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Probabilistic analysis of power dissipation in VLSI systems

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Probabilistic analysis of power dissipation in VLSI systems

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Content PROBABILISTIC ANALYSIS OF POWER DISSIPATION IN VLSI SYSTEMS by Radu Marculescu 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 (ELECTRICAL ENGINEERING) August, 1998 Copyright 1998 Radu Marculescu Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSmr PARK LOS ANGELES. CALIFORNIA 90007 This dissertation, written by under the direction of h..xs. Dissertation Committee and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillm ent of r e -' quirements for the degree of DOCTOR OF PHILOSOPHY Dean of Cradupte Studies Date ..AjgLSUg.t ........... DISSERTATION COM MITTEE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. * T o Tty parents, i t y w ife (Diana a n d son Andrei w itftg ra titiu k a n d (ove. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments Coming to an end with this thesis, I would like to thank to those people who influenced my career and made this endeavor possible. I want to apologize, in advance, for this rather impersonal way of acknowledging their real contribution to this journey started exactly five years ago. My gratitude and appreciation goes far beyond these lines. I am deeply indebted to my advisor. Professor Massoud Pedram, for his guidance and continuous encouragement throughout my Ph.D. I will always treasure the open and stimulating relationship I have had with him throughout these difficult years. He was always open to new ideas and gave me complete freedom to pursue my own research interests. He taught me many things not only about CAD (and good quality research) but also about life. I would like to thank Professors Sandeep Gupta and Michael Arbib for serving in my thesis committee and for taking from their time to review this thesis. They had always time to listen to me and gave me good advice. I would also like to thank Professors John Chôma Jr., Giovanni De Micheli and Srinivas Devadas for helping me during my last year as doctoral student. I am grateful to all my current and past colleagues in THE Low-Power group for helping me out in many occasions and for creating such a friendly environment; special thanks to Drs. Chi-Ying Tsui and Chih-Shun Ding for interesting and stimulating discussions. I wish also to thank NSF, DARPA and SRC for their financial support that made this work possible. Before coming to US, many dedicated individuals have influenced my career and stimulated my passion for research: my elementary school teacher, Ms. Coter, introduced m Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. me to the wonderful world of mathematics; the hlgh-school teachers S. Maftei, S. Oprita and T. Jaluba, with whom I had the privilege to work in Dorohoi, have trained me for a technical profession; Professors Liviu Goras and Nicolae Gheorghiu (whom I met in Iasi) opened my eyes to research and offered me an analytical perspective of the real (not necessarily engineering) world. Later, when I joined the CS department at the “Politehnica” University in Bucharest, I had the chance of meeting Professors Irina Athanasiu, Serban Petrescu and Cristian Giumale who helped and encouraged me to pursue an academic career. My friends and colleagues have played an important role in my life and profession. In Iasi, Cristian Chitaru was always close to me in discovering the (new) world. (Thank you Chitaras for your wonderful friendship and those pure moments we have spent together.) I also remember loan Rosea with whom I have had (and still have) many interesting discussions. After finishing my Master degree, I joined the CAD group at ICCE, Bucharest. There I met Marius Garbea, Tudor Niculiu, and Sorin Cotofana with whom I discussed on any possible subject that we could imagine. Even if we are not in the same CAD group anymore, I am happy that we are still enjoying doing research on our own. Outside my technical training and profession, Dan Condurache offered me the acute perspective of life (or being alive) and created for me an unforgettable Macondo. Monica & Gigi, helped me to come in US and have constantly encouraged me to move forward. My friends in LA, Haru, Araxi, Adi, Alex, Margaret & Irv, have made my life easier and more enjoyable in many occasions. Finally, I am deeply grateful to my family for everything they did for me to make this adventure possible. My parents, Elena-Graziela and Vasile, made me what I am, gave me the sense of right and wrong and taught me what love is. My wife, Diana, shared with me everything in a perfect balance of love and understanding. She is just my other half. And our son Andrei simply is. IV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Chapter 1 Introduction......................................................................................1 1.1 Motivation......................................................................................................... 2 1.2 Sources of power dissipation........................................................................... 3 1.3 Basic issues in power estimation.....................................................................6 1.4 Thesis contribution............................................................................................ 8 1.5 Thesis outline..................................................................................................13 Chapter 2 Preliminaries.................................................................................. 16 2.1 Finite order Markov chains.............................................................................17 2.2 State classification, decomposability, probability distribution.................... 21 Chapter 3 Circuit Dependent Techniques......................................................25 3.1 Introduction.................................................................................................... 26 3.2 Prior w ork.......................................................................................................30 3.3 An analytical model for dependencies..........................................................32 3.3.1 Temporal correlations......................................................................32 3.3.2 Spatial correlations........................................................................... 38 3.4 Propagation mechanisms...............................................................................42 3.4. 1 Computation of transition probabilities...........................................43 3.4.2 Propagation of transition correlation coefficients.......................... 45 3.4.3 Complexity issues............................................................................ 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.5 An axiomatic approach to conditional probability.......................................49 3.5.1 Issues in performance management.................................................49 3.5.2 Conditional independence and signal isotropy...............................52 3.5.3 Computation of transition probabilities using e-isotropic signals .59 3.5.4 Computation of correlation coefficients using e-isotropic signals .62 3.6 Practical considerations and experimental results........................................64 3.7 Summary.........................................................................................................76 Chapter 4 Circuit Independent Techniques: The Combinational Case 78 4.1 Introduction.................................................................................................... 79 4.2 Prior w ork...................................................................................................... 82 4.3 A first-order probabilistic model....................................................................82 4.3.1 Problem formulation.........................................................................83 4.3.2 Proof of correcmess..........................................................................84 4.4 Flat models..................................................................................................... 87 4.4.1 Dynamic Markov models.................................................................87 4.4.2 A practical procedure....................................................................... 94 4.4.3 Practical considerations...................................................................99 4.4.4 Experimental results........................................................................103 4.5 Hierarchical models..................................................................................... 108 4.5.1 Nonhomogeneous sequences.........................................................108 4.5.2 Micro/macro-state modeling..........................................................112 4.5.3 A Hamming distance-based criterion for microstate grouping.... 118 4.5.4 A practical procedure......................................................................121 4.5.5 Experimental results....................................................................... 123 4.6 Summary.......................................................................................................128 VI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Circuit Independent Techniques: The Sequential Case....129 5.1 Introduction...................................................................................................130 5.2 A high-order probabilistic model................................................................. 132 5.2.1 Problem formulation...................................................................... 133 5.3 The effect of finite-order statistics on FSM behavior................................ 135 5.4 Fixed and variable-order Markov sources................................................... 138 5.4.1 The order of a Markov source........................................................ 138 5.4.2 Composite sequences..................................................................... 143 5.4.3 Variable-order dynamic Markov models....................................... 145 5.5 Practical considerations and experimental results......................................146 5.6 Summary.......................................................................................................154 Chapter 6 Conclusions................................................................................156 6 .1 Thesis summary...........................................................................................157 6.2 Future Work................................................................................................. 160 References...................................................................................162 vu Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures Figure 1.1 A CMOS inverter for power analysis............................................................... 4 Figure 2. 1 Sequence characterization with STG and transition matrix......................... 19 Figure 3.1 An illustration of spatiotemporal correlations.............................................. 27 Figure 3.2 Temporal effects on any signal line x .............................................................32 Figure 3.3 A lag-one Markov Chain describing temporal effects on line x ................. 33 Figure 3.4 The paradigm of signal, conditional and transition probabilities............... 37 Figure 3.5 Two spatially correlated signal lines x and y ................................................. 38 Figure 3.6 A lag-one Markov Chain for spatial correlations between x and y .............39 Figure 3.7 Probability calculations fo r/= xj © X 2 ® X 3 ................................................49 Figure 3.8 An example involving highly correlated signals...........................................51 Figure 3.9 Switching activity overestimation by ignoring input statistics................... 52 Figure 3.10 An example to illustrate conditional independence..................................... 54 Figure 3.11 An example to illustrate pure and e-isotropy................................................58 Figure 3.12 The experimental setup..................................................................................64 Figure 3.13 Switching activities distribution in C /7 (pseudorandom inputs) ..............6 6 Figure 3.14 Switching activities distribution in C17 for two biased input sequences .67 Figure 3.15 The impact of the correlation level on sw_act estimation in f5 1 m ............ 69 Figure 3.16 Speed-up factor vs. circuit complexity......................................................... 73 Figure 3.17 Overestimation factor by ignoring input correlations..................................76 Figure 4.1 Data compaction for power estimation......................................................... 80 Figure 4.2 The tuple (Markov-Chain, target_circuit).................................................... 83 Figure 4.3 The tree DMTq for the sequence in Example 4.1......................................... 8 8 V lll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.4 Construction of the tree DMTi.......................................................................91 Figure 4.5 The tree DMT\ for the sequence in Example 4 .1......................................... 92 Figure 4.6 The vector compaction procedure................................................................. 95 Figure 4.7 DMT^ for sequence in Example 4.3 (model_size = 35)...............................97 Figure 4.8 DM Ti for sequence in Example 4.3 {model_size = 30)...............................99 Figure 4.9 Experimental setup........................................................................................103 Figure 4.10 Two sequences and their corresponding transition graphs......................... 109 Figure 4.11 The transition graph for the composite sequence S *................................. 110 Figure 4.12 Average power dissipation for C / 7..............................................................I ll Figure 4.13 A two-level hierarchy for the sequence S *.................................................114 Figure 4.14 Average Hamming distance variation.........................................................119 Figure 4.15 The experimental setup for combinational circuits....................................123 Figure 4.16 The two-level hierarchy for C6288............................................................. 125 Figure 5.1 Two sequences with the same first-order statistics....................................131 Figure 5.2 The tuple (Markov-Chain, F SM )................................................................ 133 Figure 5.3 Two STGs obtained for the signal lines of benchmark d k l7 137 Figure 5.4 Conditional entropies for a lag-one and a lag-two Markov sequences.... 142 Figure 5.5 Conditional entropies for two lag-two Markov subsequences.................. 143 Figure 5.6 Typical behavior of conditional entropies for composite sequences 144 Figure 5.7 A second-order dynamic Markov tree....................................................... 146 Figure 5.8 The experimental setup for sequential circuits..........................................147 Figure 5.9 Node-by-node analysis for benchmark planet...........................................152 IX Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables Table 3.1. C17: TCs for pseudorandom inputs...............................................................6 6 Table 3.2. C l7: sw_act for pseudorandom inputs.......................................................... 6 6 Table 3.3. C17: sw_act for biased inputs........................................................................ 67 Table 3.4. C17: sw_act for biased inputs........................................................................ 67 Table 3.5. Pseudorandom inputs; no limit (/ «») in TCs calculation........................ 71 Table 3.6. duke2 - Speed-up vs. accuracy.......................................................................72 Table 3.7. Pseudorandom inputs; / = 4 in TCs (p.W @ 20MHz; VQ[ÿ=5W)................... 74 Table 3.8. High-correlations on primary inputs (jiW @ 20MHz; Vqj^ S V ) ................74 Table 3.9. Total Power Consumption (|iW @ 20MHz; V£>/>=5V)................................ 75 Table 4.1. Total Power (p,W@20MHz) for sequences of type 1................................. 105 Table 4.2. Total Current (mA) for sequences of type 2 ...............................................106 Table 4.3. Results for Simple Random Sampling........................................................ 107 Table 4.4. Results for DMC Approach.........................................................................107 Table 4.5. Total power consumption (jiW@ 20 M Hz).................................................124 Table 4.6. Results obtained for Simple Random Sampling.........................................127 Table 4.7. Results obtained for HMM approach.......................................................... 127 Table 5.1. Total Power (p.W@ 20MHz) for input sequences of order 2......................149 Table 5.2. Total Power (p.W@ 20MHz) for composite input sequences.....................151 Table 5.3. Total Power (p.W@20MHz) for long sequences........................................ 153 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract Computer-Aided Design tools play an important role in the efficient design of high- performance digital systems. In the past, time, area and testability were the main concerns of the design community. \^ th the growing need for low-power electronics, power analysis and low-power synthesis have also become major design concerns. The objective of this thesis is to provide the theoretical foundations o f an integrated framework for power analysis of digital CMOS circuits. To this end, the problem of power estimation is addressed from a probabilistic viewpoint and the following contributions are made; 1) For the class of static power estimation techniques, the previous work done on switching activity estimation is extended to explicitly account for complex spatiotemporal correlations which occur at the primary inputs when the target circuit receives data from real applications. More precisely, using lag-one Markov Chains, two new concepts - conditional independence and signal isotropy - are brought into attention and based on them, sufficient conditions for exact analysis of complex dependencies are given. It is shown that the relative error in calculating the switching activity of a logic gate using only pairwise probabilities can be upper-bounded. Relying on the concept of signal isotropy, approximate techniques with bounded error are proposed for estimating the switching activity. XI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2) For the class of dynamic power estimation techniques, an approach which reduces the simulation time by orders of magnitude is proposed using the paradigm of sequence compaction. Given an initial input sequence, we compact this sequence into a much shorter one such that the new data represents a good approximation as far as total power consumption is concerned. To this end, a hierarchical Markov model is introduced as a flexible framework for capturing not only complex spatiotemporal correlations, but also the dynamic changes in the sequence characteristics. Furthermore, a family of variable- order dynamic Markov models is presented to handle the case of sequential circuits. These Markov models provide an effective way for accurate modeling of external input sequences that affect the behavior of finite state machines. XU Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.1 Motivation In the past, the common view on integrated circuits performance was centered around the concept of speed of the circuit This perception was motivated by the continuous demand for increased processing power which is normally required in most applications. Generally speaking, the improvements in performance can be obtained at the expense of the silicon area. As a consequence, using sophisticated CAD tools, the VLSI designer has traditionally explored the area-time implementation space, attempting to find a reasonable balance between the two (often conflicting) objectives. Despite the fact that power consumption is another metric which can quantify the performance of a circuit, until recently, power concerns were totally secondary in the design process. Of course, in certain ultra low-power applications (e.g. pacemakers, wrist watches, etc.), power dissipation has always played a major part in the design cycle. However, in the case of high-performance microprocessors or general ASICs, we have to note that these systems were just designed for maximum performance, regardless of the side effects on power consumption. In the recent years, however, this picture has changed drastically. Driven by increased levels of device integration and complexity, together with higher device speed, power dissipation has become a crucial design concern, limiting the number of devices that can be put on a chip and dramatically affecting the packaging and cooling costs associated with ASICs. Power dissipation is even a bigger concern for the class of battery-powered personal computing devices and wireless communication systems. To give some figures, for a portable multi-media terminal, the projected power budged, when implemented using Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. off-the-shelf components not designed for low-power operation, is around 40W. With modem Nickel-Cadmium batteries offering around 20W-hours/pound, such a terminal would require 20 pounds of batteries for 10 hours of operation between recharges [38]. Taking a completely different perspective, contemporary high-performance processors dissipate as much as SOW . In the near future, it is projected that a lOcm^ microprocessor, clocked at 500MHz, would consume around 315W [38]. As a consequence, the analysis and synthesis of low-power VLSI systems has become a very active area of research in the VLSI conununity. Because excessive power dissipation reduces the reliability and increases the cost of cooling and packaging systems, it is obvious that the successful research on low-power systems can be beneficial for the development of both, high- performance and portable systems. In this thesis, we focus our attention on the issue of power analysis and provide the theoretical foundations of an integrated framework for power estimation in digital circuits. CAD tools have always been the major requirement for designing state-of-the-art VLSI circuits and low-power VLSI systems are no exception to this rule. We do need a new generation of CAD tools targeting power optimization and, to this end, a methodology for accurate power analysis is a must. 1.2 Sources of power dissipation In this section we will review the main sources of power dissipation in CMOS digital circuits with particular emphasis on average power consumption. Peak power dissipation is also an important concern but, in this research, we assume that the operation of the circuit has been ensured for worst-case conditions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Let us consider the CMOS inverter in Figure 1.1. Under normal operation, the average power consumed by this inverter can be described by the following simple equation [4 7 ] which basically reflects one static iPstatic^ and one dynamic component ^P sh o rt Pleakage P dynam ic^' P avg ~ P static P short Pleakage Pdynam ic ( 1-1) DD n_ Figure 1.1 A CMOS inverter for power analysis Ideally, CMOS circuits dissipate no static (DC) power since, in the steady state, there is no direct path from Vqd to ground. Of course, this scenario can never be realized in practice since in reality the MOS transistor is not a perfect switch. Thus, there will always be leakage currents and substrate injection currents, which will give rise to a static component of CMOS power dissipation. Fortunately, for current technology, the actual contribution from this static component is several orders of magnitude below other contributors and therefore, this component is practically absent for well-designed CMOS circuits. However, we note that this component may become significant under some circumstances and consequently, logic families which experience static power dissipation should be avoided in low-power design. The dynamic component of power dissipation arises from the transient switching behavior of the CMOS device. At some point during the switching, both NMOS and PMOS devices in Figure 1.1 will be turned on and, during this time, a short-circuit exists between Vqq and ground and currents are allowed to flow. Fortunately, in well-designed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CMOS circuits, the short-circuit power dissipation is usually a small fraction (10-15%) of the total power dissipation then it is reasonable to expect that, in the design for high- performance, short-circuit power dissipation is not a major concern. ^leakage mainly caused by reverse saturation currents in diffusion regions of the PMOS and NMOS transistors as well as the subthreshold leakage current of transistors that are normally off. Unfortunately, despite the fact that in today’ s circuits Pleakage is still a small fraction (less than 1 0 %) of the total power dissipation, the subthreshold leakage is becoming increasingly important as power voltages decrease. Finally, the dominant source of power dissipation (80-90% of total power) is represented by the dynamic dissipation due to capacitance charging and discharging. This situation is modeled in Figure 1.1 where all capacitances are lumped at the output capacitance C. Assuming a gate-level implementation of the target circuit, one can calculate the average dynamic power consumption by using the formula; P dynamic = ( 1.2) where is the clock cycle frequency, Vqq is the supply voltage, C(x) and sw(x) represent the load capacitance and the switching activity, respectively, at the output of any gate X in the circuit. The product C{x)sw{x) is often referred to as the switched capacitance o f the circuit. As we can see from equation (1.2), power dissipation is linearly dependent on the switched capacitance. Generally speaking, calculation of the switched capacitance is difficult as it depends on static and dynamic parameters of the circuit which are not readily available or precisely characterized. Among the factors which have an important role in accurate estimation of the switched capacitance we mention: physical Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. capacitance, circuit structure, input pattern dependence, the delay model, statistical variation of circuit parameters. In this thesis, we assume that the physical capacitance is either calculated or taken from the library and therefore concentrate on average switching activity estimation. Summarizing our discussion about sources of power dissipation in CMOS circuits, we conclude that, for current technology, the dominant fraction of average power is attributed to ± e dynamic component caused by the switching activity at the gate outputs. For this reason, the focus of this thesis is on estimating the dynamic power consumption. 