UWB polarization measurements in multipath channels

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Content UWB POLARIZATION MEASUREMENTS IN MULTIPATH CHANNELS by SangHyun Chang A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful¯llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) May 2007 Copyright 2007 SangHyun Chang Dedication To everyone I met during my time at USC ii Acknowledgements Firstly, I would like to thank Dr. Robert Scholtz and Mrs. Scholtz for their support and encouragement. Dr. Scholtz, you are the ¯rst man I got to respect in my life. I'd like to give my sincere thank to Dr. Aluizio Prata, Jr. for the truly enjoyable discussion. I am also grateful to other faculty members at USC, Professor Kenneth Alexander, Keith Chugg, Antonio Ortega, Namgoong Won, and Charles Weber, including Jong-Seon No in Korea. My gratitude goes to the sta® of the Communication Science Institute (CSI), Milly Montenegro, Mayumi Thrasher, Gerrielyn Ramos, for their assistance and friendship. And I should thank all colleagues at UltRa Laboratory (our friendship must go forever!) and CSI, especially David K. Lee. Also It is my pleasure to acknowledge all friends from Seoul Science High School and Seoul National Uni- versity. There are two friends who provided me a turning point: Bo-Hyung Han who encouraged me to apply USC, and Joon-Yong Lee who helped me to contact Dr. Scholtz. I'm very much grateful to Bo-Hyung and Joon-Yong. I would also like to thank my best roommate Yong-Jin Cho and my best friend Suk-Hwan Uh. Finally,IwouldliketothankstheArmyResearchO±ceforsupportingmyresearch through the MURI project under under Contract DAAD19-01-1-0477. I love you, Mom and Dad. iii Table Of Contents Dedication ii Acknowledgements iii List Of Tables vi List Of Figures vii Abstract xi 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Dissertation Overview . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Electromagnetic Wave and Antenna: Preliminaries 7 2.1 De¯nition of Polarizations . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Geometrical Receiving Antenna Description . . . . . . . . . . . . . 8 3 UWB Propagation Measurement 12 3.1 Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Receiving Antenna Orientations . . . . . . . . . . . . . . . . . . . . 16 4 Wave Polarization Estimation Process 30 4.1 A UWB Multipath Channel Model . . . . . . . . . . . . . . . . . . 30 4.2 An Electric-Field Estimator . . . . . . . . . . . . . . . . . . . . . . 33 4.3 A Test of the Electric-Field Estimator . . . . . . . . . . . . . . . . 36 4.3.1 Received Waveform Estimation . . . . . . . . . . . . . . . . 36 4.3.2 Result and Discussion . . . . . . . . . . . . . . . . . . . . . 36 4.4 Frequency Selectivity on Wave Polarization . . . . . . . . . . . . . . 39 iv 5 Received Signal Energy Bounds and Distribution 50 5.1 Analytical Received Energy Bounds . . . . . . . . . . . . . . . . . . 50 5.2 Received Energy Analysis for Some Channels . . . . . . . . . . . . . 52 5.3 Empirical Received Energy Distribution . . . . . . . . . . . . . . . 53 5.4 Empirical Received Energy Bounds . . . . . . . . . . . . . . . . . . 58 6 UWB Array Propagation Measurement 62 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.2 Array Propagation Measurement Setup . . . . . . . . . . . . . . . . 63 6.3 A Wave Polarization Estimation Process with a Multipath Decom- position Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.3.1 The Sensor-CLEAN Algorithm and Combining. . . . . . . . 71 6.3.2 An Electric Field Estimator . . . . . . . . . . . . . . . . . . 77 6.4 Application to the Measured Data . . . . . . . . . . . . . . . . . . . 78 6.5 Limitation on the Estimation Process . . . . . . . . . . . . . . . . . 83 7 Conclusions and Future Work 85 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.2.1 Polarization Diversity Channel Measurement . . . . . . . . . 86 Bibliography 88 Appendix A UWB Polarization Measurements in an Anechoic Chamber . . . . . . . . 91 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 A.2 Antenna Characteristics Measurement . . . . . . . . . . . . . . . . 92 A.2.1 The Re°ection Coe±cient . . . . . . . . . . . . . . . . . . . 92 A.2.2 The Frequency Response and the Sensitivity . . . . . . . . . 92 A.3 A Scattered UWB Signal Measurement in an Anechoic Chamber . . 98 A.3.1 Measurement Equipments and Transmitting Antenna . . . . 98 A.3.2 Receiving Antenna . . . . . . . . . . . . . . . . . . . . . . . 100 A.4 Polarization Estimation of a Scattered UWB Signal . . . . . . . . . 106 A.4.1 Scattering Measurement Transfer Function System . . . . . 106 A.4.2 Wave Polarization Estimation . . . . . . . . . . . . . . . . . 108 A.4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . 111 A.4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 v List Of Tables 4.1 Descriptions of matrices in the polarization estimation process. . . . 35 5.1 Statistical information of the received signal energy at location B, C, D, and E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.2 Receivingantennaorientationsformaximizingreceivedsignalenergy at location B, C, D, and E.. . . . . . . . . . . . . . . . . . . . . . . 60 5.3 Receivingantennaorientationsforminimizingreceivedsignalenergy at location B, C, D, and E.. . . . . . . . . . . . . . . . . . . . . . . 61 5.4 Receivingantennaorientationsformaximizingreceivedsignalenergy assuming free space environment at location B, C, D, and E. . . . . 61 A.1 The antenna's coordinate system bases the receiving antenna polar- ization for receiving FD antenna orientations. . . . . . . . . . . . . 103 A.2 Geometrical description of receiving FD antenna orientations. . . . 104 vi List Of Figures 1.1 The physical channel from location A to location B. . . . . . . . . 2 1.2 A physical channel measurement setup from location A to locationB. 2 1.3 Three measurements of a physical channel for di®erent receiving an- tenna orientations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Dipole antenna model for an antenna orientation representation in coordinate systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 The geometry and the dimensions of a folded dipole antenna in pic- tures. The left picture shows the front side of the antenna, and the right picture shows the back side of the antenna. . . . . . . . . . . . 10 3.1 Floor plan of UltRa Labaratory. . . . . . . . . . . . . . . . . . . . . 13 3.2 Measurement setup picture for receiving location B. . . . . . . . . . 14 3.3 Measurement setup picture for receiving location C. . . . . . . . . . 14 3.4 Measurement setup picture for receiving location D. . . . . . . . . . 15 3.5 Measurement setup picture for receiving location E. . . . . . . . . . 15 3.6 The measured transmitted UWB monocycle pulse. . . . . . . . . . . 17 3.7 A block diagram of the measurement setup. . . . . . . . . . . . . . 17 3.8 The magnitude of measured LNA frequency response. . . . . . . . . 18 3.9 Thegeometryofthetransmittingantennaorientationintheabsolute Cartesian coordinate system, where ^ ½ tx = 1 2 [¡1;¡1; p 2] T . . . . . . . 20 3.10 The geometry of an orthogonal receiving antenna orientation in the absolute Cartesian coordinate system, where ^ ½ 1 =a z . . . . . . . . . . 21 vii 3.11 The geometry of an orthogonal receiving antenna orientation in the absolute Cartesian coordinate system, where ^ ½ 2 =-a y . . . . . . . . . 22 3.12 The geometry of an orthogonal receiving antenna orientation in the absolute Cartesian coordinate system, where ^ ½ 3 =-a x . . . . . . . . . 23 3.13 The geometry of the arbitrary receiving antenna orientation in the absolute Cartesian coordinate system, where ^ ½ a = 1 2 [1;1; p 2] T . . . . . 24 3.14 Measured waveforms at receiving location B. . . . . . . . . . . . . . 26 3.15 Measured waveforms at receiving location C. . . . . . . . . . . . . . 27 3.16 Measured waveforms at receiving location D. . . . . . . . . . . . . . 28 3.17 Measured waveforms at receiving location E. . . . . . . . . . . . . . 29 4.1 Estimated and measured waveforms for an arbitrary antenna orien- tation ^ ½ a at location B, C, D and E.. . . . . . . . . . . . . . . . . . 38 4.2 The measured frequency response of the channel at location C. . . . 43 4.3 The measured frequency response of the channel at location E. . . . 44 4.4 The measured impulse response of the channel at location C. . . . . 45 4.5 The measured impulse response of the channel at location E. . . . . 46 4.6 The polarization unit vector E EG (f)=jE EG (f)j at location C. . . . . 47 4.7 The polarization unit vector E EG (f)=jE EG (f)j at location E. . . . . 48 5.1 Received signal energy distributions at location B. . . . . . . . . . . 54 5.2 Received signal energy distributions at location C. . . . . . . . . . . 55 5.3 Received signal energy distributions at location D. . . . . . . . . . . 56 5.4 Received signal energy distributions at location E. . . . . . . . . . . 57 6.1 A diagram of the library building where the propagation measure- ment experiment was performed. . . . . . . . . . . . . . . . . . . . 64 6.2 Measurement site picture from behind the transmitting antenna. . . 65 6.3 Three orthogonal antenna orientation setup pictures. . . . . . . . . 66 6.4 The measured transmitted UWB monocycle pulse. . . . . . . . . . . 67 viii 6.5 Measured amplitude vs. time at the center of the array for the 1 st antenna orientation (top), the 2 nd antenna orientation (center), and the 3 rd antenna orientation (bottom). . . . . . . . . . . . . . . . . . 69 6.6 The °owchart of the UWB polarization estimation process. . . . . . 70 6.7 Theantenna'scoordinatesystemforthe1 st antennaorientationmea- surement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.8 Results of the decomposition algorithm for the 1 st (top), the 2 nd (center), the 3 rd (bottom) antenna orientation measurements. . . . 80 6.9 Measured amplitude vs. time at the center of the array for a spe- ci¯c antenna orientation (top) and the estimated waveform with 13 multipath components by the polarization characterization method (center) and the di®erence between the measured waveform and the estimated waveform (bottom). . . . . . . . . . . . . . . . . . . . . . 82 A.1 Pictures and the geometry of a TEM horn antenna. . . . . . . . . . 93 A.2 The re°ection coe±cients of TEM horn and FD antennas. . . . . . 94 A.3 The magnitudes of TEM horn and FD antenna sensitivities in the boresight direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 A.4 The scattering measurement setup. . . . . . . . . . . . . . . . . . . 99 A.5 The geometry of the scattered impinging wave and a receiving FD antenna model. The solid line shows the virtual sphere and the H- plane of the antenna. The antenna polarization ^ ½ a and the scattered electric ¯eld polarization ^ ½ e are on the small dotted elliptical plane, which is perpendicular to the wave impinging direction ^ k and tan- gential to the sphere for the plane wave. . . . . . . . . . . . . . . . 102 A.6 Measurement setup pictures. . . . . . . . . . . . . . . . . . . . . . . 105 A.7 ThereceivedwaveformfortheverticallypolarizedTEMhornreceiv- ing antenna with the target present. . . . . . . . . . . . . . . . . . . 112 A.8 Received signals with and without the target are plotted overlayed for the vertically polarized TEM horn receiving antenna (top). The target-scattered signal component is plotted (bottom), which is the di®erence between two signals at the top. . . . . . . . . . . . . . . . 113 ix A.9 Received waveforms of the target-scattered signal component with the receiving TEM horn antenna in the TEM measurement, where the antenna was rotated around the boresight direction by 0 ± ,¡45 ± , and¡90 ± , from the original vertically polarized antenna orientation. 115 A.10Measured and estimated waveforms of the target-scattered signal component with the 45 ± -rotated TEM horn receiving antenna (top) and the error between the measured and estimated waveform (bot- tom) in the TEM measurement. . . . . . . . . . . . . . . . . . . . . 116 A.11Received waveforms of the target-scattered signal component with V, H, and R orientations in the FD measurement. . . . . . . . . . . 117 A.12The antenna transfer function (sensitivity) magnitudes for V, H, R, and E orientations in the FD measurement. . . . . . . . . . . . . . 118 A.13Measured and estimated waveforms of the target-scattered signal componentwithEorientationoftheFDreceivingantenna(top)and the error between the measured and estimated waveforms (bottom) in the FD measurement. . . . . . . . . . . . . . . . . . . . . . . . . 119 A.14Measured and estimated waveforms of the calibration measurement setup(top)andtheerrorbetweentheestimatedandmeasuredwave- form (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 x Abstract Ultra-wideband (UWB) radio systems have attracted considerable attention as a candidate for wireless personal area network (WPAN) applications. In order to evaluate the performance of UWB radio systems, a radio link budget calculation is necessary for real channels. An important part in the link budget calculation is related to the e®ect between the receiving antenna and the electric-¯eld wave polarization impinging on the receiving antenna, for example, the variance on a received signal waveform or a received energy with respect to receiving antenna orientations. To characterize the polarization e®ect at the receiver, it is useful to establish the way to measure UWB wave polarization in real channels. In this thesis, UWB polarization measurements in indoor multipath channels are studied for a transmitted UWB impulse signal. To investigate the polarization e®ect, we propose a UWB multipath channel model with a Hertzian dipole an- tenna approximation. Based on the channel model, a wave polarization estimation process is proposed and tested on multipath signal waveforms. Waveforms were measuredforthreeorthogonalreceivingantennaorientationstore°ectpolarization sensitivity. The ¯delity of the estimation process is veri¯ed by comparison between the measured and estimated signal waveforms for a reoriented receiving antenna. With the polarization estimate, analytic received signal energy bounds with respect to receiving antenna orientations are provided in closed form, and the cor- respondingoptimalreceivingantennaorientationischaracterizedforthemaximum xi received energy. For each of the measured channels, the empirical received signal energy distribution is plotted for all possible receiving antenna orientations, and empirical received signal energy bounds are discussed for measured channels. xii Chapter 1 Introduction 1.1 Motivation How should one measure a \physical channel" from a location to another location? In order to measure a wireless physical channel as shown in Figure 1.1, we usually deploy transmitting and receiving antennas at each of transmitting location A and receiving locationB, and measure the channel response over a speci¯ed range with excitationbyatransmittedsignalusingapropersetupofmeasurementinstruments. Hence,themeasuredchannelresponsemightbecharacterizedbyatransferfunction from the transmitting antennaterminal to the receiving antenna terminal. That is, the measured channel response is not a measurement from location A to location B, but a measurement from the transmitting antenna terminal a to the receiving antenna terminal b in Figure 1.2. Figure 1.3 shows three measured channel responses in an indoor environment. Although all three measurements were conducted for the same physical channel, i.e., the same transmitting and receiving locations, each of measurements had dif- ferent receiving antenna orientations. The channel responses were measured by a digitalsamplingoscilloscopeforexcitationbyaUWBtransmittedmonocyclepulse 1 A B a b Figure 1.1: The physical channel from location A to location B. Tx Rx A B a b Figure 1.2: A physical channel measurement setup from location A to location B. with a sub-nanosecond pulse width. As shown in Figure 1.3, it is clear that varia- tion of the receiving antenna orientation can change the received signal waveform signi¯cantly, and a channel response measurement with transmitting and receiving antennasisinsu±cienttomeasurea\physicalchannel". Infact,thevarianceonthe received signal waveform with respect to receiving antenna orientations is related to the e®ect between the receiving antenna and the electric ¯eld wave polarization impinging on the receiving antenna. Therefore, in order to complete a physical channel measurement, we must characterize the polarization e®ect at the receiver. Inthisthesis, basedonthebackgroundofantennatheory, wewillestablishtheway to measure wave polarization in real UWB channels. Continuous wave (CW) time-harmonic ¯eld polarization can be characterized conventionallyaslinearlypolarized,circularlypolarized,orellipticallypolarized[7]. However, when a transmitted UWB signal propagates, especially with penetration and/orre°ectionbyanobject,thetransmittedsignalpolarizationofeachfrequency 2 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.01 0 0.01 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.01 0 0.01 Amplitude (V) 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.01 0 0.01 Time (s) ρ 1 ρ 2 ρ 3 Figure1.3: Threemeasurementsofaphysicalchannelfordi®erentreceivingantenna orientations. component may vary individually over a very wide frequency band. Hence in real channels, it is impossible to describe UWB wave polarization in the way that con- ventionalCWtime-harmonicpolarizationischaracterized. Therefore,itisusefulto characterize UWB wave polarization and reconstruct a polarization measurement procedure. 1.2 Problem Statement The problem statement in this thesis is as follows. ForexcitationbyaUWBtransmittedsignalatatransmittingantennaterminal, measure the impinging electric-¯eld at a receiving location, by using propagation 3 measurements with various receiving antenna orientations, where the electric-¯eld is a three-dimensional vector function of time and frequency. In short, characterize a transfer function which relates the impinging electric ¯eld at the receiving antenna reference point to the transmitted signal generator voltage at the transmitting antenna terminal over the transmitted UWB signal bandwidth. 