High Range Resolution Micro-Doppler Radar Theory and Its. Application to Human Gait Classification

Transcription

1 High Range Resolution Micro-Doppler Radar Theory and Its Application to Human Gait Classification DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Zachary Andrew Cammenga Graduate Program in Electrical and Computer Science The Ohio State University 2017 Dissertation Committee: Graeme Smith, Advisor Chi-Chih Chen Emre Ertin

2 Copyrighted by Zachary Andrew Cammenga 2017

3 Abstract This work advances the use of radar for target classification. High range resolution (HRR) and micro-doppler signal analysis both provide radar operators the ability to classify observed targets. However, traditional micro-doppler analysis does not account for a target moving in range and is therefore not compatible with high range resolution waveforms. We propose the use of a joint-range-time-frequency (JRTF) signature that combines the range information of a HRR radar and the frequency signature of a time-frequency transform. Complicated targets that consist of returns from multiple scatterers have been considered as the independent combination of the return from each target. This is a simplification of the problem and does not consider how the target interaction affect the micro-doppler signature. These interactions were studied and were found to be the generating factor of unique characteristics within the micro-doppler signature. The analytic expression for the JRTF signature is derived and analysis is provided for the motion of a simple pendulum. The implementation of the JRTF signature is presented and signal processing techniques for improved JRTF signature manipulation are provided. Finally, we apply the JRTF signature to the classification of human gait. Features extracted from the JRTF signature are compared to features extracted from the micro- ii

4 Doppler signature, and results show increased classification performance using the JRTF features. iii

5 For my wife and family iv

6 Vita May 2008 Holland Christian High School 2012 B.S. Electrical Engineering, University Of Michigan 2016 M.S. Electrical Engineering, The Ohio State University 2012 to present Graduate Research Associate, Department of Electrical and Computer Engineering, The Ohio State University Fields of Study: Major Field: Electrical and Computer EngineeringElectrical and Computer EngineeringElectrical and Computer Engineering v

7 Table of Contents Abstract... ii Dedication... iv Vita... v Table of Contents... vi List of Tables... ix List of Figures... xi Chapter 1: Introduction Overview Contributions Organization of Paper... 5 Chapter 2: Basic Theory Radar Basics Range Estimation Radar Range Resolution Pulse Compression vi

8 2.5. Summary of micro-doppler Theory Observation of Micro-Doppler Classification Chapter 3: Literature Critique Classification in Radar High Range Resolution Radar Early Research on Micro-Doppler High Range Resolution Micro-Doppler Classification of Human Gait in Micro-Doppler Conclusions of the Literature Survey Chapter 4: Scatterer Combination in Micro-Doppler Signatures Micro-Doppler of a Rotating Target Processing and the Time-Frequency Relationship Experimental Validation Electromagnetic Modeling and Micro-Doppler: Flat Plate Combination of Target Scatterers and Angular Dependence Chapter 5: HRR Micro-Doppler HRR Micro-Doppler Theory Experimental Processing Approach vii

9 Chapter 6: Classification Using HRR Micro-Doppler Signature Introduction to Classification Features in the JRTF signature Experimental Setup Feature Selection HRR Micro-Doppler Classification Chapter 7: Conclusions Future Works References viii

10 List of Tables Table 1: Parameters of the radar range equation Table 2: Radar and geometrical parameters for simulation and experimental validation of pendulum Table 3 Radar and geometrical parameters for simulation and experimental walking pedestrian Table 4: Radar parameters used in experiment Table 5: Radar parameters for rotating target simulations Table 6: Radar parameters for walking person simulation Table 7: Radar parameters for human activity classification Table 8: Comparison of intra and inter class separation of proposed features Table 9: List of HRR features selected for use in classification Table 10: List of narrowband features presented by Kim for classification Table 11: Results of SVM classification using decision tree and narrowband feature set Table 12: Results of SVM classification using decision tree reported by Kim Table 13: Results of Mahalanabis distance classification using narrowband feature set 125 Table 14: Results of SVM classification using decision tree and HRR feature set Table 15: Results of Mahalanobis Distance classification using HRR feature set ix

11 Table 16: Average classification rate using a single feature Table 17: Results of Mahalanobis Distance classification using only Average Velocity feature x

12 List of Figures Fig. 1: Range resolution target geometry Fig. 2: LFM signal Fig. 3: Matched filter response of a LFM signal Fig. 4: Geometry of radar and rotating target [7] Fig. 5: The micro-doppler modulation induced by rotation of a point scatterer Fig. 6: The STFT and slow time Fig. 7: Example spectrogram of a rotating target generated with STFT with a window length of 5 pulses Fig. 8: Classification procedure Fig. 9: Two-Class classification with SVM Fig. 10: Spectrogram depicting amplitude modulations Fig. 11: The two-scatterer point target scene considered. With radius R = λ and distance from radar D= 900 m Fig. 12: The phase response of a single rotating target Fig. 13: The phase response of a rotating target with a center scatterer of twice the amplitude present Fig. 14: Phasor diagram representing the target scene xi

13 Fig. 15: (a) The micro-doppler spectrogram of a single rotating target using a STFT with sliding window length of 40 (b) The micro-doppler spectrogram of a rotating target and center scatterer of twice the amplitude using a STFT with sliding window length of Fig. 16: The spectrogram results using a STFT with different sliding window sized. (a) Single rotating target. (b) Rotating target with center scattering Fig. 17: Experimental setup used observing a sphere rotating about another larger sphere Fig. 18: Comparison between (a) simulated results and (b) experimental measurements using processing consisting of both a window length of 5 and a window length of Fig. 19: Experimental results showcasing the rotating target with center scatterer Fig. 20: Geometry of flat plate aspect angle analysis Fig. 21: The amplitude (a) and phase (b) resposne of a flat plate as aspect angle changes comparing CST Microwave Studio results with theoretical calculations Fig. 22: The amplitude (a) and phase (b) resposne of a flat plate as aspect angle changes comparing noisy CST Microwave Studio results with theoretical calculations Fig. 23: The phase resposne of a flat plate as aspect angle changes comparing theoretical calculations with a model using a single point scatterer Fig. 24: The spectrogram produced from a rotating flat plate rotating concurrently with a point target at a distance of 2m from the center Fig. 25: A zoomed in look at of the aspect angles when the flat plate has an angular dependent response Fig. 26 Geometry of a radar target with rotating parts xii