1 3 Basic issues in power estimatioii Existing techniques for power estimation at gate- and circuit-level can be divided in two main classes: static and dynamic [37], [25]. Static techniques rely on probabilistic information about the input stream (e.g. switching activity of the inputs, signal correlations, etc.) to estimate the internal switching activity of the circuit. These techniques generally provide sufficient accuracy with low computational overhead. However, they cannot accurately capture factors such as slew rates, glitch generation and propagation. In addition, a major challenge in probabilistic power estimation approaches is the ability to account for internal dependencies due to the reconvergent fan-out in the target circuit. This problem, which we will refer to as the “ circuit problem” , is by no means trivial. Indeed, a whole set of solutions have been proposed, ranging from approaches which build the global OBDDs [1] and therefore capture all internal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. dependencies, to efficient techniques which partially account for dependencies in an incremental manner [32], [45], [46]. Dynamic techniques [19], [18] explicitly simulate the circuit under a “typical” input stream. Consequently, their results depend on the simulated sequence, and the required number of simulated vectors is usually high. These techniques can provide sufficient accuracy, but the price to be paid is too high. Switching activity information can be extracted by doing exhaustive simulation on small circuits; it is, however, unrealistic to rely on simulation results for large circuits. To address this problem, a Monte Carlo simulation technique was proposed in [7]. This technique uses an input model based on a Markov process to generate the input stream for simulation. The simulation is performed in an iterative fashion and the iteration terminates when some stopping criterion is met. The approach has two deficiencies. First, the required number of samples (which directly impacts the simulation run time) is approximately proportional to the ratio between the sample variance and the square of the sample mean value. For certain sequences, this ratio becomes large, thus significantly increasing the simulation run time. Second, if the sample distribution significantly deviates from the normal distribution, the simulation may terminate prematurely. Difficult distributions that cause premature termination are bi- modal, multi-modal, and distributions with long or asymmetric tails [11]. Existing statistical techniques for power estimation in sequential circuits require a large number of simulated vectors. Basically, their efficiency is lower than that for combinational circuits [8 ], [34]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In this thesis, we investigate both alternatives that is, we provide possible solutions for static and dynamic power estimation. This is because, in our vision, the designer should benefit from both types of techniques in order to asses the potential of a new design from the power consumption perspective. In addition to this, the unprecedented complexity of the current digital systems demands a combined set of tools for power estimation rather than the use of a single tool. 1.4 Thesis contribution The methodology presented in this thesis offers a possible solution to the critical need for CAD tools to estimate power dissipation during the design process in order to meet the power budget without having to go through a costly redesign effort. The results presented in this thesis advance the state-of-the-art by providing the following key contributions. • An analytical model fo r spatio-temporal correlations in digital circuits To estimate the power consumption one has to calculate the signal and transition probabilities of the internal nodes of the target circuit. Besides delay and circuit stmcture, the signal and transition probabilities at the internal nodes are very sensitive to the statistics of the input patterns that feed the circuit. In Chapter 3, we propose the first analytical model which accounts for spatiotemporal correlations under the zero-delay hypothesis. As the main theoretical contribution, we extend the previous work done on switching activity estimation to explicitly account for complex spatio-temporal correlations which occur at the primary inputs when the target circuit receives data from real applications. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The model based on spatio-temporal correlations clearly demonstrates, qualitatively and quantitatively, the importance of being accurate node-by-node (not only for the total power consumption) and identifies potential drawbacks in previous approaches when patterns feeding the inputs become highly correlated. The practical value of this model becomes relevant during optimization and synthesis for low-power when accurate power estimation is the critical step. Based on this model, we developed a set of efficient techniques for power estimation in large combinational modules. The whole idea behind efficiency is to calculate and propagate transition correlation coefficients from the circuit inputs toward the circuit outputs, and hence, eliminate the need for building the global function of the intermediate nodes. As quantitative illustration, these techniques can analyze combinational modules with 10,000 gates in about 200 CPU sec., yet producing power estimates with less than 10- 15% error, compared to circuit-level simulation. Going deeper into details, the model of spatiotemporal correlations is based on the following ideas introduced in this thesis. a) Uniform modeling o f spatial and temporal correlations using time-homogeneous Markov Chains Temporal correlations for the values of some signal x in two successive clock cycles are considered through a Markov Chain with only two states; spatial correlations for a pair of signals {x,y) are modeled by a four-state Markov Chain. By using this formalism, we were able to consider in a systematic way different kinds of dependencies in large combinational modules for both pseudorandom and highly correlated input streams. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. b) Using the conditional independence and signal isotropy concepts to develop an incremental propagation mechanism based on binary decision diagrams (BDDs) Based on conditional independence and signal isotropy concepts, we provide sufficient conditions for exact analysis of complex dependencies. More precisely, we demonstrate that the relative error in calculating the switching activity of a logic gate using only pairwise probabilities can be upper-bounded. In the general case, the conditional independence problem has been shown to be NP-complete and thus appropriate approximate techniques (with bounded error) are presented to estimate the switching activity. From a practical point of view, the use of conditional independence and signal isotropy concepts significantly speed-up the estimation process (the larger the module, the more substantial is the speed-up that can be achieved) while the quality of estimates is improved by one order of magnitude, on average, compared to other proposed approaches that ignore the input correlations. In summary, the model of spatio-temporal correlations improves the state-of-the-art in two ways: theoretically, by providing a deep insight about the relationship between the logic and probabilistic domains, and practically, by offering a sound mathematical framework and an efficient technique for power analysis. • A class o f dynamic Markov models fo r input data modeling As we have already noted, existing techniques for switching activity estimation are either simulative or non-simulative in nature. Simulative techniques are in general more accurate, but tend to be significantly slower compared to the other ones. It is simply impractical to simulate a large circuit using millions or even thousands of vectors. Yet, 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. this is currently necessary to get accurate power estimates using circuit-level or switch- level simulators. It is thus useful to develop a vector compaction technique that reduces the input vector size while preserving the statistical behavior of the original set (and hence the average power dissipation). This can make simulators a viable option for power analysis even for very large circuits as the size of the input vector set can be greatly reduced. To be more precise, having characterized the input statistics (using signal probabilities and pairwise spatiotemporal correlations), one has to generate a minimum cardinality set of input data that exhibits the same statistical behavior. To this end, having an initial sequence (assumed representative for some target circuit), we target lossy compression, that is the process of transforming an input sequence into a smaller one, such that the new body of data represents a good approximation as far as total power consumption is concerned. In Chapter 4 and Chapter 5, we provide different solutions for vector compaction problem which reduces the gap between simulative and nonsimulative approaches used in power estimation. The mathematical foundation of these techniques is probabilistic in nature and they make extensive use of concepts and results from the theory of finite Markov chains. One important feature of these techniques is the independence from the actual implementation of the target circuit, therefore one can use any of these techniques early in the design cycle when the structure of the circuit is not determined yet. As the results demonstrate, large compaction ratios of few orders of magnitude can be obtained without signifrcant loss (less than 5% on average) in the accuracy of power estimates. These achievements are possible by developing the following three models. 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a) The flat model The basic approach is presented in Chapter 4; it relies on adaptive {dynamic) modeling of binary input streams as first-order Markov sources of information; it is applicable to combinational and, under some circumstances, to sequential circuits. The model introduced in Chapter 4 is inspired from the literature on data compression [9]. However, the original model in [9] is not completely satisfactory for our purpose. We thus extend the initial formulation to manage not only correlations among adjacent bits that belong to the same input vector, but also correlations between successive input patterns. This model gives good results when used for homogeneous input sequences that is, input sequences which contain statistically similar vectors. This is because homogeneous sequences exercise the circuit such that the value of average power converges rapidly. However, non-homogeneous input sequences (that is, input sequences where the stimuli may contain a mixture of vectors, each one very different as far as average switching activity per bit is concerned) are very common in practice and they may determine erroneous power estimates depending on which component (with low or high activity) is visited more often. Two refinements of this model are the hierarchical and variable-order models. b) The hierarchical model In Chapter 4, we introduce the hierarchical Markov model as an effective way to structure the input space into a hierarchy of macro- and micro states-. at the first (high) level in the hierarchy we have a Markov chain of macrostates; at the second (low) level, each macrostate is in turn characterized by a Markov chain for all its constituent microstates. Our primary motivation for constructing this hierarchical structure is to enable a better modeling of the different stochastic levels that are present in 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sequences that arise in practice. Another important property of these models is the ability to capture different operating modes of the circuit using the first level in the hierarchical Markov model. This provides high adaptability to different operating modes of the circuit. c) The variable-order model This model is introduced in Chapter 5 and it is fully applicable to sequential circuits because it considers temporal correlations longer than one time step. Relying on the concept of block entropy, we present a technique for identifying the actual order of variable-order Markov sources of information that is, sources of information that can be piecewise modeled by Markov chains of different orders. Using the concept of dynamic Markov chain modeling, we propose an effective approach to compact an initial sequence into a much shorter one such that the steady state and transition probabilities (and Aerefore the total power consumption) in the target finite state machine (FSM) are preserved. We note that the vector compaction procedures presented in Chapter 4 and Chapter 5 perform much better than classical statistical sampling techniques proposed for power estimation. 1.5 Thesis outline We first provide a general background in the theory of finite order Markov chains. Far firom being exhaustive, we restrict ourselves to only those concepts that are needed for a better understanding of the following chapters. We then start the main part of the thesis by first exploring, from a probabilistic point of view, the issue of switching activity estimation in combinational circuits. In Chapter 3, based on lag-one Markov chains, we 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. propose a new analytical model which accounts for spatio-temporal correlations under the zero-delay hypothesis. Evaluations of the model and a comparative analysis on common benchmark circuits show that node-by-node switching activities are strongly pattern dependent and therefore, accounting for dependencies is mandatory if accuracy is a major concern. In the second part of this thesis, that is in Chapter 4 and Chapter 5, we present the theory and practice of sequence compaction for power estimation. We first note that the paradigm of sequence compaction looks very attractive because it represents a first step towards reducing the gap between static and dynamic techniques currently used in power estimation. As the main theoretical contribution, under stationary conditions, we formulate and prove the correctness of vector compaction problem and then introduce a family of dynamic Markov trees that can be used effectively in sequence compaction. More precisely, in Chapter 4, we introduce the hierarchical modeling of Markov chains as a flexible framework for capturing not only spatio-temporal correlations, but also the dynamic changes in the sequence characteristics such as different operating modes or power distributions. In Chapter 5, we deal with the special case of sequential circuits and introduce the variable-order dynamic Markov model as an effective way for accurate modeling of external input sequences that affect the behavior of the target FSM. As the experimental show, for both combinational and sequential circuits, large compaction ratios of orders of magnitude can be obtained without significant loss in accuracy for power estimates. 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Finally, Chapter 6 contains a summary of the theoretical and practical results presented in this thesis and a discussion of directions for future research. 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Preliminaries 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The purpose of this chapter is to provide a short background for the main concepts that will be used in the following chapters. The reader interested in switching theory and synthesis and optimization of digital circuits is referred to standard books published in the last few decades [17], [24], [5], [30], [10], [16]. In what follows, we present only the basic definitions and notations from the theory of finite-order Markov chains that will be used throughout this thesis. Far from being exhaustive, we restrict our attention to only those concepts that are required by our working hypotheses. For a complete documentation on the subject, the reader is referred to monographs [22], [20], [35], [44]. 2.1 Finite order Markov chains A stochastic process is defined as a family of random variables {x(t), r e T) defined on a given probability space and indexed by the parameter t, where t varies over the index set T. The stochastic process is said to be stationary when it is invariant under an arbitrary shift of the time origin. In this case, the values assumed by the random variable x(t) are called states, and the set of all possible states forms the state space of the process. A Markov process [x(t), r e T} is a stochastic process whose dynamic behavior is such that, given the present value, the future values of the process are independent of the past values. This is the so called “Markov property” and it defines a fundamental subclass of stochastic processes. We shall assume that the transitions out of state x(t) are independent of time and, in this case, the Markov process is said to be time-homogeneous. If the state space of a Markov process is discrete, the Markov process is referred to as a Markov chain (MC). In what follows, we consider only MCs with finite state space. If 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. we assume that the index set T is also discrete, then we have a discrete-parameter MC. We may assume, without loss of generality, that T = {1,2, „.} and denote the MC by Definition 2.1 (Lag-one MC) A discrete stochastic process > i is said to be a lag-one MC if at any time step n > 2 and for all states p{^n - ~ ^n-l^n-2 ~ = ® ^ l) = P(^n ~ I ~ l) (2 .1) The conditional probabilities p(jc„ = a„|x„_, = a„_,) are called single-step transition probabilities and represent the conditional probabilities of making a transition firom state x„.| to state at time step n. In homogeneous MCs these probabilities are independent of n and consequently written as p^j = p(x„ = y |x„ _ i = i) for all n > 2 . Note: Latter in this thesis, we will model the stochastic behavior of binary sequences by MCs. For any two consecutive vectors v/, vj, we can write symbolically that Pij = p(next_vector = j\present_vector = i) ; that is, the probability pij of a transition to vector V j given that the present vector is v ,-. The matrix Q, formed by placing Pÿ in row i and column j, for all i and j, is called the transition probability matrix. We note that Q is a stochastic matrix because its elements satisfy the following two properties; 0 < pÿ < I and ^ P ÿ = 1. J Although conditional transition probabilities can be used as a rough approximation to the transition probabilities, in general we need to know the probability of a transition independent of the present state. These are so-called total transition probabilities Pi^j 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and can be calculated knowing the state probabilities, that is = tc, • Pij, where jc, represents the probability that the MC is in a given state i. An equivalent description of the MC can be given in terms of its state transition graph (STG). Each node in the STG represents a state in the MC, and an edge labelled pij (from node / to node j) implies that the one-step transition probability from state i to state j is pij. Example 2.1 Let Sj and S2 be two 2-bit sequences, of length 48, as shown in Figure 2.1a. These two sequences, have exactly the same set of first-order temporal statistics that is, they cannot be distinguished as far as wordwise one-step transition probabilities are con cerned. In fact, in Figure 2.1b we provide the wordwise transition graph for these two sequences. Each node in this graph is associated to a distinct pattern that occurs in S i and S2 (the upmost bit is the most significant one, e.g. in Sj, vj = ‘1’, V 2 = ‘2’, V 3 = ‘3’,..., v^g = ‘1’). Each edge represents a valid transition between any two valid patterns and has a nonzero probability associated with it. For instance, the pattern ‘3’ in Sj and $ 2 is always followed by ‘1’ (thus the edge between nodes ‘3’ and ‘1’ has the probability 1) whereas it is equally likely to have either ‘O ’, ‘2’ or ‘1’ after pattern ‘1’. S i " ^ 0 1 0 0 0.33 0.33 0.33 0 V - 4g . . . V3V2V, 0110110110000110000001101100000000001100000000110 1101101101111101101011011010111010111010110101101 S 2 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 onoiioiioi 1011011011011011011011011011011011011 (=) (c) Figure 2.1 Sequence characterization with STG and transition matrix fOJ3 0J3 Q = 0 0 0 1 0 10 0 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Starting with different initial states and using a random number generator we may, of course, generate other sequences equivalent with S; and S2 as far as the one-step transition probabilities are concerned. We can then see the graph in Figure 2.1b as a compact, canonical, characterization of sequences S; and 8 2 . Suppose that we want to compute the occurrence probability of string v = ‘ 0 1 1 0 ’ that is, the probability that transition / — > 2 is taking place in S;. To this effect, we just use p{v) = p{v^v2) = p(v^) ■ piy^^y^j which gives us the value of 1/6 . ff we are interested in finding the two-step transition probability 0 — > / - > / in S2 , then using piy) = p ( v , v 2' '3) = • P(y^\v1v{), we get the value of 1/6. The matricial representation, equivalent with the STG, is given in Figure 2. Ic. We can easily verify that the sum of all elements on each row is I, Q thus being indeed a stochastic matrix. Definition 2.2 (Lag-^ MC) A discrete stochastic process {x„}„ ^ ; is said to be a lag-t MC if at any time step n>k+l : p{^n - “ n |- ^ n - I “ ^n-l-^n-2 ~ ~ 2) = P(^n - = ® n - l - * n -2 = ^ n - 2"'^n-k ~ ®lfc) It should be noted that any lag-/: MC can be reduced to a lag-one MC based on the following result. Proposition 2.1 [35]. If is a lag-k MC then {v„}„ ^i, where v„ = (a„, “n+*-i)> is a multivariate first-order MC. 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As a consequence, the study of lag-k MCs is practically reduced to study the properties satisfied by lag-one MCs. We will refer subsequently only to lag-one MCs but, by virtue of Proposition 2.1, all results easily translate to lag-t MCs. 2.2 State classification, decomposability, probability distribution One of the fundamental problems in MCs theory is related to the state classification process. According to whether or not a state occurs infinitely often two types of states can be distinguished. D ^nition 2.3 (Recurrent State) A state in the MC is called recurrent if the probability of returning in this state after n > 1 steps is greater than zero. Otherwise, the state is called transient. If the greatest common divisor over all such integers n is d= 1, then the state is also called aperiodic. A finite MC has at least one recurrent state. If there are no transient states, then the MC is called recurrent. In our subsequent discussion, we will consider that all states are recurrent since all transient states vanish after a small number of steps. Indeed, it can be proved [20] that if a MC starts in a transient state then, with probability 1, it stays in the set of transient states for just a finite number of steps after which it enters a recurrent class where it remains forever. On account of this property a MC with transient states will also be called absorbing. Definition 2.4 (Nondecomposable MC) A Markov chain is said to be nondecomposable if every state can be reached from every other state in a finite number of steps. Otherwise, 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the chain is called decomposable and its transition matrix can be written as a block- diagonal matrix: Ô 1 0 ... 0 Q = 0 Qz ••• 0 0 0 ... Q r , where the r sets of states contain only recurrent states and do not communicate with each other. Here the stochastic matrices Qi, Q^ , a r e the stochastic matrices associated with the recurrent classes. If r > 1 no communication is possible between these classes and, in fact, we have r distinct MCs that can be studied separately. Note'. The STG in Figure 2.lb is nondecomposable; we note that, in general, the STG associated to any input sequence of vectors is also nondecomposable. Definition 2.5 (Periodic MC). A nondecomposable Markov chain is called periodic if and only if there is some periodic state with period (/> 1. In this case, the f/-th power of Q (that is, 6 ^) can be written as a block-diagonal matrix with d matrices on the main diagonal. Otherwise, the Markov chain is said to be aperiodic. A state that is recurrent and aperiodic is said to be ergodic. If all the states of a MC are ergodic, then the MC itself is said to be ergodic. Changes of states over n > 1 time steps are ruled by probability rules simply expressed in terms of py. Let us denote by plj the probability of transition from state i to state j in exactly n steps, namely: p"j = = j\x„ = /), whatever the integer m. It is easily 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. recognized that probabilities plj (which are called the n-step transition probabilities) represent the entries of the g" matrix (called the n-step transition matrix), n >1. The g" matrix itself is still a stochastic matrix and satisfies the identity g™^" = g™ ■ g " , m, /i > 0 (g° is by definition the unit matrix I) or just the system of equations • plj, k known as the Chapman-Kolmogorov equations [35], [20]. In words, to go from i to j in (m+n) steps, it is necessary to go from i to an intermediate state k inm steps and then from k to j in the remaining n steps. By summing over all possible intermediate states k, we consider all possible distinct paths leading from i to J in (m+n) steps. Theorem 2.1 [35] For a nondecomposable MC, the equation k - Q = n has a unique solution that represents the stationary distribution of the MC. □ Note: If the Markov chain is aperiodic, then g" converges to a stable matrix when /i -> < » , and 7 C can be found to be any row of the limiting matrix. Otherwise, g" is no longer convergent in the usual sense, but in Cesaro sense [22]; that is, the limit lim ^ + 6 + +G— exists and it is equal to : t o t ! A = . where A = Urn g" '' d d #i-»oo A, ... 0 ; each 0 ... A, Aj is a stable stochastic matrix. In this case, the solution of the equation in Theorem 2.1 is I 7 * given by 7t = ^ • [x, ...Jt^] , where 7i,- is any row of the stable matrix A ,-, for each / = 1, 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 ,..., d. Note’ . Intuitively, in the case of binary sequences, the concept of stationary distribution implies that, as the observation time increases, that probability that the MC characterizing the sequence reaches each of its states converges to a stationary set of real numbers. Finding the steady-state behavior is a fundamental problem in CAD and, generally speaking, it is possible to find STGs for which the state probabilities do not exist. However, in this thesis, we will not go beyond the requirements of Theorem 2.1 for existence and uniqueness of the state probability vector. 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Circuit Dependent Techniques 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1 Introduction As we have already seen, having a gate-level implementation of the target circuit, one can calculate the average dynamic power consumption by using the well-known formula pdynamic = (-()fw(x), where is the clock cycle frequency, Vdq is the supply voltage, C(x) and sw(x) represent the load capacitance and the switching activity, respectively, at the output of any gate x in the circuit. Estimating the capacitance at the technology independent phase of the logic synthesis process is difficult as it requires estimation of the load capacitances from structures which are not mapped yet to gates in a cell library. We assume, however, that the capacitances are either estimated or taken from a library and, in addition, we ignore the interconnect capacitances. As consequence, we concentrate on estimating the average switching activity per node (gate) which is the key parameter that needs to be correctly determined because charging and discharging different load capacitances is by far the most important source of energy dissipation in digital CMOS circuits. Common digital circuits exhibit many dependencies; the most known one is the dependency due to reconvergent fan-out among different signal lines, but even structurally independent lines may have dependencies (induced by the sequence of inputs applied to the circuit) which cannot be neglected. Accounting for all kinds of dependencies is impossible even for small circuits; consequently, for real-size circuits, only some of the dependencies have been considered and even then, only heuristics have been proposed [37]. This is a consequence of the difficulty in managing complex data dependencies at acceptable levels of computational work. 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Besides dependencies described above (called also spatial dependencies), another type of correlations, namely temporal may appear in digital circuits. Let us consider a simple case to illustrate these issues. The circuit in Figure 3.1 is fed successively by three input sequences, 8 1 , 8 2 and 8 3 ; 8 % is an exhaustive pseudorandom sequence, 8 3 is also an exhaustive sequence but generated by a 3-bit counter and 8 3 is obtained by a ‘faulty’ 3- bit counter. xcy I Si (random) xcy I S2 (counted) ► c — xcy I S3 (faulty) a b z Total N i 2 2 2 6 N2 2 4 4 1 0 N3 0 2 2 4 Figure 3.1 An illustration of spatiotemporal correlations All three sequences have the same signal probability on lines x, y and c (p = 0.5), but even intuitively they are completely different. There are two other measures which differentiate these sequences, namely transition and conditional probabilities, and switching activity calculations should rely on them. In fact, these sequences exercise the circuit such that the number of transitions Ni, N2 , N3 on each signal line a, b, z (and hence the total number of transitions) becomes quite different once we feed 8 % , 8 2 , or 8 3 respectively. To accurately compute the number of transitions, we should undoubtedly account for the influence of reconvergent fanout, specifically in the previous figure, a and b cannot be considered independent signal lines. This problem could be solved (but only 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for small circuits) by expressing the function of each signal in terms of primary inputs, but even then, neglecting the correlations among primary inputs can lead to incorrect results. As we see in this example, assuming input independence for sequences $ 2 and S3 , is an unrealistic hypothesis because the patterns in each of them are temporally correlated (for example, each pattern in sequence $ 2 is obtained from the previous one by adding a binary 1). Even more than this, transitions such as 0 ^ 1 or I 0 on apparently independent signal lines (e.g. x and c in Figure 3.1) are correlated and a detailed analysis on these input streams can reveal a strong spatial relationship. Consequently, to accurately compute the switching activity, one has to account for both spatial and temporal dependencies starting from the primary inputs and continuing throughout the circuit In this chapter, we propose a new analytical model which accounts for spatiotemporal correlations under the zero-delay model. Its mathematical foundation consists of using lag-one Markov Chains to capture different kinds of dependencies in combinational circuits [27]. Temporal correlations for any signal x are considered through a Markov Chain with only two states whereas spatial correlations for any pair of signals (x,y) are modeled with a four-state Markov Chain. To summarize, the basic assumptions we use throughout this chapter are: • The target circuit is combinational and the logic value of any signal line x can only be 0 or 1 . • Under a zero-delay model, any signal line x can switch at most once within each time step. 