1.3 Related Work Ultra-wideband (UWB) systems transmit signals with -10 dB bandwidths that are at least 20% of center frequency [12]. Hence, a UWB transmitted signal sounding the propagation channel undergoes distortion over a wide frequency range. In order to characterize a UWB multipath channel, Cramer, et al., estimated time- of-arrival, angle-of-arrival, and waveform shape, based on observations of spatial arraysofUWBreceivedsignalwaveforms[11]. However,antennasinUWBsystems are signi¯cant pulse-shaping ¯lters [31], and the receiving antenna characteristics which include its transfer function, gain pattern, and antenna polarization, were considered to be a part of the multipath propagation channel. Hence, the spatial and temporal model in [11] was insu±cient for designing the receiver for a di®erent antennaorientationandforestimatingthewavepolarizationofaUWBtransmitted signal. For narrow band radios, Li and Compton estimated both the two-dimensional arrivalanglesandthepolarizationsofincomingsignalsusingtheESPRITalgorithm in [28] with a square array of crossed dipoles [20]. Many authors have investigated polarization diversity and the capacity behavior with respect to the polarization of 4 the transmitting and the receiving elements and the distance between the trans- mitting and receiving arrays [2,19,32,34]. For UWB signals, most research about polarization was related to UWB radar applications. Manyauthorsmeasuredimpulsescatteringresponsesanddetermined radar cross sections for di®erent targets [3,4,8,13]. Sarytchev, et al., developed a polarization structure of UWB radar signal using the Hilbert transform [29], and Lorber developed a time domain radar range equation to analyze a ultra-wideband short-pulse radar system [21]. Choudhury derived the transmitted electric-¯eld po- larization of resistively loaded orthogonal dipoles excited by short pulses in free space [9]. Koshelev, et al., developed a vector receiving antenna (VRA) to investi- gatethe¯eldpolarizationstructureofUWBsingle-pathradiationpulses[18],where theVRAconsistsofthreemutuallyperpendiculardipoleswithajointphasecenter. However, the past research results on UWB polarization in radar applications are di±cult to apply to UWB polarization measurements in multipath channels, since the radar research results are mainly focused on a channel relevant to a target. 1.4 Dissertation Overview In this thesis, UWB polarization measurements in indoor multipath channels are studied for a transmitted UWB impulse signal. To characterize the polarization e®ect, the description of electromagnetic wave and antenna characteristics are pre- sented in Chapter 2. In Chapter 3, a UWB polarization measurement setup is described in detail. In Chapter 4, we propose a UWB multipath channel model with a Hertzian dipole antenna approximation. Based on the channel model, a wave polarization estimation process is proposed and tested on multipath signal waveforms. 5 In Chapter 5, with a polarization estimate, analytic received signal energy bounds with respect to receiving antenna orientations are provided in closed form, andthecorrespondingoptimalreceivingantennaorientationischaracterizedforthe maximum received energy. For each of four measured channels, empirical received signal energy distribution is plotted for all possible receiving antenna orientations, and empirical received signal energy bounds are discussed for measured channels. In Chapter 6, UWB polarization measurement with a multipath decomposition process is studied with a 3£3£3 array of propagation measurements using three receivingantennaorientations. Forthemulti-pathchannel,anarraysignalprocess- ingalgorithm(Sensor-CLEANalgorithm)isappliedtothearraymeasurementdata to decompose a received waveform with a dense multi-path pro¯le into its compo- nent single-path signals. The impinging electric ¯eld is estimated at the center of the array by combining the waveform components of individual multipath signals, and the performance of the estimation was evaluated. Concluding remarks and future research directions are given in Chapter 7. 6 Chapter 2 Electromagnetic Wave and Antenna: Preliminaries 2.1 De¯nition of Polarizations To characterize the polarization of scattered UWB signals, it is useful to clarify the de¯nitions of polarization. For the following de¯nitions from IEEE Standard [16], it is assumed that an antenna is a passive linear reciprocal device. ² polarization (of an antenna): In a given direction from the antenna, the polarization of the wave transmitted by the antenna. ² polarization [of a wave (radiated by an antenna in a speci¯ed direction)]: In a speci¯ed direction from an antenna and at a point in its far ¯eld, the polarization of the (locally) plane wave that is used to represent the radiated wave at that point. ² polarization; receiving (of an antenna):Thepolarizationofaplanewave, incident from a given direction and having a given power °ux density, that results in maximum available power at the antenna terminals. 7 ² polarization vector (for a ¯eld vector): A unitary vector that describes the state of polarization of a ¯eld vector at a given point in space. In this thesis, we mainly consider the last two polarization de¯nitions, which are the polarization of a receiving antenna and the polarization of the electric-¯eld. 2.2 Geometrical Receiving Antenna Description Suppose that an absolute (global) Cartesian coordinate system with x, y, and z- coordinate axes is used to specify the position of a receiving antenna reference point in three-dimensional space. The origin of the absolute coordinate system is the receiving antenna reference point. Figure 2.1 shows an example of a dipole antenna model in the absolute Cartesian coordinate system. The orientation of the receiving antenna de¯nes a (local) antenna's coordinate system, which is the standard spherical coordinate system in [15]. The antenna's coordinate system is typically de¯ned with respect to a mechanical reference on the antenna. For example, in Figure 2.2 and [36], the antenna's coordinate systems are represented by x 0 -axis , y 0 -axis, and z 0 -axis for a folded dipole antenna. The x 0 -axis directs to the antenna boresight, the x 0 -z 0 plane is the principal E-plane and the x 0 -y 0 plane is the principal H-plane. For a vertically polarized antenna boresighted on the x- axis, the absolute coordinate axes (x, y, and z-axes) are the same as the antenna's coordinate axes (x 0 , y 0 , and z 0 -axes), respectively. Suppose that a wave impinging direction to the receiving antenna reference point is given by the direction-of-arrival ¡a R , where a (¢) represents a base vector in subscript's coordinate direction. The wave elevation angle µ and the azimuth 8 y ϕ Wave Impinging direction θ y' z x x' z' a θ a R a z' = ρ ρ ρ ρ Figure 2.1: Dipole antenna model for an antenna orientation representation in coordinate systems. 9 Figure2.2: Thegeometryandthedimensionsofafoldeddipoleantennainpictures. The left picture shows the front side of the antenna, and the right picture shows the back side of the antenna. 10 angle Á of arrival (AoA), with respect to the antenna's coordinate system, can be represented as µ = arccos(a R ¢a z 0) Á = arctan ³ a R ¢a y 0 a R ¢a x 0 ´ where a R is the radial basis in the spherical coordinate system, and a x 0, a y 0, and a z 0 are the (local) Cartesian bases of the antenna's coordinate system. The receiving antenna polarization can be represented by ¡a µ , where a µ is a basis in the spherical coordinate system. In general, a µ =(cosÁcosµ)a x 0 +(sinÁcosµ)a y 0¡(sinµ)a z 0;: (2.1) For a dipole receiving antenna, the receiving antenna orientation can be rep- resented by a direction of current on the antenna in transmitting mode ^ ½ = a z 0, as shown in Figure 2.1. The antenna orientation can also be represented by the elevation angle µ ^ ½ and azimuth angle Á ^ ½ with respect to the absolute Cartesian coordinate system, as follows. µ ^ ½ = arccos(^ ½¢a z ) Á ^ ½ = arctanf(^ ½¢a y )=(^ ½¢a x )g: (2.2) 11 Chapter 3 UWB Propagation Measurement 3.1 Measurement Setup UWB polarization measurements were conducted in UltRa Laboratory at Univer- sity of Southern California in Figure 3.1. A folded dipole transmitting antenna in Figure 2.2 and [36] was located at a ¯xed location A in Figure 3.1 for all polar- ization measurements, and wave polarization was sensed by another folded-dipole receiving antenna at four locations B, C, D, and E. Transmitting and receiving antennas each were mounted on styrofoam structures. The height of transmitting and receiving antennas were 1.07 m and 0.97 m, respectively. The ceiling height was about 6.40 m, the storage platform height was 2.74 m, and the instruments rack height was 1.98 m. As shown in Figure 3.2, 3.3, 3.4, and 3.5, locations B and C had line of sight to location A. Location D had line of sight blocked by a radar absorbing material storage box, and location E had line of sight blocked by the storage box and a metal column supporting a storage platform. The dimensions of the storage box were 0.91 m£ 1.83 m£ 1.83 m. Amonocyclepulsewithasub-nanosecondwidthproducedbyanAvetechAVE2- C-5000 monocycle pulse generator excited the transmitting antenna periodically 12 Figure 3.1: Floor plan of UltRa Labaratory. 13 TX@A RX@B Figure 3.2: Measurement setup picture for receiving location B. TX@A RX@C Figure 3.3: Measurement setup picture for receiving location C. 14 TX@A RX@D Figure 3.4: Measurement setup picture for receiving location D. TX@A RX@E Figure 3.5: Measurement setup picture for receiving location E. 15 every 2 ¹s. A 20 dB attenuated direct output measurement of the monocycle pulse generator and its FFT spectrum are shown in Figure 3.6. The input impedance of the monocycle pulse generator is 50 . The e®ective 10 dB bandwidth of the monocycle generator output is from 0.7 to 5.9 GHz. With respect to the re°ection coe±cients of the folded dipole antenna in [36], the bandwidth of these antennas corresponds to the bandwidth of the monocycle pulse in Figure 3.6. As shown in Figure 3.7, a stable clock triggered both the monocycle generator and a Tek- tronix TDS8000 digital sampling oscilloscope (DSO). The clock was generated by a Hewlett Packard HP8110A pulse generator with 0 ns delay, 50% duty cycle, 2 ns leadingedge,thehighvoltage4V,thelowvoltage0V,andtherepetitionfrequency 500 kHz. The trigger level of the DSO was set to 50% level of the trigger signal at the rising edge with a 20 dB attenuator connected to the trigger line. The receiving antenna converts the impinging electromagnetic ¯eld into a volt- age at the receiving antenna terminal. The received voltage was ampli¯ed by a low noise ampli¯er (LNA), whose frequency response gain is about 24 dB from 1 to 4 GHz, as showninFigure 3.8. The output of the LNA wassampled witha sampling rate of 80 GHz. The number of samples was 4000, and the span of the samples was 50 ns. The sampled signal was averaged 256 times to increase the signal-to-noise power ratio. 3.2 Receiving Antenna Orientations For each of four receiving locations, waveform measurements were conducted for three orthogonal antenna orientations. To describe the orientation of an antenna at a speci¯c location, we de¯ne a three-dimensional absolute (global) Cartesian coordinate system. The origin of the absolute coordinate system is at the center of 16 2 2.5 3 3.5 4 x 10 −8 −0.2 −0.1 0 0.1 0.2 Time (s) Amplitude (V) 0 2 4 6 8 x 10 9 −10 −5 0 5 10 Frequency (Hz) Amplitude (dB) Figure 3.6: The measured transmitted UWB monocycle pulse. Transmit Receive Figure 3.7: A block diagram of the measurement setup. 17 0 1 2 3 4 5 6 7 8 x 10 9 20 20.5 21 21.5 22 22.5 23 23.5 24 24.5 25 Frequency (Hz) (dB) Figure 3.8: The magnitude of measured LNA frequency response. 18 the receiving antenna. As shown in Figure 3.1, x-axis points to the south, y-axis pointstotheeast,andz-axispointstothezenith. Theabsolutecoordinatesystemis independent of antenna orientations. Antenna orientation de¯nes a (local) antenna coordinatesystem. Theantennacoordinatesystemistypicallyde¯nedwithrespect to a mechanical reference on the antenna. For example, in Figure 2.2, the antenna coordinatesystemisrepresentedbyx 0 -axis,y 0 -axis,andz 0 -axisforthefoldeddipole antenna. If we approximate the folded dipole antenna by a Hertzian dipole antenna as shown in Figure 2.1, the antenna orientation can be represented by a direction of current on the antenna in transmitting mode ^ ½ = a z 0, where a (¢) represents a base vector in the subscript's coordinate direction 1 . Then, as shown in Figure 3.9, the transmitting antenna orientation is represented by ^ ½ tx = 1 2 [¡1;¡1; p 2] T in the absolute coordinate system at the transmitting antenna location, where the su- perscript (¢) T denotes the transpose. Three orthogonal antenna orientations for polarization measurements are represented by ^ ½ 1 =a z , ^ ½ 2 =-a y , and ^ ½ 3 =-a x , respec- tively, in Figure 3.10, 3.11, and 3.12. For the validation of the estimation process, another waveform was measured with an arbitrary receiving antenna orientation ^ ½ a = 1 2 [1;1; p 2] T in Figure 3.13. The arbitrary receiving antenna orientation ^ ½ a and the transmitting antenna orientation ^ ½ tx are chosen to bring polarization sensitiv- ities of three orthogonal antenna orientations, ^ ½ 1 , ^ ½ 2 , and ^ ½ 3 . For each of four measured channels in Figure 3.14, 3.15, 3.16, and 3.17, three propagation measurements with orthogonal antenna orientations, ^ ½ 1 , ^ ½ 2 , and ^ ½ 3 , are applied to the polarization estimation process in Chapter 4. The ¯delity of the 1 TheHertziandipoleantennaapproximationprovidesanantennapatternsymmetryonthez'- axis. Inthecaseofnon-symmetricantennapatterns,twovectorsarerequiredtode¯nedantenna's orientation. 19 y y' z x x' z' ρ ρ ρ ρ tx Figure 3.9: The geometry of the transmitting antenna orientation in the absolute Cartesian coordinate system, where ^ ½ tx = 1 2 [¡1;¡1; p 2] T . 20 y y' z x x' ρ ρ ρ ρ z' 1 Figure 3.10: The geometry of an orthogonal receiving antenna orientation in the absolute Cartesian coordinate system, where ^ ½ 1 =a z . 21 y y' z x x' ρ ρ ρ ρ z' 2 Figure 3.11: The geometry of an orthogonal receiving antenna orientation in the absolute Cartesian coordinate system, where ^ ½ 2 =-a y . 22 y y' z x x' ρ ρ ρ ρ z' 3 Figure 3.12: The geometry of an orthogonal receiving antenna orientation in the absolute Cartesian coordinate system, where ^ ½ 3 =-a x . 23 y y' z x x' z' ρ ρ ρ ρ a Figure 3.13: The geometry of the arbitrary receiving antenna orientation in the absolute Cartesian coordinate system, where ^ ½ a = 1 2 [1;1; p 2] T . 24 estimation process is veri¯ed by comparison between the measured and estimated waveforms for the arbitrary antenna orientation ^ ½ a . 25 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.05 0 0.05 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.05 0 0.05 Amplitude (V) 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.05 0 0.05 Time (s) ρ 1 ρ 2 ρ 3 (a) Three orthogonal antenna measurements with antenna orientations, ^ ½ 1 , ^ ½ 2 , and ^ ½ 3 . 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.05 0 0.05 Time (s) Amplitude (V) ρ a (b) The arbitrary antenna orientation ^ ½ a measurement. Figure 3.14: Measured waveforms at receiving location B. 26 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.02 0 0.02 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.02 0 0.02 Amplitude (V) 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.02 0 0.02 Time (s) ρ 1 ρ 2 ρ 3 (a) Three orthogonal antenna measurements with antenna orientations, ^ ½ 1 , ^ ½ 2 , and ^ ½ 3 . 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.02 0 0.02 Time (s) Amplitude (V) ρ a (b) The arbitrary antenna orientation ^ ½ a measurement. Figure 3.15: Measured waveforms at receiving location C. 27 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.02 0 0.02 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.02 0 0.02 Amplitude (V) 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.02 0 0.02 Time (s) ρ 1 ρ 2 ρ 3 (a) Three orthogonal antenna measurements with antenna orientations, ^ ½ 1 , ^ ½ 2 , and ^ ½ 3 . 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.02 0 0.02 Time (s) Amplitude (V) ρ a (b) The arbitrary antenna orientation ^ ½ a measurement. Figure 3.16: Measured waveforms at receiving location D. 28 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.01 0 0.01 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.01 0 0.01 Amplitude (V) 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.01 0 0.01 Time (s) ρ 1 ρ 2 ρ 3 (a) Three orthogonal antenna measurements with antenna orientations, ^ ½ 1 , ^ ½ 2 , and ^ ½ 3 . 1.2 1.3 1.4 1.5 1.6 x 10 −7 −0.01 0 0.01 Time (s) Amplitude (V) ρ a (b) The arbitrary antenna orientation ^ ½ a measurement. Figure 3.17: Measured waveforms at receiving location E. 29 Chapter 4 Wave Polarization Estimation Process 4.1 A UWB Multipath Channel Model ForaUWBradiomultipathchannel 1 ,areceivingantennawiththeinputimpedance Z R (f) senses the impinging electric ¯eld E(f;µ;Á) of each path. The impinging electric ¯eld at frequency f is assumed to be a plane wave at the elevation µ and azimuth Á angles of arrival in the antenna coordinate system, as shown in Figure 2.1. The receiving antenna has the vector e®ective height h(f;µ;Á) for the path at the angles of arrival, and the antenna is serially connected to a load with impedance Z L (f). By [24] and [1], a receiver load voltage V L (f) can be represented by superposition of each path's signal with additive Gaussian noise as follows. V L (f)= X n · E n (f;µ n ;Á n )¢h(f;µ n ;Á n ) Z L (f) Z L (f)+Z R (f) ¸ +N(f) (4.1) 1 UWB polarization measurements in a single path channel is studied in Appendix A. 30 where the operator (¢) is vector inner product, n is a path index, and N(f) is receivernoise. Werelatethee®ectiveheighth(f;µ;Á)toaHertziandipoleantenna's e®ective height by [14]. Then, h(f;µ;Á)'¡a µ C 0 sinµ (4.2) where C 0 is an in¯nitesimal conducting wire length of the Hertzian dipole antenna, and a µ is a base vector of a spherical coordinate system representation in the (lo- cal) antenna coordinate system in Figure 2.1. In general, a µ = (cosÁcosµ)a x 0 + (sinÁcosµ)a y 0¡(sinµ)a z 0, and receiving antenna polarization is equal to¡a µ . For eachpath,theimpingingelectric¯eldE(f;µ;Á)isorthogonaltoabasevectora R of the spherical coordinate system because¡a R is the wave propagation direction, as shown in Figure 2.1. From Equation (2.1), a R = (sinµcosÁ)a x 0 +(sinµsinÁ)a y 0 + (cosµ)a z 0. Then,wecansubstituteEquation(4.2)intoEquation(4.1)withanaddi- tionalterma R C 0 cosµ,whichisorthogonaltotheimpingingelectric¯eldE(f;µ;Á), for each path as V L (f)' X n · E n (f;µ n ;Á n )¢ ¡ ¡a µn C 0 sinµ n +a Rn C 0 cosµ n ¢ Z L (f) Z L (f)+Z R (f) ¸ +N(f): (4.3) The base vectora z 0 in the local Cartesian coordinate system can be represented by a linear combination of two base vectors a µ n and a R n as a z 0 =¡a µ n sinµ n +a R n cosµ n (4.4) 31 for all n. We assume that the input impedance of the receiving antenna Z R is constant over the transmitted signal bandwidth 2 , and the load impedance Z L is constant at 50, for example, for the DSO. Then by Equation (4.3) and (4.4), Equation (4.1) can be approximated by V L (f) ' X n · E n (f;µ n ;Á n )¢C(f)^ ½ ¸ +N(f) = · E(f)¢C(f)^ ½ ¸ +N(f) (4.5) where ^ ½ = a z 0, C(f) = Z L (f) Z L (f)+Z R (f) C 0 , and E(f) = P n E n (f). Hence, when we approximate the receiving antenna by the Hertzian dipole antenna, the receiver load voltage is a scaled inner product between the impinging electric ¯eld and the antenna orientation ^ ½. Note that the load voltage is independentof the direction of theimpingingwave. Thisapproximationmaymitigatetheestimationprocesscom- plexityforresolvingeachpathwaveformandcharacterizingtheimpingingdirection of each path waveform in a multipath channel 3 . 