14 Fig. 27 Geometry of swinging pendulum Fig. 28 Fast and slow time response of a pendulum Fig. 29 Frequency signature of pendulum Fig. 30 Calculation of the Joint-Range-Time-Frequency data cube using a TFR Fig. 31: Simulation and experimental results of person walking toward radar Fig. 32 Spectrogram of each range bin contain scattering of simulated walking person. The color scale is in db intensity and common to all sub-plots Fig. 33 Experimental spectrogram of each range bin contain scattering of walking person. The color scale is in db intensity and common to all sub-plots Fig. 34: OS CFAR of the JRTF signature Fig. 35: Sorted frequency values for OS CFAR Fig. 36: The geometry considered showing a pedestrian walking perpendicular to the radar line of sight which is located at the origin looking down the y axis Fig. 37: Number of frequency components (indicated by image intensity) as a function of range and time Fig. 38: Joint Range-Time-Frequency Cube visualized using colors to represent range bin Fig. 39: Experimental setup where A, B, C, D are paths people walk in the four experiments Fig. 40:Near-field geometry of walking person xiii

15 Fig. 41: Radar results of single person walking towards radar. (a) Depicts the traditional spectrogram. (b) Depicts the number of frequency components in each range bin. (c) Depicts the two dimensional portrayal of the joint range-time-frequency data cube Fig. 42: Radar results of single person walking perpendicular to the radar. (a) Depicts the traditional spectrogram. (b) Depicts the extracted number of frequency components in each range bin. (c) Depicts the two dimensional portrayal of the joint range-timefrequency data cube Fig. 43: Radar results of two people walking towards radar. (a) Depicts the traditional spectrogram. (b) Depicts the extracted number of frequency components in each range bin. (c) Depicts the two dimensional portrayal of the joint range-time-frequency data cube Fig. 44: Radar results of two people walking perpendicular to the radar. (a) Depicts the traditional spectrogram. (b) Depicts the extracted number of frequency components in each range bin. (c) Depicts the two dimensional portrayal of the joint range-timefrequency data cube Fig. 45: (a) Rotating target geometry with stationary target in the center. (b) Rotating target geometry with stationary target behind Fig. 46: Simulated spectrogram representation of target scene corresponding to geometry (a) and (b) Fig. 47: Simulated spectrogram of walking person considering only torso, hand, and foot motion xiv

16 Fig. 48: Simulated two dimensional representation of the JRTF signature of walking person considering only torso, hand and foot motion. Where the color represents the range bin the target is in Fig. 49: Experimental Setup for activity classification Fig. 50: Comparison of the mean and standard deviation of the max range extent of the target, a rejected feature. The mean value is given by the square and the standard deviation given by the top and bottom bar Fig. 51: Comparison of the mean and standard deviation of the mean number of scatterers, a selected feature. The mean value is given by the square and the standard deviation given by the top and bottom bar Fig. 52: Classification rate as a function of number of features used Fig. 53: Range-time signature of a person running. The color scale represents the number of frequency components present at that point in range-time Fig. 54: Comparison of the mean and standard deviation of the density of scattering centers, a selected feature. The mean value is given by the square and the standard deviation given by the top and bottom bar Fig. 55: Comparison of the mean and standard deviation of the density of scattering centers in front of the torso, a selected feature. The mean value is given by the square and the standard deviation given by the top and bottom bar Fig. 56: Comparison of the mean and standard deviation of the density of scattering centers behind the torso, a selected feature. The mean value is given by the square and the standard deviation given by the top and bottom bar xv

17 Fig. 57: Comparison of the mean and standard deviation of the average velocity, a selected feature. The mean value is given by the square and the standard deviation given by the top and bottom bar Fig. 58: Comparison of the mean and standard deviation of the range extent to scatterer ratio, a selected feature. The mean value is given by the square and the standard deviation given by the top and bottom bar Fig. 59: Suggested decision tree using SVM Fig. 60: Error rate of k-nn classification for multiple k values Fig. 61: Two dimensional feature space of Average Velocity and Bandwidth of each classified activity Fig. 62: Contribution of individual features to classification rate Fig. 63: Effects of the number of training data points used and the classification performance of the minimum Mahalanobis distance using HRR micro-doppler features Fig. 64: Effects of training data on the classification performance of SVM1 using HRR micro-doppler features Fig. 65: Average classification rates for different SNR levels xvi

18 Chapter 1: Introduction 1.1. Overview This dissertation explores the use of radar measurements of targets that exhibit micro-motions through the combination of high range resolution and micro-doppler signature analysis techniques to create a joint signature that can be used for feature extraction and classification. The identification of what a radar is observing is extremely important in order to help the radar operator, or cognitive system, to make useful decisions about future actions. Over the years the sensitivity of radar systems has improved leading to superior target detection and localization performance. Many different approaches have been used to identify the radar s target, but that continues to be a difficult task. Identifying the radar s target however, has continued to be a challenging task and many different approaches have been attempted. Contemporary radar design provides capabilities significantly beyond the classical detection and ranging. It is now possible to form radar images, using synthetic aperture radar (SAR) and inverse SAR (ISAR) methods [1], undertake sophisticated electronic scanning and beam control activities [2] and to measure characteristic signatures of targets to assist in automatic target recognition [3]. Here we concern ourselves with the study of targets joint range and Doppler radar signatures. 1