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Under these hypotheses, we present theoretical and practical evidences showing that conditional independence is a concept powerful enough to overcome difficulties arising from structural dependencies and external input dependencies [28]. More precisely, based on conditional independence and signal isotropy, we give a formal proof showing that the statistics taken for pairwise correlated signals are sufficient enough to characterize larger sets of dependent signals. The practical value of these results becomes particularly evident during optimization and synthesis for low-power [4]; a detailed analysis presented here illustrates the importance of being accurate node-by-node (not only for the total power consumption) and identifies potential drawbacks in previous approaches when patterns feeding the inputs become highly correlated. To support the potential impact of this research, experimental results are presented for common benchmark circuits. The chapter is organized as follows. First, we review the prior work relevant to our research. In Section 3.3 we present in detail the analytic model for switching activity estimation which accounts for spatiotemporal correlations. In Section 3.4 we present global and incremental propagation mechanisms for transition probabilities and transition coefficients calculation. We also discuss the algorithmic complexity for the proposed propagation mechanisms. In Section 3.5 we improve the results given in Section 3.4 by providing an enhanced propagation mechanism based on conditional independence and signal isotropy. In Section 3.6 we give some practical considerations and report our results on common benchmark circuits. Finally, we summarize our main contribution. 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2 Prior work Most of the existing work in pseudorandom testing and power estimation relies on probabilistic methods and signal probability calculations. One of the earliest works in computing the signal probabilities in combinational circuits is presented in [36]. While the algorithm is simple and general, its worse case time complexity is exponential. For tree circuits which consist of simple gates, the exact signal probabilities can be computed during a single post-order traversal of the network [15]. An algorithm, known as the cutting algorithm, which computes lower and upper bounds on the signal probability of reconvergent nodes is presented in [39]. The bounds are obtained by cutting the multiple- fanout reconvergent input lines and assigning an appropriate probability range to the cut lines and then propagating the bounds to all the other lines of the circuits by using propagation formulas for trees. The algorithm runs in polynomial time in the size of the circuits. Ercolani et al. present in [12] a procedure for propagating the signal probabilities from the circuit inputs toward the circuit outputs using only pairwise correlations between circuit lines and ignoring higher order correlations. The signal probability of a product term is estimated by breaking down the implicant into a tree of 2-input AND gates, computing the correlation coefficients of the internal nodes and hence the signal probability at the output. Similarly, the signal probability of a sum term is estimated by breaking down the implicate into a tree of 2-input OR gates. People working in power estimation have also considered the issue of signal probability estimation. An exact procedure based on Ordered Binary-Decision Diagrams (OBDDs) [6 ] which is linear in the size of the corresponding function graph (the size of the graph, of 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. course, may be exponential in the number of circuit inputs) can be found in [33]. Using an event-driven simulation-like technique, the authors describe a mechanism for propagating a set of probability waveforms throughout the circuit. Unfortunately, this approach does not take into account the correlations that might appear due to reconvergent fan-out among the internal nodes of the circuit The authors in [1] use symbolic simulation to produce exact boolean conditions for switching at a particular node of the circuit. However, this approach is expensive in terms of computational cost (time and space requirements). Recently, a few approaches which account for correlations have been proposed. Using an event-driven probabilistic simulation technique, Tsui et al. account in [45] only for first-order spatial correlations among probabilistic waveforms. Kapoor in [21] suggests an approximate technique to deal with structural dependencies, but on average the accuracy of the approach is modest. In [40] the authors rely on lag-one Markov Chains and account for temporal correlations; unfortunately, they assume independent transition probabilities among the primary inputs and use global OBDDs to evaluate the switching activity (severely limiting the size of the circuits they can process). In what follows, we introduce a new model which extends over the previous work by taking into account spatiotemporal correlations at the primary inputs of the target circuit and by providing a general mechanism for handling them inside the circuit. 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 An analytical model for dependencies We adopt the conventional probability model which consists of the triplet (A, 2, p), where A represents the sample space, £ denotes the class of events of interest and p is the probability measure associated to £. 33.1 Temporal correlations Let us consider first a combinational logic module fed in turn by the input vectors vj, V 2 ,— , Figure 3.2 Temporal effects on any signal line x While the input vectors Vj, V 2 > — » applied at the primary inputs of the circuit at time steps 1,2,..., n, the logic value of any line x may be 0 or 1. Hence, under the zero-delay model, X may switch at most once during each clock cycle. Let be a random variable which describes the state of line x at any time n. If {x„ is modeled as a lag-one Markov chain (Figure 3.3), then its behavior, over the state set Q = {0,1}, can be described through the transition matrix Q [35]: 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Q = Po,o Pl,0 X X Po, \ P l , l (3.1) Figure 3.3 A lag-one Markov Chain describing temporal effects on line x Every entry in the Q matrix represents the conditional probability of signal line x and may be viewed as the one-step transition probability to state j at step n from state i at step n-l. Definition 3.1 (Conditional Probabilities) We define the conditional probabilities of any signal line x as: = 0 (3.2) where n We note that g is a stochastic matrix that is, every column adds to unity: Po.o+Po. 1 = 1 Pf.o+Pf.i = 1 (3.3) A lag-one Markov Chain has the property that one-step transition probabilities do not depend on the ‘history’, i.e. they are the same irrespective of the number of previous steps. The process is homogeneous and stationary: indeed, because any combinational circuit is a memoryless device, having a homogeneous and stationary distribution at the primary inputs is a sufficient condition for homogeneity and stationarity to hold throughout 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the circuit [35]. Because the process is homogeneous, then the probability distribution of the chain k may be expressed as: Î C = (3.4) where % is the initial distribution vector. Because the process is also stationary, then equation (3.4) becomes: n = Qn (3.5) Theorem 3.1 The signal probabilities may be expressed in terms of conditional probabilities as follows: p(x = 0 ) = Pf.o pix = I) = Po.i Pf.o+Po.i Pf.o+Po.i Proof: We may write equation (3.5) explicitly as: (3.6) p{x = 0 ) j>(x= I) X X Po,o Pl.O X X Po. 1 P i. I pix = 0 ) p{x= 1 ) _ or p{x = 0) = p§op(x = 0) + pf^oPix = I) and pix = I) = p§^iPix = 0) + pf , p(x - I), where p(x = 1 ) represents the signal probability of line x. On the other hand we have that pix = 0) = 1 - pix = 1), respectively pix = 1) = 1 - pix = 0) and then equation (3.6) follows immediately. ■ Definition 3.2 (Transition Probabilities) We define the transition probabilities of any signal line x as: Pixi^j) = p((x„ = j) n (x„_ 1 = 0 ) Vf, y = 0 , 1 (3.7) 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Signal, conditional and transition probabilities associated to any signal line x are not independent measures. The following two theorems describe quantitatively the relationship between them. Theorem 3.2 Transition probabilities may be expressed in terms of conditional probabilities as: Proof: Using equation (3.2) and stationarity of the process, we have: = pix = ï)pf j for any values i, j = 0,1. From equation (3.6), the above formulas are straightforward. ■ Theorem 3.3 Conditional probabilities may be expressed in terms of transition probabilities as: ^ P(^o-»o) ^ P(^o->i) P(Jfo-»o) + P(-»^o->i) p{xq^(^ + p{xq^{) ^ pjxi^o) ^ p(xi _ i) P(j:i -»o ) + p(x i_ ,i) p(x,_,o) + p(J^i_>i) Proof: It suffices to use the following two identities obtained from equation (3.8): P(^0 -^o) + P ( - < ^ 0 ->l) = p(Xi_,o) + P(Xi_,i) = ■ Pf.O + Po.l Pf.O+Po.l 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Example 3.1 Suppose that the signal line x takes a set of binary values described by the following string " a a b a h a a a b b ", where a, b e {0, I}. To compute the signal, condi tional and transition probabilities, we have to do the following calculations: • Signal probability calculations 6 4 p(x„ = a) = —, p{x„ = 6 ) = — (because, out of 1 0 symbols, a is occurring 6 times and b is occurring 4 times). • Conditional probability calculations X 3 /7(jc„ = a|x„_, =a) = Pa^ = - (symbol a appears 6 times and half of the time it is followed by another a). p(jc„ = 6 |x„_, = a) = p^j, = I (symbol a appears 6 times and half of the time it is followed by b). p(x„ = 6 |x„_, = b) = pIj, = I (symbol b appears 4 times and once it is followed by another b). /j(x„ = a|jc„_ J = 6 ) = pI^ = - (symbol b appears 4 times and 3 times it is followed by a). Based on these probabilities we can define the stochastic matrix Qas: Q = 3 3 6 6 1 3 4 4 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ■ Transition probability calculations X 6 3 3 P^^a-^b) = • Pajb " ÏÔ 6 ~ immediately from equation (3.8) and equation (3.6)) = P(x = b)pl^ = ^ ' ^ = ^^Pljb- Relying on Theorem 3.1, Theorem 3.2, and Theorem 3.3, the relationship between signal, conditional and transition probabilities can be illustrated as in Figure 3.4. Pq, O' Po, 1 Plo^Pii^ "Ÿ ^ p(x = 0 )"\ I V p(x = 1 ) J f P(xq^ q ), P(Xo_»i)>^ V pjXi _,o), p(Xi _^i)J Figure 3.4 The paradigm of signal, conditional and transition probabilities As we can see, we need less information to compute the signal probabilities, but the ability to derive anything else is severely limited. On the other hand, once we get either conditional or transition probabilities we have all we need to characterize that particular signal. Definition 3.3 (Switching Activity) For any signal line x, the switching activity is defined as: sw{x) = p(xo_»i) + p(x, ^o ) = 2 (3.10) As we can see, the switching activity depends only on two consecutive time steps an thus, using lag-one Markov chains is sufficient for estimating the switching activity. 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Note: we should point out that equation (3.10) reduces to the well-known formula jw(jc) = 2 • p(x= 1) • [1 - p{x =1)1 only if events are temporally uncorrelated. As long as we deal with temporally correlated signals, the exact relationship in equation (3.10) should be used. For instance, in Example 3.1, jw(x) = 3 /1 0 + 3 /1 0 ;t2 (6/10) ( 4 / 1 0 ). 33.2 Spatial correlations This type of correlations has two important sources: • Structural dependencies due to reconvergent fanout in the circuit; • Input dependencies that is, spatial and/or temporal correlations among the input signals which are ± e result of the actual input sequence applied to the target circuit. Referring to the combinational module in Figure 3.5, the lines x and y are obviously correlated because of the reconvergent fanout; on the other hand, even independent signal lines like the primary inputs of this module may also become correlated due to a particular input sequence (as is the case with sequences Si and S3 in Figure 3.1 when the structurally independent lines x and c become correlated). Figure 3.5 Two spatially correlated signal lines x and y To take into account the exact correlations is practically impossible even for small circuits. To make this problem more tractable, we allow only pairwise correlated signals. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. which is undoubtedly an ^proximation, but provides good results in practice. Consequently, we consider the correlations for all 16 possible transitions of a pair of signals (x,y) and model them as a lag-one Markov Chain with 4 states (denoted by 0, 1,2,3 which stand for the encoding (X ), 01, 10, 11 of (x,y)). Pool P ii P n 1 /»I3 P?i Figure 3.6 A lag-one Markov Chain for spatial correlations between x and y Definition 3.4 (Pairwise Conditional Probabilities) We define the conditional probability o f pair of signals (x,y) as: = 1 = y„ - 1 = j)) (3.11) where a, 6 = 0, l,2 ,3 ,a being encoded as ij and b as H. It basically describes the probability that the pair of signals (x, y) goes from state ij at time n- 1 to state kl at time step n. Ercolani et al. consider in [12] structural dependencies between any two signals in a circuit, through the signal correlation coefficients (SCs): sc, - where i, y = 0,1. Assuming that higher order correlations of two signals to a third one can be neglected, the following approximation is used: s c ,? = = SC ,SC SSC Jl (3.,3, 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Proposition 3.1 For every pair of signals {x,y) the following equations hold: X 1 Vj = 0, 1 (3.14) y=o, 1 X SCfyp{x = i) = 1 Vy = 0, 1 . I = 0,1 The set of 4 equations and 4 unknowns SCf^ i, j — 0, I is indeterminate; the matrix of the system has the rank < 3 in all non-trivial cases (i.e. when none of the signal probabilities is I). Proof: From the definition of SC, we get: X = = = 0 % (y = j)) = 1 7 = 0 .1 • ' y = 0 . 1 The second equation follows in a similar manner. ■ Our approach is more general: to capture the spatial correlations between signals, for each pair of signals (x,y) and for all possible transitions, we consider instead transition correlation coefficients (TCs). Definition 3.5 (Transition Correlation Coefficients) We define the TCs for any two signals x,yas where i,j, k, 1 = 0, 1 . Note: If the signals a and b in Figure 3.1 are spatially correlated then, based on TCs defined above, we have: i^i-»o) ~ i)f ->o)^^i. 1 0 - 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Definition 3.6 We define the TCs among three signals x, y, z as: where iJ , k,l,m ,n = 0,\. Neglecting higher order correlations, we therefore assume that the following relation holds for any signals x, y, z and any values i, j, k. I, m, n = 0, 1 : TC S'l.n = (3.17) Proposition 3.2 For every pair of signals (x,y) the following equations hold: ~ ^ Vi, A : = 0, I (3.18) ;./ = o.l S = I V/, 1 = 0,1. i, k = 0,1 The above set of 8 equations and 16 unknowns i, j, k, 1 = 0, I is indeterminate; the matrix of the system has the rank < 7 in all non-trivial cases (i.e. none of the transition probabilities is 1 ). Proof: Similar to the proof for Proposition 3.1, but using the definition of TCs. ■ The last two propositions are very important from a practical point of view. The set of equations involving SCs may be solved knowing only SC[i for example, and this was the approach taken in [12] (although, no similar analysis appeared in their original paper). In the more complex case involving TCs, we need to know at least 9 out of 16 coefficients in order to deduce all other values. 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4 Propagation mechanisms Having already described the analytical model for dependencies, we present subsequently the mechanism for propagating spatiotemporal correlations from the primary inputs throughout the target circuit. To this end, in what follows we ignore the higher order correlations; that is, correlations between any number of signals are expressed only in terms of pairwise correlation coefficients. Definition 3.6 and equation (3.17) may be easily extended to any number of signals. Based on the above assumption, we use an OBDD-based procedure for computing the transition probabilities and for propagating the TCs throughout the network. The main reason for using the OBDD representation for a signal is that it is a canonical representation of a Boolean function and it offers a disjoint cover which is essential for our purposes. Depending on the set of signals with respect to which we represent a node in the boolean network, two approaches may be used: • a global approach: for each node, we build the OBDD in terms of the primary inputs of the circuit; • an incremental approach: for each node, we build the OBDD in terms of its immediate fanin and propagate the transition probabilities and the TCs through the boolean network. The first approach is more accurate, but requires much more memory and run time; indeed, for large circuits, it is nearly impractical. The second one offers good results whilst being more efficient as far as memory requirements and running time are concerned. However, the propagation mechanisms we present subsequently are equally applicable to both global and incremental approaches. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4.1 Computation of transition probabilities L e t/b e a node in the boolean network represented in terms of n (immediate fanin or primary input) variables X/, x„;/m ay be defîned through the following two sets of OBDD paths; - Ill - the set of all OBDD paths in the ON-set off, - Ilo - the set of all OBDD paths in the OFF-set of f. Some of the approaches reported in the literature (e.g. [1]), use the XOR-OBDD of / a t two consecutive time steps to compute the transition probabilities. We consider instead only the OBDD of / and through a dynamic programming approach, we compute the transition probabilities more efficiently. Based on the above representation, the event /switching from value i to value/ (/, j = 0, 1), may be written as: n /,_», = u u n^Cifc, (3.19) Jt6 rijjCe fljjt = 1 * * where are the values of variable on the paths k and k ’ respectively (that is for path K , x i ^ = for path n \ where y^ = 0, 1 , 2 , and 2 stands for don’t care values) for each k= 1,2,..., n. In other words, this event basically represents the union over all possible switchings from a path (/i, 12,..., /„) to a path (Ji. y^,..., /,). Thus, the probability of the event ‘ /switches from value i to value y ' (i, y = 0 , 1 ) may be expressed as: Pifi^j) = p( k J ^ n ix . ] (3.20) Vjt e e H yt = i t ^kJ 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Applying the property of disjoint events (which is satisfied by the collection of paths in the OBDD), the above formula becomes; pC /.^y) = X Z .) (3.21) nerijjt'eriy * = 1 * * Since the variables xj^ may not be spatially independent of one another, the probability of a path to ‘switch’ from (/j, to (/% , j„) cannot be simply expressed as the product of transition probabilities of individual variables. Instead, we will use the following result which holds if we neglect higher order correlations. Proposition 3.3 If equation (3.17) is true for any three signals in the set {xi, X 2,..., x„}, then: * = 1 * * Proof: Follows directly from equation (3.17) by induction on the number of variables. ■ According to this result, the transition probability of the signal /f o r any values i, y = 0, 1 satisfies the following: Theorem 3.4 The transition probability of a signal/from state i to state j (/, j = 0, I) is: I z n (3-23) Tte n.rfe 1 * * lSk L 7=1 Using the definition of conditional probability and the expression of X y , we get P(^xfi) K P i x C \ f i ) K 7=1 pix) A A p{x) 7 = 1 The construction of xy’s was done such that x and fjS have disjoint supports then, according to Definition 3.9, they are logically independent; equivalently, we get K ^ P( O / , ) = Y Ï P(/, j which is true iff / , are logically independent i.e. their supports are y=I mutually disjoint: 5«p(/, ) n Supif^^ = 0 or 5, n 5,^ = 0 for any y, 1<K. Thus, the set of signals built above has at least k signals conditionally independent with respect to x iff C has at least k mutually disjoint sets. To conclude, CIP is NP-complete. ■ 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. One may extend the notion of conditional independence with respect to a single signal to that with respect to a subset of signals. The disadvantage is that, even if we find such a set, we may not express the probability of complex events in terms of probabilities of pairs of events as it is the case with conditional independence with respect to a single signal. Thus, from a computational point of view, this does not seem to be useful. In the following, we will use instead an approximation of conditional independence which holds for correlated inputs. Definition 3.10 (Signal Isotropy) Given the set of n signals {xi, X 2 ,..., x„}, we say that the conditional independence relation is isotropic, if it is true for all signals Xj, X 2 ,..., x„; more precisely, taking out all x,’s, one at a time, the subset of the remaining (n - I) signals is conditionally independent with respect to the taken x ,-. Returning to our example in Figure 3.10, given the set of signals [a, b, c} we have that {a, b) is conditionally independent with respect to c, but the sets [a, c] or {b, c] are not conditionally independent with respect to b, or a, respectively; it follows that conditional independence is not isotropic in this particular case. Intuitively, the concept of isotropy as defined above, is restrictive by its very nature and it is hardly conceivable that a set of signals taken randomly from a target circuit will satisfy Definition 3.10. Our goal, however, is not to use this concept as it is, but to make it more practical and therefore we propose the following approximation. Definition 3.11 (e-Isotropy) The property of conditional independence for a set of n signals is called e-isotropic if there exists some e (e > 0 ) such that; 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. n l^ iS n , i*i ________________________ J P( O JCi •*/) lS;Sn,y>/ < e for any / = 1, 2,..., n (3.34) Differently stated, e-isotropy is an approximation of pure isotropy within given bounds of relative error. A natural question may arise now: how often it is appropriate to consider e-isotropy as an approximation of pure isotropy? To answer this question, we consider in Figure 3.II several common situations involving the set of signals {u, v, w} and the relative position of their logic cones (each cone illustrates the dependence of signals v, w on the primary inputs). Whilst the isotropy is completely satisfied only in (b), the e-isotropy concept is applicable in all other cases; more precisely, the conditional independence relation is partially satisfied in (a) with respect to w, in (c) with respect to u and V and in (d) with respect to u and v. u V (b ) u V (c) u V (d) Figure 3.11 An example to illustrate pure and e-isotropy Based on the previous definition, we get the following: Theorem 3.7 Given an e-isotropic set of signals the probability of the composed signal p( O X y ) can be estimated within e-relative error as: 7 = 1 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 X n X;) = (3.35) (nH Proof: From the definition of e-isotropy, we get (V i = 1 , 2,..., n): n XXy|x,) __________ I P ( P i X | x , ) probability as: <e which may be re-written using the definition of conditional n X X fX y) n — 2.( V l - e < ^ ~n ~ ‘ -----------< 1 + E for each/ = 1 , 2 , . . . , n P(PX y) J= I Multiplying all inequalities we get exactly the above claim. ■ This proposition provides a strong result: given that n signals are e-isotropic, the probability of their conjunction may be estimated within e-relative error using only the probabilities of pairs of signals, thus reducing the problem complexity from exponential to quadratic. This is similar to the concept of pairwise correlation coefficient introduced in [12] and generalized in [27]. However, the approach in [27] does not provide sufficient accuracy for highly correlated signals as we shall see later. 3,53 Computation of transition probabilities using e-isotropic signals If the e-isotropy property is satisfied. Theorem 3.7 may be easily extended to boolean functions represented by OBDDs. L et/b e a boolean function of n variables xj, X 2 ,..., x„ 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. which may be defined through the ON- and OFF-sets as in Section 3.4. In the global approach,/is represented in terms of the primary inputs, while in the incremental approach it depends only on its immediate fanin variables. Based on this representation, we give the following result. Theorem 3.8 G iven/a boolean function of variables X;, X 2 ,..., the following hold: a) If the set {-Cy}isys„ (where each variable is ei±er direct or complemented) is e- isotropic, then ± e signal probability p (f= i) with i = 0 , I may be expressed within e- relative error as: 2 ( n = ii)j XZ = 0 = X -------------- n ------- (3.36) Ite n , r n \ „ r[p ('C t= 4 ) Vfc= 1 / where is the value taken by variable in the cube n e IT ,-. b) If the set j is e-isotropic, then the transition probability p (/}_^) (with 1 , 7 = 0 , 1 ) may be expressed within e-relative error as: = S I (3.37, J t e n . i t ’ e r i y / n \ „ t", where z^,7 ^ are the values taken by the variable in the cubes 7i e IT ,-, and 7 C ’ e Ily. Proof: a) We may defîne the event ‘ /takes the value /’ as: ( / = 0 = u n ( - > ^ i f c = i*) Jte n,.*= I 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the probabilistic domain, this becomes; P(f = 0 = ^ p( O ~ 4)) Ken, t = 1 because the paths tc e IT ,- are disjoint. Since the input variables are e-isotropic, we may use Theorem 3.7 to express the probability of each cube, getting exactly equation (3.36). b) Proof similar to case a) but considering instead the event ‘ / switches from i to f expressed as: ' .V n,.V n, The above result may be reformulated using signal and transition correlation coefficients; it can be used in signal probability and switching activity estimation, given that the e- isotropy conditions are met. Corollary 3.1 Given a set of signals {xy}, as in Theorem 3.8 and a boolean function / o f variables the following hold within e-relative error: a)p (/ = 0 = % [ 1 7 = (3.38) Ken , I ^ k < l < n 4 -1 I n b ) p ( / , ^ , ) = I s ( n w ) ) Ke n.K ’e Hy 1 S i< /Sn t = l Proof: a) and b) Using the definition of transition probability and Theorem 3.8, we easily get the above inequalities. ■ For the incremental approach, this result can be extended to the calculation of correlation coefficients (5Cy or TCs) between two signals in the circuit. In practice, this 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. becomes an important piece in the propagation mechanism of probabilities and coefficients through the boolean network. 3.5.4 Computation of transition correlation coefficients using e-isotropic signals Theorem 3.9 Given a set of signals a boolean function / of variables fiysn ’ and x a signal from the circuit, if {jci, xj,..., x„, x} is a set as in Theorem 3.8, then the correlation coefficients (SCs and TCs) can be expressed within e-relative error^ as: -T -7 « f a) 3 C \f = - 7 ^ ----------------------------- ^ (3.40) s( n p if = i) s X ( n _ J t e n j j f e n , l S t < / S n '' * = i V b) T C „ ,j,--------------- X ÿ - ; --------------------------------- where/,y,p, q = 0,1. Proof: a) and b) follow directly from the defînition of SCs and TCs and using the events = I and X - j simultaneous’ and ‘ / switches from / to j and x from p to ^ simultaneously’, respectively. ■ These results lead to a new heuristic algorithm for signal and transition probability estimation under input streams which exhibit spatiotemporal correlations. We may thus see equation (3.39) as the improvement of equation (3.23) by using the notions of conditional independence and signal isotropy. Compared to the heuristic proposed in 1. This e is the maximum over all values that occur during the incremental propagation process. 