2 The folded dipole antenna in [36] has a UWB antenna, whose re°ection coe±cient is below -10 dB for the transmitted signal bandwidth. 3 In [5], a multipath decomposition algorithm was tried to estimate UWB wave polarization by using 3-dimensional (3 £ 3 £ 3) array propagation measurements. In Chapter 6, the UWB polarization array measurement is discussed in detail. 32 4.2 An Electric-Field Estimator For data measured by the DSO in Chapter 3, the time domain representation of the load voltage can be given by v k (t)=s k (t)+n k (t); (4.6) where v k (t) is received signal voltage, s k (t) is noiseless signal voltage, and n k (t) is receiver noise for the k th receiving antenna orientation ^ ½ k (, or k = 1;2, or 3). The variable t is a time index out of 4000 samples, where t 1 = 116:7000 ns < t 2 = 116:7125 ns < ::: < t 4000 = 166:6875 ns, as described in the measurement setup of the DSO. Using Equation (4.5) and (4.6), the equivalent frequency domain repre- sentation of the load voltage can be given in linear time-invariant form by V k (f)=S k (t)+N k (f)=[E(f)¢C(f)^ ½ k ]+N k (f); (4.7) where V k (f), S k (f), and N k (f) are the Fourier transform of v k (t), s k (t), and n k (t), respectively, and S k (f) = [E(f)¢C(f)^ ½ k ]. The variable f is a frequency index out of 4000 samples, where the frequency resolution is 20 MHz and the span is 80 GHz (2-sided). For matrix calculations, we de¯ne a received signal matrix V, a noiseless signal matrix S, a noise matrix N, an electric ¯eld matrix E, an impedance matrix C, and an antenna orientation matrix R in Table I. For example, the element in the 33 2 nd row and the 500 th column of E, [E] (2;500) =E(f 500 )¢a y . 4 Then, Equation (4.7) can be represented by V =S+N=CR T E+N: (4.8) Assuming that the receiver noise n k (t) is white Gaussian and orthogonal to the signal s k (t) for all k in Equation (4.6), the optimal estimate of the electric ¯eld matrix e is given by e E=(CR T ) ¡1 e S (4.9) where e S is the output of a Wiener ¯lter operator for smoothing Gaussian noise in V row by row 5 . Therefore, the estimate of the electric ¯eld E(f) is represented by e E(f i )=E i ,whereE i denotesthei th columnoftheelectric¯eldmatrixE. Notethat weneedatleastthreeantennaorientationmeasurementsfortheestimationprocess, and any three antenna orientations are appropriate only if the antenna orientation matrix R is non-singular. The e®ects of cables and the LNA are ignored in this exposition. 4 For a matrix A, the notation [A] (i;j) denotes the element of A in the i th row and the j th column. 5 By [23], the optimal estimate of S k (f) in Equation (4.7) is e S k (f) = S s k (f) S s k (f)+S n k (f) V k (f)= S v k (f)¡S n k (f) S v k (f) V k (f) = S v k (f)¡¾ 2 n k S v k (f) V k (f) where S(f) denotes power spectral density, and ¾ 2 n k is noise variance. For the smoothing process, the noise variance estimate was given by the average of power spectral density from -40 to -20 GHz and from 20 to 40 GHz. 34 Table 4.1: Descriptions of matrices in the polarization estimation process. description notation size de¯nition received sig. matrix V 3£ 4000 [V] (i;j) =V i (f j ) noiseless sig. matrix S 3£ 4000 [S] (i;j) =S i (f j ) noise matrix N 3£ 4000 [N] (i;j) =N i (f j ) electric ¯eld matrix E c 3£ 4000 E c =C(f)[E x ;E y ;E z ] T impedance matrix C 4000£ 4000 C=diag(C(f 1 );:::;C(f 4000 )) antenna ort. matrix R 3£ 3 R=[^ ½ 1 ;^ ½ 2 ;^ ½ 3 ] 35 4.3 A Test of the Electric-Field Estimator 4.3.1 Received Waveform Estimation Withtheestimateofimpingingelectric¯eld e E(f),areceivedsignalforanarbitrary antenna orientation can be estimated by by using Equation (4.5). The received signal estimate e V(f) can be represented by e V(f)= e E(f)¢C(f)^ ½; (4.10) where ^ ½ is the arbitrary receiving antenna orientation. In this section, the ¯delity of the estimation process is veri¯ed by comparison between the measured and estimated waveforms for the arbitrary antenna orienta- tion ^ ½ a described in Chapter 3. For each of four measured channels in Figure 3.14, 3.15, 3.16, and 3.17, three propagation measurements with orthogonal antenna ori- entations, ^ ½ 1 , ^ ½ 2 , and ^ ½ 3 , are applied to the received waveform estimation for the receiving antenna orientation ^ ½ a . 4.3.2 Result and Discussion InFigure4.1,Measuredandestimatedwaveformsforthearbitraryantennaorienta- tion ^ ½ a areshownforeachoffourchannels. Itisclearthatthewaveformestimation shows a close waveform pattern to the measured waveform, not only for the line of sight channel from location A to location B and C, but also for the blocked lied of sight channel from location A to location D and E. Although the received signal at location E had the smallest dynamic range of the received signal voltage and the distance between the transmitting and receiving antenna was farthest, the 36 waveform estimate has estimated waveform pattern close to the actual measured waveform. Thenormalizedmeansquaredestimationerror, thatis, (kr(t)¡e r(t)k 2 =kr(t)k 2 ), between measured signal waveform r(t) and estimated signal waveforme r(t) at lo- cation B, C, D and E were 0:1034, 0:1262, 0:1916 and 0:2054, respectively, for thearbitraryreceivingantennaorientation ^ ½ a . Estimationerrorsmaybecausedby approximationsandinaccurateexperimentalcon¯gurations. Forexample,threeor- thogonal antenna orientation measurements must be conducted for a ¯xed location of the antenna reference point. However, when we change the antenna orientation, the antenna's location and orientation were calibrated manually and deviations of the location and orientation were unavoidable. For example, location deviation of 0:06 m is about a wavelength error for the wave at 5 GHz, which might cause a large error. 37 1.2 1.25 1.3 1.35 1.4 x 10 −7 −0.05 0 0.05 Time (s) Amplitude (V) meas est (a) Location B. 1.25 1.3 1.35 1.4 1.45 1.5 x 10 −7 −0.02 0 0.02 0.04 Time (s) Amplitude (V) meas est (b) Location C. 1.25 1.3 1.35 1.4 1.45 1.5 x 10 −7 −0.02 0 0.02 Time (s) Amplitude (V) meas est (c) Location D. 1.4 1.45 1.5 1.55 1.6 1.65 x 10 −7 −0.01 0 0.01 Time (s) Amplitude (V) meas est (d) Location E. Figure4.1: Estimatedandmeasuredwaveformsforanarbitraryantennaorientation ^ ½ a at location B, C, D and E. 38 4.4 Frequency Selectivity on Wave Polarization Inprevioussections,apolarizationmeasurementprocedurewasproposedandtested for the transmitted UWB signal, where the impinging wave polarization is a func- tion of frequency. The UWB channel model and the UWB polarization measure- ment procedure can be generalized and applied for any transmitted signal of arbi- trary bandwidth. Then, how wide should the signal bandwidth be in order to require the polar- ization measurement procedure in section 4.2? How much does the transmitted signal polarization of each frequency component vary individually over the fre- quency band? Or, how narrow should the signal bandwidth be to not require the polarization measurement procedure in section 4.2? These questions are related to frequency selectivity of the signal polarization over the signal bandwidth. If the signal polarization of each frequency component is characterized to have approx- imately equal gain and linear phase, the signal polarization can be modeled and measured in narrow-band fashion. Otherwise, the signal polarization should be as a function of frequency. Coherence bandwidth is a statistical measure of the range of frequencies over which the channel can be considered \°at" (i.e., a channel which passes all spectral componentswithapproximatelyequalgainandlinearphase)[26]. Whentheband- widthoftheinputisconsiderablylessthanthecoherencebandwidth,thechannelis usually referred to as °at, or °at fading. When the bandwidth is much larger than the coherence bandwidth, the channel is said to be frequency-selective [35]. Note that°ator frequency-selectivepropertyis nota characteristicof the channelalone, but of the relationship between the transmission's bandwidth and the coherence bandwidth. In [26], the coherence bandwidth B c is a de¯ned relation derived from 39 the root mean square (RMS) delay spread ¾ ¿ . For example, if the coherence band- widthB c is de¯ned as the bandwidth over which the frequency correlation function is above 0.9, then the coherence bandwidth is approximately B c '[50¾ ¿ ] ¡1 . If the de¯nition is relaxed so that the frequency correlation function is above 0.5, then the coherence bandwidth is approximately B c '[5¾ ¿ ] ¡1 . To investigate frequency selectivity on wave polarization, we need to consider the characteristics of a transfer function which relates the impinging electric ¯eld atthereceivingantennareferencepointtothetransmittedsignalgeneratorvoltage at the transmitting antenna terminal over the transmitted signal bandwidth. InEquation(4.5),thereceiverloadvoltageV L (f)canberepresentedbyascaled inner product between the impinging electric ¯eld E(f) and the receiving antenna orientation ^ ½ with additive Gaussian noise N(f), as follows. V L (f)' h E(f)¢C(f)^ ½ i +N(f); where the receiving antenna is approximated by a Hertzian dipole antenna. The receiverloadvoltageV L (f)representationcanbegeneralizedforanarbitrarytrans- mitted signal V G (f) which drives the transmitting antenna. The e®ects of cables or anLNAareignoredinthisexposition. In[24], thevectortransferfunctionH EG (f) is de¯ned and relates the electric ¯eld at the receiving antenna reference point to the transmitted signal generator voltage by E(f)=V G (f)H EG (f): (4.11) The vector transfer function H EG (f) has an input of the transmitted signal V G (f) which is a scalar with the scale of voltage [V], and an output of the impinging 40 electric ¯eld E(f) which is a three dimensional vector at the receiving reference point with the scale of voltage per meter [V/m]. By Equation (4.5) and (4.11), the receiver load voltage is represented by V L (f)=V G (f) h H EG (f)¢C(f)^ ½ i +N(f): (4.12) The vector transfer functionH EG (f) can be measured by making three orthog- onal receiving antenna measurements, as described in previous chapters. A vector network analyzer (VNA) can be useful for the propagation measurements, where the port 1 and port 2 are calibrated at the transmitting antenna terminal and the receiving antenna terminal, respectively. VNA measurements can calibrate the ef- fectoftheLNA,cables, andsoon. Hence, thetransmittedsignalcanbeconsidered as V G (f)=1 over the measured frequency span, that is, E(f)=H EG (f) by Equa- tion (4.11). Then, using Equation (4.12), the load voltage is represented in the frequency domain by V L (f)= h H EG (f)¢C(f)^ ½ i +N(f): (4.13) Additionally, H EG (f) = H x (f)a x +H y (f)a y +H z (f)a z =E EG (f) h EG (t) = h x (t)a x +h y (t)a y +h z (t)a z = e x (t)a x +e y (t)a y +e z (t)a z =e EG (t): 41 In this section, we will investigate frequency selectivity on wave polarization, i.e., E EG (f)=jE EG (f)j, over frequency band where the transmitting and receiving an- tennas are matched at 50 from approximately 1 to 4 GHz [36]. The vector transfer function H EG (f) was measured by applying VNA propa- gation measurements to the estimation procedure in section 4.2. The transmitting antenna orientation was 1 2 [1;¡1; p 2] T at location A in Figure 3.1. At location C and E, three orthogonal antenna measurements were conducted with the receiving antenna orientations ^ ½ x = [0;0;1] T , ^ ½ y = [0;1;0] T , and ^ ½ z = [¡1;0;0] T . The port 1 and port 2 of the VNA were calibrated at the transmitting antenna terminal and the receiving antenna terminal, respectively. The VNA was set up to sweep from 0.05 to 8.05 GHz in 1601 steps, that is, frequency resolution was 5 MHz. The number of measurements averaged was 16 per point. Figure4.2and4.3showmeasuredS 21 (f)parameters,orthereceiverloadvoltage V L (f), in the frequency domain at location C and E for three antenna orientations, respectively. Figure 4.4 and Figure 4.5 show the receiver load voltage in the time domain by using the inverse Fourier transform of V L (f) for three antenna orien- tations, where the sampling frequency is 8 GHz. It is shown that in Figure 4.2 and 4.3 that there exists frequency selectivity even for the channel frequency band- widthless than 500 MHz, whichis smaller than the required ultra wide bandwidth. In Figure 4.6 and 4.7, the polarization unit vector E EG (f)=jE EG (f)j is shown for location C and E, respectively. It is clear that the polarization unit vector has a signi¯cant variance even over any 100 MHz bandwidth from 1 to 4 GHz. In this section, we investigated frequency selectivity of wave polarization. The degree of frequency selectivity of wave polarization in Figure 4.6 and 4.7 is close to the degree of frequency selectivity in the indoor propagation channel in Figure 4.2 and 4.3. Therefore, any transmitted signal polarization which undergoes frequency 42 1 1.5 2 2.5 3 3.5 4 x 10 9 −80 −60 −40 1 1.5 2 2.5 3 3.5 4 x 10 9 −80 −60 −40 Amplitude (dB) 1 1.5 2 2.5 3 3.5 4 x 10 9 −80 −60 −40 Frequency (Hz) ρ x ρ y ρ z Figure 4.2: The measured frequency response of the channel at location C. 43 1 1.5 2 2.5 3 3.5 4 x 10 9 −80 −60 −40 1 1.5 2 2.5 3 3.5 4 x 10 9 −80 −60 −40 Amplitude (dB) 1 1.5 2 2.5 3 3.5 4 x 10 9 −80 −60 −40 Frequency (Hz) ρ x ρ y ρ z Figure 4.3: The measured frequency response of the channel at location E. 44 0 0.5 1 1.5 x 10 −7 −5 0 5 x 10 −4 0 0.5 1 1.5 x 10 −7 −5 0 5 x 10 −4 Amplitude 0 0.5 1 1.5 x 10 −7 −5 0 5 x 10 −4 Time (s) ρ x ρ y ρ z Figure 4.4: The measured impulse response of the channel at location C. 45 0 0.5 1 1.5 x 10 −7 −2 0 2 x 10 −4 0 0.5 1 1.5 x 10 −7 −2 0 2 x 10 −4 Amplitude 0 0.5 1 1.5 x 10 −7 −2 0 2 x 10 −4 Time (s) ρ x ρ y ρ z Figure 4.5: The measured impulse response of the channel at location E. 46 1 1.5 2 2.5 3 3.5 4 x 10 9 0 0.5 1 E x (f)/|E(f)| 1 1.5 2 2.5 3 3.5 4 x 10 9 0 0.5 1 E y (f)/|E(f)| 1 1.5 2 2.5 3 3.5 4 x 10 9 0 0.5 1 Frequency (Hz) E z (f)/|E(f)| Figure 4.6: The polarization unit vector E EG (f)=jE EG (f)j at location C. 47 1 1.5 2 2.5 3 3.5 4 x 10 9 0 0.5 1 E x (f)/|E(f)| 1 1.5 2 2.5 3 3.5 4 x 10 9 0 0.5 1 E y (f)/|E(f)| 1 1.5 2 2.5 3 3.5 4 x 10 9 0 0.5 1 Frequency (Hz) E z (f)/|E(f)| Figure 4.7: The polarization unit vector E EG (f)=jE EG (f)j at location E. 48 selective indoor multipath channel propagation should be measured and modeled in the way introduced in Chapter 4 due to the frequency dependency. The de- gree of frequency selectivity of wave polarization may be di®erent in some radar applicationswhichfocusonasinglepathchannel,transmitter!target!receiver. 49 Chapter 5 Received Signal Energy Bounds and Distribution 5.1 Analytical Received Energy Bounds For the estimated received signal e V(f) with 4000 samples in Equation (4.10), the received signal energy E at the load (or the DSO with the load impedance Z L = 50) can be represented by E(^ ½)= 1 Z L 4000 X i=1 e V 2 (f i )¢f = 1 Z L 4000 X i=1 k e E(f i )¢C(f i )^ ½k 2 ¢f; where the received signal energy is a function of an antenna orientation ^ ½, the sampling frequency ¢f is [50 ns] ¡1 = 20 MHz, and the operator k¢k denotes the norm. ByusingParseval'stheorem,wecanalsorepresentthereceivedsignalenergy as E(^ ½)= 1 Z L ° ° °C e E T ^ ½ ° ° ° 2 ¢f = 1 Z L ° ° °e e T c ^ ½ ° ° ° 2 ¢t=C 0 ^ ½ T (e e c e e T c )^ ½; (5.1) where each row of e e c is the inverse Fourier transform of each row of C e E, and C 0 = ¢t Z L . The received signal energy has a symmetry for the antenna orientations ^ ½ and¡^ ½. 50 The3£3matrixe e c e e T c hasacompleteorthonormalsetofeigenvectorsfq 1 ;q 2 ;q 3 g with corresponding eigenvalues ¸ 1 ;¸ 2 ; and ¸ 3 , becausee e c e e T c is real and symmet- ric[33]. Expandtheantennaorientation ^ ½intermsofeigenvectorsas ^ ½= P 3 i=1 ® i q i , where ® i =q T i ^ ½. Then, Equation (5.1) can be represented by E(^ ½) = C 0 ^ ½ T (e ee e T ) 3 X i=1 ® i q i =C 0 3 X i=1 ® i ^ ½ T (e ee e T )q i = C 0 3 X i=1 ® i ¸ i ^ ½ T q i ³ *(e ee e T )q i =¸ i q i : ´ = C 0 3 X i=1 j® i j 2 ¸ i : ³ *® i =q T i ^ ½: ´ (5.2) Note that P 3 i=1 j® i j 2 = 1, since 1 = k^ ½k 2 = ^ ½ T ^ ½ = ( P 3 i=1 ® i q i ) T ( P 3 i=1 ® i q i ) = P 3 i=1 j® i j 2 . Using Equation (5.2), the received signal energy is bounded by C 0 ¸ min =C 0 min i ¸ i ·E(^ ½)·C 0 max i ¸ i =C 0 ¸ max (5.3) Hence, if we set ® i 0 = 1, and all the other ® i = 0 for i 6= i 0 , then E(^ ½) = C 0 ¸ i 0 and ^ ½ =§q i 0 . Then, the maximum received signal energy is E(^ ½) = C 0 ¸ max for a pair of antenna orientations ^ ½ =§q i max , where i max = argmax i ¸ i . Similarly, the minimumreceivedsignalenergyisE(^ ½)=C 0 ¸ min forapairofantennaorientations ^ ½=§q i min , where i min =argmin i ¸ i . Equation(5.2)and(5.3)showthattherearethreeorthogonalreceivingantenna orientations, where one antenna orientation among three orthogonal antenna ori- entations is the best antenna orientation in terms of the maximum received energy and another among three orthogonal antenna orientation is the worst antenna ori- entation in terms of the minimum received energy. The best and worst receiving 51 antenna orientations always exist and are not the same, only if the received energy is not same for all possible receiving antenna orientation, which is a realistic as- sumption in multipath channels. It is interesting to note that even in multipath environments the best receiving antenna orientation is orthogonal to the worst re- ceiving antenna orientation. In the following section, some properties of received signal energy with respect to the receiving antenna orientation will be investigated for some propagation channel examples. 