19 A target s characteristic radar signature can be estimated in different signal processing domains. Typical examples of signatures include: polarimetric signatures that are based on the target s response to the polarisation of EM radiation [4]; reflectivity signatures that represent how much of the incident radiation is backscattered by the target [5]; high range resolution (HRR) observations of targets, that ultimately lead to radar imaging, that estimate the target s physical structure as a signature [6]; and finally, micro-doppler signatures that provide information on the micro-motions of the target sub components [7]. The latter two signatures, HRR and micro-doppler, are the dominant choice among researchers working on automated target recognition and there are many publications available that show how successful recognition can be achieved using the these signatures. However, the two signatures are conventionally thought of as orthogonal to one and other and classification techniques should be based on either HRR or micro-doppler. More recently, research has begun considering how the HRR and micro-doppler signatures might be used together to provide a joint signature that contains more target information. It was demonstrated that concatenating the HRR signatures from different target aspects could lead to improved target recognition performance [8] and that the micro-doppler signatures from different target aspects carried unique target information [9]. A common conclusion of multiaspect target recognition is that increasing the orthogonality of the aspects increases the recognition performance. It follows from these conclusions that a joint HRR micro-doppler signature, as opposed to the conventional narrowband micro-doppler signature that contains the entire target in a single range bin, 2

20 would be valuable for recognition since the two signatures are also orthogonal. For the case of a human target, several methods that combine high range resolution with micro- Doppler have been proposed [10] [19] Contributions The following broad areas have been contributed to by the research described here Scatterer interactions in micro-doppler signatures Mechanical vibration or rotation of structures in a target, known as micromotions, will induce a frequency modulation on the backscattered signal and generate signature effects potentially unique to the target and its micro-motion. These modulations are termed the micro-doppler signature of the target. Micro-Doppler signatures of complicated targets that contain multiple micro-motions have been assumed as the independent combination of each individual micro-motion signature considered by itself. This assumption does not take into consideration the impact of the combination of these micro-motions on the total micro-doppler signature. These interactions have been considered and it is shown that when multiple micro-motions exist they can add constructively and destructively resulting in features in the final micro-doppler signature. The primary contribution is the identification of destructive combination of the phase response in a micro-doppler signature when more than one scatterer is in a single range bin. This result is displayed theoretically through the derivation of the phase response of a rotating target when in the presence of a stationary target. The theoretical mathematics are supported by simulated and experimental data, which is then examined 3

21 using a Short Time Fourier Transform (STFT). The effects on the output of the STFT by the destructive combination of the scattering centers is considered for various length of STFT windows. In addition to the study of the rotating point scatter, the interaction of a rotating target with an angle dependent return is considered. The angle dependence of a moving target was studied through the observation of a rotating flat plate. The results of this study have shown that targets can exhibit micro-doppler signatures that are angular dependent and only visible to the radar over a limited angular extent High Range Resolution Micro-Doppler Theory Development The theoretical derivation of the joint-range-time-frequency data cube allows for simultaneous consideration of HRR and micro-doppler signatures in radar. The presented derivation shows the effects of a HRR signal spreading the Doppler information over multiple range bins leading to a range-time surface that also shows Doppler characteristics. The HRR micro-doppler technique is then applied to a simple pendulum target using an ideal point scatter model and the three dimensional visualization of the displayed result. The technique is then expanded to consider more complicated scattering through the analysis of human gait using both simulation and experimental verification. The main contributions in the area of HRR micro-doppler theory are: the development of the theory of HRR micro-doppler from first principles, and the development of signal processing for generating a range-time-frequency signature from experimental data. 4

22 Classification Using HRR micro-doppler features Classification using micro-doppler signatures has been shown that it can be effective. Similarly, success has been found using HRR for classification. It has been widely assumed that the combination of these into a joint signature could provide increased information about the target. However, this assumption has not previously been tested. This research shows the results of classification using a joint range-time-frequency signature by classifying different human activities. The main contribution of this research is the extraction of features from the jointrange-time-frequency. These extracted features were used to classify human activities and then compared to classification results that used the targets micro-doppler signature to extract features Organization of Paper Chapter 1 provides an introduction to the research and clarifies the contributions of the research and outlines the dissertation. Chapter 2 introduces basic radar theory. Fundamental radar theory including range estimation, range resolution, pulse compression, and Doppler frequency are discussed. In addition, the basic of micro-doppler motion models and processing is presented. Chapter 3 explores the existing studies on micro-doppler, high range resolution, and classification of human gait. In addition to these topics, an in depth analysis of research that considerers both micro-doppler and high range resolution is presented. 5

23 In chapter 4 the cause of modulation that is apparent in micro-doppler signatures containing multiple micro-motions is presented. A theoretical approach to the interaction of multiple scatterer is given. It is shows the interaction between the scattering centers causes characteristic patterns within the micro-doppler signature. In addition to the interaction of these scatterers, the angle dependence of targets and its effect on the micro- Doppler signature is also considered. Chapter 5 presents the combination of micro-doppler and high range resolution into a joint-range-time-frequency signature (JRTF). This JRTF signature is analytically derived. This analytic derivation is expanded to an experimental implementation and the signal processing is described. Chapter 6 presents the results of experimentation using the JRTF signature for classification. The extraction of features from the JRTF signature and the justification of the performance of selected features is given. The extracted features are used in the classification of human activities and compared to the classification results of other researchers. Chapter 7 discusses the conclusions of the dissertation and suggests future work that could build on the research presented. 6

24 Chapter 2: Basic Theory This chapter presents the fundamentals of radar and introduces the basics of micro-doppler processing. This material in this section is summarized from standard radar texts[20], [21]. References will only be provided when the source material is from outside fundamental radar theory Radar Basics A radar operates by transmitting a Radio Frequency (RF) signal and receiving the echoes of that signal that are backscattered by the target. A radar uses the received waveform to provide information about the target such as the direction to the target or the range to the target. The radar is able to determine range to the target by measuring the time delay between transmission and reception. Depending on the type of radar it is also possible to tell other characteristics of the target such as its radial velocity by measuring the Doppler shift of the returned waveform. Radar is a popular sensor because of the fact that it is able to determine range and velocity measurements to a target rapidly and even under challenging circumstances such as poor weather or darkness. With these upsides, there is a constant push to expand the capabilities of radar sensors to be able to extract more information about the target. One such sought after improvement is the ability to identify a detected target. Doing so relies 7