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Section 3.4, this new approach based on conditional independence has also the advantage of supplying bounds for error estimation provided that the input signals are e-isotropic. This bounding value could not be justiOed by using the spatiotemporal hypothesis alone. Finally, the model introduced above provides a way to improve the run time requirement as shown below. Proposition 3.5 If is a correlation coefficient (SC or T O at level j (given by a topological order from inputs to outputs of the circuit), then it is related to Cj ,[(0 < I <f) [■ üh1 by a proportionality relationship expressible as C, « = (Cy_;) , where n represents the average fan-in value in the circuit. Moreover, if then -> I (the signals on level j-l behave as uncorrelated). Proof: First part follows from Theorem 3.9 and Corollary 3.1 if the conditions required in Theorem 3.8 are satisfied. The second part follows immediately, more precisely, (Cj_i) — > 1 when / -> « > and this represents the condition of uncorrelated signals in our approach. ■ In other words, we do not need to compute the coefficients which are beyond some level / in the circuit; instead, we may assume them equal to 1 without decreasing the level of accuracy. Also, the larger the average fanin n o f the circuit, the smaller value fo r I may be used. It is worthwhile to note that the conditional independence relationship, more specifically the concept of e-isotropy, is essential for this conclusion. The approach based on spatiotemporal correlations only, does not provide a sufficient rationale for such 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a limitation. This is actually a very important heuristic to use in practice and its impact on run time is huge; limiting the number of calculations for each node in the boolean network to a fixed amount (which depends on the value set as threshold for /) reduces the problem of coefficients estimation from quadratic to linear complexity. 3.6 Practical considerations and experimental results All experiments are performed using the SIS [42] environment on an Ultra SPARC 2 workstation with 64 Mbytes of memory; the working procedure is shown below. Input sequence generation , Comparison Switching activity estimation Switching activity extraction Zero-delay gate-level simulation Input sequence analysis (iiq>ut coirelatioiis extraction) Figure 3.12 The experimental setup To generate pseudorandom inputs we have used as input generator a maximal-length linear feedback shift register modified to include the all-zero pattern; these registers are based on primitive polynomials that is, they randomly generate all distinct patterns that correspond to a given polynomial before repeating the sequence. Purely random generators do not exist, therefore the primitive polynomials used, give us multiple correlations among primary inputs. The length of the input register was set equal to the number of inputs of the circuit under analysis, thereby creating a pseudorandom source; when the length of this register became too large, we tried to keep the time/space requirements at a reasonable level 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and hence, for these cases we generated only a signifîcant part of the exhaustive sequence (up to 2 ^ input patterns). As the standard measure for power estimation, we have used the average switching activity at each node of the circuit calculated as in equation (3.10). In our experiments, we were mainly interested in measuring the accuracy of the model in estimating the switching activity locally (at each internal node of interest) and globally (for the entire circuit), given a set of inputs with spatiotemporal correlations. The analysis part of the experiment may be skipped if the user specifies directly the characteristics of the input stream (transition probabilities and correlation coefficients). To illustrate the main features of our approach, we consider in Figure 3.13 the ISCAS circuit C l7, fed by the sequence generated with the primitive polynomial p(x) = I © jc © x^. Due to the deterministic way in which we generate the input sequence, structurally independent signal lines become correlated as is the case with the inputs 1 & 2,2 & 3,3 & 6 , 6 & 7; in turn, the fan-out points on the input lines add additional correlations. For an accurate analysis of the switching activity, we have to account for all these dependencies. In Table 3.1 we list the transition probability coefficients for this particular input sequence; we mention that in this case, all signal correlation coefficients (5Cs) are equal to 1, therefore they cannot make a difference alone. In Table 3.2 we present the estimated and exact values of the switching activity per clock cycle. In Figure 3.13, the color code is used 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to reflect the switching activity at the output of the gates, i.e. darker gates are more active (in particular, gate 23 is the most active one). ? 1 — \ _ u _ r 6 Figure 3.13 Switching activities distribution in C17 (pseudorandom inputs) Table 3.1. C17: TCs for pseudorandom ij.u 1 & 2 2&3 3&6 6&7 0 0 .0 0 2 .0 0 0 0 2 .0 0 0 0 10000 1.7143 0 0 .0 1 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0.4444 0 0 .1 0 2 .0 0 0 0 2 .0 0 0 0 10000 1.7778 0 0 .1 1 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0 1 .0 0 2 .0 0 0 0 2 .0 0 0 0 2 .0 0 0 0 21857 0 1 .0 1 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0 1 .1 0 2 .0 0 0 0 10000 10000 1.7778 0 1 .1 1 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 1 0 .0 0 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 1 0 .0 1 2 .0 0 0 0 10000 2 .0 0 0 0 1.7778 1 0 .1 0 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0.4444 1 0 .1 1 2 .0 0 0 0 10000 10000 1.7143 1 1 .0 0 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 1 1 .0 1 2 .0 0 0 0 10000 10000 1.7778 1 1 .1 0 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 0 .0 0 0 0 1 1 .1 1 2 .0 0 0 0 10000 10000 21857 Table 3.2. C 17; rw act for Node Estimated sw_act Exact sw_act 1 0.5000 03000 2 0.5000 03000 3 0.5000 03000 6 0.5000 03000 7 0.5625 03625 1 0 0.3750 0.3907 1 1 01500 01649 1 6 0.5000 03236 1 9 0.5687 03236 2 2 01978 03125 23 0.6006 0.5625 It should be pointed out that the actual values for all coefficients in Table 3.1 represent the characteristics of the input stream; if we select another primitive polynomial to generate the inputs, we may obtain a completely different set of transition correlation coefficients. This dependency is even more salient if we consider ‘biased inputs’ (i.e. the switching activity is not 0.5). To generate such sequences, we used a simple functional generator 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. based on the random function in C language. For each bit, we set up a specific threshold r e [0 , 1 ] and generated a set of random numbers in [0,1]. If these numbers exceeded the threshold t, then the output of the generator is set to I; otherwise the output is 0. We give in Table 3.3 (f = 0.25 for all bits) and Table 3.4 (r e [0.1,0.5]), the values obtained for two such sequences, and in Figure 3.14 the new distribution of switching activity among the internal nodes of the circuit. As we can see, gates 22 and 19 become in turn the most active gates in these two experiments. We note that in these cases we have spatial dependencies among all primary inputs. Figure 3.14 Switching activities distribution in C17 for two biased input sequences Table 3.3. C 17: sw_act for biased Table 3.4. C 17: sw_act for biased inputs Node Estimated sw_act Exact sw_act 1 0.5625 05625 2 0.3750 05750 3 0.3125 05125 6 0.5625 05625 7 0.5000 05000 1 0 0.3125 05125 II 0.3125 05125 1 6 0.4569 05000 1 9 0.3367 05125 2 2 0.4960 05000 23 0.4527 0.4375 Node Estimated sw_act Exact sw_act I 0.0625 0.0625 2 0.1250 0.1250 3 05500 05500 6 05000 05000 7 I .O O O O I .O O O O 1 0 0.1250 0.1250 II 05500 05500 1 6 0.1295 0.1250 1 9 0.7262 0.7500 2 2 05040 0.1250 23 0.4597 05000 To further assess the impact of correlations, we considered the benchmark f51m and the following two scenarios: 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a) Low Correlations: the input patterns are generated by a linear feedback shift register which implements the primitive polynomial; p (x) = 1 © x © © x*. b) High Correlations: the input patterns are generated using the state lines of an 8 -bit counter. In order to do a fair comparison between the existing estimation techniques (including the ones which use global OBDDs) and our technique, we had to choose a small sized circuit. The estimated values in both cases were compared against the exact values of switching activity obtained by exhaustive simulation; all internal nodes and primary outputs have been taken into consideration (see Figure 3.15)*. 1. In Figure 3.15, on the x axis, we report the absolute error of switching activity that is. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S o » n a r i o « : Q l o b a l / N o c o r r . 2 5 r 20 - 5 . bll4 0 I1 6 0.18 oiz Ô .Ô 2 0.04 d~06 0 '% ^ S c f M t f l o b : Q t o b a l / N o a p a t f o f c w p o r ml c o r r . % my] 0.35 0.4 045 0.5 S c e n a r i o a : G l o b a l / T a m p o r a f c o r r . % 0 .6 2 0.04 Ô.06 b"oe a i Ô .1 2 0.14 0 Z 16 b .is o!a Ufa e r r o r S c e n a r i o a : I n c r e m e n t a l / S p a t f o t e m p o r a l c o r r ô .04^ô !Œ 5 ! 3 5 cf.ii o.i4 a b s o l u t e e r r o r S c e n a r i o b : Q t o b a y T e m p o r a l c o r r . 30, 25 10 a b s o l u t e e r r o r 005 S c e n a r i o b : I f w r e m e n t a l / S p a t t o t e m p o r a i c o r r . 8& 50- 20 10 o36^ oio7 oT» oios o J S c e n a r i o a : I n c r n i e n l a l / S p a t l o t e m p . c o f r . / l » o t r o p y 30" 10 T 5 . 1 ^ S c a n f l o b : I n c f w t i i i t a l / S p a t l o t w n p . e o r r ./ l s Q tr o p y 4 0 3 5 I S 1 C ■ Figure 3.15 The impact of the correlation level on sw^act estimation in f51m 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In Scenario (a), all approaches are quite accurate but we point that only considering spatiotemporal correlations and signal isotropy ensures the highest accuracy (100% of the nodes estimated with error less than 0.1). As the results of Scenario (b) show, the level of correlation on the primary inputs strongly impacts the quality of estimation. Specifically, it makes completely inaccurate the global approach based on input independence (despite the fact that internal dependencies due to reconvergent fan-out are accounted by building the global OBDD). As expected, this is visible mostly in Scenario (b), where less than 20% of the nodes are estimated with precision higher than 0.1. On the other hand, even if temporal correlations are taken into account, but the inputs are assumed to be spatially uncorrelated, only 80% of the nodes are estimated with error less than 0.1. Accounting for spatiotemporal correlations provides excellent results for highly correlated inputs (100% nodes estimated with precision 0.1), but the mean error in the hypothesis of conditional independence is anyway smaller (90% of the nodes are estimated with error less than 0.05). These results clearly demonstrate that power estimation is a strongly pattern dependent problem, therefore accounting for dependencies (at the primary inputs and internally, among the different signal lines) Is mandatory if accuracy is important. From this perspective, accounting for spatiotemporal correlations and using the conditional independence and signal isotropy concepts seems to be the best candidate to date. Using some ISCAS'85 benchmarks, we further performed the following experiments: a) One set of experiments to validate the model based only on spatiotemporal correlations without conditional independence. 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. b) Another one to assess the impact of the limiting technique from Proposition 3.5. c) The last one to validate the model based on spatiotemporal correlations with conditional independence and signal isotropy. Once again, the switching activity values and power consumption were estimated at each internal node and primary output and compared with the ones obtained from logic simulation. We found that power estimation for the entire circuit is not a real measure to use in low-power design and power optimization where the switching activity at each node has to be accurately estimated. a) Experiments to validate the model based only on spatiotemporal correlations In the following, we give the error values for pseudorandom inputs, using the incremental approach without conditional independence. In reporting the error, we compared our switching activity estimates with the results of logic simulation at every internal node and primary output. All benchmarks were mapped with Ub2genlib. Table 3.5. Pseudorandom inputs; no limit^ (/ — > °q) in TCs calculation Circuit MAX MEAN RMS STD Time (sec.) C17 0.0565 0.0119 0.0238 0.0226 0.04 C432 0.0716 0.0133 0.0222 0.0179 30.17 C499 0.0334 0.0047 0.0072 0.0055 88.17 C880 0.1131 0.0158 0.0306 0.0264 45.38 Cl 355 0.0393 0.0027 0.0051 0.0044 61.76 CI908 0.0353 0.0044 0.0082 0.0069 67.15 C3540 0.1765 0.0276 0.0429 0.0318 491.19 C6288 0.2046 0.0204 0.0443 0.0396 616.22 a. tis the lim it used in Proposition 3.5. 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. b) Experiments concerning run time improvement Heuristic proposed in Section 3.5.3 is important in practice not only for substantially reducing the run time but also for keeping the same level of accuracy as the case when the threshold limit is set to infinity (that is, we did not use any limitation in TCs calculation). In the following, we present a detailed analysis for the benchmark duke2 which exhibits a typical behavior; in the first case the limit was set to infinity, in the second one the limit was 4. To report error, we used standard measures for accuracy: maximum error (MAX), mean error (MEAN), root-mean square (RMS) and standard deviation (STD); we excluded deliberately the relative error from this picture, due to its misleading prognostic for small values. Table 3.6. duke2 - Speed-up vs. accuracy Low Correlations High Correlations Error NO limit (/ — > oo) Limit / = 4 NO limit (/ — > oo) Limit 1 = 4 MAX 0.0744 0.0710 0.0299 0.0299 MEAN 0.0133 0.0161 0.0056 0.0055 RMS 0.0223 0.0269 0.0085 0.0083 STD 0.0182 0.0219 0.0065 0.0063 Time (sec) 131.08 40.83 130.55 42.14 As we can see, the quality of estimation is practically the same in both cases whilst the run time is significantly reduced in the second approach. It should be pointed out, that this limiting technique works fine also for partitioned circuits which is an essential feature in hierarchical analysis for power estimation. Running extensively our estimation 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. tool on circuits of various sizes and types (ISCAS’85 benchmarks, adders, multipliers), we observed the following general tendency for the speed-up. S p e ^ -u p vs. circuit complexity 20, 18- 16- J4- @12 - 10 - 10Ô0 2060 3060 4060 50Ô0,6QÔ0 70Ô0 80ÔO 9000 number of gates Figure 3.16 Speed-up factor vs. circuit complexity We can see that the speed-up is about 3-5-5 times for less complex circuits, but it may become 20 - 5 - 30 times for large examples; we estimated the power consumption for multipliers on 16 bits (2048 gates) and 32 bits (9124 gates) and the run times were 43.90 and 195.13 sec. We present in Table 3.7 the results obtained for the set of ISCAS’85 benchmarks using the limit / = 4 in TCs calculation. By comparing these results with those given in Table 3.5, we can see that the quality of the estimates remained basically the same while the run time was significantly improved. 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.7. Pseudorandom inputs; / = 4 in TCs calculation (|iW @ 2QMHz; Vnif=5V) Circuit MAX MEAN RMS STD Total Power Time (sec.) C432 0.1916 0.0281 0.0465 0.0374 3372.57 11.82 C499 0.0624 0.0134 0.0184 0.0126 7645.56 10.57 C880 0.0691 0.0135 0.0211 0.0164 6391.83 18.35 C1355 0.0225 0.0041 0.0051 0.0030 6797.92 4.24 C1908 0.1315 0.0091 0.0206 0.0185 7435.34 12.69 C3540 0.2010 0.0307 0.0509 0.0407 16356.82 60.86 C6288 0.0890 0.0142 0.0241 0.0196 46846.48 29.10 c) Experiments to validate the conditional independence and signal isotropy The experiments were performed on large ISCAS’85 examples using pseudorandom and highly correlated inputs (obtained from counted sequences of length 2^®); all results reported here, have been derived using the value / = 4 as the limit for coefficients calculations. To report error, all estimations were verified against exhaustive simulation performed with SIS logic simulator. To show the impact of using signal isotropy concept, for high-correlation scenario, we also present in Table 3.8 the results obtained if signal isotropy is not used. WITH conditional independence (e-isotropy) WITHOUT conditional independence Circuit MAX MEAN RMS STD Total power MAX MEAN RMS STD Total power C432 0.2538 0.0225 0.0585 0.0545 306.88 0.8499 0.1058 0.2274 0.2032 1390.07 C499 0.1566 0.0421 0.0760 0.0634 2283.03 0.4254 0.0387 0.0933 0.0851 2543.99 C880 0.0175 0.0013 0.0040 0.0038 263.13 0.7853 0.0471 0.1630 0.1571 482.60 C1355 0.1930 0.0227 0.0520 0.0469 1865.81 0.4722 0.0516 0.1252 0.1144 1961.76 C1908 0.3907 0.0294 0.0868 0.0820 3156.83 0.4903 0.0459 0.1000 0.0892 3201.10 C3540 0.0279 0.0279 0.0030 0.0030 166.25 0.5463 0.0280 0.0365 0.0365 207.38 C6288 0.1773 0.0231 0.0521 0.0471 8843.72 0.5639 0.1092 0.1995 0.1685 19428.91 As we can see, by using conditional independence and signal isotropy, the accuracy for node-by-node analysis improves, on average, by an order of magnitude; on the other hand. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. by not using conditional independence at all, the total power consumption is overestimated by 100% on average. To further asses the impact of the correlation level, we considered the following experiment. We estimated the total power consumption for a set of ISCAS circuits using random inputs (the low correlation scenario in Table 3.9). After that, we sorted the input vectors (therefore we kept the same signal probability on the inputs) and we applied the resulting file to the same set of circuits (the medium correlation scenario). Finally, we considered as input a counted sequence with the same length and we estimated once again the total power consumption (the high correlation scenario in Table 3.9). All the values of power consumption are reported in p.W at 20 MHz. Circuit Low Correlations Medium Correlations High Correlations Circuit 1 : C432 3372.57 2046.93 306.88 Circuit 2: C499 7645.56 5068.32 2283.03 Circuit 3: C880 6391.83 4781.47 263.13 Circuit 4; Cl 355 6797.92 4272.74 1865.81 Circuit 5: CI908 7435.34 4307.76 3156.83 Circuit 6: C3540 16356.82 8464.05 166.25 Circuit 7: C6288 46846.48 40748.45 8843.72 As we can see, there is a significant difference in all cases: not only the switching activities at each internal node were completely different as the level of inputs correlation changes, but also the values of total power consumption. For example, for C432, the total power estimated under low correlated inputs was 3372.57 |iW, while this value for strongly correlated inputs was 306.88 jiW (there is a factor of 11 between the two). The same behavior has been observed for other circuits. To give a more meaningful measure to this experiment, we give in Figure 3.17 the error made in medium and high correlation 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. scenarios by ignoring the correlations on the circuit inputs (o-medium correlations and *- high correlations). Circuit Number Figure 3.17 Overestimation factor by ignoring input correlations As we can see, once again the level of correlations has a significant impact on our predictions; whilst for medium correlations one may overestimate in general about twice, for high correlations the overestimate is about 10 (e.g. Circuit 6: C3540, Circuit3: C880). To conclude, input pattern dependencies (in particular highly correlated inputs) is an extremely important issue in power estimation. From this perspective, power analysis needs analytical models to overcome this difficulty. 3.7 Summary In this chapter, we proposed an original approach for switching activity estimation in combinational logic modules under pseudorandom or highly biased input streams. Using the zero-delay hypothesis, we have derived a probabilistic model based on lag-one Markov Chains which supports spatiotemporal correlations among the primary inputs and internal lines of the circuit under consideration. The main feature of our approach is the 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. systematic way in which we can deal with complex dependencies that may appear in practice. From this perspective, the new concepts of conditional independence and signal isotropy are used in a uniform manner to fulfill practical requirements for fast and accurate power estimation. Under general assumptions, the conditional independence problem has been shown to be NP-complete; consequently, efficient heuristics have been provided for transition probabilities and correlation coefficients calculation. As a distinctive feature, the approach presented in this chapter improves the state-of-the-art in two ways; theoretically by providing a deep insight about the relationship between the logical and probabilistic domains, and practically by offering a sound mathematical framework and an efficient technique for power analysis. 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 Circuit Independent Techniques: The Combinational Case 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1 Introduction As we have seen in the previous chapter, the key element in the process of power estimation is accurate and fast estimation of the average switching activity. Static techniques (like the probabilistic power estimation approach presented in Chapter 3) are in general faster, but less accurate than those based on exhaustive simulation (also called dynamic techniques). Usually, the input correlations and the reconvergent fan-out in the target circuit make things very complicated and simplifying assumptions (like input independence or the zero-delay model) become mandatory. On the other hand, general simulation techniques provide sufficient accuracy, but at high computational cost; it is simply expensive to simulate large circuits using millions or even thousands of input vectors and therefore, the length of the simulation sequence is another important issue that must be considered. From this perspective, the research presented in this chapter shifts the focus from the “ circuit problem ” to the “input problem ” and proposes an original solution for power estimation based on the paradigm of sequence compaction. This kind of technique is appealing because it is practically independent of the actual implementation of the target circuit and it can therefore be used early in the design cycle when the structure of the circuit has not been determined yet. The basic idea is illustrated in Figure 4.1. Given an input sequence Lq (Figure 4.1a), to evaluate the total power consumption of a target circuit, we first derive the Markov model of the input sequence and then, having this compact representation, we generate a much 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. shorter sequence L , equivalent with L q, which can be used with any available simulator to derive accurate power estimates (Figure 4. lb). out circmt (initial sequence L q) (initial sequence L q ) (a) c u c u it (compacted sequence L « Lq) Figure 4.1 Data compaction for power estimation The key element in this schema is the actual Markov model used to represent the initial input sequence. The construction of this Markov model relies on adaptive {dynamic) modeling of binary input streams as first-order Markov sources of information. Once this first-order Markov model is constructed, the sequence compaction procedure we propose subsequently is applicable to combinational and, under some circumstances, sequential circuits. The adaptive modeling technique itself (best known as Dynamic Markov Chain or DMC modeling) was introduced recently in the literature on data compression [3], [9] as a candidate to solve various compression problems. Traditionally, data compression techniques were aimed to lossless compression, that is the ability to encode a body of data, transmit it eventually over a communication line and finally, decode it uniquely, without any loss of information during this process. Our objective in this chapter is slightly different: having an initial sequence (assumed representative for some target circuit), we target lossy compression [43], that is the process of transforming an input sequence into a 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. shorter one, such that the new body of data represents a good approximation as far as total power consumption is concerned. From the very beginning, the DMC modeling technique looked very promising and indeed, in most practical situations, has been more effective than any other compression technique available to date. However, the original model introduced in [9] is not completely satisfactory for our purpose. In this chapter, we thus extend the initial formulation to manage not only correlations among adjacent bits that belong to the same input vector, but also correlations between successive input patterns. The results presented in this chapter represent an important step toward reducing the gap between dynamic and static techniques commonly used in power estimation. We also point out that the present approach can be used without any difficulty to generate benchmark data for power estimation, that is, input sequences with different lengths that satisfy a set of user-prescribed characteristics in terms of word-level transition or conditional probabilities. This chapter is organized as follows: first, we review the prior work relevant to our research. Section 4.3 formalizes the power-oriented vector compaction problem and discusses different parameters which makes this approach effective in practice. In Section 4.4 we introduce the basic concepts of DMC modeling technique and present a DMC- based procedure for vector compaction. To provide a better adaptability of the basic DMC modeling technique to the dynamic changes in the sequence characteristics, in Section 4.5, we introduce the hierarchical modeling of Markov chains as a flexible framework for sequence compaction. Finally, we conclude by summarizing our main contribution. 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2 Prior work As described in the introductory part of this chapter, a number of issues are important for power estimation and low-power synthesis: the input statistics which must be properly captured and the length o f the input sequences which must be applied are two such issues. Generating a minimai-length sequence of input vectors that satisfies these statistics in not trivial. More precisely, LFSRs which have traditionally found use in testing or functional verification [2], are of little or no help here. The reason is that more elaborate set o f input statistics must be preserved or reproduced during sequence generation for use by power simulators. One such attempt is [31] where authors use deterministic FSMs to model user- specified input sequences. Since the number of states in the FSM is equal to the length of the sequence to be modeled, the ability to characterize anything else but short input sequences is limited. A more elaborate and effective technique was presented in [26] where, based on stochastic sequential machines (SSMs), the authors succeed in compacting large sequences without significant loss in accuracy. In what follows, we present a new approach based on DMC modeling. This new framework is specifically tailored for power-oriented data compaction. 4.3 A first-order probabilistic model We use the theory of discrete-parameter time-homogeneous Markov chains to derive a probabilistic model suited for combinational circuits. As shown in Figure 4.2, we model the ‘tuple’ {inputjsequence, target_circuit) by the ‘tuple’ {Markov_chain, target_circuit). 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Markov_chain represents the Markov chain that models the inputjsequence and targetjcircuit is the combinational circuit where power consumption has to be determined. arkov chain Logic target circuit Figure 4.2 The tuple {Markov-Chain, targetjcircuit) Let denote a random variable associated to primary inputs of the circuit shown in Figure 4.2; is then the probability that the input is at time step n. We are interested in defining the probabilities p(x„) and p(jc^„.i) because, as we can see in Figure 4.2, they completely capture the characteristics of the input data that feeds the target circuit. 4.3.1 Problem formulation Using the probabilities p(x„) and p ix ^ ^ -ù ia Figure 4.2, the vector compaction problem can be formulated as follows: given a sequence of length L q , find another sequence of length L < L q (consisting of a subset of the initial sequence), such that the average transition probability on the primary inputs is preserved wordwise, for two consecutive time steps. More formally, the following condition should be satisfied: | p ( % _ i ) - p ' ( % - i ) | < e (4.1) 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where p and p* are the probabilities in the original and compacted sequences, respectively. □ This condition simply requires that the joint transition probability for the primary inputs is preserved within a given level of error, for any two consecutive vectors. We want to prove now that indeed, by having satisfied equation (4.1), it is guaranteed to produce a new sequence which is asymptotically close to the original one. 43.2 Proof of correctness As usual, Q represents the stochastic matrix associated to the original input sequence, i.e. Pij = P(V;jv,), where v ,-, vj are any two consecutive vectors in the initial sequence. To produce an equivalent sequence from a reference one, one should preserve the word-level transition probabilities. This essentially becomes the problem of preserving both conditional and state probabilities because p, _»y = Jt, • p^j, where tc ,- is the fth component of the state probability vector (that is, p(j,-)) and pi represents the transition probability of going from vector v ,- to vector v y . (From Theorem 2.1, it can be seen that 7t is the left eigenvector that corresponds to the eigenvalue A , = 1 in the general equation k Q = X-k). To complete the proof, we note that every stochastic matrix has 1 as simple eigenvalue and all other eigenvalues have absolute values less than one. This is in fact a consequence of the Perron-Frobenius theorem [41] which states that for every matrix with non-negative entries, there exists a simple \ positive eigenvalue greater than the absolute value of any 1. This means that the multiplicity of the root X = 1 in the equation n ■ Q = X ■ n is one. 