5.2 ReceivedEnergyAnalysisforSomeChannels Suppose that all electric ¯elds impinge on the receiving antenna in same direc- tion. Regardless of receiver noise, the minimum received signal energy is null for a pair of antenna orientations ^ ½ = q i min =§a R , because the e®ective height is also null for the antenna orientations, that is, h = ¡a µ C 0 sin(0) = 0. Note that the wave impinging direction is¡a R . In Equation (5.3), the minimum eigenvalue ¸ min would be null, and the eigenvalueq i min with the minimum eigenvalue would be the corresponding antenna orientation. When an electric ¯eld impinges on the same direction and the electric ¯eld is linearly polarized, there exists another pair 1 of antenna orientations that null the received signal energy. These antenna orientations are parallel to the wave incidentplane,andareorthogonaltoq i max . Althoughthee®ectiveheightisnonzero, polarization mismatch causes null received signal energy. Hence, in Equation (5.3), two eigenvalues other than ¸ max are nulls, and the corresponding eigenvectors are orthogonal with each other. For both of the cases above, with the existence of additive white Gaussian noise, the minimum received energy would be the variance 1 By the symmetry, there always exist a pair of antenna orientations which satis¯es energy condition, that is, ^ ½ and¡^ ½. 52 ofthenoisescaledbythetimeinterval,whichistheintervalofthetimeintegration, assuming high signal-to-noise ratio. It is worth noting a narrow band case. When the transmitted signal has narrow bandwidth, the electric ¯eld is not a function of frequency over the transmitted signal bandwidth. Hence, the electric ¯eld vector E(f) in Equation (4.5) is not a function of frequency, and an electric ¯eld matrix e can be represented by a single column matrix with the size of 3 £ 1 in the frequency domain. Therefore, the electric ¯eld estimation process can be the process estimating three complex number elements in the single column matrix, and the process in the narrow band case is simpler than the process in UWB case. 5.3 Empirical Received Energy Distribution Fortheestimateoftheimpingingelectric¯eld,thereceivedsignalwaveformandthe received signal energy can be estimated for any arbitrary receiving antenna orien- tation. Figure 5.1, 5.2, 5.3, and 5.4 show received signal energy distributions for all possible receiving antenna orientations. The receiving antenna orientation can be representedbytheelevationangleµ ^ ½ andazimuthangleÁ ^ ½ ofthereceivingantenna orientation in the absolute Cartesian coordinate system, where µ ^ ½ = arccos(^ ½¢a z ) and Á ^ ½ = arctanf(^ ½¢a y )=(^ ½¢a x )g, as in Equation (2.2). There always exist a pair of antenna orientations, that is, ^ ½ and -^ ½, which satis¯es an energy condition by the non-negativity of the received energy, as follows. E(^ ½)=E(µ ^ ½ ;Á ^ ½ )=E(180 ± ¡µ ^ ½ ;Á ^ ½ +180 ± )=E(¡^ ½): 53 Figure 5.1: Received signal energy distributions at location B. In Figure 5.1, 5.2, 5.3, and 5.4, the received energy distribution is plotted when ^ ½ points at the upper half sphere, that is, ^ ½¢a z >0 or 0<µ ^ ½ <90 ± . For each of four channels, the brighter region represents the higher received energy, and the darker regionrepresentsthe lowerreceivedenergy. Thevariationof the channelmakesthe energy distribution with respect to the receiving antenna orientation variable. All offour¯gureshaveaverybrightregionandtwodarkregions. Thesethreeextremes correspond to the receiving antenna orientations of eigenvectors in Equation (5.2). The mean of the received energy, mean ^ ½ [E(^ ½)], which is averaged over receiving antenna orientation with a degree resolution in the spherical coordinate system has the largest value at location B, as shown in Table 5.1. The further the distance 54 Figure 5.2: Received signal energy distributions at location C. 55 Figure 5.3: Received signal energy distributions at location D. 56 Figure 5.4: Received signal energy distributions at location E. 57 Table 5.1: Statistical information of the received signal energy at location B, C, D, and E. sensing statistical parameter location mean ^ ½ [E(^ ½)] var ^ ½ [E(^ ½)] max ^ ½ [E(^ ½)]¡min ^ ½ [E(^ ½)] B 2.086£10 ¡28 J 4.067£10 ¡14 9.299 dB C 1.791£10 ¡29 J 1.282£10 ¡14 5.287 dB D 5.742£10 ¡30 J 7.238£10 ¡15 5.961 dB E 5.894£10 ¡31 J 3.922£10 ¡15 3.743 dB betweenthe transmitter location and the receiverlocation, the smaller the received energy mean. The received energy variance with respect to the receiving antenna orientation, var ^ ½ [E(^ ½)], and the di®erence between the maximum received energy and the minimum received energy, max ^ ½ [E(^ ½)]¡min ^ ½ [E(^ ½)], have the same order statistics of the receiving locations as the mean received energy, mean ^ ½ [E(^ ½)]. AtlocationE,receivedenergyatelevationanglearound50 ± hasalargevariance overazimuthanglesinFigure5.4. Thisillustratesthefactthatareceivingantenna orientation may have a considerable e®ect on the performance of a indoor UWB radio receiver. For example, a user holds a UWB radio embedded hand held device (or PDA) at location E. The antenna at the device is in the receiving mode, and tilted at an elevation angle µ ^ ½ =50 ± . If the user's body rotates horizontally to vary the azimuth angle Á ^ ½ , then at some angles, the user might have a low quality of service caused by insu±cient received energy captured in the UWB receiver. 5.4 Empirical Received Energy Bounds By using the empirical energy distribution in Figure 5.1, 5.2, 5.3, and 5.4, max- imum and minimum received energies at location B, C, D and E can be found with corresponding receiving antenna orientations. The grid search results, with 1 degree resolution in the spherical coordinate system, are shown in Table 5.2 and 58 5.3. Received waveforms have maximum energies at location B, C, D and E for antenna orientations (µ ^ ½ ;Á ^ ½ ) = (42 ± ;53 ± ), (8 ± ;175 ± ), (7 ± ;245 ± ), and (14 ± ;312 ± ), re- spectively. Received waveforms have minimum energies at location B, C, D and E forantennaorientations(µ ^ ½ ;Á ^ ½ )=(78 ± ;309 ± ),(98 ± ;173 ± ),(84 ± ;31 ± ),and(91 ± ;37 ± ), respectively. Theempiricalresultsofreceivingantennaorientationandreceiveden- ergyinTable5.2and5.3areconsistentwiththeanalyticalresultsoftheeigenvalue and eigenvector using Equation (5.2) and (5.3). As a reference, the best receiving antenna orientations are described in Ta- ble 5.4, assuming that the propagation channel is a free space channel with holding the geometry of the transmitter con¯guration described in Chapter 3 and each re- ceiverlocationatlocationB,C,D,andE.Thedi®erencebetweenthebestantenna orientation in Table 5.2 and 5.4 for each channel shows the e®ect of the multi- path channel, even though at the blocked line-of-sight location E the best receiving antenna orientation for the measured multipath channel in Table 5.2 is coinciden- tally close to the best receiving antenna orientation for the free space channel in Table 5.4. As shown in Table 5.1, 5.2, and 5.3, at the receiving location B on the line-of- sightofthetransmittinglocationA,thedi®erencebetweenmaximumandminimum received signal energy is about 9.3 dB, which is a signi¯cant variation the receiver might experience for a ¯xed static multipath channel. Even for the blocked line-of- sight receiver location E, the di®erence between maximum and minimum received signal energy is about 3.7 dB. At location B, the large amount of received signal energyiscomefromdirectline-of-sightpathpropagationandgroundre°ection. On the other hand, at location E, received signal energy is come from other multipath componentsthandirectline-of-sightpathanddirectgroundre°ection,sincetheline of sight is blocked by the storage box and a metal column. Hence, the di®erence 59 Table 5.2: Receiving antenna orientations for maximizing received signal energy at location B, C, D, and E. sensing maximum received signal energy location ^ ½ µ ^ ½ Á ^ ½ E(^ ½) B 2 4 0:405 0:535 0:740 3 5 42 ± 53 ± 6.60£10 ¡14 J C 2 4 ¡0:130 0:012 0:991 3 5 8 ± 175 ± 1.89£10 ¡14 J D 2 4 ¡0:051 ¡0:111 0:993 3 5 7 ± 245 ± 1.06£10 ¡14 J E 2 4 0:158 ¡0:176 0:972 3 5 14 ± 312 ± 4.87£10 ¡15 J between the energy variation at location B (9.3 dB) and at location E (3.7 dB) shows the e®ect of polarizations of line-of-sight signal and ground re°ection signal, which is likely to be linearly polarized. 60 Table 5.3: Receiving antenna orientations for minimizing received signal energy at location B, C, D, and E. sensing minimum received signal energy location ^ ½ µ ^ ½ Á ^ ½ E(^ ½) B 2 4 0:618 ¡0:758 0:210 3 5 78 ± 309 ± 7.76£10 ¡15 J C 2 4 ¡0:984 0:124 ¡0:131 3 5 98 ± 173 ± 5.60£10 ¡15 J D 2 4 0:856 0:508 0:101 3 5 84 ± 31 ± 2.69£10 ¡15 J E 2 4 0:801 0:599 ¡0:022 3 5 91 ± 37 ± 2.06£10 ¡15 J Table 5.4: Receiving antenna orientations for maximizing received signal energy assuming free space environment at location B, C, D, and E. sensing maximum energy assuming free space location ^ ½ µ ^ ½ Á ^ ½ B 2 4 ¡0:0531 ¡0:6341 0:7714 3 5 40 ± 85 ± C 2 4 0:1261 ¡0:4721 0:8725 3 5 29 ± 285 ± D 2 4 0:1628 ¡0:3430 0:9251 3 5 22 ± 295 ± E 2 4 0:1502 ¡0:2106 0:9660 3 5 15 ± 305 ± 61 Chapter 6 UWB Array Propagation Measurement 6.1 Introduction Inprevioussectionand[6],theimpingingelectric¯eldwasestimatedbyusingthree orthogonal antenna measurements. The number of measurements was minimized to estimate the three-dimensional electric ¯eld at the receiving antenna reference point. Received waveform for an arbitrary receiving antenna orientation could be estimated accurately for a receiving antenna which can be approximated by a Hertzian dipole antenna. However, in non-Hertzian dipole receiving antenna case, the received waveform cannot be estimated in the way in [6], since the antenna e®ective length is a complex function of wave impinging direction. Therefore, in order to apply the electric ¯eld estimation result to non-Hertzian dipole antenna receiver,theangle-of-arrivalandthewaveformshapeofeachmultipathcomponents must be characterized for the multipath propagation channel. In this chapter, UWB polarization measurements with a multipath decomposi- tionprocessisstudiedwitha3£3£3arrayofpropagationmeasurementsusingthree receiving antenna orientations. In order to estimate the impinging electric ¯eld at a receiving location in a multipath channel, an array signal processing algorithm, 62 the Sensor-CLEAN algorithm in [11], is applied to the array measurement data to decompose a received waveform with a dense multi-path pro¯le into its component single-path signals. The Sensor-CLEAN algorithm can provide the propagation channel characterization: time-of-arrival, angle-of-arrival, waveform shape and po- larization of each resolvable waveform in the composite received signal. Based on a modi¯ed channel propagation model which includes polarization characteristics, the electric ¯eld at the center of the array is estimated by combining each of three orthogonal antenna measurement's multipath waveform components by using the estimated channel characteristics. The performance of the estimation process will be evaluated. 6.2 Array Propagation Measurement Setup An array of propagation measurements was conducted in the lobby of Seaver Sci- ence Library building at University of Southern California. As shown in Figure 6.1 and 6.2, a 3£3£3 cube of virtual array data with 30.48 cm spacing was mea- sured by moving the receiving antenna to 27 positions. The array measurements were repeated 3 times, each time using a di®erent receiving antenna array element orientation as shown in Figure 6.3. In the ¯rst test, the antenna was vertically polarized; in the second test, the antenna was horizontally polarized with the same boresight as the ¯rst; in the third test, the antenna was horizontally polarized with the top facing the original boresight. Hence, receiving antenna orientation vectors were [0;0;1] T , [¡1;0;0] T , and [0;1;0] T , respectively. The antenna was supported by a PVC pipe connection structure for three values of the height, and the bottom ofthesupportingstructurewasembeddedinoneof3£3gridofholeswith30.48cm 63 Transmitter Receiving array Vending Machine Sofa and Desk Vending Machine Bookshelf and desk Table 24" absolute coordinate system y x z N Figure 6.1: A diagram of the library building where the propagation measurement experiment was performed. 64 Figure 6.2: Measurement site picture from behind the transmitting antenna. 65 Figure 6.3: Three orthogonal antenna orientation setup pictures. spacing on the plywood. Therefore, the total number of measurements per array location was 81. With a given angle-of-arrival of the impinging wave, the impinging wave polar- ization may be characterized by two antenna orientation measurements. However, ifthewaveimpingingdirectioniseitherthetoporbottomdirectionofoneantenna, and the wave polarization is orthogonal to the other antenna polarization, the re- ceived waveform does not give information of the electric ¯eld magnitude because of a weak signal-to-noise ratio (SNR). Hence, three orthogonal antenna orientation measurements were conducted for each array position. Thetransmittingandreceivingantennaswerelinearlypolarizeddiamonddipole antennas [30]. The gain of the antenna peaks at about 0.5 dBi just above 2.0 GHz. The 3 dB gain bandwidth runs from about 1.4 GHz to 3.0 GHz. The diamond dipole is about 75% e±cient around its 2 GHz center frequency with about a 3.0:1 VSWR. The transmitting antenna was rotated 45 ± on the boresight axis from the vertically polarized orientation, and was 60 inches above the °oor. The center of the array was 152.4 cm above the °oor. 66 1.045 1.05 1.055 1.06 1.065 1.07 1.075 x 10 −6 −1 −0.5 0 0.5 1 Time (s) Amplitude (Volts) 0 1 2 3 4 5 x 10 9 0 10 20 30 Frequency (Hz) Amplitude (dB) Figure 6.4: The measured transmitted UWB monocycle pulse. 67 A monocycle pulse with a sub-nanosecond width excited the transmitting an- tennaperiodicallyeverymicrosecondandthereceivedwaveformwasmeasuredwith a digital sampling oscilloscope (DSO). A 20 dB attenuated direct output measure- mentofthemonocyclepulsebyDSOisshownwithitsFFTspectruminFigure6.4. The e®ective 10 dB bandwidth of the pulser output is from 0.42 GHz to 2.36 GHz. A stable clock triggered both the transmitting pulser and the digital sampling os- cilloscope. The clock was generated by a signal generator with 0 ns delay, 50 ns width, 2 ns leading edge, the high voltage 4 V, the low voltage 0 V, and the repeti- tion frequency 1.00 MHz. The trigger level of DSO was set to 90 mV with a trigger line 10 dB attenuator connected. The sampling rate of the measured waveform was 20 GHz and each sample was averaged over 256 sweeps to achieve a higher SNR. Typicalmeasuredpro¯lesareshowninFigure6.5fordi®erentantennaorientations. 6.3 AWavePolarizationEstimationProcesswith a Multipath Decomposition Algorithm In this section, a polarization estimation process utilizes the array propagation measurement data. The °owchart of the polarization estimation process is shown in Figure 6.6. For the ¯rst step of the polarization estimation process, the Sensor- CLEAN algorithm is applied to each set of the three antenna orientation measure- ments separately, whereby the algorithm provides three sets of output estimates. The Sensor-CLEAN algorithm is an iterative multipath decomposition algorithm which decomposes a received waveform with a dense multi-path pro¯le into each 68 1025 1030 1035 1040 1045 1050 1055 1060 −0.1 0 0.1 1025 1030 1035 1040 1045 1050 1055 1060 −0.1 0 0.1 Amplitude (Volts) 1025 1030 1035 1040 1045 1050 1055 1060 −0.1 0 0.1 Time (ns) Figure 6.5: Measured amplitude vs. time at the center of the array for the 1 st antennaorientation(top),the2 nd antennaorientation(center),andthe3 rd antenna orientation (bottom). 69 Make array measurements for 3 different receiving antenna orientations Sensor-CLEAN #2 Sensor-CLEAN #1 Sensor-CLEAN #3 Group decomposed path waveform shape by time-of-arrival and angle-of-arrival through three antenna orientations Estimate and combine the electric fields for each path Figure 6.6: The °owchart of the UWB polarization estimation process. componentsingle-pathsignal. TheSensor-CLEANalgorithmcanprovidetheprop- agation channel characteristics in each antenna orientation measurement: time-of- arrival, angle-of-arrival, waveform shape in the composite received signal. For the second step, a decomposed multipath element in one set is combined with other elements by searching in the other two sets with a time-of-arrival constraint T w , an elevation angle-of-arrival £ w and a direction-of-arrival constraint A w . For the last step, we estimate the impinging electric ¯eld by using some antenna properties and the Gram-Schmidt process. 70 6.3.1 The Sensor-CLEAN Algorithm and Combining In Equation (4.1), the UWB multipath channel model was provided by V L (f)= X n · E n (f;µ n ;Á n )¢h(f;µ n ;Á n ) Z L (f) Z L (f)+Z R (f) ¸ +N(f): The polarization vector of the receiving antenna ^ p is de¯ned by ^ p = h(f;µ;Á) jjh(f;µ;Á)jj (also, see the de¯nition in Chapter 2), and the receiving antenna transfer function H(f;µ;Á) is de¯ned as the sensitivity of the antenna [27] by a relationship as ^ pH(f;µ;Á) = h(f;µ;Á) Z L (f) Z L (f)+Z R (f) for the elevation and azimuth angle-of-arrival, µ and Á, respectively 1 . Then, the time domain representation of Equation (4.1) is given by v L (t)= X n [e n (t)¢ ^ p n ]¤h(t;µ n ;Á n )+n(t) (6.1) When the Sensor-CLEAN algorithm is applied to array measurement data, the Sensor-CLEAN algorithm decomposes a received waveform with a dense multi- pathpro¯leintoeachcomponentsingle-pathsignal. TheSensor-CLEANalgorithm with post-processing, the Wave-Map algorithm [10], provides estimates of each component's time-of-arrival, angle-of-arrival, and waveform shape in the following channel model, which is modi¯ed from Equation (6.1). v L (t)= N X n=1 s rec;n (t¡¿ n ;µ n ;Á n )+n(t) (6.2) where ¿ n is the time-of arrival of the n th out of N multipath components at an elevation angle µ n and an azimuth angle Á n , s rec;n (t) is the n th received impulse 1 Note that the angle-of-arrival is de¯ned in the antenna's coordinate system in Chapter 2. 71 waveform which can be given by s rec;n (t¡¿ n ;µ n ;Á n )=[e n (t)¢^ p n ]¤h(t;µ n ;Á n ), and n(t) is receiver noise. The time-of-arrival of a multipath component is de¯ned by ¿ n = argmax t [s rec;n (t;µ n ;Á n )]. The Sensor-CLEAN algorithm with the Wave-Map algorithm in [10] is generalized to the continuous time case and summarized by the following steps . 1. Input : Measured pulse response waveform s (0) j (t) at the j th element, 1 · j · M, from M di®erent sensors; the sensor position vector d j from the center of the array to the position of the j th sensor element; loop gain factor °; the relaxation window half-width T p ; a detection threshold T det which is used to control the stopping time of the algorithm; the Wave-Map algorithm parameters of T w and A w are the temporal and directional window size for combining the Sensor-CLEAN algorithm detected signal output which comes from the same path component. 2. Initialize : Set the Sensor-CLEAN algorithm iteration counter i to i=0. Set the initial detection list P (0) and the initial multipath component list S 0 to theemptylist. Constructthedelay-and-sumbeamformerassociatedwiththe direction-of-arrival¡^ a R such that B (i) (t;¡^ a R )= M X j=1 s (i) j ³ t+ d j ¢(¡^ a R ) c ´ where c is the speed of light. 3. Signal Detection : If max (t;¡^ a R ) ¯ ¯ B (i) (t;¡^ a R ) ¯ ¯ < T det , then set the number of iterations I =i and Go to step 7. Otherwise, iÃi+1. 