25 on the use of radar signal processing techniques such as high range resolution processing and micro-doppler signal analysis Range Estimation Radar waveforms are categorized into two main classes: continuous waveform (CW) and pulsed wave [20]. The CW is used in many radar applications, but the focus of this document will be the use of the pulsed waveform. The range to a target can be calculated through the use of the time delay corresponding to when the transmitted signal returns to the radar. The range to a single target is given by R = c t d 2 where t d is the round trip time between when the signal was transmitted and when it was received. The maximum unambiguous detection range can be determined by the pulserepetition frequency (PRF). In a pulsed system the time interval between pulses is the pulse repetition interval (PRI). This PRI consists of the pulse width τ and the listening time of the receiver. The PRF can be calculated from the PRI by using the relationship PRF = 1 PRI. ( 2 ) Unambiguous range measurements can be acquired only when the t d to a target is shorter than the PRI. If the t d is not shorter than the length of the PRI this can cause an ambiguity. To avoid ambiguities, the PRF should be selected appropriately. The maximum unambiguous range, R u, is given by ( 1 ) 8

26 R u = c 2PRF. ( 3 ) The maximum detection range is a function of the and signal-to-noise ratio (SNR). If the reflected waves are not detectable above the noise floor a target detection is not possible. The SNR is the ratio between the received power P r and the noise power P n. The equation for received signal power, termed the radar range equation, is given by P r = P tg t G r λ 2 σ (4π) 3 R 4 L s. The parameters used in ( 4 ) are listed in Table 1. ( 4 ) P t G t G r λ σ R Table 1: Parameters of the radar range equation. L s Power Transmitted Gain of the transmit antenna Gain of the receive antenna Wavelength RCS of the target Range to the target System Loss In radar systems, the primary source of noise in the receiver is thermal noise [20]. The power of the thermal noise given P n = kt 0 FB ( 5 ) Where k is Boltzmann s constant, T 0 is the standard temperature, F is the noise figure, and B is the instantaneous bandwidth. The SNR can now be given as the ratio of P r and P n producing SNR = P r P t G t G r λ 2 σ = P n (4π) 3 R 4 L s kt 0 FB. ( 6 ) If the minimum SNR for target detection is set, SNR min, then the maximum detection range can be determined as 9

27 4 P t G t G r λ 2 σ R max = (4π) 3 R 4 L s kt 0 F B SNR min. ( 7 ) 2.3. Radar Range Resolution When considering a radar system, the ability to distinguish targets in range is valuable. The measure of how close targets can be to one another and be identified as separate targets is termed range resolution. High range resolution, the ability to separate closely spaced targets, provides the ability to separate the backscatter from multiple targets. With high enough range resolution it is even possible to decompose a target backscatter into its individual scattering centers. This ability has been shown to provide unique information about a target which has been used in classification [6]. Fig. 1: Range resolution target geometry. When multiple targets are present, with range to the first target given by R and range to the second target given by R + δr as seen in Fig. 1, the difference in their range δr determines if the radar can resolve them. The difference in the time delay between the return from the two targets is given by t d2 t td1 = t delta = 2(R + δr) 2R c c = 2δR c. ( 8 ) For a pulsed radar the minimum t delta that can be resolved is the pulse length, τ. Setting t delta = τ and solving for δr, the equation for the minimum δr becomes 10

28 δr = cτ 2. ( 9 ) Waveform modulation and appropriate signal processing can improve this δr. This dissertation considers HRR signals. These signals are considered to be signals that allow for resolution of scattering centers within a target and result in a complicated target to be span more than one point in range. For example, the target example case presented in Fig. 1 would be considered to be a HRR case if the two targets were resolvable in range and target 1 and target 2 are the scattering return from different parts of the same object. Signals that do not allow for this separation are termed narrowband signals Pulse Compression Waveform modulations that allow pulse compression are a practical solution to increase the range resolution of the radar. Pulse compression uses a long pulse with a modulation and compresses the received signal by decoding the modulation [20]. One of the most common modulations used in radar is a linear frequency modulation (LFM). The pulsed LFM can be expressed as x(t) = cos (2πf c (t τ 2 ) πb τ (t τ 2 ) 2), 0 t τ. ( 10 ) where f c is the carrier frequency and t is time. A picture of the LFM signal is shown in Fig

29 Fig. 2: LFM signal Pulse compression occurs when the received signal of the radar passes through a matched filter. The matched filter is used to compress a modulated pulse like the LFM into a single point and was developed to maximize the SNR of the received signal. The output, y(t), of matched filter in the time-domain is given by the convolution of a signal x(t) with its time reversed conjugate y(t) = x(τ)x (t τ)dτ. ( 11 ) The matched filter output of a LFM signal with a bandwidth of 1 MHz is calculated and shown in Fig. 3. It takes on the form of a sinc(x) function where sinc(x) is 12

30 defined as sinc(x) = sin(x). The absolute value of the peak response is normalized to 0 x db. Fig. 3: Matched filter response of a LFM signal. While the range resolution of an unmodulated pulse is inversely proportional to the pulse length, the range resolution of a pulse compressed signal is decided by the width of the highest peak. Using the half power point (-3 db) as the metric for this width, the range resolution of a pulsed compressed signal is given by δr 3dB c 2B. ( 12 ) where B is the signal bandwidth. A high range resolution corresponds to a low δr. Therefore, a radar signal with a high bandwidth has high range resolution and is not dependent on τ. 13