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. other eigenvalue. This result is very important because it makes it possible to analyze the effect of perturbation of matrix Q on the eigenvectors that correspond to the eigenvalue 1. To this end, let’s assume that the newly generated sequence is characterized by the matrix G* = [pij] where p^j = p^j + (£,y represents the error induced by perturbations) and |e ^ y | < 1. We can write Q* = Q + z B where e = max\z^j\ and b^j = ^ . Because Q* characterizes a sequence of vectors, it is also a stochastic matrix and therefore, as stated in Theorem 2.1, it has an eigenvalue X = 1. But, from the theory of algebraic functions [48] we have that, for any eigenvector % o f Q corresponding to the simple eigenvalue k = 1, there exists an eigenvector iz* of Q* corresponding to the simple eigenvalue X * = I , such that |tc - 7 c * B = 0(e) (read as ‘zero o f epsilon'), where 0(e) is any power series in e (convergent for sufficiently small e) having the form k^e + k2^ +.... As a consequence, since \p^j - p*] = 0(e), it is easy to see that |p, - P*_+J = 0(e). ■ Summarizing, we have that: Corollary 4.1 If the stochastic matrix Q is properly preserved during the compaction process, then the transition probabilities of the newly generated sequence are asymptotically close to the original ones, that is |p, _»y- P*-» J = 0 (e). □ Proof: Follows immediately from the above discussion. ■ We have thus proved that we can asymptotically reproduce an initial sequence by preserving its matrix Q. From a practical point of view, let’s see the implications of the 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. above corollary on total power consumption in a target circuit when the original input sequence is approximated by a new one. CoroUary 4.2 If P and P* are the values of the total power consumption which are obtained for two sequences satisfying the conditions in Corollary 4.1, then we have that |/>-P*( = 0(e). □ Proof: We have that P = ^ P/-», where is the output capacitance i.j.k of gate k and n* is the number of transitions at the output of gate k when vector v ,-, followed by vector vj, is applied at the input of the circuit. Assuming that the input sequence is approximated by another input sequence such that the new set of transition probabilities satisfies = 0(e), then the error made in the value of total power consumption is given by \P_P*\<1^. v l ^ . Y • Q • 4 = 0(E) (4.2) i,j,k m Corollary 4.2 basically shows that, if the new sequence is asymptotically close to the original one, then the same type of asymptotic relationship holds for the total power values. We have, therefore, proved that a first-order probabilistic model is sufficient to perform sequence compaction for power estimation in combinational circuits. The remaining portion of this section describes how can we efficiently do compaction in practice. 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4 Flat models Without loss of generality, in what follows we restrict ourselves to finite binary strings, that is, finite sequences consisting only of O ’s and I's. The set of events of interest is the set 5 of all finite binary sequences on b bits. A particular sequence S\ in S consists of vectors v;, V 2 ,..., v„ (which may be distinct or not), each having a positive occurrence probability. Indices 1, 2,..., n represent the discrete time steps when a particular vector is applied to a target circuit. 4.4.1 DynamicMarkov models Imposing a total ordering among bits, such a sequence may be conveniently viewed as a binary tree (called DMTq from Dynamic Markov Tree o f order zero) where nodes at level j correspond to bit y (1 <y < h) in the original sequence; each edge that emerges from a node is labelled with a positive count (and therefore with a positive probability) that indicates how many times the substring from the root to that particular node, occurred in the original sequence. For clarity, let’s consider the following example. 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Example 4.1 For the following 4-bit sequence consisting of 8 non-distinct vectors: (vj, V 2 , V 3 , V 4 , V 5 , V 5 , V 7 , V g) = (0000, (X X )1, 1(X)1,1IC X ), 1(X)1, 11(X ), 1001, 11(X)) the construction of the tree D M T q is shown step-by-step in Figure 4.3a. 000 first bit in vj second bit in w , is processeii is processed rc a n fa_ _ is processed 000 foaiifa bit in w , bit in g o I - 3 fourth bit in vj is processed second bit invj processed processed is processed (a ) Figure 4.3 The tree D M T q for the sequence in Example 4.1 Obviously, the whole Markov tree that models this sequence must have four levels because the original sequence is a 4-bit sequence. Without loss of generality, we assume a left-to-right order among bits that is, the leftmost bit in any vector vj to vg is considered as being bit number one (and consequently represented at level one in D M T q as shown in Figure 4.3a), the next bit is considered as being bit number two and so on. Every time a vector is completely scanned (this corresponds to reaching the level four in the tree), we 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. come back to the root and start again with the next vector in the sequence. While the input sequence is scanned, the actual counts on the edges are dynamically updated such that, for this particular example, they finally become as indicated in Figure 4.3b. The Markov tree in Figure 4.3b contains in a compact form all the spatial information about the original sequence vj, V 2 ,..., vg. We point out that this sparse structure is possible only by using the dynamic (adaptive) fashion of growing the tree DMTq just illustrated. Another approach would have been to consider a static binary tree capable to model any 4- bit sequence and just to update the counts on the edges while scanning the original sequence. By doing so, we would end up with the obvious disadvantage of having 15 instead of 9 nodes in the structure for the same amount of information; this reason alone is sufficient for considering from now on only dynamically grown Markov models. Proposition 4.1 If p is a probability function from S to [0,1], then: p(v) = ^ p(vx) (4.3) x e S for all V in S, where vx represents the event corresponding to the joint occurrence of the strings v and x. The above condition, simply states that the sum of the counts attached to the immediate successors of node v equals its own value p(v). As we can easily see in Figure 4.3, equation (4.3) is satisfied at every node in this representation k In addition, based on the counts of the terminal edges, we may easily compute the probability of occurrence for a particular vector in the sequence. For instance, the probability of occurrence for string 1. This is actually similar to KirchofF’s law for cuirents. 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ‘lOOr is 3/8 (because the count on the terminal edge that corresponds to ‘I(X)r is 3 and the length of the sequence is 8) while the occurence probability of the string ‘1111’ is zero, ‘1111’ being a ‘forbidden’ vector for this particular sequence. For power estimation purposes, it follows that not only a particular vector v ,- in a given sequence is important, but also its relative position in that sequence matters. More precisely, different permutations of vectors belonging to the same initial set (Vj, V 2 ,— , v„), define completely different input sequences. Coming back to the structure just introduced, we observe that D M Tq alone cannot capture this property; we say that D M Tq has no memory and therefore the relative order of vectors in the initial sequence is irrelevant in the construction of D M Tq. In Figure 4.3b for instance, the value of 3/8 is the probability that we find the particular string (state) ‘1001’ in the original sequence, but this gives us no indication about the sequencing of this vector relative to another one, say 0001’. To properly solve the vector compaction problem, we refine now the above structure by incorporating first-order memory effects. Specifically, we consider a more intricate structure, namely a tree called DMTi {Dynamic Markov Tree o f order 1). Example 4.2 For the same sequence in Example 4.1, suppose we want to construct its corresponding tree DMTj. We begin as in D M Tq and for each leaf that represents a valid combination in the original sequence, we construct a new tree (having the same depth as 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D M Tq) which is meant to preserve the co n text in which the next combination occurs (Figure 4.4). (a) (b) Figure 4.4 Construction of the tree OMT^ For instance, the vector V 2 = 0001 follows immediately after v; = 0000; consequently when we reach the node that corresponds to vj (the leftmost path in Figure 4.4a), instead of going back to the root and therefore ‘forgetting’ the context, we start building a new tree (rooted at the current leave of DM Tq) as indicated in Figure 4.4a. The newly constructed tree will preserve the context in which V 2 = 0001 occurred that is, immediately after vj = 0000 (this is denoted by V j — > V 2 ). After processing the pair (vi,V 2 ), we come back to the root and continue with the pair (v2 ,vg) as shown in Figure 4.4b. In fact, all vectors except the first and the last are processed exactly twice, once in the upper D M Tq and next in the lower subtree. What is important to note here is that a ll vector pairs in the original sequence are processed, that is, none of them is skipped during the construction of DMT^. This is the theoretical basis for accurate modeling of the input 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sequences as first-order Markov sources of information. Similarly, continuing this process for all leaves in D M Tq, we end up by building the whole tree D M T i as shown in Figure 4.5. upper subtree lower subtree Figure 4.5 The tree D M Ti for the sequence in Example 4.1 In Figure 4.5, the upper subtree (levels 1 to 4) represents D M Tq, that is, it sets up the state probabilities for the sequence; the lower subtrees (levels 5 to 8), give the actual sequencing between any two successive vectors. To keep the counts in these subtrees consistent, while we traverse the lower subtrees and update the counts on their edges, we also accordingly increment the counts on the paths in the upper subtree. In practice the counts of these two subtrees may differ by one (as it can be seen on the rightmost path at the border between the upper and lower subtree in Figure 4.5), due to the finite length of the sequences. A practical solution to this issue is to consider the input sequence as being cyclic. Obviously, DMT^ provides more information than D M Tq. To give an example, the string ‘lOOr can follow only after ‘0001’ or ‘1100’, information that cannot be gathered 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. by analyzing D M Tq alone. We also note that counts on the edges allow a quick calculation of the transitions probabilities that characterize any particular sequence. To this end, we use the following result: Proposition 4.2 [35] We write the probability of a vector string v = viV 2 ...v„ as follows: p(v) = p(V|) ■ p(v2 |vi) • ... - p(v„jv,V2 ...v„_,) (4.4) where the conditional probabilities are uniquely defined by: p{tiv) = p{yx) / p(v). For example, if we want to calculate the transition probability ‘1001’ ‘1100’ we have p(v) = pfviVj) = p(v,) • pCvjjvj) = 3 /8 , which is exactly the count on the path ‘ 10011100’ in the tree DMT% divided by the sequence length. Theorem 4.1 Any sequence in S can be modeled as a first-order Markov source using the tree DMT\ and the probability function p. We call this process Dynamic Markov Chain (DMC) modeling. Proof: If V = V j v^is a string in the structure DAfTj such that vj is in the upper tree and V 2 is in the lower tree, then p (v2 \v\)=p (v) / p (vi). Thus, the parameters stored on the edges of DMTi structure provide the conditional probabilities that characterize the lag-one Markov chain for the sequence in S. ■ Theorem 4.2 The structure DMTi parameters p are equivalent to a stochastic sequential machine. Sketch o f proof: DMTi tlefines a Markov source (based on Theorem 4.1). Any Markov source is characterized by a stochastic matrix. According to the decomposition Theorem 1 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. given in [26], this matrix is uniquely associated to a stochastic machine (a finite-state machine with randomly generated inputs). Thus, DMTi equivalent to a SSM. ■ Generally speaking, the theory of stochastic sequential machines is far more developed than the theory of DMC modeling. However, the DMC modeling technique based on DMTi seems to be more effective as it offers a much more compact structure and generally outperforms the compaction techniques based on stochastic machines. The structure DMTi just introduced is general enough to capture completely spatial correlations and first-order temporal correlations. Indeed, the recursive construction of DMTI by considering successive bits in the upper and lower subtrees completely captures the word-level (spatial) correlations for each individual input vector in the original sequence. Furthermore, cascading lower subtrees for each path in the upper subtree, gives the actual sequencing (temporal correlation) between two successive input patterns. 4.4.2 A practical procedure A practical procedure to construct DMTi und generate the compacted sequence is given in Figure 4.6a. During a one-pass traversal of the original sequence (when we extract the bit- level statistics of each individual vector vi,V 2 ...,v„ and also those statistics that correspond to pairs of consecutive vectors (viV 2 > , (v2 V 3 ),...,(v„.2 V„.i),(v„.iv„)), we grow simultaneously the tree DMTi- We continue to grow DMTj as long as the number of nodes in the Markov model is smaller than a user-specified threshold {model_size), otherwise we just generate the new sequence up to that point and discard (flush) the model. A new 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Markov model is started again and the process is continued up to the end of the original sequence. procedure DMC {input JUe, ratio, madel_fize) { imtial_ftate = new_state 0 ; symbol = tead_input {inputJUe)-, update_tree {symbol upper_free. initial_ftatey, c r tjta te = last_^tate {symbol upperJree)-, while (!EOF {input JUe)) { sym bol = read.input {inputJUe)-, If {number_of_ftates < m odeljize) ( update_tree {symbol lowerjrees, crt_ftate); update_tiee {symbol upper_tree, initialjtate); crt_ftate = Iast_state {symbol upper_tree); } else { genenile_seq {upper_tree. lowerjtrees, ratio); flush_model {upper_tree. lowerjrees); in itia ljta te = Dcw_statt (); symbol = lead.input {inputJUe); update_tree {symbol upperjtree, initialjtate); c rtjta x e = last_state {symbol upper_tree); ] } genetate_seq {upper_tree, lowerjrees, ratio); .seq {upperjree. lower jre e s . ratio) { c rtjy m b o l = generate.random 0: lower_tree_node = last_node (upperjtree. crtjym bol); upper_treejiode = root (upperjree); do { for each bit in the current vector { generate * 0’ or ‘ I’ to maxiinize the decrease in absolute error. f t ( O' is generated) { lower_tree_node = left (lowerjreejtode); upperjreejwde = left (upper_tree_node); I eke ( lower_tree_node = right (lowerjreejtode); upperjreejwde = right (upperjreejiode); } \ lower_tree_node = upperjtreejuide; upper_tree_node = root (upperjree); ) while there are still vectors to be generated; (a) (b) Figure 4.6 The vector compaction procedure The generate_seq procedure called by the DMC program is detailed in Figure 4.6b. Each generation phase is driven by the user-specified compaction parameter ratio-, that is, in order to generate a total of L = Lçjratio vectors, we have to keep the same compaction ratio for every dynamically grown Markov model. For generation, we use a modified version of the dynamic weighted selection algorithm [14]. In that approach, a similar structure with DMTq is built; more precisely, a full tree having on the leaves the symbols that need to be generated. The counts on the edges are dynamically changed and the symbols are generated according to their probability distribution. For this, a single random number generator is required in order to divide the interval [0,1] into subintervals that 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. correspond to symbols’ probabilities. At each level, the random number is compared to the left probability; if lower, a zero value is generated; if greater, a one value is generated and the number is decreased by the left probability. We use this strategy only to generate the first vector. After that, to ensure a minimal level of error, we use an error controlling mechanism in a greedy fashion. More precisely, at each level in the lower Markov tree, in order to decide whether a zero or one has to be generated, we compute the transition probabilities for both alternatives and choose the one that minimizes the absolute error accumulated up to that point. Simultaneously, the upper tree is parsed from the root to the leaves, according to the bits generated in the lower subtree. The procedure is then resumed until the needed number of vectors is generated. Example 4.3 Assume that we are given the following 3-bit sequence consisting of 17 non- distinct vectors: ( V i , V 2, V 3, V4, V5, Vg, V7, Vg, Vg, V j i , V 12, V 13, V 14, V 15, V 16, V 1 7 ) = ( 001, 100, 001, 110,111,111,101,110, 011, 000, 101, 001, 100, 000, 110, 110,011); our objective is to compact this sequence with a compaction ratio of 2. We start building the Markov model that characterizes the initial sequence. For clarity, the construction of the tree DMT% is shown in Figure 4.7 and Figure 4.8 for two different scenarios. First, in Figure 4.7, we assume that the parameter model_size is set by the user 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to the value 35; this means that the model can be grown dynamically (without any need for flushing) until this limit is reached. Figure 4.7 DMT; for sequence in Example 4.3 {modeljsize = 35) Once we built the Markov tree in Figure 4.7, we start the procedure generate_seq with parameter ratio = 2 and generate a subset of 8 vectors which best approximate the original sequence. If we assume that x = 0.23 is the first randomly generated number, based on the tree in Figure 4.7, since 0.23 < 7/17 we take the left edge, generate a value 0 and x remains unchanged. At the second level, x = 0.23 < 5/17 so again we generate a ‘0’ and leave x unchanged. Now x = 0.23 > 2/17 so a ‘1’ is generated and x becomes x = 0.23 - 2/17 = 0.11. For the lower subtree rooted at the node denoted by the vector ‘001’ (that is, we parse the upper subtree according to the already generated bits 0, 0, 1), to produce the second vector, we use the error controlling mechanism. Specifically, at node in Figure 4.7, the only choice is to take the right edge, generating a ‘l’. Next, at node A / 2 , the absolute error made for the transition probabilities becomes 12/17-1/81 +11/17-01 = 0.066 if we take the left edge, and 12/17-01 +11/17 - 1/81 = 0.183 if we take the right edge (8 is the length of the sequence to be obtained). The first choice is preferred and therefore a ‘0’ 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is generated. At the last level, at node the decision is quite simple as we have only one descendent. Thus, after the first vector ‘001’, we generate ‘101’ as the second vector. The generation procedure continues for the lower subtree rooted at the node denoted by the vector ‘101’ until the desired length L = L(Jratio (that is, 9 vectors) is achieved. Despite its locality, this decision strategy performs very well in practice; as we will see in the experimental part, the overall level of error is very small in all practical cases. In the second scenario, illustrated in Figure 4.8, the model_size parameter is set by the user as being 30 therefore the tree in Scenario 1 cannot be grown as such because the limit of 30 nodes is reached before the whole sequence is scanned. As a consequence, once we reach this limit (this actually happens immediately after processing the subsequence v^, V 2 ,..., V g ), we stop growing the tree and call generatejseq procedure with parameter ratio = 2 (Figure 4.8a). This will produce a subsequence of 4 vectors which best approximate the first ‘segment’ (vi, V 2 ,..., vg) of the original sequence. After that we flush the model (keeping only the very last processed vector v g ) and start a new Markov tree as shown in Figure 4.8b. When the whole sequence is exhausted, based on this new Markov tree, we generate a new subset of 4 vectors which best approximate the second ‘segment’ ( v g v^Q , vji,..., vyj) of the original sequence. We also note that this strategy does note allow ‘forbidden’ vectors that is, those combinations that did not occur in the original sequence, will not appear in the final 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. compacted sequence either. This is an essential capability needed to avoid 'hang up' states during simulation for power estimation. r ~ f (a) • 0 (b ) Figure 4.8 DMTj for sequence in Example 4.3 {modeljsize = 30) In general, by alternating the generation and flushing phases in the DMC procedure, the complexity of the model can be effectively handled. The issue of accuracy in the context of these repeated flushes is discussed in the subsequent section. 4.43 Practical considerations 4.43.1 Complexity related issues The DMC modeling approach offers the significant advantage of being a one-pass adaptive technique. As a one-pass technique, there is no requirement to save the whole sequence in the on-line computer memory. Starting with an initial empty tree DMTi, while 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the input sequence is scanned incrementally, both the set of states and the transition probabilities change dynamically making this technique highly adaptive. Input sequences having a large number of bits b are very common in practice; the success of DMC models for sequence compaction when b is large is based on two key observations: • The larger the value of b is, the sparser the structure of DM Ti will be. The DMC modeling technique exploits this observation by starting with an initially empty model and dynamically growing (’on-demand’) the Markov tree that characterizes the input sequence. By doing so, one can expect to build much smaller trees than the ones otherwise obtained by using a static model based on an initial full tree. • Biased sequences which usually occurs in practice as candidates for power estimation, contain a relatively small number of distinct patterns which arise in many different contexts in the whole sequence. Therefore, a probabilistic model is ideally suited for modeling them. A natural question still remains: when should the growing process be halted? If it is not halted, there is no bound on the amount of memory needed. On the other side, if it is completely halted we lose the ability to adapt if some characteristics of the source message change. A practical solution is to set a limit on the number of states in the DMC [9], [29]. When this limit is reached, the Markov model is flushed and a new model is started. Although this solution may appear too drastic, in practice it performs very well. The intuition behind this property is the capability of DMC model to adapt very fast to changes that occur while the input is scanned. A less extreme solution to limit model 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. growing is also possible; we can keep a backup buffer that retains the last p vectors emitted by the source and whenever the model should be discarded, we may reuse this information to avoid starting the new model from the scratch. 4.43,2 Accuracy related issues Let us consider first the case when the number of nodes allowed in the model is sufficiently large to process the whole sequence. In this case, we may use two compaction strategies. The first is to monitor the current values of the transition probabilities and compare them with the transition probabilities of the original sequence. When the difference between the two sets of probabilities becomes sufficiently small, the generation process is halted. This way, we are able to satisfy any user specified error level for the transition probabilities. The second strategy is to set the compaction ratio upfront, perform compaction, and then compute the error induced by compaction a posteriori. In this second scenario, the user may define the largest value of the compaction ratio based on the stationarity hypothesis which should be satisfied on any segment of the original sequence that is compacted. More precisely, the shortest subsequence where the stationarity hypothesis must be satisfied limits the highest compaction ratio that can be achieved. No matter what strategy is used, if the initial sequence has the length Lq and the newly generated sequence has the length L<L q, then the outcome of this process is a compacted sequence, equivalent to the initial one as far as total power consumption is concerned; we say that a compaction ratio of r = Lq/L was achieved. lOl Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If the number of cells allowed in the model (model_size parameter in DMC procedure) is too small to allow the processing of the whole sequence, then we must resort to flushing. We should note that in this case the same compaction ratio has to be used for all subsequences and thus its value has to be set upfront. To see how the flushing technique affects the accuracy, suppose that during the building of the Markov model, flushing occurs after the first nj vectors, then after the next ri2 vectors, and so on. If the number of flushes is/, then + rij + ... + rif= n. We have the following result: Theorem 4.3 If the i-th subsequence is approximated with an error less than £ ,-, then the accuracy for the whole sequence is: / E = ( 1 / n ) - ^ n , • e , < m a x ( e ,) (4.5) I = 1 Proof: We have that: / / n. . X " t ■ P K V m - l) Z 7 ■ - 1 ) X % - |) = — ----- :----------- andp*(x„x„_,) = ^ n r where r is the compaction ratio and pi (p*) is the probability distribution in subsequence i that corresponds to the original (compacted) sequence. If the i-th subsequence is approximated with an error less than e,-, then the accuracy for the whole sequence is 8 = i= I 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Therefore, as long as the models for partial DMCs accurately capture the transition probabilities for the initial subsequences, the transition probabilities for the entire sequence are preserved up to some e. However, the non-homogeneous sequences that may arise in practice (e.g. sequences with bi-modal distribution) can have very different transition probabilities for each subsequence. In such cases, if flushing is done properly so as to distinguish between subsequences with different transition behavior, then the overall accuracy can be significantly improved. This is a very interesting problem and it will be investigated later in this chapter. 4.4.4 Experimental results The overall strategy is depicted in Figure 4.9. Starting with an t-bit input sequence of length n, we perform a one-pass traversal of the original sequence and simultaneously build the basic tree DMTj; during this process, the frequency counts on edges of DMT^ are dynamically updated. Initial sequence of length Lq Compacted sequence of length L«Lq Comparison Generate compacted sequence One-step DMC modeling; build DMT^, dynamically update counts on the edges Gate-level logic simulation total power estimation Gate-level logic simulation total power estimation Figure 4.9 Experimental setup 1 0 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The next step in Figure 4.9 does the actual generation of the output sequence. If the initial sequence has the length Lq and the new generated sequence has the length L, then we say that a compaction ratio of r = L(/L was achieved. Finally, a validation step is included in the strategy; for short sequences we used the commercial tool PowerMill [18] whilst for long sequences we resorted to an in-house gate-level logic simulator developed under SIS. In Table 4.1 and Table 4.2, we provide the real-delay results for two types of initial sequences. Sequences of type 1 are large input streams having the same initial length Lq =100,000 and being then prime candidates for compaction; type 1 refers to biased sequences obtained by doing bit-level logical operations on ordinary pseudorandom sequences. The sequences of type 2 (having the length of 4,(X X ) vectors) are highly biased sequences obtained from real industry applications. As shown in Table 4.1, sequences of type 1 were compacted with two different compaction ratios (namely r = 50 and 100); we give in this table the total power dissipation measured for the initial sequence (colunm 3) and for the compacted sequence (columns 4, 5). In the last column, we give the time in seconds (on a Sparc 20 workstation with 64 Mbytes of memory) necessary to read and compress data with DMC modeling. Since the compaction with DMC modeling is linear in the number of nodes in the structure DMTi, the values reported in the last column are far less than the actual time needed to simulate the whole sequence. During these experiments, the number of states allowed in the Markov model was 20,000 (about 560 Kbytes). 