72 4. Detected Signal Storage : Estimate the time-of-arrival e t (i) , the direction-of- arrival¡e a (i) R , and the detected waveform e w (i) (t) at the i th iteration by using the following equations. ( e t (i) ;¡e a (i) R ) = arg max (t;¡^ a R ) ¯ ¯ B (i¡1) (t;¡^ a R ) ¯ ¯ ; (6.3) e w (i) (t) = ° · 1 M rect ³ t 2T p ´ ¸ B (i¡1) (t+ e t (i) ;¡e a (i) R ): (6.4) Appendf e t (i) ;¡e a (i) R ;e w (i) (t)g at the i th iteration to the detection listP. P (i) = P (i¡1) [ © f e t (i) ;¡e a (i) R ;e w (i) (t)g ª : 5. Update the Residual Waveforms: Removethedetectedwaveform e w (i) (t)from measure waveforms of each sensor. That is, for8j2f1;2;:::;Mg, s (i) j (t)=s (i¡1) j (t)¡e w (i) ³ t¡ e t (i) ¡ d j ¢(¡e a (i) R ) c ´ : 6. Iterate the Sensor-CLEAN algorithm : Go to step 3. 7. The Wave-Map algorithm initialization : Set the signal detection index list as C 0 = f1;2;:::;Ig. Set the Wave-Map algorithm iteration counter n to n = 0. Construct a beamformer using the detection list P (I) , the residual beamformer after the Sensor-CLEAN algorithm iteration, and the detected waveforms as follows. B 0 n (t;¡^ a R )=B (I) (t;¡^ a R )+ M X j=1 X i2Cn e w (i) ³ t¡ e t (i) ¡ d j ¢(^ a R ¡e a (i) R ) c ´ : 73 8. Path Detection : If C n = ; or max (t;¡^ a R ) ¯ ¯ B 0 n (t;¡^ a R ) ¯ ¯ < T det , then set the number of iterations N =n and STOP. Otherwise, nÃn+1. Note that the number of the Wave-Map algorithm iterations N is the number of resolvable multipaths. 9. Multipath Component Characterization and Storage : Estimate the time-of- arrival ¿ n , the direction-of-arrival¡^ a R;n , and the waveform shape s rec;n (t) of the n th detected multipath component as follows. e s rec;n (t) = X i2Wn e w (i) (t); e ¿ n = e t n ; ( e t n ;¡e a R;n ) = arg max (t;¡^ a R ) ¯ ¯ B 0 (n¡1) (t;¡^ a R ) ¯ ¯ ; (6.5) where W n = n l2C n¡1 ¯ ¯ ¯ ¡ j e t (l) ¡ e t n j<T w ¢ ^ ¡ jje a (l) R ¡e a R;n jj<A w ¢ o : (6.6) Equation (6.6) shows that the detected signals that are within a distance T w in time and A w in the direction of arrival are assumed to come from the same path component to be combined by the Wave-Map algorithm. Append fe s rec;n (t);e ¿ n ;¡e a R;n g at the n th iteration to the multipath component listS. S n =S n¡1 [ © fe s rec;n (t);e ¿ n ;¡e a R;n g ª 10. Update the Signal Detection Index List : Remove combined detected signals fromfurtherconsiderationbyupdatingthesignaldetectionindexlistasC n = C n¡1 ¡W n . 74 11. Iterate the Wave-Map algorithm : Go to step 8. (End of the Sensor-CLEAN algorithm with the Wave-Map algorithm.) For the three antenna orientation array measurements, the Sensor-CLEAN al- gorithm with the Wave-Map algorithm is applied to each set of the three antenna orientation measurements separately, whereby the algorithm provides three sets of output estimates, that is the three sets of the multipath component list S. The three sets of the multipath component listS correspond to the following equations. The 1 st measurement : v (1) L (t)= N (1) X n=1 s (1) rec;n (t¡¿ (1) n ;¡e a (1) R;n )+n (1) (t): (6.7) The 2 nd measurement : v (2) L (t)= N (2) X n=1 s (2) rec;n (t¡¿ (2) n ;¡e a (2) R;n )+n (2) (t): (6.8) The 3 rd measurement : v (3) L (t)= N (3) X n=1 s (3) rec;n (t¡¿ (3) n ;¡e a (3) R;n )+n (3) (t): (6.9) For the next step, a decomposed multipath element in one set is combined with other elements by searching in the other two sets with a time-of-arrival constraint T w andadirection-of-arrivalconstraintA w (sameastheWave-Mapalgorithmpara- meter T w and A w ). The combined multipath waveforms are assumed to come from the exitation of the impinging electric ¯eld on the same path over the same time interval. Hence,forexample,ifthecombinedmultipathcomponentwaveformsfrom three measurements are s (1) rec (t¡¿;¡e a R ), s (2) rec (t¡¿;¡e a R ), and s (3) rec (t¡¿;¡e a R ), then the relationship of s (i) rec (t¡¿;¡e a R ) = [e(t)¢ ^ p (i) ]¤h(t;µ (i) ;Á (i) ) holds for i = 1;2, 75 and 3 (See Equation (6.1)). Therefore, Equation (6.7), (6.8), and (6.9) can be represented by The 1 st measurement : v (1) L (t)= N X n=1 [e n (t)¢ ^ p (1) n ]¤h(t;µ (1) n ;Á (1) n )+n (1) (t): (6.10) The 2 nd measurement : v (2) L (t)= N X n=1 [e n (t)¢ ^ p (2) n ]¤h(t;µ (2) n ;Á (2) n )+n (2) (t): (6.11) The 3 rd measurement : v (3) L (t)= N X n=1 [e n (t)¢ ^ p (3) n ]¤h(t;µ (3) n ;Á (3) n )+n (3) (t): (6.12) Note that the polarization unit vector of the receiving antenna ^ p (i) can be given by theelevationandazimuthangle-of-arrival,µ (i) andÁ (i) ,by ^ p (i) =(cosÁ (i) cosµ (i) )a x 0+ (sinÁ (i) cosµ (i) )a y 0¡(sinµ (i) )a z 0 in Equation (2.1). The angle-of-arrivals for the an- tennasensitivityinEquation(6.10),(6.11),and(6.12)correspondtotheequivalent direction-of-arrival, that is, ¡e a R;n = sinµ (1) n cosÁ (1) n ^ a (1) x +sinµ (1) n sinÁ (1) n ^ a (1) y +cosµ (1) n ^ a (1) z = sinµ (2) n cosÁ (2) n ^ a (2) x +sinµ (2) n sinÁ (2) n ^ a (2) y +cosµ (2) n ^ a (2) z = sinµ (3) n cosÁ (3) n ^ a (3) x +sinµ (3) n sinÁ (3) n ^ a (3) y +cosµ (3) n ^ a (3) z For the last step, we estimate the impinging electric ¯eld e n (t) by the Wiener ¯lteringandtheGram-SchmidtprocessinEquation(6.10),(6.11),and(6.12),which will be described in the next subsection. 76 6.3.2 An Electric Field Estimator The impinging electric ¯elde n (t) is estimated multipath component by component using three sets of the multipath component list S. Firstly, e n (t)¢ ^ p (i) n can be estimated for i=1;2; and 3 by the Wiener ¯ltering in [25] and [23]. e n (t)¢ ^ p (i) n =F ¡1 ( V (i) L (f)H ¤ (f;µ (i) n ;Á (i) n ) jH(f;µ (i) n ;Á (i) n )j 2 +C (i) ) (6.13) where the superscript ( ¤ ) denotes the complex conjugate,F ¡1 is the inverse Fourier transform,C (i) isasmoothingconstantrelatedtothevarianceofn (i) (t),andV (i) L (f) is the Fourier transform of v (i) L (t), for i=1;2; and 3. Theleft-handsideinEquation(A.12),e n (t)¢^ p (i) n ,isaprojectionoftheimpinging electric¯eldontheantennapolarizationvector. Thepolarizationunitvectorofthe receiving antenna n ^ p (1) n ;^ p (2) n ;^ p (3) n o of and the impinging electric ¯eld vector e n (t) are in the same plane, which is perpendicular to the direction-of-arrival. Hence, the Gram-Schmidt process can solve the electric ¯eld e n (t) as follows. e n (t)= ^ p (1) n [e n (t)¢ ^ p (1) n ]+ ^ p (2 0 ) n k ^ p (2 0 ) n k " e n (t)¢ ^ p (2 0 ) n k ^ p (2 0 ) n k # ; (6.14) where ^ p (2 0 ) n = ^ p (2) n ¡(^ p (1) n ¢^ p (2) n )^ p (1) n . The Gram-Schmidt process in Equation (6.14) candeterminetheelectric¯eldaslongasanytwoantennapolarizationsin n ^ p (1) n ;^ p (2) n ;^ p (3) n o are selected to span the wave plane perpendicular to the direction-of-arrival. With the given electric ¯eld estimate, the channel model in Equation (6.1) can give a received waveform estimate v (a) L (t) for an arbitrary antenna of the antenna 77 sensitivity h (a) (t;µ (a) ;Á (a) ) with an arbitrary orientation at an arbitrary position d (a) as follows. e v (a) L (t)= N X n=1 · e n ³ t¡ d (a) ¢(¡^ a R ) c ´ ¢ ^ p (a) n ¸ ¤h (a) (t;µ (a) n ;Á (a) n ); (6.15) assuming that resolved N number of planar waves by the Wave-Map algorithm impinge on the arbitrary position d (a) . 6.4 Application to the Measured Data In this section, the UWB polarization estimation algorithm was applied to the array measurement data. For the Sensor-CLEAN algorithm with the Wave-Map algorithm, The width of the relaxation window was 2T p = 34 samples = 0.77 ns, with the loop gain factor ° = 0:2 and a detection threshold of T det = 0:01 volts. The Wave-Map algorithm parameters of T w and A w were T w = 6 samples = 0.3 ns and A w = cos(15 ± ). The maximum of the delay-and-sum beamformer output was searched in Equation (6.3) and (6.5) over the direction-of-arrival grid at 2 ± increments in azimuth, and the following 36 elevation angles: 20 ± , 30 ± , 40 ± , 45 ± , 50 ± , 55 ± , 60 ± , 65 ± , 70 ± , 72 ± , 74 ± , 76 ± , 78 ± , 80 ± , 82 ± , 84 ± , 86 ± , 88 ± , 90 ± , 92 ± , 94 ± , 96 ± , 98 ± , 100 ± , 102 ± , 104 ± , 106 ± , 108 ± , 110 ± , 115 ± , 120 ± , 125 ± , 130 ± , 135 ± , 140 ± , 150 ± , and 160 ± . (The elevation and azimuth angle-of-arrivals in this section are de¯ned in the antenna's coordinate system for the 1 st antenna orientation as shown in Figure 6.7, where the antenna's coordinate system is de¯ned as ^ a 0 x = ^ a y , ^ a 0 y = ¡^ a x , and ^ a 0 z = ^ a z in the absolute coordinate system in Figure 6.1.) The number of detected multipath components were N (1) = 49, N (2) = 34, and N (3) = 21 in Equation (6.7), (6.8), and (6.9) for the 1 st , 2 nd , and 3 rd antenna 78 x' y' z' Figure 6.7: The antenna's coordinate system for the 1 st antenna orientation mea- surement. orientation, respectively. The detected multipath components were combined into N = 13 associated combinations with T w = 6 samples and A w = cos(15 ± ) in the channel model of Equation (6.10), (6.11), and (6.12). The superimposed 13 decomposed waveform elements of the multipath decomposition algorithm result are plotted in Figure 6.8 for each of three antenna orientation measurements in Equation (6.10), (6.11), and (6.12) (Please compare Figure 6.8 with Figure 6.5.). Fortheelectric¯eldestimation,thesensitivityofthereceivingantennaH(f;µ;Á) was measured by the Network Analyzer in the anechoic chamber for any combi- nation of angle-of-arrivals, µ and Á, where µ 2 f15 ± , 30 ± , 45 ± , 60 ± , 75 ± , 90 ± g and 79 1025 1030 1035 1040 1045 1050 1055 1060 −0.1 0 0.1 1025 1030 1035 1040 1045 1050 1055 1060 −0.1 0 0.1 Amplitude (Volts) 1025 1030 1035 1040 1045 1050 1055 1060 −0.1 0 0.1 Time (ns) Figure6.8: Resultsofthedecompositionalgorithmforthe1 st (top),the2 nd (center), the 3 rd (bottom) antenna orientation measurements. 80 Á2f0 ± , 15 ± , 30 ± , 45 ± , 60 ± , 75 ± , 90 ± g. It is assumed that the antenna is isotropic, and the sensitivity for intermediate angle-of-arrivals were approximated by piece- wise cubic spline interpolation of the measured 2 . The smoothing constant C (i) in the polarization characterization method was chosen to be 5:5£10 ¡8 for all i=1, 2, and 3. The ¯delity of the polarization estimation process with the multipath decom- position algorithm is veri¯ed by a waveform measurement for an arbitrary antenna orientation. The measured wveform is compared with the estimated waveform by using Equation (6.15). The arbitrary antenna orientation has the antenna coordi- nate system bases, ^ a 0 x = [ 1 p 2 ; 1 p 2 ;0] T , ^ a 0 y = [¡ 1 2 ; 1 2 ;¡ 1 p 2 ] T , and ^ a 0 z = [¡ 1 2 ; 1 2 ; 1 p 2 ] T . The arbitrary antenna orientation is same as the one when from the 1 st vertically polarized orientation, the antenna is rotated 45 ± on the boresight axis and the re- ceiverstructureisrotated45 ± atthetop. Thetestlocationofthereceivingantenna wasatthecenterofthearrayandthereceivedwaveformisshownatthetopofFig- ure 6.9. It is noted that the received waveform at the top of Figure 6.9 is di®erent from any measured waveform in Figure 6.5. Theestimatedwaveformbythepolarizationestimationprocessfor13multipath components is in the center part of Figure 6.9. The bottom trace is the di®erence betweenthemeasuredwaveformandtheestimatedwaveform. Theestimatedwave- form shapes were matched to the corresponding part of the received waveform so that there were no °ipped waveform estimates. The paths whose times-of-arrival wereearlierthan1040nswereestimatedclosertothemeasureddatathanthepaths whose times-of-arrival were later than 1035 ns. In Figure 6.9, there is no combined multipathcomponentfrom1033nsto1046ns, eventhoughtherearedetectedmul- tipath component by the multipath decomposition algorithm out of three antenna 2 See MATLAB help of 'interp1' with an option 'cubic'. 81 1025 1030 1035 1040 1045 1050 1055 1060 −0.1 0 0.1 1025 1030 1035 1040 1045 1050 1055 1060 −0.1 0 0.1 Amplitude (Volts) 1025 1030 1035 1040 1045 1050 1055 1060 −0.1 0 0.1 Time (ns) Figure6.9: Measuredamplitudevs. timeatthecenterofthearrayforaspeci¯can- tennaorientation(top)andtheestimatedwaveformwith13multipathcomponents by the polarization characterization method (center) and the di®erence between the measured waveform and the estimated waveform (bottom). orientation measurements, as shown in Figure 6.8. Note that the number of com- bined multipath components with the parameters T w and A w is N = 13, whereas the detected multipath by the multipath decomposition algorithm are N (1) = 49, N (2) =34, andN (3) =21forthreemultipathdecompositionalgorithm. Combining mismatch is caused by non-optimality of the multipath decomposition algorithm, whichincludesthedependencyofthealgorithmparameters. However,althoughthe estimated waveform from 1023 ns to 1027 ns is not similar to any of the waveforms in Figure 6.5, the estimation waveform result was relatively reliable. 82 6.5 Limitation on the Estimation Process Although the Sensor-CLEAN algorithm and the Wave-Map algorithm was useful to decompose the multipath pro¯le and estimate the time-of-arrival and angle-of- arrival, thealgorithmcouldnotprovidethewaveformshapeestimatesofaresolved componente±cientlyenoughforthepolarizationestimationinthesetupusedhere. Limitations on the estimation process is described as follows. ² As with most indirect algorithms, the solution generated by the Sensor- CLEAN algorithm is a function not only of the data, but of the input pa- rameters as well [10]. For example, the detected waveform is removed by a ¯nitetimeintervalgatingwiththeparameterT w . However,whenamultipath signal component has a late time-of-arrival, the signal component might have undergonemoredistortionbychannelobjectssothatthetimedurationofthe signal component might be wider than the time duration of the line-of-sight signal component. Hence, the ¯nite time interval gating needs to be provided with an adaptive width parameter which corresponds to the time duration of each signal component. In reality, it is di±cult to implement the adaptive width parameter optimally. ² The Sensor-CLEAN algorithm in section 6.3 is not originally designed for estimating each multipath waveform shape, but for estimating the time-of- arrival and the angle-of-arrival in [10]. The inter-multipath interference in the multipath wavform shape estimation is unavoidable, so that the Sensor- CLEAN algorithm becomes suboptimal. 83 ² The non-optimality of the multipath decomposition algorithm can make the combiningprocessfuzzy. Forexample,theinaccurateestimationofthemulti- path waveform shape, the improper ¯nite time interval gating, low resolution todecomposeadjacentsignalcomponents,andsoon. Intheprevioustest,the number of combined multipaths were N = 13 out of N (1) = 49, N (2) = 34, and N (3) = 21 where N (i) is the number of decomposed signals for the i th antenna orientation measurement. ² In the propagation measurements, all sensors in the array need to capture all common multipath signals of the planar electromagnetic wave for the best performance of the decomposition algorithm. Hence, if the size of the array is small, it is more likely for the sensors to undergo the common signals of the planar wave at the cost of the angular resolution of the array. ² The PVC antenna supporting structure for the virtual array measurement might interfere with the channel measurement and degrade the result. Ad- ditionally, the receiving antenna sensitivity, which is a function of wave im- pinging direction, was quantized spatially by every 15 degrees. More antenna sensitivity measurements with a narrower spatial resolution can decrease ap- proximation errors of the estimation. 84 Chapter 7 Conclusions and Future Work 7.1 Conclusions The (¯rst) polarization estimation process in Chapter 4 for UWB multipath chan- nels has low calculation complexity and requires only three propagation measure- ments. That is, by measuring three linearly independent polarizations, we can characterize the e®ect between the impinging electric ¯eld and receiving antenna. Since the polarization estimation process was veri¯ed for the antenna which can be approximated to the Hertzian dipole antenna, the estimation process is applicable for any electronically small antenna (antenna that ¯ts within a ball of ¸=2¼ ra- dius), generally. The estimation process is applicable to any type of UWB signal and narrow-band signal. Receiving antenna orientation can have a considerable e®ect on the performance of an indoor UWB radio receiver. Characterizing the polarization e®ect of the electric ¯eld and a non-Hertzian dipole receiving antenna in a multipath channel remains an elusive goal. 85 7.2 Future Work 7.2.1 Polarization Diversity Channel Measurement Theimpingingelectric¯eldonthereceivingantennacanbeestimatedbymeasuring three linearly independent polarizations, as described in previous chapters. With a given transmitted signal, the estimation process can measure a transfer function from the transmitting antenna terminal to the receiving antenna reference point. Bythereciprocitytheorem,theestimationprocesscanberepresentedasmeasuring atransferfunctionfrom thereceivingantennaterminalto thetransmittingantenna reference point. Let's go back to the question in Chapter 1. How can one measure a \physical channel" from a location to another location in the sense of polarization diversity? Can we measure a transfer function from the transmitting antenna reference point to the receiving antenna reference point? Forascatteringmeasurement, physicalchannelhasapropagation path, thatis, the transmitting antenna ¡! target ¡! the receiving antenna. The polarization diversity of the single path channel can be represented by a polarization scattering matrix, which is de¯ned as [17] E s (f)= ¹ S(f)¢E i (f); where the polarization scattering matrix ¹ S(f) relates the scattered electric ¯eld E s (f) and the incident electric ¯eld E i (f) at frequency f. The polarization scat- teringmatrix ¹ S(f)maybeconsideredasa\physicalchannel"representationforthe scattering measurement. The polarization scattering matrix ¹ S(f) in the scattering measurement can be modeled by a vector. Because E(f) can be decomposed into 86 two independent directions of polarizations (there is no component in the direction of propagation), the polarization scattering matrix ¹ S(f) is a 2£2 complex matrix: 2 6 4 E s V (f) E s H (f) 3 7 5 = 2 6 4 S VV (f) S HV (f) S VH (f) S HH (f) 3 7 5 2 6 4 E i V (f) E i H (f) 3 7 5 (7.1) where each of the scattered electric ¯eldE s (f) and the incident electric ¯eldE i (f) has linearly independent vector components E V (f) and E H (f). However, the rep- resentation in Equation (A.8) cannot be applied to a multipath channel case, since in the multipath channel the impinging electric ¯eld on the receiving antenna may have more than two dimensions of diversity. In future research, UWB polarization diversity channel measurement will be investigated in multipath channels. 