31 2.5. Summary of micro-doppler Theory Doppler Fundamentals Before micro-doppler can be discussed the basics of Doppler must be reviewed. When a radar transmits an electromagnetic signal to a target, the signal interacts with the target and returns back to the radar. Changes in the properties of the returned signal reflect the characteristics of interest for the target. When the target moves with a constant velocity, the carrier frequency of the returned signal will be shifted. This is known as the Doppler effect [22]. The theory of special relativity predicts that the received frequency will be [20]: f r = ( 1 + v/c 1 v/c ) f c This equation can be simplified because the velocity being observed is much ( 13 ) smaller than the speed of light. This simplification leads to the well-known equation for the Doppler frequency shift: f d = 2v c f c = 2v λ ( 14 ) The Doppler shift can also be considered specifically in relation to a radar system. When a radar sends out a signal towards a moving target it will impart a Doppler shift corresponding to the motion in the direction of the radar. This Doppler shift is seen as the rate of change of phase in the radar return. The two-way phase is given by φ = 2π 2R λ and thus: f d = 1 dφ 2π dt = 1 2π d dt (4πR λ ) = 2 dr λ dt = 2v λ = 2vf c c ( 15 ) 14

32 The ability to distinguish between multiple targets in Doppler is important to providing the ability to separate multiple moving targets based on their Doppler characteristics. When multiple targets are contained in a range cell Doppler resolution, δ d, allows for the velocity of the targets to be distinguished. Doppler resolution is given by δ d = 1 T dwell where T dwell is the dwell time defined as T dwell = 15 N PRF where N is the number of pulses observed and PRF is the pulse repetition frequency. Therefore, a large dwell time results in a fine Doppler resolution corresponding to a greater ability to separate targets in frequency Micro-Doppler In addition to the bulk motion of a target, there are many targets that contain structures that exhibit micro-motion such as vibrations or rotations. There are many different targets that exhibit micro-motions including the rotations of a helicopter rotor and the oscillating motion of a person s limbs as they walk. Micro-motions produce their own frequency modulations that induce an additional Doppler frequency in addition to the Doppler from bulk translational motion. If the target exhibits only constant translational velocity the Doppler shift is time-invariant. However, if the target also exhibits micro-motion the Doppler shift is time-varying. The micro-motions yield features that can be extracted from a target s signature that are distinct and are absent from the response of targets without micro-motions. One basic example of a micro-motion that leads to visible micro-doppler effects is a rotating target. The basic geometry of the rotating target case is represented in Fig. 4 which depicts a point scatterer P rotating about a center point P 0.

33 Fig. 4: Geometry of radar and rotating target [7] target [7]. Victor Chen provides a derivation of the micro-doppler response for the rotating Assume the target rotates about its axes x, y, and z with an angular velocity vector ω = (ω x, ω y, ω z ) T or a scalar angular velocity Ω = ω. The corresponding initial rotation matrix R Init from [7] is presented as cos φ sin φ cos ψ sin ψ 0 (16 ) R Init = [ sin φ cos φ 0] [ 0 cos θ sin θ] [ sin ψ cos ψ 0] sin θ cos θ where φ is the rotation around the z-axis, ψ is the rotation around the x-axis, and θ is the rotation around the y-axis. Point scatterer P is initially located at r 0 = (x 0, y 0, z 0 ) T corresponding to the initial range of the target given by r(t) at time t 0 where r(t) is given by r(t) = t dc 2 where t d corresponds to the time delay to the target and c is the speed of light. 16 ( 17 )

34 The initial motion of the rotating target can then be described by R Init r 0 and the unit vector of the rotation becomes ω = (ω x, ω y, ω z ) = R Init ω. Using the fact that ω f micro Doppler = 2f c [ω r ] radial the micro-doppler response from a rotating target is given by f micro Doppler = 2f c (Ωω r ) radial ( 18 ) Using this equation to model the behavior of a rotating target the results can be seen in Fig. 5. Fig. 5: The micro-doppler modulation induced by rotation of a point scatterer. This response of the rotating point target shows a sinusoidal response in the frequency domain because of the periodic motion of the scatterer. 17

35 2.6. Observation of Micro-Doppler A conventional approach for investigating the frequency spectrum of a discrete signal is through the use of the Discrete Fourier Transform (DFT). The DFT is given by N 1 X(f) = x[t] e 2πift N t=0 18 ( 19 ) Where X(f) is the DFT of the time dependent signal x[t] and can be calculated through the use of the Fast Fourier Transform (FFT) algorithm described in [23]. The FFT is a time independent method for observing the frequency content of a signal. One consideration when trying to observe a target that exhibits micro-doppler motion is that the micro-doppler signature is a time varying attribute and the spectral components of the received signal vary with time. As such, the DFT is inappropriate for observing a micro-doppler signature. In order to observe the micro-doppler phenomenon, it requires the use of time-frequency analysis. The most common of which is the Short Time Fourier Transform (STFT) given by S x (t, ω) = x(τ)h(t τ)e jωτ dτ ( 20 ) where h(t) is a sliding analysis window. The width of the sliding window relates to how the signal is represented. It determines whether there is high frequency resolution or high time resolution. A longer window provides higher frequency resolution while a shorter window provides higher time resolution. Collecting the radar return from a target over multiple pulses ( slow time ) and using the STFT allows for the time-frequency spectrum to be observed. This relation can be described pictorially in Fig. 6 where fast time corresponds to the down range distance to the target and slow time corresponds to

36 the pulse number. The output of the STFT is referred to as a spectrogram. An example spectrogram depicting the micro-doppler signature of a rotating point scatter observed using a STFT with a sliding window of 5 pulses is depicted in Fig. 7. Fig. 6: The STFT and slow time. Fig. 7: Example spectrogram of a rotating target generated with STFT with a window length of 5 pulses. One limitation of time-frequency analysis is time-frequency uncertainty. There is a tradeoff between time resolution and frequency resolution. In order to achieve high time resolution, the window used in the STFT must be short. However, if a short window 19

