intra-coded significance bits to multiple sensors. However, there is no theoretical justification if such method is optimal or if there is a theoreti cal limit on the coding of the significance bits. In order for the other sensors to share these intra-coded significance bits, other means are needed to enable this noticing that encoding is done independently. The next open problem is “P2: Can communications during encoding do anything good?”. Our guess is that communications during encod ing not for the sake of obtaining side-information may help to achieve a better overall rate-distortion performance for a group of sensors. To shed some light on this problem (P2), taking a look at Figure 8.2, instead of sending the intra-coded significance bits to the fusion center, a node broadcasts this information. Neighboring sensors can utilize or share this information through the broadcast channel to extract the refinement bitplanes following the similar steps in the set-partitioning decoding algorithms. The benefits are two-fold. First, after the intra- coded significance bits are broadcast, each neighboring sensor is no longer required to send this information, this is a significant saving when a large number of neighboring 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. #........ ' - 4 ----------- ► Siginificane Bits Refinement Bits Figure 8.2: Do encoding communications help? sensors are presented. Second, in essence, encoding execution on neighboring sen sors is replaced by the decoding execution, and this reduces processing delays quite dramatically noticing that decoding under set-partitioning is at least twice as fast as the encoding processing. One of side-effects under this approach is that some syn chronization is required, i.e. neighboring sensors can only start to encode its readings when the broadcast significance bits are received. Likewise, at the fusion center, the decoding can start when its reference node’s readings are available. The other prob lem with this significance bit sharing approach is that correlation estimation is harder. However, since accurate crossover probability during encoding is not really critical, this problem can be addressed to a satisfactory level in practice. On the detection problem, this thesis is focused on distributed detection algorithms. These algorithms make use of field diversity to ensure robust detection. Detection the ory has been well developed for the case where a single node is used, in particular, 163 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CFAR type detector and ROC curve of a detection algorithm are practically very use ful. However, in the distributed detection problem, there are multiple nodes involved and multiple stages involved for a detection scheme, namely, individual detection, and detection predicate fusion. Our third open problem is then “P3: how to derive a ROC curve for a given distributed detection algorithm, and for practical purposes, how to devise CFAR type distributed detection algorithms?” A general methodology is the key to this problem. So far only some individual instances can we succeed. We believe that these open problems could significantly affect the efficiency of a sensor network. Nevertheless, currently, some practical sensor systems are already in operation, and it has been proven that they can be powerful tools in many applications. From a different point of view, when sensor systems grow in scale, we believe these open problems will be more valuable. 164 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reference List [1] A. Aaron and B. Girod, “Compression with side information using Turbo codes, ” Proc. Data Compression Conference, Apr. 2002. [2] H. Abrach, S. Bhatti, J. Carlson, H. Dai, J. Rose, A. Sheth, B. Shucker, J. Deng, and R. Han, “MANTIS: System Support For MultimodAl NeTworks of In-situ Sen sors,” 2nd ACM International Workshop on Wireless Sensor Networks and Appli cations (WSNA) 2003, pp. 50-59. [3] R. R Brent, F. G. Gustavson and D. Y. Y. 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Tang, Caimu
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Compression, correlation and detection for energy efficient wireless sensor networks
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