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Prata, Jr., \Broadband printed circuit board folded di- pole antenna," in IEEE Antennas and Propagation Society Symposium, vol. 1, pp. 771-774, Montery, CA, Jun. 2004. 90 Appendix A UWB Polarization Measurements in an Anechoic Chamber A.1 Introduction In this thesis, we consider UWB polarization measurements in multipath channels. For the ¯rst step of the research, UWB polarization measurements in a single path channel is studied. Firstly, the characteristics of three antennas we used are described. Secondly, bistatic UWB impulse radar range measurements are introduced. The measurements were conducted to characterize the polarization of a scattered UWB signal in a shielded anechoic chamber. Thirdly, a polarization estimationprocessforascatteredUWBsignalisintroducedandwasappliedtothe measurement data, based on a vector transfer function model of the transmitting antenna, the propagation channel, and the receiving antenna. The ¯delity of the estimation process is veri¯ed by comparison between the measured and estimated waveforms for a reoriented receiving antenna. 91 A.2 Antenna Characteristics Measurement A.2.1 The Re°ection Coe±cient The re°ection coe±cient and the frequency response measurements were carried out to characterize antennas in a shielded anechoic test chamber 1 using a Hewlett Packard HP8720D vector network analyzer (VNA). Two TEM horn antennas and a broadband printed circuit board folded dipole (FD) antenna [36] were tested, which are shown in Figure 2.2 and A.1. In the scattering measurement, TEM horn antenna # 1wasusedasatransmittingantenna,andTEMhornantenna # 2andthe FDantennawereusedasreceivingantennas. Forbothtypesofre°ectioncoe±cient and insertion loss measurements, the VNA was set up to sweep from 0.05 to 16.05 GHz in 1601 steps, using an IF bandwidth of 3000 Hz with a sweep time 3.2472 s. Calibration of the VNA was performed at the connection point(s). It is shown that the re°ection coe±cients jS 11 (f)j of the TEM horn antennas and the FD antenna are less than -9 dB for the frequency band higher than 1.7 GHz and the frequency band 1.5 - 4.2 GHz, respectively as shown in Figure A.2. A.2.2 The Frequency Response and the Sensitivity Frequency response S 21 (f) measurements were conducted by the VNA to char- acterize the boresight directional sensitivities of TEM horn antennas and some directional sensitivities of the FD antenna. The sensitivity H R (f;µ;Á), which acts 1 UltRa Lab at University of Southern California incorporates a radio frequency shielded ane- choic test chamber with inside dimensions of 9.144 m £ 4.572 m £ 4.572 m and with NRL ¯re-rated broadband pyramidal type of absorbers. The usable frequency range of the chamber is from 300 MHz to 18 GHz. The quiet zone performance was proposed to be lower than -35 dB for higher frequency than 1 GHz, and the chamber was manufacured by Advanced ElectroMagnetics, Inc. 92 (a) Pictures of a TEM horn antenna z' y' x' 13.5 cm 6.2 cm 19.5 cm 18.0 cm 4.0 cm 16.2 cm (b) The geometry and the dimension of a TEM horn antenna Figure A.1: Pictures and the geometry of a TEM horn antenna. 93 0 1 2 3 4 5 6 7 8 x 10 9 −30 −25 −20 −15 −10 −5 0 Frequency (Hz) |S 11 (f)| (dB) horn1 horn2 fd Figure A.2: The re°ection coe±cients of TEM horn and FD antennas. 94 as the receiving transfer function of the antenna, has units of length, and is de¯ned in [27] by V R (f) = E(f)¢H R (f;µ;Á) = E(f)¢H R (f;µ;Á)^ ½ a (A.1) where V R (f) is the Fourier transform of the voltage amplitude at the receiving antenna terminal, E(f) is the impinging electric ¯eld at the receiving antenna reference point, and ^ ½ a is the receiving antenna polarization, which consists of a ^ µ¡component and a ^ Á¡component 2 . In Equation (A.1), the sensitivity is rep- resented as a product of the receiving polarization ^ ½ a and the scalar sensitivity H R (f;µ;Á) when the polarization is matched to a wave arriving at an elevation angle µ and an azimuth angle Á. When the transmitting and receiving antennas are connected to the calibrated connection points of port 1 and port 2, the frequency response from port 1 to port 2 is represented in [27] by S 21 (f)= (j2¼f)´ 0 4¼cZ 0 H T (f) e ¡j2¼fR a =c R a H R (f;µ;Á) (A.2) where H T (f) and H R (f;µ;Á) denote the transmitting and receiving antenna sen- sitivities, Z 0 = 50 corresponds to the identical source and load impedances, ´ 0 =377 is the intrinsic impedance of free space, R a is the distance between the transmitting and receiving antenna reference points, and c = 3£10 8 m/s is the speedoflight. Equation(A.2)assumesthatthetransmittingandreceivingantenna 2 The sensitivity H R (f) is a function of the antenna e®ective length L R (f), namely H R (f) = L R (f) Z0 Z R +Z0 where the output impedance of the antenna is Z R when the antenna is loaded with impedance Z 0 . 95 polarizations are matched, and the transmitting antenna is boresighted on the re- ceiving antenna. The sensitivity of the receiving antenna is the receiving antenna transfer function, when the source and load impedances are matched at Z 0 . The sensitivities of two TEM horn antennas and the FD antenna can be solved using the measured S 21 (f) data of all (three) antenna combinations. Three S 21 (f) measurements were conducted with boresight transmission and reception: 1) the TEM horn antenna # 1 to the TEM horn antenna # 2, 2) the TEM horn antenna # 1 to the FD antenna, and 3) the TEM horn antenna # 2 to the FD antenna. Then by Equation (A.2), the boresight sensitivities of TEM horn and FD antennas can be solved. There were more S 21 (f) measurements for various FD receiving antenna directions: every 5 ± azimuth angle Á, that is, Á 2 © = f0 ± ;5 ± ;:::;355 ± g, and the elevation angle µ 2 £ = f20 ± ;30 ± ;40 ± ;50 ± ;60 ± ;70 ± ;80 ± ;90 ± g. Assuming that the FD antenna sensitivity H R (f;µ;Á) has a symmetry on the H-plane, that is, H R (f;µ;Á) = H R (f;180 ± ¡µ;Á), the sensitivities for any integral µ and Á were interpolatedusingtheLagrangeinterpolatingpolynomialofdegree2withthemea- suredS 21 (f)data 3 . ForallS 21 (f)measurements,thedistancebetweentwoantenna 3 For a receiving antenna direction in the upper half sphere (µ <90 ± ), let µ 1 ;µ 2 ;µ 3 , and µ 4 be 4 elevation angles near the elevation antenna direction µ in £. Let Á 1 ;Á 2 ;Á 3 , and Á 4 be 4 azimuth angles near the azimuth antenna direction Á in ©. The ¯rst step in estimating the sensitivity H R (f;µ;Á) is to adjust the elevation angle µ. For each value of i in f1;2;3;4g, the estimate of H R (f;µ;Á i ) is the arithmetic mean of two Lagrange interpolation estimates: 1) Lagrange interpolation estimate of H R (f;µ;Á i ) by H R (f;µ 1 ;Á i ), H R (f;µ 2 ;Á i ), and H R (f;µ 3 ;Á i ), and 2) Lagrange interpolation estimate of H R (f;µ;Á i ) by H R (f;µ 2 ;Á i ), H R (f;µ 3 ;Á i ), and H R (f;µ 4 ;Á i ). The second step in estimating the sensitivity H R (f;µ;Á) is to adjust the azimuth angle Á. The estimate of H R (f;µ;Á) is the arithmetic mean of two Lagrange interpolation estimates: 1) Lagrange interpolation estimate of H R (f;µ;Á) by H R (f;µ;Á 1 ), H R (f;µ;Á 2 ), and H R (f;µ;Á 3 ), 2) Lagrange interpolation estimate of H R (f;µ;Á) by H R (f;µ;Á 2 ), H R (f;µ;Á 3 ), and H R (f;µ;Á 4 ). Therefore, with the symmetry on the H-plane, the sensitivity H R (f;µ;Á) for any µ and Á can be interpolated by the Lagrange interpolating polynomial of degree 2. 96 0 1 2 3 4 5 6 7 8 x 10 9 −70 −65 −60 −55 −50 −45 −40 −35 −30 −25 −20 Frequency (Hz) [m (dB)] horn1 horn2 fd Figure A.3: The magnitudes of TEM horn and FD antenna sensitivities in the boresight direction. reference points was 1.016 m with polarization match, and an Orbit AD-20R and BIB-100G antenna positioner was used to adjust the direction of the FD antenna. The total number of S 21 (f) measurements was 3+(360=5)£8 = 579. Figure A.3 shows the sensitivities of antennas. 97 A.3 A Scattered UWB Signal Measurement in an Anechoic Chamber A.3.1 MeasurementEquipmentsandTransmittingAntenna Ascatteringexperimentwasconductedintheshieldedanechoicchambertoprohibit interference from the outside areas and minimize re°ections created by scattering of energy by absorbing materials on the inside the chamber. A scattering target was placed R 1 = 4:189 m from a transmitting antenna and R 2 = 4:064 m from a receivingantennatobeinthefar-¯eldregion. FigureA.4depictsthatbistaticangle is 14:036 ± . The target and antennas were supported on a virtually electromagnetic wave transparent styrofoam structures of equal height. Amonocyclepulsewithasub-nanosecondwidthproducedbyanAvetechAVE2- C-5000 monocycle pulse generator excited the transmitting antenna periodically every 2.5 ¹s. The received scattered signal's voltage was measured with a Hewlett Packard HP54750A digital sampling oscilloscope (DSO). A 20 dB attenuated di- rect output measurement of the monocycle pulse generator by the DSO and its FFT spectrum are shown in Figure 3.6. The output and input impedance of the monocycle generator and the DSO are matched at Z 0 =50 . The e®ective 10 dB bandwidthofthemonocyclegeneratoroutputisfrom0.7to5.9GHz. Withrespect to the re°ection coe±cients of antennas in Figure A.2, the bandwidth of antennas con¯rms to the bandwidth of the monocycle pulse in Figure 3.6. A stable clock triggered both the monocycle generator and the DSO. The clock was generated by a Hewlett Packard HP8110A pulse generator with 0 ns delay, 50% duty cycle, 2 ns leadingedge,thehighvoltage4V,thelowvoltage0V,andtherepetitionfrequency 98 4.064 m Top View (floor RAM omitted for clarity) 2 4.189 m 1.016 m Tx Rx copper plate 0.45 m 3.048 m 0.711 m 9.144 m 4.572 m 4.064 m 2 Tx & Rx copper plate 0.41 m 0.8636 m 0.711 m 4.572 m Side View Radar absorbing material (RAM) Monocycle Pulse Generator DSO LNA Pulse Generator (Trigger) 14.036 o The absolute Coordinate system x y z Figure A.4: The scattering measurement setup. 99 400 kHz. The trigger level of the DSO was set to 50 mV at the rising edge with a 20 dB attenuator and a blocking capacitor connected in the trigger line. The TEM horn antenna # 1 was used as a transmitting antenna, which was carefully boresighted on the target centroid by a laser pointer to maximize received signal strength and to provide the most uniform incident ¯eld intensity across the target. The transmitting antenna was put into a styrofoam structure to be rotated by ¡60 ± on the boresight from a vertically polarized antenna orientation. For example, when the perfect conductor is placed at the location of the target in Fig- ure A.4, the scattered electric ¯eld polarization at the receiving antenna reference point would be approximately ^ ½ e = [ 0; p 3 2 ; 1 2 ] T . The target was a copper plate with dimensions of 0.45 m£ 0.41 m, which was placed to make a normal vector on the copper plate bisect the bistatic angle between the transmitting and receiving antennas. Chamber shielding with absorbing materials on the inside the chamber reduces outside interference and attenuates any undesired scattered signals signi¯cantly other than the desired target-scattered signal, that is, the transmitting antenna! target! receiving antenna re°ection. Therefore, in the presence of the target, the received electromagnetic ¯eld can consist of the target-scattered signal, attenuated undesired scattered signal, and the ¯eld directly coupled from the transmitting TEM horn antenna. A.3.2 Receiving Antenna Two types of the scattering measurement were conducted with the receiving TEM hornantenna # 2(TEMmeasurement),andthereceivingFDantenna(FDmeasure- ment). Forbothmeasurements, themeasureddatawillbeprocessedforestimating 100 thepolarizationofthescatteredimpingingwaveinthenextchapter,assumingthat a scattered wave impinging direction ^ k is known. In the scattering measurement with the receiving TEM horn antenna # 2 (TEM measurement), the TEM horn antenna # 2 was boresighted on the target centroid, aligned with the known scattered wave impinging direction ^ k. For estimating the polarizationofthetarget-scatteredwave, acoupleofreceivedwaveformsweremea- sured with vertically and horizontally polarized receiving antennas. Another re- ceivedwaveformwasmeasuredwiththeantennarotated¡45 ± aroundtheboresight from the original vertically polarized antenna orientation, for the validation of the target-scattered wave polarization estimator in the TEM measurement. On the other hand, in the scattering measurement with the receiving FD an- tenna (FD measurement), the same target-scattered wave was measured with the same target present at the same receiving antenna reference point as in the TEM measurement, for estimating the target-scattered wave polarization. However, the target-scatteredwavedidnotimpingetotheboresightofthereceivingFDantenna. ReceivedwaveformsweremeasuredwiththreeorthogonalFDantennaorientations, which are referred as vertical, horizontal, and radial antenna orientations (V, H, andRorientations). FigureA.5showstheantennaorientationsintheantenna'sco- ordinate system with the target-scattered wave impinging direction ^ k=[¡1;0;0] T . The absolute coordinate system is de¯ned consistently with the chamber geometry to facilitate understanding the actual spatial plan as depicted in Figure A.4. Three Cartesian bases of the antenna's coordinate system of V, H, and R orientations are shown with the corresponding receiving antenna polarization in Table A.1. Three antennaorientationsareorthogonalsuchthat ^ x 0 V = ^ y 0 H =¡^ z 0 R , ^ y 0 V = ^ z 0 H = ^ y 0 R ,and ^ z 0 V = ^ x 0 H = ^ x 0 R , wherethesubscriptdenotesthecorrespondingantennaorientation. 101 y θ x' z' y' x z ρ ρ ρ ρ e ρ ρ ρ ρ a k 2π - ϕ (a) V orientation. y θ z' y' x z ρ ρ ρ ρ e k ϕ x' ρ ρ ρ ρ a (b) H orientation. y θ z' y' x z ρ ρ ρ ρ e k 2π - ϕ x' ρ ρ ρ ρ a (c) R orientation. y θ x' z' y' x z ρ ρ ρ ρ e ρ ρ ρ ρ a k 2π - ϕ (d) E orientation. Figure A.5: The geometry of the scattered impinging wave and a receiving FD antenna model. The solid line shows the virtual sphere and the H-plane of the antenna. The antenna polarization ^ ½ a and the scattered electric ¯eld polarization ^ ½ e are on the small dotted elliptical plane, which is perpendicular to the wave impinging direction ^ k and tangential to the sphere for the plane wave. 102 Table A.1: The antenna's coordinate system bases the receiving antenna polariza- tion for receiving FD antenna orientations. ^ x 0 ^ y 0 ^ z 0 ^ ½ a V orientation 2 4 0:6124 0:7071 ¡0:3536 3 5 2 4 ¡0:6124 0:7071 0:3536 3 5 2 4 0:5000 0 0:8660 3 5 2 4 0 0 1 3 5 H orientation 2 4 0:5000 0 0:8660 3 5 2 4 0:6124 0:7071 ¡0:3536 3 5 2 4 ¡0:6124 0:7071 0:3536 3 5 2 4 0 0:8944 0:4472 3 5 R orientation 2 4 0:5000 0 0:8660 3 5 2 4 ¡0:6124 0:7071 0:3536 3 5 2 4 ¡0:6124 ¡0:7071 0:3536 3 5 2 4 0 ¡0:8944 0:4472 3 5 E orientation 2 4 0:8095 0:4132 0:4172 3 5 2 4 ¡0:5567 0:7660 0:3124 3 5 2 4 ¡0:1868 ¡0:4924 0:8501 3 5 2 4 0 ¡0:5012 0:8653 3 5 For the validation of the target-scattered wave polarization estimator in the FD measurement, another received waveform was measured with the reoriented antenna, whose orientation is represented as E orientation in Table A.1. E orien- tation was chosen to have the receiving antenna polarization di®erent from those of previous measurements (see the mutual inner product column in Table A.2). Supposing when the perfect conductor was placed at the location of the target, the receiving polarization of E orientation would be about perpendicular to the scatteredwavepolarization. Therefore, thereceivedsignalforEorientationisfrom the cross-polarized scattered ¯eld component, and may be expected to have a low SNR. To attain the FD receiving antenna orientations, the antenna positioner was utilizedprecisely. Theantennacablewas¯xedtoastyrofoamstructurebuiltonthe positioner,asshowninFigureA.6. Thepositionerisanazimuth-over-elevationtype antenna positioner, which consists of a gantry which provides the µ rotation and an azimuth positioner which provides the Á rotation [15]. To locate the receiving 103 Table A.2: Geometrical description of receiving FD antenna orientations. antenna AoA mutual ^ ½ a inner product arctan £ orientation µ Á V H R E (^ ½ a ¢^ y) (^ ½ a ¢^ z) ¤ V 60 ± 315 ± 1.0000 0.4472 0.4472 0.8653 0 ± H 128 ± 51 ± 0.4472 1.0000 -0.6000 -0.0613 63.4349 ± R 128 ± 309 ± 0.4472 -0.6000 1.0000 0.8353 -63.4349 ± E 101 ± 325 ± 0.8653 -0.0613 0.8353 1.0000 -30.0812 ± antenna at the desired receiving antenna reference point and adjust the antenna orientation,allexperimentproceduresintheFDmeasurementwereconductedwith thestyrofoamstructuretilted(µrotation)androtatedataproperangle(Árotation) by the positioner 4 . For all measurements in both of the TEM measurement and the FD measurement, receiving antenna was located at the same antenna reference point with di®erent orientations. Thereceivingantennaconvertsthescattered¯eldintoavoltageatthereceiving antennaterminal. Sincethevoltageatthereceivingantennaterminalwasweak,the voltage was ampli¯ed by a low noise ampli¯er (LNA). In Figure 3.8, the frequency response of the LNA is shown to be about 24 dB gain from 1 to 4 GHz. The outputoftheLNAwassampledbytheDSOwithasamplingrateof200GHz. The numberofsampleswas4000,andthespanofthesampleswas20nscenteringabout the scattered signal. This sampled signal was ensemble-averaged over 256 times to increase the signal-to-noise power ratio (SNR). For each antenna orientation, the received waveform was measured in two cases, with and without the target, to 4 When the antenna orientation was adjusted by the positioner, there were 3 sequential move- ments of the antenna orientation adjustment: 1) tilting antenna ® ± at the joint between the antenna terminal and the cable connector, 2) rotating the upright styrofoam structure ¯ ± on the (horizontal) x-y plane by the positioner (Á rotation), and then 3) tilting the styrofoam structure ° ± toward the target by the positioner (µ rotation). V orientation was set up with ®=0 ± , ¯ =¡45 ± , and ° =30 ± , H orientation was with ®=¡90 ± , ¯ = 45 ± , and ° = 30 ± , R orientation was with ® = ¡90 ± , ¯ = ¡45 ± , and ° = 30 ± , and E orientation was setup with ®=¡50 ± , ¯ =¡40 ± , and ° =30 ± . 104 (a) Transmitting (left) and receiving (right) TEMhornantennastructuresintheTEMmea- surement. (b) TransmittingTEMhorn(left)andreceiving FD (right) antenna structures in the FD mea- surement. (c) Target from the receiving antenna (d) FD antenna structure Figure A.6: Measurement setup pictures. 