37 length is used the length of the Fourier transform is small and the accuracy in frequency is limited. This tradeoff must be weighed when considering micro-doppler analysis. There are other time-frequency representations besides STFT that can provide a time-frequency view of the micro-doppler signature. Two such transforms are the Wigner-Ville Distribution (WD) and the Wavelet Transform (WT). The WD is given by W x (t, ω) = x (t + τ 2 ) x (t τ 2 ) e jωτ dτ ( 21 ) where x* (t) is the complex conjugate of x(t). Various interesting properties of the WD [24] such as preservation of time and frequency support, instantaneous frequency, group delay, etc., make the WD a useful tool for signal analysis [25]. The WT attempts to overcome the time-frequency tradeoff is given by WT x (b, a) = 1 a t b x(t)g ( a ) dτ ( 22 ) The WT is the convolution of the signal x(t) and an analysis window g(t). The analysis window and g(t) and scalar a such that a is inversely proportional to the frequency f. This provides high spectral resolution, low temporal resolution, at low frequencies and low spectral resolution, high temporal resolution, at high frequencies. In contrast, the STFT has fixed spatial and temporal resolution at all frequencies [25]. In [25] Kadambe concludes that all time-frequency transforms contain cross terms when there is more than one component in the input signal. These cross terms can interfere with the analysis of multicomponent signals. However, Kadambe shows that the cross terms for the STFT and the WT occur at the intersection of the respective transforms of the two signals under consideration. This property of the STFT and WT cross terms make them ideal for multicomponent signal analysis. Because of the benefits 20

38 of the cross term properties of the STFT and the uniform spectral and temporal resolution this dissertation employs the use of the STFT over other time-frequency transforms Classification A radar classifier is a machine learning technique that uses training data to provide a basis of target identification. The classifier uses the training data as the basis for the classification algorithm. The test data is then fed into the classification algorithm which is then used to make predictions about the test data. This process is depicted in the use of training data is depicted in Fig. 8. Fig. 8: Classification procedure. There are many different methods that can be used in the classification algorithm stage of the classifier. This dissertation focuses on the use of two algorithms; the support vector machine and the minimum Mahalanobis distance. These methods are described further. A support vector machine (SVM) is a classifier defined by using a separating hyperplane and they have been shown to have strong classification abilities [26]. Through the use of given training data, the margin-based classifier outputs an optimal hyperplane 21

39 which creates a decision boundary. The optimal hyperplane is the one that maximizes the separation between the different classes of the training data. Fig. 9 illustrates the decision boundary as a solid line between of two linearly separable classes. The two dotted lines show the margin between the classes which is maximized by the SVM technique. Fig. 9: Two-Class classification with SVM. The Mahalanobis distance is the measure of the distance between a point x i and a distribuition with mean x and covariance matrix C x [27]. MD i = (x i x )C x 1 (x i x ) T ( 23 ) It is a multidimensional way to measure how many standard deviations x i is from the distribution. The prediction of the classifier for x i corresponds to the class that corresponds to the minimum Mahalanobis distance, MD i. 22

40 Chapter 3: Literature Critique This chapter discusses the literature on the current state and history of micro- Doppler radar and target classification using radar Classification in Radar A radar obtains information about a target by comparing the transmitted signal with the received signal, which is backscattered from the target. The present of a received signal indicates the presence of a target, but knowing a target is present is of little use by itself. Something more must be know. Radar provides the location of a target as well as its presence. In addition to these characteristics it can also provide information about what type of target it is from a characteristic radar signature. The use of this information is known as target classification [28]. For practical utilization, most of the target classification methods require automatic processing. Some method of signal recognition or pattern recognition must be applied to be able to correctly estimate the type of target. A target s characteristic radar signature can be estimated in different domains of signal processing. Typical examples of signatures include the following: polarimetric signatures that are based on the target s response to different polarizations of EM radiation [4]; reflectivity signatures that are based on how much of the incident radiation is reflected by the target [5]; high range resolution (HRR) observations of targets, that ultimately lead to radar imaging, that attempt to estimate the target s physical structure as 23

41 a signature [6]; and finally, micro-doppler signatures that provide information on the micro-motions of the target sub components [7]. The latter two signatures, HRR and micro-doppler, are the dominant choice among researchers working on automated target recognition and there are many publications available that show how successful recognition can be achieved using these signatures High Range Resolution Radar Contemporary radar design provides capabilities significantly beyond the classical detection and ranging. It is now possible to form radar images, using synthetic aperture radar (SAR) and inverse SAR (ISAR) methods [1], undertake sophisticated electronic scanning and beam control activities [2] and to measure characteristic signatures of targets to assist in automatic target recognition [3]. Here we concern ourselves with the study of targets radar signatures. HRR is a well-developed technique that has been around very soon after the inception of radar, but improvement to the technique is still being pursued today. HRR radar provides the ability to deconstruct the scattering centers of a target by allowing them to be separated from each other in range. Each scattering center observed by the radar has a return at a specific range this creates a range profile of the environment [29]. The use of HRR radar allows for the returns from each scattering center to be resolved from one another. By measuring the responses from a variety of targets and using the HRR information, it has been shown that classification of targets is possible using this range [6], [30] [34]. Classification results vary, but successful classification rates of 70-85% are typical. The results of classification using HRR methods is promising, but some 24

42 challenges remain. Because a target RCS changes rapidly with respect to aspect angle, classification can be difficult. However, there have been attempts to create robust approaches in classification that are less dependent on viewing angle [35]. Another limitation of HRR classification techniques is that non-rigid targets such as people can be difficult to classify because of their lack of rigidity [36]. However, it has been shown that other techniques such as micro-doppler analysis have been effective at non-rigid target classification if the target exhibits Doppler characteristics [37] Early Research on Micro-Doppler Some of the initial research into micro-doppler predates the introduction of the term micro-doppler. Much of the earliest work concentrated specifically on Jet Engine Modulation (JEM)[38], [39]. JEM classifies aircraft by the micro-doppler signature return from the inside of their engines. Early JEM technique involved measuring the frequency of the returns from the moving blades inside the engine chassis and determining the frequency of the main spectral components. In order to do this accurately an observation time of 25 ms was required, but provided accurate classification. These observation times limit the amount of scan time which was too slow for most air defense applications. A separate but related area in early studies examined the rotation patterns of ballistic missiles[40]. Ballistic missile motion consists of a translational motion combined with the micro-motion of coning. These studies relied on the extraction of the phase of the target and allowed for the extraction of the micro-motion signature which provided a signature of the coning motion after removing the translational motion. 25