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.1. Total Power (jiW@20MHz) for sequences of type 1 Circuit No.of Inputs Power for initial $eq. Estimated power for r = 50 Estimated power for r= 100 Time for DMC (sec) C432 36 1816.32 1838.89 1779.60 42 C499 41 3697.84 3546.65 3622.26 48 C880 60 3314.07 3229.85 3329.31 75 C1355 41 3205.27 3044.20 3109.18 48 C3540 50 10876.22 9910.08 10687.32 61 C6288 32 110038.69 114199.50 109077.42 37 s344 9 751.58 748.54 719.53 10 s386 7 818.11 844.58 848.80 8 $838 34 1052.05 1061.73 1091.14 41 $1196 14 3687.47 3702.32 3580.63 16 $9234 36 9192.75 9157.31 9209.75 43 Avg.% enor 2.80 2.93 As we can see, the quality of results is very good even when the length of the initial sequence is reduced by 2 orders of magnitude. Thus, for C432 in Table 4.1, instead of simulating 100,000 vectors with an exact power of 1816.32 nW, one can use only 2000 vectors with an estimate of 1838.89 jiW or just 1000 vectors with a power consumption estimated as 1779.60 pW. This reduction in the sequence length has a significant impact on speeding-up the simulative approaches where the running time is proportional to the length of the sequence which must be simulated. The sequences of type 2 were compacted for two compaction ratios (r = 5 and r =10) using PowerMill; to asses the potential of efficiency of the approach, for both original and compacted sequences, we report also the actual running time required by PowerMill to provide power estimates. The number of nodes allowed for the Markov model 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. construction, was 5,000 (about 140 Kbytes); the CPU time for DMC modeling was below 3 seconds in all cases. As it can be seen in Table 4.2, the average relative error is below 5% while the speed up in power estimation is about one order of magnitude on average. For example, using the original sequence of 4000 vectors, PowerMill takes about 1186 seconds to estimate a total current of 0.4135 mA for C432. On the other side, using the sequence generated with DMC of only 400 vectors (r = 10), PowerMill can estimate a total current o f0.4404 mA in only 1 2 0 seconds. Table 4.2. Total Current (mA) for sequences of type 2 Initial sequence Compacted sequence Circuit No. of Inputs Current (mA) Time to simulate (sec) Current (mA) r = 5 Current (mA) r= 10 Time to simulate (sec) r= 10 C432 36 0.4135 1186 0.4352 0.4404 120 C499 41 0.8188 2675 0.8337 0.8290 235 C880 60 0.7907 2289 0.8324 0.8023 274 CI355 41 1.1375 2993 1.1549 1.1461 284 C1908 33 1.2976 4034 1.2821 1.2833 367 C3540 50 3.4490 9467 4.0500 3.8580 1082 C6288 32 14.5749 88032 14.8020 15.9315 5005 Avg. % err 4.85 4.80 Finally, we compare our results with simple random sampling of vector pairs from the original sequences. In simple random sampling, we performed 1,000 simulation runs with 0.99 confidence level and 5% error level on each circuitL We report in Table 4.3 the maximum and average number of vector pairs needed for total power values to converge I. This means that the probability of having a relative error larger than 5% is only 1% . 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [37]. We also indicate the percentage of error violations for total power values, using the thresholds of 5%, 6% and 10%. Using different seeds for the random number generator (and therefore choosing different initial states in the sequence generation phase), we ran a set of 1,000 experiments for the DMC technique. In Table 4.4, we give the DMC results for the same thresholds as those used in simple random sampling. Table 4.3. Results for Simple Random Sampling Number of vector pairs Error violations Circ. Max. Avg. >5% >6% >10% C432 3300 2176 1.1 0.7 0.4 C499 1500 862 1.4 1.3 0.2 C880 3990 2705 1.8 0.4 0.7 C1355 1380 814 1.7 1.0 0.2 C1908 1620 846 1.9 1.3 0.2 C3540 2340 1446 2.0 1.3 0.4 C6288 7470 5422 1.4 1.4 0.3 Table 4.4. Results for DMC Approach Error violations Circ. No. of vect. >5% >6% >10% C432 2000 6.7 1.9 0.0 C499 800 0.3 0.0 0.0 C880 2000 1.4 0.1 0.0 C1355 800 0.2 0.0 0.0 C1908 800 1.9 1.2 0.0 C3540 1000 0.9 0.0 0.0 C6288 2000 0.0 0.0 0.0 Once again, the results obtained with DMC modeling technique score very well and prove the robustness of the present approach. As we can see, using fewer vectors, the accuracy of DMC is higher than the one of simple random sampling in most of the cases. 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5 Hierarchical models In this section, we introduce the hierarchical modeling of Markov chains as a flexible framework for capturing not only complex spatiotemporal correlations, but also the dynamic changes in the sequence characteristics such as different circuit operating modes or varying power distributions. 4.5.1 Nonhomogeneous sequences As the experimental results show, for homogeneous input sequences, the model presented so far performs very well. Large compaction ratios of few orders of magnitude can be obtained without significant loss (less than 5% on average) in the accuracy of power estimates. However, for input sequences that exhibit widely different transition behaviors over time, the overall accuracy can suffer because the probabilistic model used to represent the input sequence is a flat model', that is, it models the average behavior of the input sequence but does not adapt very well to changes in the input characteristics. For real sequences which may contain a mixture of stimuli with very different switching activities, a compaction technique with higher adaptability is obviously needed. To illustrate the significance of this issue, we consider an example. Example 4.4 Let S; be a 5-bit sequence as shown in Figure 4 .10a; next to it, we represent the word-level transition graph that corresponds to this sequence. Each state in this graph corresponds to a distinct pattern in the sequence and each edge represents a valid transition between any two patterns that occur in the sequence; the label of each edge captures the conditional probability of transition from the source node to the destination 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. node. This particular set of inputs behaves like a pseudorandom sequence because any vector is equally likely to be followed by any other remaining pattern. The total number of bit flippings in the whole sequence is 36; then, dividing this value by the sequence length. A s 5 ) - T - 0 we get an average value of 3 transitions per time step, oioio 1 = = OH" ,1010 0.33 0.33 y V 033^0.33 ^III îîîoî I yYXvKV J ^ I S mm (b) Figure 4.10 Two sequences and their corresponding transition graphs In Figure 4.10b we consider another sequence S2 , which is completely deterministic and highly correlated. It has an average value of 1.33 transitions per step, thus producing less activity compared to S;. Suppose that we duplicate 25 times the original sequence Sj and 100 times the sequence S2 , getting two new sequences S, and Sj, respectively. Based on s[ and S j, we construct now a new sequence S* which is formed by concatenating s[ and Sj in the order S, @ 8 3 @ S, ; the transition graph representation of this new ‘macrosequence’ is given in Figure 4.11. 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.11 The transition graph for the composite sequence S* The question now becomes, how will the average power consumption look like as a function of time, when S* is applied to any circuit? Obviously, the circuit now has two very different modes of operation: one where much activity is generated at the primary inputs, and a second one where about one single input bit toggles at every time step. In Figure 4.12 we can see the effect of these two different regimes on average power consumption for benchmark C l7. Starting initially with S, , after 300 time steps the value of average power stabilizes around 1 lOpW; then, when the characteristics of the input sequence change, the power value goes down toward 70p.W and finally, due to the increase of the switching activity at the primary inputs, it comes up towards 90pW. This type of behavior is very common in practice. More precisely, only homogenous input sequences (which contain statistically similar vectors) will exercise the circuit such that the value of average power will converge rapidly. A typical example is a set of pseudorandom vectors where the average power value stabilizes after applying only tens no Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of vectors. However, in practical applications, the stimuli may contain a mixture of vectors, each one very different as far as average switching activity per bit is concerned. Benctunatlc C17 ISOi 100 « I & a I 50 100 200 300 400 Time Slap 500 600 700 800 900 Figure 4.12 Average power dissipation for C 17 A compaction procedure based on random walks in graphs where some pairs of vectors have very small transition probabilities, has the potential drawback of ‘hanging’ into a small subset of states. As a consequence, an erroneous power value can be obtained depending on which component (with low or high activity) is visited more often. The same type of phenomenon is observed for statistical methods when the distribution is very different from a normal one (e.g. bi-modal distributions [11]), or when one selects from the initial sequence only the first few hundred vectors. Thus, in practice, to compact large sequences that contain non-homogeneous power behaviors, a technique with high adaptability is needed. We use hierarchical Markov models to structure the input space into a hierarchy of macro- and micro-states: at the first (high) level in the hierarchy we have a Markov chain 1 1 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of macrostates; at the second (low) level, each macrostate is in turn characterized by a Markov chain for all its constituent microstates. Our primary motivation for this hierarchical structure is to enable a better modeling of the different stochastic levels that are present in sequences that arise in practice. Another important property of such models is to capture the different operating modes of a circuit using the first level in the hierarchical Markov model, thus providing high adaptability to different operating modes of the circuit. After constructing the hierarchy of the input sequence, starting with some macrostate, a compaction procedure with a specified compaction ratio is applied to compact the set of microstates within that macrostate. Next, the control returns to the higher-level in the hierarchy and, based on the conditional probabilities that characterize the Markov chain at this level, a new macrostate is entered and the process repeats until the end of the sequence (last macrostate) is reached. By doing so, we combine the advantages offered by the hierarchical model with the flexibility of DMC modeling. 4JL2 Micra/macro-state modeling Having the transition graph associated to a vector sequence, our task now is to partition the transition graph associated with a vector sequence into subgraphs that correspond to different behaviors (in particular, different power consumptions). To do this, we provide first some useful definitions and results. 1 1 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Définition 4.1 (Weighted transition graph). A weighted transition graph is a directed graph where any edge from state s^ to state sj is labelled with a conditional probability Pÿ= and a weight w ,y * associated with the transition j,- — > sj. Definition 4.2 (Weight of a random walk). The weight of a random walk in a weighted transition graph is given by: ^ ' (4.6) S j .S j where p(s{) is the I’ th component of the state probability vector and w ,y is the weight associated to the transition j,- Sj. Definition 4.3 ((e, 5)-property). A weighted transition graph is said to have the (£, 5)- property if there exists a grouping {Sj, 52,...^^} on the set of states (ft, sj,-., of the transition graph satisfying: • (8-criterion): V s i e 5^, S j e 5/ then p ( s ,ljy ) < e and p (s y ls ,) < e ; • (5-criterion): V Sf^, for any two states Si, sj e s^, sj connected by an edge, 3 such that I - w ,y I < 5. Also, if A : < /, W ^’s are such that W f^ < W ). J^'s are called macrostates whereas Si e are called microstates within the macrostate Sf^. The intuitive reason for the above definition is that the conditional probabilities from any microstate in to another microstate in S[ (,S i^ # Si) are negligible (e-criterion), and all transitions among microstates belonging to the same macrostate have similar weights (5- 1. W e shall see later in this section the meaning of these weights for our particular application. 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. criterion). For instance, in Figure 4.11 microstates ‘10’, ‘21’, ‘26’, ‘29’ form the macrostate s[ (with high activity), while ‘15’, ‘30’, ‘31’ form Sj (with low activity). In general, a particular microstate may appear in more than one macrostate since not only the vector itself, but also the context in which it appears is important (as in Definition 4.3, the weight value for a transition determines whether the microstates belong to the macrostate or not). Therefore, the grouping of states is done such that the transitions are clustered according to their associated weights. From what has been defined so far, it is possible to hierarchically structure the input space. Specifically, instead of considering the input sequence as a flat sequence of vectors we can see it as a structured multi-level discrete stochastic process called Hierarchical Markov Model (HMM). The HMM generalizes the familiar Markov chain concept by making each of its macrostates a stochastic model on its own, i.e. each macrostate is a Markov model as well. For instance, the graph in Figure 4.11 can be represented hierarchically as shown below, where the macrostate s | identifies ± e high activity mode and Sj the low activity one. Macro ' state Si* Macro state S ^ 3Ù \< r Figure 4.13 A two-level hierarchy for the sequence S* 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Note'. It should pointed out that, in general, the high-level Markov chain is not autonomous; that is, some conditional probabilities may be different from 1. For example, if the initial sequence can be structured as: 5j @ S2 @ @ S2 @ Si (having thus three modes of operation), then in the high-level Markov chain we have ^(5 3 1 ^2 ) = P (*S il*S 2 ) = 0.5 (because it is equally likely to go from S2 to either or Si). The initial problem of compacting a flat input sequence becomes now equivalent to that of compacting a hierarchy of subsequences. Since the vectors from each macrostate are gathered using the same 0-criterion, the compaction is done now inside each macrostate. This way, the ‘hang-up’ problem mentioned in Section 4.5.1 is avoided, that is all macrostates are guaranteed to be visited (as their transition probabilities ‘scale-up’ after hierarchization). For instance, in Figure 4.11, the transition probabilities between s[ and S2 equal 0.001 and 0.007 respectively; in the hierarchical organization shown in Figure 4.13, these probabilities become 1. We now present some useful results for HMM characterization. Theorem 4.4 If the state probability of each macrostate and the state probabilities for all microstates within a macrostate are preserved, then the state probability distribution for the initial (flat) sequence is completely captured. □ Proof: Let s,- be any state from the original sequence. Then, its probability is given by: P(f,) = ■ P(St) (4.7) where Pt(^,) denotes the state probability of r,- in macrostate Sj^. ■ 11s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Thus, the state probabilities are the same if they are preserved inside each macrostate and also the probability distribution for macrostates is correctly captured. Theorem 4.5 Having a hierarchical model satisfying the e-criterion, if the transition probabilities of the microstates are preserved within each macrostate and if the state probabilities of the macrostates are correctly captured, then the transition probabilities of the initial sequence are reproduced with an error less than or equal to £. □ Proof: The transition probability between two states j,- and sj can be expressed as: /'(Vy) = X + X Pi^i^jSkSi) = '^Pki^iSj) • p(S^) + S f, Sj € S j 6 SjG Sj + X Pkl(^i^j'^ ■ where p^is^Sj) is the transition probability between microstates 5 ,- and sj inside macrostate Sf^ and p^i(s^sj) denotes the transition probability between those states if they belong to macrostates S[, respectively. Since our assumption is that the hierarchy satisfies the e- criterion, we have that pi^iÇs^Sj) < e and hence: P(SiSj)^'^Pk(^iSj) -piS^) + e- 2 f (% ) = '^PtiSfSj) - p(S^) + E. ■ S ^ S j, 5 / S j Thus, if the macrostate probability distribution and the transition probabilities inside each macrostate are preserved, then the actual transition probabilities are preserved up to some error e. Theorem 4.6 If the hierarchy satisfies the (e, 6)-property (as in Definition 4.3), then the weight of a random walk in the flat model satisfies: 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - W i + e - ^ % X % ) W yy + 6 (4.8) Si€St.SjeS, where p(5jt) is the probability of being in macrostate is the weight associated to macrostate as in Definition 4.3 and piS/^i) = p(Sf^ p(5/ISjt) is the transition probability from macrostate Sjt to macrostate 5/. □ Proof: Let W be the weight of the initial sequence. Then, using Definition 4.2 and Theorem 4.5, we have; W < ^ ^p^is^sj) ■ p(Sfc) + S V / ) • J j.J y .ît Sj.Sj S * . jye Spk^l On the other hand, if S p sy are in the same macrostate 5^ and the hierarchy satisfies the 5 - criterion, then there is some W jt such that [w^j - < 5. Hence, we also have: M ' ^ • P(^k) • ( + 5) + e ■ ^ p ( % ) • or equivalently J ,- . J y S j X j.J y j , e S^,SjS S^k*l W < • W i^ + e- ^ p(Si^Si) ■ W y y + 6 which is exactly the above claim. ■ St ij, Sj SjS S/^.Sje Sj,k*l In other words, a random walk on the HMM preserves up to some error the average weight of the original sequence. The first term in the above sum represents the average weight per macrostate, whereas the second accounts for the weight of transitions among them. This general formulation applies immediately to our problem defined in Section 4.3.1. In fact, if the input sequence is hierarchically structured. Theorem 4.5 guarantees that 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. equation (4.1) is still satisfied. Moreover, Theorem 4.6 guarantees that the average power value is maintained. Practically, this is very important because the hierarchical model has the advantage of being highly adaptive as opposed to a flat processing of the input which does well ‘on average’. 4,53 A Hamming distance-based criterion for microstate grouping In practice, it is hard to determine the weight Wy for each individual transition. Specifically, an exact procedure would require detailed information about the circuit (e.g. capacitive loads, internal structure) and employ a fast simulation procedure to derive the exact power consumption for each pair of vectors from the original sequence. In practice, such an attempt may be unsatisfactory due to the simulative overhead and the requirement to have detailed circuit information. Therefore, we adopt a different, circuit-independent criterion to structure the input space. As suggested in Example 4.4, what we need is an indicator of the level of activity at the primary inputs of the circuit. To this end, we must correctly identify the different stochastic levels (subsequences) in the initial input sequence and then apply the (e,5)-grouping of microstates based on their associated weights. For instance, in Example 4.4, we need to correctly distinguish the two power modes which are present in the initial sequence S*. Once we identify these different subsequences in the initial sequence S*, our compaction procedure works fine regardless of the power consumption values that would arise from the application of these two subsequences to the target circuit. To structure the input space, we propose to use the average Hamming distance between two consecutive vectors because, from our 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. investigations, it is a reliable indicator of the level of activity. However, the framework we provide is open to other, more elaborate, weighting functions. In the particular situation described in Example 4.4 (see Figure 4.13), based on the Hamming distance criterion, the input sequence can be roughly classified into “high activity” and “low activity” macrostates'. While this kind of partitioning into high and low activity modes can always be used, in practice it is better to have a more refined model. For instance, if the set of all possible values for the Hanuning distance is divided in three equally-sized regions that correspond to low, medium and high activity, then we can identify more than two modes of operation. A more refined model might be required in some applications where a large number of operational modes exist (e.g. an initialization mode, an active mode, a standby mode, and a sleep mode). Benchmark C17 2.5 1.5 100 200 700 300 400 Time 500 Time Step 600 800 900 Figure 4.14 Average Hamming distance variation We also note that the average Hanuning distance may not always capture the specific behavior of different groups of bits in the input sequence. For instance, a criterion based 1. If more than 3 out of S bits change, we are in the high activity mode; otherwise we are in the low activity one. 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. on average Hamming distance may not distinguish between the two cases where either the most significant bits (MSB) are more active than the least significant bits (LSB) or vice versa. The two cases, however, induce different power modes (even if the average Hanuning distance is the same). To handle this situation, we refine the high-level Markov model as follows. Instead of having only low and high activity macrostates, we define four macrostates: - low activity LSB & low activity MSB - low activity LSB & high activity MSB - high activity LSB & low activity MSB - high activity LSB & high activity MSB and, in this way, we structure the input space to have a stronger correlation between the behavior of the input sequence and the power modes induced by this sequence in the target circuit. In the current implementation, the user has the freedom to specify the number of groups of bits to be considered, as well as the number of macrostates for each group. Thus, at the expense of a more complex model, we are able to find a better hierarchization of the input sequence which closely follows the power modes of the target circuit. To detect the changes in the input sequence, a variable-size sliding window is used to compute the average value of the predictor function (in our case, the Hamming distance). At every time step, the average Hamming distance for all vectors starting with the first and ending with the current vector is computed. Next, we decide whether the behavior of the sequence has changed (that is, we are in the same macrostate or not) by comparing the average Hamming distances at steps p k and {p + \) k, where k is a fixed parameter 1 2 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (window size) and p is an integer. If the difference between these average Hamming distances is larger than the 8 parameter set by the user, then we start with a new macrostate. We note that the size of the chosen window should not be too small (due to the fragmentation, the high-level Markov chain becomes similar to the flat model) or too large (the low-level Markov chain becomes similar to the flat model). However, our experience shows that a window size k of 50-100 vectors works very well in practice. We note that the e-criterion (if satisfied) is already accounted for by this procedure since all macrostates are guaranteed to be visited (due to the scaling of conditional probabilities in the high-level model). This does not result in an incorrect value for the total power consumption, as is the case for the flat model. 4.5.4 A practical procedure We describe now a practical procedure for constructing DMTi and generating the compacted sequence. First, based on average Hamming distance criterion explained in Section 4.5.3, the vectors of the original sequence are assigned to macrostates. During the second traversal of the original sequence (when we extract the bit-level statistics of each individual vector and also those statistics that correspond to pairs of consecutive vectors (V 1V 2 ), (v2 V 3 ),...,(v„.2 V„_i),(v„.iV„)), we grow simultaneously the trees DAfTj inside each macrostate (the low-level of the hierarchy) and also the DMTi tree for the sequence of macrostates (the high-level of the hierarchy). Vectors within each macrostate are sequenced together in the same DMT\. If the input sequence satisfies the (£, 5)-property, the transitions introduced this way do not significantly change the characteristics (average 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. weight and transition probabilities) of the model. We continue to grow the trees at both levels of hierarchy as long as the Markov model is smaller than a user-specified threshold, otherwise we just generate the new sequence up to that point and discard (flush) the model. A new Markov model is started again and the process is continued until the original sequence is completely processed. For the generation phase itself, we still use a modified version of the dynamic weighted selection algorithm as described in Section 4.4.2. In general, by alternating the generation and flush phases in the DMC procedure, the complexity of the model can be effectively handled. The only remaining issue is to determine how many vectors must be generated inside each macrostate before a transition to another macrostate is performed. In general, if a subsequence of length L ,- is assigned to macrostate S[, after compaction with ratio r , it has to be reduced to L/r. We note that inside all macrostates the same compaction ratio should be used, otherwise the composition of the sequence (as far as power consumption is concerned) may be totally different than that of the initial one. On average, each macrostate 5 ,- should be visited (p(5,) • M) times where M is the length of the ‘macrosequence’ (i.e. the length of the initial sequence of macrostates). Thus, each time a macrostate 5 ,- is visited, a number of L /{r ■ p{S^) M ) vectors needs to be generated. Since the compaction is done only at the microstate level, the length of the macrosequence is preserved (the generation procedure stops when M macrostates are obtained). It is also noted that this strategy does not allow “forbidden” vectors which means that those 1 2 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. combinations that did not occur in the original sequence, will not appear in the final compacted sequence either. 4,5,5 Experimental results The overall strategy (which is very similar to that in Figure 4.9) is depicted in Figure 4.15. Initial sequence of length A g C om pacted sequence of length L*Lq Com parison G enerate a com pacted sequence L_____ Gate-level logic simulation; total power estimation Gate-level logic sim ulation; total power estim ation l\vo-step D M C m odeling; build DMT,'s for top and low level models an d update them dynam ically Figure 4.15 The experimental setup for combinational circuits Starting with an input sequence of length Lq, we perform a one-pass traversal of the original sequence to assign microstates to macrostates. Next, we simultaneously build the trees OMT^ for the entire hierarchy (macro- and microstates). During this process, the frequency counts on DAfTj’s edges are dynamically updated. Finally, a validation step is included in the strategy; we resorted to an in-house gate- level logic simulator developed under SIS. The total power consumption of some ISCAS’85 benchmarks has been measured for the initial and the compacted sequences, making it possible to assess the eHectiveness of the compaction procedure. In Table 4.5, we provide the real-delay results for a set of highly biased sequences (of length 4,000) which contains three types of sequences: a high activity sequence, followed by a low activity sequence, and finally by a pseudorandom sequence. As shown in 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.5, the initial sequences were compacted with two different compaction ratios (namely r = 5 and 10); we give in this table the total power dissipation measured for the initial sequence (column 2) and for the compacted sequence (columns 3-6). The time in seconds (on a Sparc 20 with 64 Mbytes of memory) necessary to read and compress data with DMC modeling was below 5 seconds in all cases, for both the flat and the hierarchical models. Since compaction with DMC modeling is linear in the number of nodes in the DMTi structure, this value is far less than the actual time needed to simulate the whole sequence. During these experiments, the number of nodes allowed in the Markov model was 20,000 on average (around 500 Kbytes of memory). The sequences satisfied the e-criterion for e = 0.001, while the parameter 5 in Definition 4.3, was set to 0.05 (# of input bits). (This corresponds to having up to 20 macrostates in the hierarchical model.) Table 4.5. Total power consumption (p.W@20 MHz) Flat model Hierarchical model Giro. Exact power Estimated power for r = 5 Estimated power for r= 10 Estimated power for r — 5 Estimated power for r= 10 C432 1810.02 1491.58 1230.77 1888.42 1906.84 C499 4390.79 5341.74 6126.58 4497.01 4591.46 C880 3788.22 4504.40 2803.14 3851.42 4006.19 C1355 3783.35 3065.71 4333.81 3910.45 3933.25 C1908 6352.07 4565.87 7094.39 6145.49 6493.44 C3540 14471.46 9005.19 3527.65 15056.43 15021.08 C6288 104158.25 81100.59 82652.47 98112.01 96295.36 Avg.