105 extract the target-scattered signal component from the received signal where the target-scatteredsignal,attenuatedundesiredscatteredsignals,andthe¯elddirectly coupled from the transmitting TEM horn antenna are coexistent. A.4 PolarizationEstimationofaScatteredUWB Signal A.4.1 Scattering Measurement Transfer Function System Before a polarization estimation process is introduced, the scattering measurement is considered from a transfer function point of view in the frequency domain. The e®ects of LNA and connection cables are ignored in this exposition. The scattering measurement system is described by a function of the transmitting antenna sensi- tivity, the scattering property of the target, and the receiving antenna sensitivity. When the input monocycle pulse in Figure 3.6 with Fourier transform V P (f) drives the transmitting antenna, the Fourier transform of the received signal's voltage V T (f) being sampled by the DSO can be modeled as V T (f)=V P (f) " ¡ H EG (f)¢H R (f) ¢ + M X i=1 ³ H i EG (f)¢H i R (f) ´ # +N T (f): (A.3) The¯rstpartinbracketscorrespondstothereceivedsignalcomponentprovidedby scattering from the target, and the second part in the brackets corresponds to the receivedsignalcomponentcausedbyallMothersignalpathsbetweenthetransmit- ting and receiving antennas. A vector transfer functionH (i) EG (f) relates the electric ¯eld at the receiving antenna reference point to the voltage at the transmitting 106 antenna terminal for a propagation path, and H (i) R (f) is a corresponding receiving antenna sensitivity. Receiver noise is represented by N T (f). To extract the target-scattered signal component from the received signal, a received waveform without the target is measured and represented with a receiver noise N B (f) as V B (f) = V P (f) M X i=1 ³ H i EG (f)¢H i R (f) ´ +N B (f): (A.4) Then, subtractionEquation(A.3)from(A.4)givesadesiredtarget-scatteredsignal component formula as V S (f) = V T (f)¡V B (f) = V P (f) h H EG (f)¢H R (f) i +N S (f) (A.5) where N S (f)=N T (f)¡N B (f). By the de¯nition of H EG (f), V S (f)= h E s (f)¢H R (f) i +N S (f) (A.6) where E s (f) is the scattered electric ¯eld which is impinging to the receiving an- tenna. The incident electric ¯eld to the target in the far-¯eld region generated by the transmitting antenna is given by [27] E i (f)=V P (f) (j2¼f)´ 0 4¼cZ 0 H T (f) e ¡j2¼fR 1 =c R 1 (A.7) 107 where R 1 is the distance between the transmitting antenna and the target. Then, withagivenpolarizationscatteringmatrix 5 ¹ S(f),avectortransferfunctionH EG (f) fromthetransmittingantennaterminaltotheelectric¯eldatthereceivingantenna is given by H EG (f)= ¹ S(f)¢ ( (j2¼f)´ 0 4¼cZ 0 H T (f) e ¡j2¼fR 1 =c R 1 ) : (A.8) Finally,substitutingEquation(A.8)into(A.5), thescatteringmeasurementsystem is described by a function of the transmitting antenna sensitivity, the scattering property of the target, and the receiving antenna sensitivity. V S (f)=V P (f) " H R (f)¢ ¹ S(f)¢ ( (j2¼f)´ 0 4¼cZ 0 H T (f) e ¡j2¼fR 1 =c R 1 )# +N S (f): A.4.2 Wave Polarization Estimation Suppose that a received waveform is measured with a known target-scattered wave impinging direction to the receiving antenna. After the ¯rst measurement, the an- tenna orientation only is changed, and another received waveform voltage is mea- sured again. The goal of this process is to develop a wave polarization estimation process,whichisvalidatedbycomparingthethirdmeasuredwaveformvoltagewith 5 The polarization scattering matrix is de¯ned as [17] E s (f)= ¹ S(f)¢E i (f) Because E(f) can be decomposed into two independent directions of polarizations (there is no component in the direction of propagation ^ k), the polarization scattering matrix ¹ S(f) is a 2£2 complex matrix: · E s V (f) E s H (f) ¸ = · S VV (f) S HV (f) S VH (f) S HH (f) ¸ " E i V (f) E i H (f) # whereE s (f)andE i (f)arethescatteredandincidentelectric¯elds,eachwithlinearlyindependent vector components E V (f) and E H (f). 108 the estimator for a di®erent antenna orientation from two antenna measurements with di®erent orientation. It is assumed that the receiving antenna sensitivity H R (f;µ;Á) is known for all antenna orientations. With a di®erent antenna orienta- tions, the impinging wave has di®erent elevation and azimuth angle-of-arrival and a di®erent receiving antenna polarizations. That is, when the antenna orientation is changed, the magnitude and direction of H R (f;µ;Á) are changed. Whentheelectromagneticwavepropagatesoverachannel,especiallywhenwith penetration and/or re°ection by an object, the electric ¯eld polarity is a function of time. Hence, even for the linearly polarized UWB transmitted signal's electric ¯eld, the signal undergoes channel ¯ltering so that the polarity becomes non-linear andtime-varying. Hence, thescatteredelectric¯eldE s (f)atthereceivingantenna reference point is represented by a equivalent time-domain representation ¹ e s (t). Then,withtwoantennaorientationmeasurements,thetime-domainrepresenta- tions of the received waveform at the DSO with measurement noise are represented by Equation (A.6) as v 1 (t)=[(¹ e s (t)¢ ^ ½ 1 )¤h R (t;µ 1 ;Á 1 )]+n 1 (t) (A.9) v 2 (t)=[(¹ e s (t)¢ ^ ½ 2 )¤h R (t;µ 2 ;Á 2 )]+n 2 (t) (A.10) where v k (t) is the k th measured voltage at the DSO at time t, h R (t;µ k ;Á k ) is the time-domain representation of the scalar sensitivity H R (f) for a wave arriving at elevation angle µ k and an azimuth angle Á k , ^ ½ k is receiving antenna polarization, and n k (t) is noise or artifacts from other multipath (k =1 or 2). The receiving antenna polarizations are perpendicular to the wave impinging direction. Hence, for the plane wave, two receiving antenna polarizations and the impinging wave polarization are in the same plane which is perpendicular to the 109 wave impinging direction ^ k, as depicted in Figure A.5. Therefore, by the Gram- Schmidt orthogonalization process, the impinging electric ¯eld is given by ¹ e s (t)= ^ ½ 1 (¹ e s (t)¢ ^ ½ 1 )+ ^ ½ 2 ¡(^ ½ 1 ¢ ^ ½ 2 )^ ½ 1 k^ ½ 2 ¡(^ ½ 1 ¢ ^ ½ 2 )^ ½ 1 k ¢ · ^ ½ 2 ¡(^ ½ 1 ¢ ^ ½ 2 )^ ½ 1 k^ ½ 2 ¡(^ ½ 1 ¢ ^ ½ 2 )^ ½ 1 k ¢¹ e s (t) ¸ (A.11) where ^ ½ 1 and ^ ½ 2 are antenna polarizations, and (¹ e i (t)¢ ^ ½ 1 ) and (¹ e i (t)¢ ^ ½ 2 ) can be estimated by Equation (A.9) and (A.10). Estimation of (¹ e s (t)¢ ^ ½ 1 ) and (¹ e i (t)¢ ^ ½ 2 ) uses a deconvolution process adapted from [25], ¹ e s (t)¢ ^ ½ k =F ¡1 ½ V k (f)H ¤ R (f;µ k ;Á k ) jH R (f;µ k ;Á k )j 2 +C ¾ (A.12) where the superscript ( ¤ ) denotes the complex conjugate,F ¡1 is the inverse Fourier transform, and V k (f) is the Fourier transform of v k (t), for k =1 or 2. The smooth- ing parameter C is selected so that the e®ective bandwidth of the deconvolved response (¹ e s (t)¢ ^ ½ k ) in Equation (A.12) just falls within that of the measurement system. For validation of the result in Equation (A.11), the third waveform v 3 (t) for a di®erentantennaorientation,wherethereceivingantennasensitivityis h R (t;µ 3 ;Á 3 ) and the receiving polarization is ^ ½ 3 , is estimated and compared with the measured waveform. Using Equation (A.6), (A.11), and (A.12), the estimate is e v 3 (t)=(¹ e s (t)¢ ^ ½ 3 )¤h R (t;µ 3 ;Á 3 ): (A.13) This polarization estimation method and the waveform estimation technique are applied and tested with measured data in the following section. 110 A.4.3 Results and Discussion In this section, the UWB signal polarization estimation process is applied to the scattering measurement data. In Figure A.7, a received waveform for the vertically polarized TEM horn receiving antenna with the target present is shown with a 50 nsspan 6 . Thedirectlycoupledcomponentfromthetransmittingantennawassam- pled around 1.44£10 ¡7 s, and the target-scattered signal component was sampled around 1.67£10 ¡7 s. To extract the target-scattered signal component from the received signal, a received waveform without the target was subtracted from the received waveform with the target present. Figure A.8 shows an example of the ex- tractionfortheverticallypolarizedTEMhornreceivingantenna. Forallscattering measurements,thedirectlycoupledcomponentdisturbedthescatteringcomponent insigni¯cantly (as shown in Figure A.8) because the di®erence between the propa- gation path length of the scattered ¯eld and that of the directly coupled ¯eld was much longer than the spatial extent of the directly coupled ¯eld component. IntheTEMmeasurement,receivedwaveformsweremeasuredwiththereceiving TEMhornantenna # 2,andtheantennawasrotatedaroundtheboresightdirection by 0 ± , ¡45 ± , and ¡90 ± , from the original vertically polarized antenna orientation. The target-scattered signal component for each antenna orientation is plotted in Figure A.9. The signal's voltage in the ¡45 ± -rotated case had a larger peak than those in the 0 ± and ¡90 ± -rotated cases. By using signal waveforms in the 0 ± and ¡90 ± -rotatedcases, thepolarizationestimationprocess can calculate theestimated waveform in the¡45 ± -rotated case. However, since the scattered wave had under- gone the same receiving antenna sensitivity with only di®erent receiving antenna 6 The ripples around 1.4£10 ¡7 s in Figure A.7, which is about 0.05£10 ¡7 s earlier than the peak, is the response of the ripples in monocycle pulser output. Figure 3.6 shows that the ripples around 2.2£10 ¡8 s is about 0.05£10 ¡7 s earlier than the peak. 111 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 x 10 −7 −1 −0.5 0 0.5 Time (s) Amplitude (V) FigureA.7: ThereceivedwaveformfortheverticallypolarizedTEMhornreceiving antenna with the target present. 112 1.665 1.67 1.675 1.68 1.685 x 10 −7 −0.2 −0.1 0 0.1 0.2 Time (s) Amplitude (V) w/ target w/o target 1.665 1.67 1.675 1.68 1.685 x 10 −7 −0.2 −0.1 0 0.1 0.2 Time (s) Amplitude (V) scatter component Figure A.8: Received signals with and without the target are plotted overlayed for the vertically polarized TEM horn receiving antenna (top). The target-scattered signal component is plotted (bottom), which is the di®erence between two signals at the top. 113 polarizations for all cases, no frequency-domain calculation of antenna sensitivity is required. That is, the received waveform estimation by Equation (A.13) can be simpli¯ed to e v 3 (t)=(^ ½ 3 ¢ ^ ½ 1 )v 1 (t)+(^ ½ 3 ¢ ^ ½ 2 )v 2 (t) (A.14) when the two receiving antenna polarization ^ ½ 1 and ^ ½ 2 are orthogonal. Then, byEquation(A.14), the measured andestimated waveformsinthe¡45 ± - rotated case are shown in Figure A.10. The peak voltage in the error was about 21.8 dB less than the peak voltage in the measured data. In the FD measurement, four received signals with di®erent antenna orienta- tionsweremeasuredwiththeFDreceivingantenna: three(orthogonal)V,H,andR orientations for the polarization estimation, and E orientation for the validation of the polarization estimatior. The target-scattered signal components for V, H, and R orientation measurements show a similar envelope in Figure A.11. However, the receivedwaveformforRorientationhas a°ippedpolarityfrom otherwaveformson account of the corresponding antenna polarization. For each antenna orientation, corresponding antenna sensitivity is shown in Figure A.12, including the antenna sensitivity for E orientation. Using Equation (A.13) with two received waveforms among three orthogonal measurements, the polarization estimation process can calculate the estimator of the received waveform for an arbitrary antenna orien- tation. Equation (A.13) in the estimation process is unable to be simpli¯ed as Equation (A.14), since the scattered wave had undergone di®erent antenna sensi- tivity for each antenna orientation measurement. Hence, for the FD measurement, the polarization estimation process calculated the estimator of the received wave- form for E orientation, using received waveform for V and R orientations with 114 1.665 1.67 1.675 1.68 1.685 x 10 −7 −0.2 −0.1 0 0.1 0.2 0.3 Time (s) Amplitude (V) 0 ° −45 ° −90 ° Figure A.9: Received waveforms of the target-scattered signal component with the receiving TEM horn antenna in the TEM measurement, where the antenna was rotated around the boresight direction by 0 ± , ¡45 ± , and ¡90 ± , from the original vertically polarized antenna orientation. 115 1.665 1.67 1.675 1.68 1.685 x 10 −7 −0.2 0 0.2 Time (s) Amplitude (V) measured estimated 1.665 1.67 1.675 1.68 1.685 x 10 −7 −0.2 0 0.2 Time (s) Amplitude (V) error (measure − est) Figure A.10: Measured and estimated waveforms of the target-scattered signal component with the 45 ± -rotated TEM horn receiving antenna (top) and the error betweenthemeasuredandestimatedwaveform(bottom)intheTEMmeasurement. 116 1.66 1.665 1.67 1.675 1.68 1.685 1.69 x 10 −7 −0.1 −0.05 0 0.05 0.1 0.15 Time (s) Amplitude (V) V H R Figure A.11: Received waveforms of the target-scattered signal component with V, H, and R orientations in the FD measurement. the smoothing constant C = 5:5£10 ¡8 . To validate the estimation process, the measured and estimated waveforms for E orientation are compared in Figure A.13. The peak voltage in the error was about 6.71 dB less than the peak voltage in the measured data. A calibration measurement of a received signal was introduced to assess the measurement accuracy. The transmitting TEM horn antenna was located at the target position, and the receiving TEM horn antenna was located at the same re- ceivingantennareferencepoint. Withpolarizationmatch,thetransmittingantenna was excited by the transmitted monocycle pulse, and the received waveform was 117 0 1 2 3 4 5 6 7 8 x 10 9 −65 −60 −55 −50 −45 −40 −35 −30 −25 Frequency (Hz) |H(f,θ,φ)| [m (dB)] V: H( f, 60 ° ,315 ° ) H: H( f,128 ° , 51 ° ) R: H( f,128 ° ,309 ° ) E: H( f,101 ° ,325 ° ) Figure A.12: The antenna transfer function (sensitivity) magnitudes for V, H, R, and E orientations in the FD measurement. 118 1.66 1.665 1.67 1.675 1.68 1.685 1.69 x 10 −7 −0.05 0 0.05 Time (s) Amplitude (V) meas est 1.66 1.665 1.67 1.675 1.68 1.685 1.69 x 10 −7 −0.05 0 0.05 Time (s) Amplitude (V) error (meas − est) Figure A.13: Measured and estimated waveforms of the target-scattered signal component with E orientation of the FD receiving antenna (top) and the error between the measured and estimated waveforms (bottom) in the FD measurement. 119 5.75 5.8 5.85 5.9 5.95 6 6.05 x 10 −8 −1 −0.5 0 0.5 Time (s) Amplitude (V) meas est 5.75 5.8 5.85 5.9 5.95 6 6.05 x 10 −8 −1 −0.5 0 0.5 Time (s) Amplitude (V) error (meas−est) Figure A.14: Measured and estimated waveforms of the calibration measurement setup(top)andtheerrorbetweentheestimatedandmeasuredwaveform(bottom). measured. InFigureA.14, themeasuredcalibrationsignaliscomparedwiththees- timatedcalibrationsignal, whichisaconvolutionbetweenthemeasuredmonocycle pulse signal and a measured impulse response by the VNA from the transmitted pulse exciting point to the DSO measurement point. The propagation path in the calibration measurement without the target was shortest among all the measure- ments including the TEM and FD measurements, and the directional TEM horn antennas were employed to entail high gain without polarization mismatch loss. Hence, the estimation accuracy of the calibration measurement would be the best that we can achieve by the polarization estimation process, and the error in the calibration measurement results from unavoidable experimental limit of accuracy. 120 To quantify the error energy of UWB pulse signal estimation, a pulse width de¯nition is proposed. The minimum x dB-energy pulse width (or the minimum y %-energy pulse width, where y = 10 ¡x=10+2 ) of a UWB pulse signal is de¯ned as the minimum partial time interval of the pulse signal, where y % of the signal energy is contained. For the calibration measurement estimation in Figure A.14, the minimum 90 %-energy pulse width (the minimum 10 dB pulse width) is from 5.811£10 ¡8 to 5.898£10 ¡8 s, and the normalized estimation error over the pulse width 7 is 0.1193. The minimum 85 %-energy pulse width is from 5.825£10 ¡8 to 5.870£10 ¡8 s, and the normalized estimation error over the pulse width is 0.1013. For the FD measurement estimation in Figure A.13, the minimum 90 %-energy pulse width is from 1.671£10 ¡7 to 1.684£10 ¡7 s, and the normalized estimation erroroverthepulsewidthis0.2888(3.84dBabovethecalibrationestimationerror). The minimum 85 %-energy pulse width is from 1.672£10 ¡7 to 1.676£10 ¡7 s, and the normalized estimation error over the pulse width is 0.1647 (2.11 dB above the calibration estimation error). The minimum 85 %-energy pulse width is close to the time width between the second zero crossing points from the signal peak. The estimation error in the TEM measurement was smaller than the estimator error in the FD measurement. Compared to the polarization estimation process in the TEM measurement with Equation (A.14), the polarization estimation process in the FD measurement with Equation (A.13) requires more complex frequency- domain calculations of antenna sensitivities, where some of antenna sensitivities were interpolated with antenna sensitivities derived from Equation (A.2). Mea- sured data of the FD measurement have lower SNR than that of the TEM mea- surement, which is caused by using lower gain receiving antenna. To improve the 7 The normalized estimation error over the pulse width T p is de¯ned as e e = ( R Tp r 2 (t)dt) ¡1 R Tp kr(t)¡ ^ r(t)k 2 dt, where r(t) is a measured signal, ^ r(t) is a estimate of r(t) , and the integrations are over the pulse width T p . 121 estimator result in the FD measurement, we must repeat S 21 (f) measurements in Equation(A.2)withdi®erenttransmittingorreceivingantennapositionsandaver- age the measured S 21 (f) over the repeated measurement data to acquire accurate antenna sensitivities. A reference antenna with known characteristics can be used to simplify antenna sensitivity measurements for a better result. A.4.4 Conclusion In the anechoic chamber measurement, two types of measurements were conducted for the polarization estimation process. The ¯rst measurement, the TEM horn measurement, is based on a UWB radar system channel scale model. The direc- tional TEM horn antenna was used to maximize received signal strength at the boresight. The second measurement, the FD measurement, is based on a simpli- ¯ed UWB wireless communication system channel model. The simpli¯ed channel is a single-path channel extracted from a UWB multipath channel. Hence, even withtheknowledgeofthetransmitterlocation, thewavein the single-path channel does not always impinge in the receiving antenna boresight direction. Therefore, threeorthogonalantennaorientationmeasurementswereconducted, andtheomni- directionalFDantennawasusedinthesecondmeasurement. Basedontheresultof the FD measurement, the polarization estimation process in the single-path chan- nel may be extended to the polarization estimation process in a UWB multipath channel with a proper decomposition algorithm. The accuracy of the estimation algorithm should depend on the e®ective band- width of the transmitted pulse, antenna bandwidth, antenna transfer function, and soon. ForahigherfrequencyUWBsignalthanourtestsetup,forexample3.1-10.6 GHz frequency band signal ¯tting FCC Mask [12], the algorithm should be valid, 122 but a more re¯ned receiving antenna placement technique for di®erent orientations will be necessary. 123