43 The work done in JEM and ballistic missile procession shows the value of exploiting the effects of targets micro-motions on the received echoes and provided impetus for further research. However, the implications on high resolution in range and Doppler are not specific to the motors of airplane engines or ballistic missiles. Since the basic theoretical foundation and the development of the research JEM has spread in many directions such as: helicopter analysis, SAR, human gait recognition. These have been examined by primarily monostatic radars. More recently multistatic geometries have been reported Introduction to Micro-Doppler The fundamental work in the micro-doppler area started with the analytical analysis of the topic and the exploration of target scenes containing simple micromotions. In [41] Chen introduces the idea of micro-doppler in radar. In this article the math of micro-doppler is described and contrasted with a target that exhibits only bulk Doppler with no micro-motions. Chen uses the assumption of a pulsed radar signal modeled as a sampled Continuous Wave (CW) radar. This assumption limits the analysis of targets that remain in a single range bin during the observation time, limiting analysis to narrowband signals. Subsequently the paper focuses on two main types of micromotion, vibration and rotation. This produces a foundation for describing micro-doppler content of echoes. It is supported by [7], [41] [46], which further extend the theory of micro-doppler and propose the capability for unique micro-doppler signatures that are characteristic of their observed targets. Together these articles make a comprehensive foundation of micro-doppler work using narrowband signals. 26

44 Micro-Doppler of Helicopters The work on helicopters is similar to the work on JEM and ballistic missiles because of the priority given to rotational micro-motions. The research reported on the micro-doppler signature of a helicopters focuses on a number of key areas [47] [53]. The most important being measurement of the rate at which the main rotor propellers are rotating in comparison to the tail rotor as well as the number of propeller blades of both rotors. With this information classification can occur. In addition to the classification SAR images have been able to produced showing the rotor blade characteristics [49], [50].To date research in this field has led to effective systems which provide accurate classification between the different helicopter classes Micro-Doppler in SAR There is substantial work in the micro-doppler field that looks closely at the field of SAR imaging in radar [49], [54] [67]. The micro-doppler research pertaining to SAR tends to focus on different things than other fields of micro-doppler research. When SAR is the main focus the micro-doppler signature is not viewed as a positive characteristic. This is because the majority of the micro-doppler characteristics in a SAR image can be attributed to vibrations if the surveying aircraft. Therefore, the research is typically aimed at removing the micro-doppler characteristics because they can cause a blurring, and spreading in the SAR image. The blurring results in an unfocused image, while the spreading causes a streaking that distorts the image. The micro-doppler characteristics then need to be determined so they can be removed and improve the image. 27

45 Multistatic Micro-Doppler The object of this research is to alleviate the angular dependence of a micro- Doppler signature, and illuminate otherwise occluded micro-doppler scattering centers[64], [68] [74]. The current literature in this area provides some simple target models in a multistatic geometry of rotating and vibrating targets[64], [73]. There is,however, a very limited amount of data from real collections [70] [72]. The conclusions from these sources state that there are benefits to use of multistatic geometries, but there is also added complexity and an increase in the quantity of data storage required. Some of the benefits include a robust signature that contains multiple aspect angles of the target which can see motion that is occluded from the monostatic viewpoint, as well as viewing the motion from both cross-polarizations and co-polerizations which is shown to highlight different features in the target. The complexity comes from the quantity of data represented by the additional channels of collection and often contains redundant information that does not improve the micro-doppler analysis High Range Resolution Micro-Doppler There are some research papers that contain a combination of both ultra-wideband (UWB) high resolution radar and micro-doppler analysis. One of the earliest papers that considers UWB and micro-doppler is by Shirodka [75]. Shirodka presents a research paper focused on through-wall heartbeat detection. In this paper Shirodka focuses on the challenges of through wall radar operation and heart movement detection. These challenges are met by using a 2-5 GHz Stepped Frequency Continuous Wave (STFC) radar and performing Moving Target Indication (MTI) processing in order to filter out 28

46 stationary clutter including the wall response. The research paper concludes that the system designed is capable of heart beat detection and that the UWB used would not be necessary. Because of the focus of this paper Shirodka does not consider the target identification abilities of a UWB radar system. In [76] Li continues in the field of using UWB and micro-doppler for heartbeat detection in complex environments, such as through the rubble of a collapsed mine. Using a similar approach to Shirodka, Li also was able to achieve success in this area. The HRR aspects of these methods are used solely for localization in range while the micro-doppler aspects are used to differentiate the heartbeat from inanimate objects. Smith in [10] presents the joint range-time frequency representation (JRTFR). The JRTFR is designed to output the data in a 3D domain representing range, frequency, and time allowing both the high resolution, and the micro-doppler features to be observed. Smith uses this tool to analyze a person walking down a hallway. This article stresses the ability to view both in a high range while viewing the micro-doppler of a target. It concludes that the use of a high resolution radar increases the features observable using micro-doppler, but does not discuss the specific effects on the micro- Doppler signature or how they might be used in classification. In [16] Tahmoush discusses some of the difficulties when working with a wideband system which allows for HRR. Tahmoush stresses that when using a wideband system there is variation in the Doppler shift between the center frequency used and the minimum and maximum frequency extents. Tahmoush discusses this relationship and quantifies the error in frequency measurement seen as a function of center frequency and 29