% err. 23.64 31.45 3.55 4.74 As we can see in Table 4.5, the quality of results is very good even when the length of the initial sequence is reduced by one order of magnitude. Thus, for C432 in Table 4.5, 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. instead of simulating 4,000 vectors with an exact power of 1810.02pW, one can use only 800 vectors with an estimate of I888.42pW or just 400 vectors with a power consumption estimated as 1906.84|xW. This reduction in the sequence length has a significant impact on speeding-up the simulative approaches where the running time is proportional to the length of the sequence which must be simulated. For instance, the average speed-up obtained for the set of benchmarks and sequences in Table 4.5 with PowerMill was 11. By comparison, if one uses the flat model (i.e. a single OMT^ is built for the whole sequence), for the same benchmarks the relative error in power prediction is more than 20%, on average. The primary reason for this inaccuracy is the lack of adaptability which characterizes the flat model when it is applied to multi-modal sequences. We present a typical case (Figure 4.16), that is the two-level hierarchy for benchmark C6288 in Table 4.5. In this diagram, for each macrostate, we indicate the number of microstates included in it; for instance, the largest macrostate contains 1560 microstates, while the smallest has only 80 microstates. 1/7 1320 2/7 1/3 1560 5/6 4/7 1/3 200h 840 1/6 Figure 4.16 The two-level hierarchy for C6288 As we can see, for this input sequence and the particular values of parameters e and S, we have identified five power modes (macrostates), each one containing a very different 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. number of micro-states. We also note that the conditional probabilities at the highest level of the hierarchy scale-up after hierarchization. Finally, we compare our results generated by HMM with simple random sampling of vector pairs from the original sequences. In simple random sampling, we performed 1,000 simulation runs with 0.99 confidence level and 5% error level on each circuit*. We report in Table 4.6 the maximum and average number of vector pairs needed for total power values to converge. We also indicate the percentage of error violations for total power values, using the thresholds of 5%, 6% and 10%. Using different seeds for the random number generator (and therefore choosing different initial states in the sequence generation phase), we run a set of 1,000 experiments for the HMM technique. In Table 4.7, we give the results obtained with the hierarchical model, for the same thresholds used in simple random sampling. 1. This means that the probability of having a relative error larger than 5% is only 1 % . 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.6. Results obtained for Simple Random Sampling Number of vector pairs Error violations (%) Circuit Max. Avg. >5% >6% >10% C432 3300 2176 1.1 0.7 0.4 C499 1500 862 1.4 1.3 0.2 C880 3990 2705 1.8 0.4 0.7 C1355 1380 814 1.7 1.0 0.2 C1908 1620 846 1.9 1.3 0.2 C3540 2340 1446 2.0 1.3 0.4 C6288 7470 5422 1.4 1.4 0.3 Table 4.7. Results obtained for HMM approach Error violations (%) Circuit Number of vectors >5% >6% >10% C432 800 2.6 0.6 0.0 C499 800 0.1 0.0 0.0 C880 800 2.8 0.9 0.0 Cl 355 800 0.1 0.0 0.0 C1908 800 0.1 0.0 0.0 C3540 1200 1.6 0.3 0.0 C6288 3600 1.5 0.0 0.0 Once again, the results obtained with the HMM modeling technique score very well and prove the robusmess of the present approach. As we can see, using fewer vectors, the accuracy of HMM is higher than the one of simple random sampling in most of the cases. Noteworthy examples are benchmarks C499, C l908, C6288 where less than 60% of the maximal number of vector pairs needed in random sampling for convergence are sufficient for HMM to achieve higher accuracy and confidence levels. 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.6 Summary The objective of this chapter was to address the vector compaction problem from a probabilistic point of view. Based on dynamic Markov Chain modeling, we proposed an original approach to compact an initial sequence into a much shorter equivalent one, which can be used after that with any available simulator to derive power estimates in the target circuit. The mathematical foundation of this approach relies in Markov models; within this framework, a family of dynamic Markov trees is introduced and characterized as an effective and flexible way to model complex spatiotemporal correlations which occur during power estimation. A major improvement was obtained by introducing the hierarchical modeling of Markov chains as a flexible framework for capturing not only complex spatiotemporal correlations, but also the dynamic changes in the sequence characteristics such as different circuit operating modes or varying power distributions. The experimental results show that large compaction ratios of few orders of magnitude can be obtained without significant loss in accuracy in total power estimates. 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Circuit Independent Techniques: The Sequential Case 1 2 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.1 Introduction The approaches presented in Chapter 4 are applicable only to combinational circuits because they consider only first-order temporal effects (i.e. pairs of consecutive vectors) to perform sequence compaction. In the case of FSMs, this is not sufficient for accurate estimation of transition probabilities and power consumption. In this chapter, we present a solution for compacting an initial sequence of length Lq, assumed representative for the data applied to the target circuit, into a much shorter sequence L such that the steady-state and transition probabilities on the signal lines are almost the same. Information about the steady-state and transition probabilities is very important because both of them completely characterize the behavior of the FSM. In particular, they have immediate application in power estimation area because, for sequential circuits, temporal correlations longer than one time step can affect the overall behavior of the FSM and therefore, result in very different power consumptions. This point is illustrated by a simple example. Example 5.1 Let Sj and S2 be two 4-bit sequences of length 26, as shown in Figure 5.1a. These two sequences have exactly the same set of first-order temporal statistics as shown in Figure 5.1b. In this figure, we provide the wordwise transition graph for these two sequences where the topmost bit is the most significant bit (e.g. in Sj, vj = V 2 = ‘ 1’, V 3 = ‘2’,..., V 2 6 = ‘9’). Suppose now that and $ 2 are fed to the benchmark s8 from the mcnc'9l sequential circuits suite. Looking at different internal nodes of the circuit, it can be noted that the total number of transitions made by each node is very different when the 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.5 circuit is simulated with Sj or 8 2 - Moreover, the total power consumption at 20 MHz is 384p,W and 476^tW, respectively, showing a difference of more than 24% even for this short sequence. This raises the natural question, “why does this difference appear?”, despite the fact that Sj and 8 2 have the same 8 TG shown in Figure 5.lb. ‘'26 10100011000001010001100000 O O O IO l 10100000001011010000 S[ 00011000011000001100001100 10110110110111011011011011 10100011000011000001010000 000101lOlOOl10100000001000 S2 00011000011000011000001100 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 (a) Figure 5.1 Two sequences with the same first-order statistics The reason for this different power dissipation is that 8 ^ and 8 2 have a different set of second-order statistics; that is, the sets of triplets (three consecutive patterns) are quite different. For instance, the triplet (1,2,7) in 8 2 does not occur in 8 % ; the same observation applies to the triplet (5,2,3) in 8 2 - The conclusion is that having the same set of one-step transition probabilities does not imply that the sets of second-order or higher-order statistics are identical and, as was just illustrated by this small example, higher order statistics can make a significant difference in F8 M’s total power consumption. The initial problem of compacting an initial input sequence can be now cast in terms of power as follows: can we transform a given input sequence into a much shorter one, such that the new body of data is a good approximation of the initial sequence as far as total power consumption is concerned? In this chapter, we will answer this question by introducing a 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. family of variable-order dynamic Markov models that provide an effective way for accurate modeling of external input sequences that affect the behavior of the target FSM. This chapter is organized as follows: First, we define the vector compaction problem for sequential circuits. After that, assuming the stationarity hypothesis, we present the main results on the effects of finite-order statistics on FSM behavior. Based on the concept of Markovian source of information, in Section 5.4, we present an efficient way to compact composite sequences that is, sequences generated with Markov sources of information having different orders. In Section 5.5, we give some practical considerations and present our experimental results on common sequential benchmarks. Finally, we conclude by summarizing our main contribution. 5.2 A high-order probabilistic model The focus is now turned from the input sequence to the actual target circuit and to the investigation of the effect of input statistics on its transition probabilities (primary inputs and present state lines). As shown in Figure 5.2, we model the ‘tuple’ {input_sequence, target_circuit) by the ‘tuple’ {Markov_chain, FSM), where Markov_chain models the input_sequence and FSM is the sequential machine where the transition probabilities have to be determined. In what follows, x„, will denote the random variables associated to 1 3 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the inputs and state lines of the target sequential machine; p ( x ^ ^ is the probability that, at time step n, the input is x„ and the state is ( input \ sequenceJ target circuit Logic [arkov chain Figure 5.2 The tuple {Markov-Chain, FSM) We are interested in defining the joint probabilities p ( x ^ ^ and p(x^s^n-i^n~0 because, as shown in Figure 5.2., they completely capture the characteristics of the input (primary inputs and present state lines) that feeds the next state and the output logic of the target circuit. 5.2.1 Problem formulation Having defined the joint random variables and \ the vector compaction problem for sequential circuits becomes essentially the vector compaction problem for combinational circuits as presented in Section 4.3.1: given a sequence of length Iq, find another sequence of length L<Lq (consisting of a subset of the initial sequence), such that the average joint transition probability on the primary inputs and present state lines is preserved wordwise. for two consecutive time steps. More formally, the following condition should be satisfied: 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. f l ) - f l ) I ^ ^ where p and p* are the probabilities in the original and compacted sequences, respectively. □ This condition simply requires that the joint transition probability for inputs and states is preserved within a given level of error for two consecutive time steps. We note that, because the vector compaction problem is given in terms of joint probabilities p ( x ^ ^ and the proof of correcmess in Section 4.3.1 remains also valid for the sequential case. Once again, the stationarity condition an the primary inputs and state lines of the FSM is essential for this result. Finally, we point out that in our approach, we do not make any attempt to compact the subsequence in the original sequence that is used for the initialization of the target FSM. More precisely, if the original sequence S consists of two components, one being the initialization sequence of the target FSM (usually 50-100 vectors) and the other being a regular set of vectors that excite the FSM (S^g), then S = Sj„ii @ Sreg- We do consider for compaction only the Sr^g component. (Usually, and thus S^eg is the prime candidate for compaction). This is primarily because we do not want to alter the initializability properties of the FSM and also because we need stationary behavior on the state lines of the FSM to have our compaction procedure work. 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 The effect of finite-order statistics on FSM behavior We will investigate now the conditions that can guarantee that equation (5.1) is satisfied. Under the general assumptions of stationarity and ergodicity [23], the following results can be proved. Theorem 5.1 If the sequence feeding a target sequential circuit has order k, then a lag-/t Markov chain which correctly models the input sequence, also correctly models the k-step conditional probabilities of the primary inputs and internal states, that is: Proof: Since is a lag-A: Markov chain, we can first prove that, for any n > fc+1, the following holds: P(^n^n-k\^n-l^n-2--X„-k) = P(-^n\^n-l^n-2— ^n-k) ' P(.^n-k\^n-l^n-2 --> ^n-k> (5.3) Let I- -Jf»_*•?„_*) be the joint transition probability for inputs and states over k consecutive time steps. Then we have: p(x„S„...X„_kSn.k> = i ^ (5.4) [ 0 otherwise For the first alternative we have: p(x„Sn...Xn_kS„.k) = P ( ^ n ^ n - l - ^ n - k ^ n - k > = P ( ^ n ^ n - k \ ^ n - l - ^ n - k ) ' P ( ^ n - I ' ■- ^ n - k ) (5-5) which becomes using the first relation, P ^ ^ n ^ n - k \ ^ n - l ^ n - 2 ' " ^ n - k ^ ~ P ^ ^ n \ ^ n - l ^ n - 2 " " ^ n - ' P ^ ^ n - k \ ^ n - l ^ n - 2 ' '" ^ n - k ^ ' equivalently. 1 ^ » - 2 1 ^ « - 2 - 1 3 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dividing both sides by p(Xn-\^n-2— ^n-k^n-k)’ and using the fact that p { X n S „ . . . X n _ t 5 n - k ) = I I — ‘/»U„_ j ... . We obtain exactly = P(XnK-i^n-l-^n-k)- For the second alternative, if is not a valid sequence, then p ( X r ^ n ^ n - l ^ n - l ^ n - 2 ^ n - 2 - " ^ n - k ^ n - k ) = 0 th is C O ncludes OUT prO of. ■ We note therefore that, preserving order-t statistics for the inputs implies also that order-fe statistics will be captured for inputs and state lines. As long as the order k correctly models the input sequence, we caimot produce new transitions of the FSM and therefore ‘forbidden’ subsequences of order k. However, by using DMTf^ we may introduce new subsequences of order higher than k r, this does not matter for the functionality o f the FSM as long as we are guaranteed by Theorem 5.1 that the conditional probabilities (and thus steady state probabilities) for inputs and state lines are preserved. In particular, the first-order statistics of the joint probabilities defined in equation (5.1) are implicitly preserved. However, modeling a A:-order source with a lower order model may introduce accumulative inaccuracies. From a practical point of view, ± is means that underestimating a high-order source, it is possible not to correctly preserve even the first- order transition probabilities and thus violate the requirement in equation (5.1). In terms of power consumption, this will adversely affect the quality of the results as we can see from the following example. Example 5.2 Once again the sequences Sj and S2 in Example 5.1 are considered and we assume that they feed the benchmark dkl7. It will be illustrated that indeed, if the input 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sequence has order two, then modeling it as a lag-one Markov Chain will not preserve the first-order joint transition probabilities (primary inputs and internal states) in the target cir cuit. We simulated the benchmark dkl7 (starting with the same initial state 7 9 ') for both sequences and we present in Figure 5.3 (Figure 5.3a is for Sj and Figure 5.3b is for S2 ) the wordwise transition graphs obtained for the signal lines 0.67 0 J 3 Figure 5.3 Two STGs obtained for the signal lines (x^„) of benchmark d kl7 The benchmark dkl7 has 2 primary inputs and 3 flip-flops, therefore, in Figure 5.3, any node is decimally encoded using 5 bits. For instance, the initial state 7 9 ' corresponds to the binary code ‘1001T that is, ‘10’ for primary inputs and ‘O il’ for state lines. Starting from state ‘19’ and applying ‘11’ on the primary inputs, the present state becomes ‘O il’, therefore we enter the node ‘ 11011’ = 27 in Figure 5.3. As we can see, because Sj can be modeled as a first order Markov source, while S2 must be modeled as a second order Markov source, the corresponding transition graphs are quite different. From a practical point of view, this means that if one high-order source is underestimated (for instance, assuming that second- or higher-order temporal correlations are not important), then even the first-order transition probabilities in the target circuit may not be preserved. In terms of 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. power consumption, this adversely affects the quality of the results as shown and discussed in Example 5.1. Corollary 5.1 If the sequence feeding the target circuit has order one, then a lag-one MC will suffice to model correctly the joint transition probabilities of the primary inputs and internal states in the target circuit, that is: (5.6) Proof: Follows from Theorem 5.1 if it = 1. ■ Thus, if is a lag-one Markov chain, then preserving the first-order statistics on the inputs is enough for preserving the joint transition probabilities for inputs and state lines. In other words, if the sequence feeding the target circuit can be accurately modeled as a first-order Markov chain, then a first-order power model can be successfully applied because the whole ‘history’ of the input is limited to only two consecutive time steps. In this particular case, the compaction problem for both FSMs and combinational circuits becomes essentially the same; that is, all that is needed is to efficiently model the input sequence as a first-order Markov model. 5.4 Fixed and variable-order Markov sources 5.4.1 The order of a Markov source The next step is to determine the order of a Markov source, because, as proved in Theorem 5.1, knowing the correct value of k, is essential for FSM analysis. To this end, based on the 1 3 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. probability of finding a ‘block’ of vectors in any sequence in S (denoted by p(v„v„_i...v,)), we introduce the following entropy-like quantities. Definition 5.1 (Block entropy). The block entropy of length n is defined as: ^/i = X ••''l) (5.7) w h e r e n > 1. P u n itio n 5.2 (Conditional entropy). The conditional entropy associated with the addition of a new vector to the left of an already existing block <v„,v„.i,...,vi> is defined as: = (5.8) f o r n > 1 a n d / iq = Definition 5.3 (Source entropy). The entropy o f a source, or the uncertainty per step, is defined as: h = lim — = lim h„ (5.9) n ^oo n n — » o o and often referred to as metric entropy or entropy rate [35]. For stationary and ergodic processes, Khinchin [23] has shown that in equation (5.7) is monotonically increasing, H ^n is monotonically decreasing and the limit in equation (5.9) exists. We can now justify the term ‘conditional’ used for the entropy in equation (5.8). Indeed, we have: I. Indices 1,2,... n designate the sequencing among vectors that occur within a block of length n. 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.10) (5.11) ^n = ^ n + l-^ n = X ‘ Iogp(v„^. , V „ . . . V , ) - - % - V i ) logKv„v„_,...v,) The first term of the above summation can be written as: X /'C''„ + i|v„v„_t...Vi).p(v„v„_i...Vi)[Iogp(v„^,|v„v„_,...v,)+ = -V ,) + Iogp(v„v„_,...v,)] = X /^K+l|''i.''„_,...Vi) p(v„V„_i...Vi)- logp(v„^i|v„v„_i...v,) + + X /^K + i|v„v„_,...v,)p(v„v„_j...Vi)logp(v„v„_,...v,) In turn, the second term in equation (5.11) can be written as: X X + 1 |‘'n^n- 1 ” ''l> = X ^Kv„-i-Vi)-logp(v„v„_i...v,) (".".-I "l) which is exactly the second term in equation (5.10). As a consequence, equation (5.10) reduces to: K = X ^K+i|v„v„_i...Vi)p(v„v„_,...v,)-logp(v„^,|v„v„_i...v,) (5.12) which illustrates the mean uncertainty of a new vector v„+i, given the knowledge of vectors v„, v^.j, v„_ 2 ,..., Vj. Therefore the conditional entropy /i„ is a measure of the predictability for the whole process. Proposition 5.1 For any lag-k Markov chain, it holds that: = Hi^ + (n-k) ( H ^ ^ i- H ^ ) , '^ n > k (5.13) 14 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Proof: We first show that = Vn>k . From equation (5.12) we can deduce that: (y.-* n) which is exactly A f c because ^ "i) Now, we can also write: ^k + n-l - ^n~^n-l From here by summation (after reducing the appropriate terms) we get: (n-k)hi^ = using equation (5.8) we get the above claim. ■ Due to statistical correlations extending over a range of only k iterations, for Markov sources of order k, the conditional uncertainty is decreasing [23] until it eventually reaches its limit value hfor n = k. The memory effects of the source are reflected by this saturation point which represents that value of n when the limit h is reached exactly or with a good approximation. Example 5.3 In Figure 5.4 we have the typical behavior for a lag-one and a lag-two Markov sequences, that have been generated by using a first and a second order recursive 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. relations, respectively. The two sources have the same order-0 and order-1 statistics, but different order- 2 statistics. TV(<calta*lmteriar(lag-i M iik w tau ca Typfcil bahwtar lor ■ lao-2 MMtov nute* 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0 T k n e m W p Figure 5.4 Conditional entropies for a lag-one and a lag-two Markov sequences In both cases, the n-step conditional entropy of the source reaches its limit h after few tens of vectors. In the first case, the limit is A = h^, while in the case of the lag-two Markov source, the limit is given by A 2 as was expected according to the above discussion. Finally, we note that there may exist different subsequences of order k within a given initial sequence. For instance, we may have an initial sequence S which consists of two very different second order subsequences, and 8 % , generated with a second order Markov source. In Figure 5.5, we plot the variation of the conditional entropies as function of time for such a situation. In this case, the block entropy in Definition 5.1 (and implicitly the source entropy in Definition 5.3) is different for these subsequences (despite the fact that both of them exhibit the same type of temporal correlations, that is second order temporal correlations) and this is enough for our approach to make the distinction. 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Typical behivlor lor • compoait» source (lsg-2 lolowed by anptfwr ü q -2) 1.6 1 . 4 0.6 0 .4 200 3 0 0 100 4 0 0 7 0 0 9 0 0 6 0 0 8 0 0 1000 Figure 5.5 Conditional entropies for two lag-two Markov subsequences We observe that, indeed, after time step 500, the conditional entropies H q, hi stabilize to different values for Sj and 8 2 - 5.4.2 Composite sequences In practice, lag-k Markov chains (with fixed k) may not be a good enough approximation for a given sequence. For example, a real sequence can be a mixture of sequences, each generated from a different order Markov source. We call such a sequence a composite sequence to emphasize its non-homogeneous characteristics. When the sequence changes its behavior due to the change in order, the stationarity and convergence hypotheses from equation (5.9) no longer hold. For instance, if we consider two mixed sequences, the first containing a lag-one followed by a lag-two Markov subsequence, and the second a lag-two 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. followed by a lag-one Markov subsequence, the behavior of conditional entropies is the one depicted in Figure 5.6. h O 0 5 100 200 300 400 500 600 700 800 900 1000 T k M M p Figure 5.6 Typical behavior of conditional entropies for composite sequences In the first case, after a sufficiently large number of steps, Aj and A 2 do not remain the same. Actually, in the long run, the limit in equation (5.9) becomes hi, thus detecting an order 2 for the source. In this case, the convergence to Aj is violated and thus the sequence changes its order from 1 to 2. In the second case, after already being stabilized to the stationary values, the conditional entropies hi, A 3 , A 4 ,... tend to increase so that the order of the second half of the sequence can no longer be considered 2 . This discussion provides a starting point for determining the order of subsequences in a dynamic fashion. The only requirement that has to be satisfied is the stationarity of each subsequence and, in this case, the entire sequence is called piecewise stationary. This is the basic hypothesis of all our subsequent experiments. To test the condition of piecewise stationarity, in practice we can use again the metric entropy in conjunction with support from the simulation of the STG of the machine. We also note that, for sequential circuits. 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. we consider piecewise stationary sequences that exhibit only a single power mode. However, for sequences that have multiple power modes, the hierarchization procedure in Chapter 4 can be still applied but, in this case, we have to consider the average Hamming distance on the primary inputs and state lines of the FSM to build the hierarchical model. This implies that knowledge about the behavior and state encoding of the FSM is available to the user. 5.4,3 Variable-order dynamic Markov models From results shown in the previous sections, we need an efficient way to model lag-Æ Markov chains that could characterize the input sequences that feed the target circuit. The structure DMTi (introduced in Chapter 4) is general enough to completely capture the correlations among all bits of the same input vector and also between successive input patterns. Indeed, the recursive construction of DMT\ by considering successive bits in the upper and lower subtrees completely capture the word-level (spatial) correlations for each individual input vector in the original sequence. Furthermore, cascading lower subtrees for each path in the upper subtree, gives the actual sequencing (first-order temporal correlation) between successive input patterns. However, conceptually it has no inherent limitation to be further extended to capture temporal dependencies of higher orders. For instance, if we continue to define recursively DMT2 (starting with DMT{), we can 1 4 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. basically capture second-order temporal correlations. For any sequence where v ,-, vp are three consecutive vectors (that is, v ,- vj — > v^), the tree DMT2 looks like in Figure 5.7. Figure 5.7 A second-order dynamic Markov tree The following result, gives the theoretical basis for using the dynamic Markov trees to capture high-order temporal correlations. Theorem 5.2 The general structure DMT^ and its parameters completely capture spatial and temporal correlations of order k. Proof: Let v = vq v^... be a string in DMT^ (the substring v ,- belongs to the i-level tree, DMTi). We have thatp (vjt I v^-i ^k-2- ^0 ) = P (^o W / P (vq vi - and thus the lag- k Markov chain characterizing the input can be fully modeled by the DMTj^ structure. ■ 5.5 Practical considerations and experimental results The DMC modeling approach offers the significant advantage of being a one-pass adaptive technique. As a one-pass technique, there is no requirement to save the whole sequence in the on-line computer memory. Starting with an initial empty tree DMTf^, 1. A T is the maximum possible order of the Markov model of the input sequence. 