Asset Metadata

Creator Chang, SangHyun (author)

Core Title UWB polarization measurements in multipath channels

School Andrew and Erna Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 02/14/2007

Defense Date 01/18/2007

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

Tag antenna,OAI-PMH Harvest,polarization,ultra-wideband,UWB

Language English

Advisor Scholtz, Robert A. (committee chair), Alexander, Kenneth S. (committee member), Prata, Aluizio, Jr. (committee member)

Creator Email sanghyun@ieee.org

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m245

Unique identifier UC1142501

Identifier etd-Chang-20070214 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-158323 (legacy record id),usctheses-m245 (legacy record id)

Legacy Identifier etd-Chang-20070214.pdf

Dmrecord 158323

Document Type Dissertation

Rights Chang, SangHyun

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email uscdl@usc.edu

Abstract (if available)

Abstract Ultra-wideband (UWB) radio systems have attracted considerable attention as a candidate for wireless personal area network (WPAN) applications. In order to evaluate the performance of UWB radio systems, a radio link budget calculation is necessary for real channels. An important part in the link budget calculation is related to the effect between the receiving antenna and theelectric-field wave polarization impinging on the receiving antenna, for example, the variance on a received signal waveform or a received energy with respect to receiving antenna orientations. To characterize the polarization effect at the receiver, it is useful to establish the way to measure UWB wave polarization in real channels.