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. while the input sequence is incrementally scanned, both the set of states and the transition probabilities dynamically change making this technique highly adaptive. Also, by using this data structure, we can easily account for the conditional entropies and detect the order of the Markov source. Under stationarity conditions, the order is detected as the minimum k such that % - < e, for some e > 0 and any n = k + 1,..., K where K is the maximum order of the source to be detected. After that, if either this condition becomes violated or the stationarity hypothesis does not hold, the model is flushed and restarted. As in the combinational case, for each grown tree, the generation phase is driven by the user- specified compaction parameter ratio r. We also note that this strategy does note allow ‘forbidden’ vectors that is, those combinations that did not occur in the original sequence, will not appear in the final compacted sequence either. This is an essential capability needed to avoid ‘hang-up’ (‘forbidden’) states of the sequential circuit during simulation process for power estimation. The overall strategy is shown in Figure 5.8. Initial sequence of length to Compacted sequence of length L*Lq Comparison Gate-level logic simulation; total power estimation G ate-level logic simulation; total pow er estimation Adaptively find the order of the source; generate compacted sequence____ D M C m odeling; build D A fT * and dynamically update counts on the edges Figure 5.8 The experimental setup for sequential circuits We assume that the input data is given in the form of a sequence of binary vectors. Starting with an input sequence of length Lq, we perform a one-pass traversal of the 1 4 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. original sequence and simultaneously build the basic tree £>MT^; during this process, the frequency counts on DMTfrs edges are dynamically updated. During this process, the order of the source is also determined. The generation step is done using a modified version of the weighted selection algorithm described in Chapter 4. Finally, a validation step is included in the strategy; for this purpose, we have used an in-house gate-level logic simulator developed under SIS. The total power consumption of some sequential benchmarks has been measured for the initial and the compacted sequences, making it possible to assess the effectiveness of the compaction procedure. In Table 5.1, we provide the gate-level power simulation results for different initial sequences having the total length of 10,000 vectors; these sequences were produced using a second order information source based on Fibbonacci series. As shown in Table 5.1, the sequences were compacted with two different compaction ratios (namely r = 10 and 20) using three Markov models: one of order zero (that is, assuming temporal independence on the primary inputs), one of order one based on DMT^ and another matching the actual characteristics of the sequence (that is, based on DMT2). This table shows the total power dissipation measured for the initial sequence (column 3) and for the compacted sequence using all models (columns 4-9). Using a Sparc 20 workstation with 64 Mbytes of memory, the time necessary to read and compress data was less than 10 seconds for all models. Since the compaction with DMC modeling is linear in the number of levels in the DMFf^ structure, these time values are far less than the actual time needed to simulate the whole 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sequence. During these experiments, the maximum number of nodes allowed in the Markov model was 200,000. Table 5.1. Total Power (|iW@20MHz) for input sequences of order 2 Estimated power for r= 10 Estimated power for r= 20 Circuit No of inpV FFs Power for initial seq. Order 0 Order 1 Adaptive Order 0 Order 1 Adaptive si 1 9 6 14/18 7025.31 6740.16 6668.31 7027.58 6667.81 6414.11 7006.11 S 1 4 2 3 17/74 5624.64 5197.43 5203.94 5577.11 5068.73 5084.82 5458.20 s5l0 19/6 2073.55 2210.85 1940.11 1980.47 2277.32 1877.72 1974.88 S 5378 35/164 13826.55 11955.22 13304.01 13666.80 11885.03 12838.12 13684.05 s820 18/5 4120.95 3864.65 3668.70 4077.06 3703.84 3561.86 404721 S 9234 36/211 12531.45 13004.71 13037.10 12247.65 13239.04 13070.32 12187.86 s953 16/29 1158.04 1199.03 1122.11 1156.47 1207.44 1102.16 1149.19 A.vg.% err. 6.47 5.84 1.43 8.41 8.24 2.04 bbaia 4/4 747.25 689.54 821.74 745.57 685.09 823.67 745.67 bbtas 2/3 337.74 284.40 362.00 337.43 284.15 323.36 337.39 dkl7 2/3 1439.65 1203.75 1306.32 1437.21 1203.56 1286.46 1436.34 donfile 2/5 3020.63 2242.27 2874.66 3016.54 2207.00 2852.06 3010.74 me 3/2 295.99 234.90 238.07 293.32 219.47 246.30 292.79 planet 7/6 8552.95 7314.99 6505.96 8442.30 7105.18 6565.44 8151.24 shiftreg 1 /3 144.60 91.43 124.27 144.49 91.27 122.56 144.63 Avg.% err. 19.65 12.69 0.41 21.03 1229 0.95 As we can see, for the model having the order 2, the quality of results is very good even when the length of the initial sequence is reduced by more than one order of magnitude. Thus, for si 196 in Table 5.1, instead of simulating 10,000 vectors with an exact power of 7025.31 jiW, one can use only 1,000 vectors (r = 10) with an estimate of 7027.58 pW or just 500 vectors (r = 20) with power consumption of 7006.11 pW. This reduction in the sequence length has a significant impact on speeding-up the simulative approaches where the running time is proportional to the length of the sequence which must be simulated. On the other hand, using a zeroth- or first-order model, the quality of the results can be seriously impaired. For instance, in the case of benchmark planet, for r 149 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. = 20, we can erroneously predict a total power of 6505.96p.W and, for r = ID, a value of 6565.44p,W for a first-order model (more than 23% error). The value of the error can be even worse for a zero-order model (e.g. benchmark donfile, where the error is more than 25%). This is because for a sequence generated with a second-order source, a model that ignores temporal correlations or considers only pairs of consecutive vectors cannot correctly preserve even the first-order transition probabilities for the primary inputs and state lines (that is, the probabilities V i) i" our notation). Table 5.2 shows the results obtained for composite sequences of length 10,000. These sequences have been generated using different generators and exhibit temporal correlation of various orders (order 2, followed by order 1 and finally, once again order 2). This hybrid character of the sequences makes a significant difference in terms of total power consumption for the analyzed benchmarks. As we can see, compared to Table 5.1, the most dramatic increase in the level of error occurs for the zero-order model; in this case, the temporal independence assumption is seriously impairing the estimation accuracy. The error for the first-order model is, on average, around 10%, while the adaptive modeling technique provide accurate results, even for a compaction ratio of r = 20. ISO Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 5.2. Total Power (p.W@20MHz) for composite input sequences Estimated power for r= 1 0 Estimated power for r = 2 0 Circuit No. of inpV FFs Power for initial seq. Order 0 Order 1 Adaptive Order 0 Order 1 Adaptive si 196 14/18 5272.36 7212.58 5395.73 5237.65 7245.41 5501.05 4972.52 S1423 17/74 3964.48 5771.69 417321 4048.88 5836.40 4046.63 3834.35 s510 19/6 1520.46 2232.08 1814.65 1527.91 2215.03 1953.39 1501.10 S5378 35/164 11566.16 14251.17 11711.97 11282.86 14325.13 11871.03 10990.86 s820 18/5 3131.51 4158.82 3368.83 3096.87 4178.75 3431.04 3196.87 S9234 36/211 10046.05 1279724 9688.30 9573.48 12951.02 9672.07 9288.65 s953 16/29 764.92 1188.94 796.02 755.66 118626 805.89 731.35 A vg.% err. 38.29 6 . 2 0 1.82 38.80 8.04 4.18 bbara 4/4 771.16 790.23 842.87 769.97 806.44 749.96 766.41 bbtas 2/3 405.29 333.21 427.52 403.45 339.82 443.14 402.84 dkl7 2/3 1392.67 1165.08 1258.88 1390.21 1135.61 1248.20 1388.09 donfile 2/5 3203.97 2455.90 3100.94 3197.36 2463.37 2997.86 3190.14 m e 3/2 335.97 289.37 298.76 332.92 295.64 285.17 328.27 planet 7/6 8619.93 7753.93 7234.10 8401.28 7605.67 7097.61 8030.61 shiftreg 1/3 149.59 100.62 140.30 149.29 97.36 139.68 148.97 A vg.% err. 16.65 8.71 0 . 6 6 17.28 9.76 1.65 As for comparing our results with simple random sampling technique for FSM circuits, we are unable to do so for the following reason. Our compaction technique starts with a finite input sequence and a user-specified compaction ratio or error level, and performs compaction by DMC modeling and sequence generation. We must therefore compare our results with statistical sampling techniques which work on finite populations (i.e. an input sequence with fixed length). Although sampling techniques for combinational circuits under a given input sequence have been developed and published in the literature (hence, we could produce results presented in Chapter 4), such techniques for FSM circuits are not known. Note that Monte Carlo Simulation (which is based on a simple random sampling strategy) assumes an infinite population and it synthesizes the ISI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. input sequence used for sampling based on a Markov model of bitwise activities. Hence, it cannot be used for our comparison purpose. Finally, we give in Figure 5.9 the node-by-node analysis of the switching activity for benchmark planet for a compaction ratio of 5. O n k r Z m o d W 100 3 1 O J O S . 1 0.15 AOfOlui* tm ir for MtcMrtg aciM y 025 0 3 5 120 100 3 1 0JO02 0004 0006 Figure 5.9 Node-by-node analysis for benchmark planet Using a lower order model than the actual order of the input sequence can significantly impair the ability of correctly estimating the switching activity on a node-by-node basis. While for the first order model the absolute error^ achieves a maximum value of 0.272 and a mean value of 0.057, it decreases to 0.019 and 0.003 respectively if a second order model is used. These results were typical for the whole set of benchmarks that we have analyzed. Based on these results, we can therefore conclude that for FSMs the adaptive technique is the only one appropriate for correctly modeling the input sequence. In Table 5.3, we provide the gate-level power simulation results for a set of different 1. The absolute error is defined as where is the switching activity obtained using the compacted sequence. 152 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. initial sequences having a length of 200,000 vectors. To generate these results, we use two different strategies. First, we compact the input sequences with three fixed compaction ratios and indicate the estimated values of total power consumption (columns 4-5). In the second scenario (columns 6-7), we consider a fixed threshold of 5% for the mean error of the transition probabilities. We monitor the current values of the transition probabilities and compare them with the transition probabilities in the original sequence. When the difference between the two sets of probabilities becomes sufficiently small, the generation procedure is halted. In this way we are able to satisfy any user specified error level for the transition probabilities. Using a Sparc 20 workstation with 64 Mbytes of memory, the time necessary to read and compact data was less than a few tens of seconds in all cases. Table 5.3. Total Power (|iW@20MHz) for long sequences Estimated power for fixed compaction ratio Fixed error level (5% mean error for transition probabilities) Circuit No of inpV FFs Power for initial seq. r= 10 r= 100 Estimated power Number of vectors required si 196 14/18 5724.28 5723.42 5702.51 5714.14 2242 SI423 17/74 3544.75 3548.86 3548.02 3563.84 8984 s510 19/6 1685.12 1692.89 1307.42 1690.10 8002 S5378 35/164 9161.58 9139.20 9355.99 9169.60 11884 s820 18/5 3609.23 3608.02 3604.97 3616.75 5962 S9234 36/211 9570.30 9569.93 9568.40 9577.91 23922 s953 16/29 817.91 817.60 816.81 817.42 4922 Avg. % err. 0.12 3.62 0.20 bbara 4/4 777.10 777.10 776.29 776.71 1404 bbtas 2/3 422.18 422.19 42226 423.23 84 dkI7 2/3 1381.27 1381.16 1380.02 1359.41 84 donfile 2/5 3250.47 3250.43 3249.96 3184.50 84 me 3/2 346.03 346.00 345.69 335.66 162 planet 7/6 7857.02 7859J7 7880.29 7881.93 2524 shiftreg 1/3 150.86 150.85 150.76 145.93 144 Avg. % err. 0.01 0.10 1.54 As we can see, the average error in total power prediction is below 2% for all 1 5 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. benchmarks while the achieved compaction ratio varies between 8 (s9234) and 90 1196) for large circuits and 80 {planet) and 2380 {bbtas) for small ones. This reduction in the sequence length has a significant impact on speeding-up the simulative power estimation approaches where the running time is proportional to the length of the sequence which must be simulated. 5.6 Summary In this chapter, we addressed the issue of sequence compaction for power estimation in sequential circuits. More precisely, we proposed an original approach to compact an initial sequence into a much shorter equivalent sequence, which can be used with any available simulator to derive power estimates in the target circuit. Our contribution is important in two ways: first, it shows the effects of finite-order statistics of the input sequence on FSMs behavior, second, based on the vector compaction paradigm, it provides an original solution for power estimation problem in FSMs. From the original results provided here under stationarity conditions, two are noteworthy for probabilistic FSM analysis: • If the sequence feeding the target circuit has order k, then a lag-k Markov chain model of the sequence will suffice to model correctly the joint transition probabilities of the primary inputs and internal states in the target circuit. • If the input sequence has order two or higher then modeling it as a lag-one Markov Chain cannot exactly preserve the first-order joint transition probabilities (primary inputs and internal states) in the target circuit. 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In addition to this, we introduced and characterized a family of variable-order dynamic Markov models which provide an effective way for accurate modeling of external input sequences that affect the behavior of FSMs. Results obtained on standard sequential benchmarks, show that using this framework, large compaction ratios can be achieved without significant loss in the accuracy of total and node-by-node power estimates. 1 5 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 Conclusions 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.1 Thesis summary Power dissipation is widely recognized as one of the main concerns in the design process of digital CMOS circuits. In this thesis, we basically focused on average power analysis and produced a set of techniques that can be used to measure the dynamic power consumption in combinational and sequential circuits, at different levels of abstraction. Because we use extensively concepts from the theory of finite-order Markov chains, the overall flavor of our investigations is probabilistic in nature. While Chapter 1 and Chapter 2 provide a general introduction and background information. Chapter 3 to Chapter 5 contain the original contribution of our work. We note that some of the original results presented in Chapter 3 to Chapter 5 may have also application beyond the area of low- power systems. In what follows, we briefly summarize their content. In Chapter 3, we introduced the spatio-temporal model for switching activity analysis in combinational logic modules. Assuming the zero-delay hypothesis, we extended the previous work done on switching activity estimation to explicitly account for complex spatiotemporal correlations which occur at the primary inputs when the target circuit receives data from real application. More precisely, using lag-one Markov Chains and two new concepts - conditional independence and signal isotropy - we provide sufficient conditions for exact analysis of complex dependencies. From a practical point of view, we have shown that the relative error in calculating the switching activity of a logic gate using only pairwise probabilities can be upper-bounded. We also proved that the conditional independence problem is NP-complete and thus, relying on the concept of signal isotropy. 157 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. we proposed approximate techniques with bounded error for estimating the switching activity. Based on the spatio-temporal model, we developed a set of efficient techniques for power estimation in large combinational modules. The whole idea behind efficiency, is to calculate and propagate transition correlation coefficients from the circuit inputs toward the circuit outputs, and hence, eliminate the need for building the global function of the intermediate nodes. As quantitative illustration, these techniques can analyze combinational modules with 10,000 gates in less than 200 CPU sec., yet producing power estimates with less than 10-15% error, on average, compared to circuit-level simulation. The model based on spatio-temporal correlations clearly demonstrates, qualitatively and quantitatively, the importance of being accurate node-by-node (not only for the total power consumption) and identifies potential drawbacks in previous approaches when patterns feeding the inputs become highly correlated. In Chapter 4, we first introduced the paradigm of vector compaction for power estimation and provided different alternatives to solve the vector compaction problem for combinational circuits. The vector compaction paradigm is a very attractive solution for power estimation not only because it reduces the gap between static and dynamic techniques used in power estimation but also because it is practically independent of the actual implementation of the target circuit and it can therefore be used early in the design cycle when the structure of the target circuit has not been determined yet. As the main contribution, assuming stationarity conditions, we formulated and proved the correctness of vector compaction problem. We also introduced and characterized a 1 5 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. family of dynamic Markov trees which can be used as an effective and flexible tool in modeling complex spatiotemporal correlations which occur during power estimation. In the second part of Chapter 4, we introduced the hierarchical modeling of Markov chains as a flexible framework for capturing not only complex spatiotemporal correlations, but also the dynamic changes in the sequence characteristics such as different circuit operating modes or varying power distributions. To this end, we structure the input space into a hierarchy of macro- and micro-states: at the first (high) level in the hierarchy we have a Markov chain of macrostates; at the second (low) level, each macrostate is in turn characterized by a Markov chain for all its constituent microstates. Our primary motivation for constructing this hierarchical structure is to enable a better modeling of the different stochastic levels that are present in sequences that arise in practice. As the experimental results show large compaction ratios of few orders of magnitude can be obtained without significant loss in accuracy in total power estimates. Finally, in Chapter 5, we extended the techniques presented in Chapter 4 to deal with sequential circuits. We first analyzed the effects of finite-order statistics of the input sequence on FSM’s behavior. Second, based on the vector compaction paradigm, we proposed an original solution for power estimation in FSMs. To this end, we introduce and characterize a family of variable-order dynamic Markov models which provide an effective way for accurate modeling of external input sequences that affect the behavior of FSMs. Relying on the concept of block entropy, we also presented a technique for identifying the actual order of variable-order Markov sources of information that is. 159 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sources of information that can be piecewise modeled by Markov chains of different orders. Results obtained on standard sequential benchmarks, show that using this framework, large compaction ratios can be obtained without significant loss in the accuracy of total and node-by-node power estimates. In summary, in this thesis we have presented a set of techniques which offer a certain degree of flexibility and can be used in practice to estimate the power consumption. However, it is the designer’s responsibility to choose the technique which is more suitable for his application and to interpret the results. 6.2 Future Work Although the body of this thesis was devoted only to power estimation in digital CMOS circuits, there is room for new investigations and extensions of the present research. Regarding the analytic model proposed in Chapter 3, the two most important extensions would be to consider sequential circuits and the real-delay hypothesis. Generally speaking, for the next generation of power estimation technique, it would be beneficial to raise the level of abstraction as much as possible. However, accurate techniques for power estimation at gate-level (like the one proposed in Chapter 3) will always have a place in the design flow so the two extensions mentioned above are very important. Regarding the set of techniques proposed in Chapter 4 and Chapter 5, there are a number research directions too. First, the restriction of input space only to binary symbols 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. should be relaxed to also accommodate sequences that allow multiple symbols (e.g. Z) because, normally, such symbols are likely to occur in designers’ specifications. In addition to this, the hierarchical model proposed for combinational circuits can be extended to be applicable to sequential circuits too. This will make the approach much more sensitive to the dynamic changes in the sequence characteristics which may occur in real hardware. Finally, the high-order modeling of the input data presented in Chapter 5 can be applied to steady-state analysis of FSMs. 161 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. References [1] A. Ghosh, S. Devadas, K. Keutzer, and J. White, ‘Estimation of Average Switching Activity in Combinational and Sequential Circuits,’ in Proc. ACM/IEEE Design Automation Conference, pp. 253-259, June 1992. [2] P. H. Bardell, W. H. McAnney, and J. Savir, ‘Built-in Test for VLSI: Pseudorandom Techniques,’ John Wiley & Sons, 1987. [3] T. Bell, J. Cleary and I. Witten, ‘Text Compression,’ Prentice-Hall, 1990. [4] L. Benini and G. De Micheli, ‘ Automatic Synthesis of Low-Power Gated-Clock Finite-State Machines,’ in IEEE Trans, on CAD, vol. 15(6), pp. 630-643, June 1996. [5] R. Brayton, G. Hachtel, C. McMullen, and A. Sangiovanni-Vincentelli, ‘ Algorithms for VLSI Logic Synthesis,’ Kluwer Academic Publishers, 1984. [6] R. E. Bryant, ‘Symbolic Boolean Manipulation with Ordered Binary-Decision Diagrams,’ in ACM Computing Surveys, Vol. 24, No. 3, pp. 293-318, Sept. 1992. [7] R. Burch, F. N. Najm, P. Yang, T. Trick, ‘ A Monte Carlo Approach for Power Estimation,’ IEEE Transactions on VLSI Systems, V ol.l(l), pp. 63-71, Mar. 1993. [8] T. L. Chou and K. Roy , ‘Statistical Estimation of Sequential Circuit Activity,’ in Proc. lEEE/ACM Intl. Conference on Computer-Aided Design, pp. 34-37, Nov. 1995. [9] G. V. Cormack and R. N. Horspool, ‘Data Compression Using Dynamic Markov Modeling,’ in Computer Journal, Vol. 30, No. 6, pp. 541-550,1987. [10] S. Devadas, A. Ghosh, and K. Keutzer, ‘Logic Synthesis,’ McGraw-Hill, New York, 1994. [11] C.-S. Ding, C.-T. Hsieh, Q. Wu and M. Pedram, ‘Stratified random sampling for power estimation,’ in Proc. lEEE/ACM Intl. Conf. on Computer-Aided Design, pp. 577- 582, Nov. 1996. [12] S. Ercolani, M. Favalli, M. Damiani, P. Olivo, and B. Ricco, ‘Testability Measures in Pseudorandom Testing,’ in IEEE Trans, on CAD, vol. 11(6), pp. 794-800, June 1992. [13] M. R. Garey and D. S. Johnson, ‘Computers and Intractability. A Guide to the Theory of NP-Complemess,’ Freeman, New York, 1979. [14] J. W. Green and K. J. Supowit, ‘Simulated Annealing without Rejected Moves,’ in Digest, of Intl. Conference on Computer Design, pp. 658-663, Oct. 1984. [15] L. H. Goldstein, ‘Controllability/Observability Analysis of Digital Circuits,’ in IEEE Trans. Circuits and Systems, vol. CAS-26, pp. 685-693, Sept. 1979. [16] G. Hachtel and F. Soraenzi, ‘Logic Synthesis and Verification Algorithms,’ Kluwer Academic Publishers, Boston. 1996. 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [17] J. Hartmanis and H. Steams, ‘ Algebraic Stracture Theory of Sequential Machines,* Prentice-Hall, 1966. [18] C. X. Huang, B. Zhang, A.-C. Deng, and B. Swirski, ‘The Design and Implementation of PowerMill,’ in Proc. Intl. Symp. on Low-Power Design, pp. 105-110, April 1995. [19] C. M. Huizer, ‘Power Dissipation Analysis of CMOS VLSI Circuits by means of Switch-level Simulation,' in IEEE European Solid State Circuits Conference, pp. 61-64, 1990. [20] M. losifescu, ‘Finite Markov Processes and Their Applications,’ John Wiley & Sons, 1980. [21] B. Kapoor, ‘Improving the Accuracy of Circuit Activity Measurement,’ in Proc. ACM/IEEE Design Automation Conference, pp. 734-739, June 1994. [22] G. Kemeny, J. L. Snell, and A. W. Knapp, ‘Denumerable Markov Chains,’ D. Van Nostrand Company, Inc., 1966. [23] A. I. Khinchin, ‘Mathematical Foundations of Information Theory,’ Dover Publisher, New York, 1957. [24] Z. Kohavi, ‘Switching and Finite Automata Theory,’ McGraw-Hill, New York, second edition, 1978. [25] E. Macii, M. Pedram, and F. Somenzi, ‘High-Level Power Modeling, Estimation and Optimization,’ in Proc. ACM/IEEE Design Automation Conference, pp. 504-511, June 1997. [26] D. Marculescu, R. Marculescu, and M. Pedram, ‘Stochastic Sequential Machine Synthesis Targeting Constrained Sequence Generation,’ in Proc. ACM/IEEE Design Automation Conference, pp. 696-701, June 1996. [27] R. Marculescu, D. Marculescu, and M. Pedram, ‘Switching Activity Analysis Considering Spatiotemporal Correlations,’ in Proc. lEEE/ACM Intl. Conference on Computer-Aided Design, pp. 294-299, Nov. 1994. [28] R. Marculescu, D. Marculescu, and M. Pedram, ‘Efficient Power Estimation for Highly Correlated Input Streams,’ in Proc. ACM/IEEE Design Automation Conference, pp. 628-634, June 1995. [29] R. Marculescu, D. Marculescu, and M. Pedram, ‘Vector Compaction Using Dynamic Markov Models,’ in lEICE Trans, on Fundamentals, Vol. E80-A, No. 10, pp. 1924-1933, Oct. 1997, Japan. [30] G. De Micheli, ‘Synthesis and Optimization of Digital Circuits,’ McGraw-Hill, New York, 1994. 1 6 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [31] J. Monteiro and S. Devadas, ‘Techniques for Power Estimation of Sequential Logic Circuits Under User-Specified Input Sequences and Programs,’ in Proc. Intl. Workshop on Low-Power Design, pp. 33-38, April 1994. [32] R N. Najm, ‘Transition Density: A New Measure of Activity in Digital Circuits,’ IEEE Transactions on CAD, Vol. 12(2), pp. 310-323, Feb. 1993. [33] P. N. Najm, R. Burch, P. Yang, and I. Hajj, Probabilistic Simulation for Reliability Analysis of CMOS VLSI Circuits,’ IEEE Transactions on CAD, Vol. 9(4), pp. 439-450, April 1990. [34] F. N. Najm, S. Goel, and I. Hajj, “ Power Estimation in Sequential Circuits,’ in Proc. ACM/IEEE Design Automation Conference, pp. 635-640, June 1995. [35] A. Papoulis, ‘Probability, Random Variables, and Stochastic Processes,’ McGraw- Hill, 1984. [36] K. Parker and E. J. McCluskey, ‘Probabilistic Treatment of General Combinational Networks,’ in IEEE Trans, on Computers, vol. C-24, pp. 668-670, June 1975. [37] M. Pedram, ‘Power Minimization in IC Design: Principles and Applications,’ in ACM Transactions on Design Automation o f Electronic Systems, vol. 1(1), pp. 1-54, Jan. 1996. [38] J. Rabaey and M. Pedram, Low-Power Design Methodologies,’ Kluwer Academic Publishers, 1996. [39] J. Savir, G. S. Ditlow, and P. H. Bardell, Random Pattern Testability,’ in IEEE Trans, on Computers, vol. C-33(l), pp. 79-90, Jan. 1984. [40] P. Schneider and U. Schlichtmann, ‘Decomposition of Boolean Functions for Low- Power Based on a New Power Estimation Technique,’ in Proc. 1994 Int'l Workshop on Low-Power Design, pp. 123-128, April 1994. [41] E. Seneta, ‘Non-Negative Matrices,’ John Wiley & Sons, New York, 1973. [42] E. M. Sentovich, K. J. Singh, C. Moon, H. Savoj, R. K. Brayton, and A. Sangiovanni- Vincentelli, ‘Sequential Circuit Design Using Synthesis and Optimization,’ in Proc. Inti Conference on Computer Design, pp. 328-333, Oct. 1992. [43] J. Storer, Data Compression: Methods and Theory,’ Computer Science Press, 1988. [44] K. Trivedi, ‘Probability and Statistics with Reliability, (Queueing, and Computer Science Applications,’ Englewood Cliffs, NJ: Prentice-Hall, 1982. [45] C.-Y. Tsui, M. Pedram, and A. M. Despain, ‘Efficient Estimation of Dynamic Power Dissipation Under a Real-Delay Model,’ in Proc. lEEE/ACM Intl. Conference on Computer-Aided Design, pp. 224-228, Nov. 1993. [46] C.-Y. Tsui, J. Monteiro, M. Pedram, S. Devadas, and A. M. Despain, Power Estimation in Sequential Logic Circuits,’ in IEEE Transactions on VLSI Systems, Vol. 3(3), pp. 404-416, Sept. 1995. 164 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [47] N. Weste, K. Eshraghian, ‘Principles of CMOS VLSI Design: A Systems Perspective,’ Addison Wesley Publishing Company, 1988. [48] J. H. Vrilkinson, ‘The Algebraic Eigenvalue Problem,’ Clarendon Press, 1988. 165 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. IMAGE EVALUATION TEST TARGET (Q A -3 ) r/ J / / 1 . 0 l.l C u Z 8 L â Uà l A O 1 .2 5 1 . 4 s H 2 £ 1 . 8 1. 6 150mm «s>> * > / o / /y /A P P L IE D ^ IIVMGE . Inc 1653 East Main Street Rochester. NY 14609 USA . ^ ^ 5 % = Phone: 716/482-0300 Fax: 716/288-5989 0 1993. Applied Image. Inc.. A U Rights Reserved Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly firom the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter &ce, while others may be fi’ om any type of computer printer. 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Asset Metadata

Creator Marculescu, Radu (author)

Core Title Probabilistic analysis of power dissipation in VLSI systems

School Graduate School

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Degree Conferral Date 1998-08

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag computer science,engineering, electronics and electrical,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c17-389545

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Document Type Dissertation

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