47 bandwidth. This relationship between bandwidth and center frequency leads Tahmoush to conclude that to limit error higher center frequencies should be used to provide a small fractional bandwidth. Tahmoush also discusses the effects of the wideband width when considering the micro-doppler signature of a person. He uses the abilities of wide bandwidth to separate parts of the body with a range resolution of 0.2m. Fogle in [19] takes an in depth look at how to use HRR micro-doppler to observe the micro-doppler signature of a single walking person. Fogle focuses on breaking down the person s micro-doppler signature into individual scattering centers that corresponds to the limbs of the person, and following the scattering from the limbs as they move in range. Fogle achieves the ability to extract the motion of the individual body parts by observing the time-frequency spectrum and then running a tracking algorithm that assumes prior knowledge of the expected frequency response of the walking person. This research is a good insight of what HRR micro-doppler analysis can provide for the observation of a single, complex target and how a complex target can be broken into its individual components, but relies heavily on the prediction of an accurate motion model of the target and is thus limited by this model. The HRR micro-doppler signature may also be used to help extract target signatures in situations where the scene is complicated and contains multiple, closely spaced targets. The traditional narrowband micro-doppler analysis does not allow for a wide signal bandwidth. If multiple targets are present the micro-doppler signature then the targets response would be overlaid in the time-frequency domain, for example the group of people class of [77]. However, through the use of a wideband signal, researchers 30

48 have shown the ability to separate the scatterering from the different targets resulting in the separation of their micro-doppler signatures [11], [17], [18]. By separating the scattering from multiple people, their micro-doppler signatures can be independently estimated resulting in the ability to classify the individual targets separately rather than relying on a collective signature, The field of research combining high resolution and micro-doppler analysis has yet to be fully explored. The existing literature provides tools for processing and observing the returns, expected error sources, and an application in heartbeat detection. The literature agrees that the high resolution capabilities increases the abilities to observe features in micro-doppler signatures, but the specific effects seen in the micro-doppler signatures are not discussed Classification of Human Gait in Micro-Doppler Perhaps the most widely studied approach towards micro-doppler is human gait classification. The idea is that people can be identified by the motion of their limbs as they perform various activities. There is a substantial number of publications on this topic and many interesting results have been presented [13], [19], [37], [44], [78] [96]. Much of the work done in gait recognition involves experimental data collection of people performing various activities[19], [37], [44], [78] [81], [85] [88], [91] [94]. These activities include walking, running, crawling, and riding a bicycle. The data collections are then used to separate the movement of the individual limbs of the person. The main scatterers that appear in the micro-doppler signatures are the arms, legs, and torso. The limb motion is cyclical and they vary in frequency. This results in sinusoids 31

49 that vary in extent and frequency in the time-frequency representation. The amplitude of the observed scattering also varies for each limb with extremities such as the foot often exhibiting the weakest response while the torso tends to have the largest amplitude response. After target detection there has been ample work done in the field of classification of the movement trying to determine what activity the person is performing. There have been multiple approaches to this problem and many researchers have approached the classification of the motions with the intent to extract features from the micro-doppler signature that can be indicative of the underlying activity. Such features in include; stride rate, stride length, and cadence velocity which helps characterize the shape size and frequency of the time-frequency representation [48], [80], [97] [100]. These approaches are promising, but are limited in their scope because they rely on Doppler information alone and do not utilize the range information and HRR techniques that could help provide an increase in the accuracy of classification or access to new features sets. Another approach to the classification of human activities is to use features that can be extracted using statistical approaches that may not be directly connected to the physical motion. These features are selected through the use of transforms, or support vector machines that pick features that provide high classification rates using the given data [101] [105]. Results using these techniques can be very promising, but since the features do not relate to the physical motion differences in the data set can lead to difficulty in repeatability. 32

50 In addition to research that makes measurements of the human gait and then uses the estimated gait parameters measurements in attempt for classification, there is more limited research that tries to model the radar returns of a human[37], [83], [89], [90]. Typically, the modeling consists of locating major scattering points of the human and modeling them as point scatterers. However, there are more advanced approaches that model limbs as ellipsoids and other simple geometrical shapes to account for the change in scatterer amplitude based on aspect angle. The field of micro-doppler that focuses on the study of human gait is wide spread. Although there has been some success in the field with classification, typically boasting above 80% successful classification rates, there is still work to be done to make classification rates and repeatability better. Classification rates of 90% average with a worst case of 80% have been implemented in battlefield situations and have been shown to be successful [106]. It should be noted that all of the literature found in this area only attempts observation and classification of the pedestrians using the Doppler characteristics without considering the HRR characteristics simultaneously Conclusions of the Literature Survey There has been substantial research in the field of radar to try and determine what a detected target might be. Many have looked at using HRR or micro-doppler signatures to extract information about the target that can then be used for target classification/identification. The use of micro-doppler signatures has been applied to many different target types that exhibit micro-motions. However, when considering the 33

51 micro-doppler signature of a target the range information that can be provided using a HRR signal has only been considered in a limited capacity. The use of HRR micro-doppler techniques has shown that it is possible to extract the physical location properties of individual scatterers within the micro-doppler signature as well as allowing for the separation of the micro-doppler signatures of multiple targets. When performing these operations, the theory behind the HRR micro-doppler signature was not discussed and potential classification benefits of using the HRR micro-doppler signature were not shown. When considering using the HRR micro-doppler signature for classification, the use of features that correspond to a physical attribute of the target will allow for the attempt at classification to provide insight on the target and provide repeatability. 34

52 Chapter 4: Scatterer Combination in Micro- Doppler Signatures In this chapter we report on investigations into the phase behavior caused by the combination of point scatters located in a single resolution cell. It is shown how the scatterers can cause variations in the micro-doppler signatures as represented by the spectrogram. Further, the phase of simple targets is calculated analytically and computed using a CST simulation. This shows that, again, the phase varies as a function of viewing angle in a complex manner that imposes a characteristic on the resulting micro-doppler signature. Such behavior is characteristic of target type and hence carries information that can, potentially aid automated target recognition. Mechanical vibration or rotation of structures in a target, known as micromotions, will induce a frequency modulation on the backscattered signal and generate micro-doppler signatures unique to the target and its micro-motion. Micro-Doppler signatures are often viewed through the use of time-frequency representations such as the STFT which produce a spectrogram of the target. These spectrograms give a pictorial representation of the micro-doppler information and can contain many different characteristic traits. One such characteristic trait that can be seen in the spectrogram of a complex target is amplitude modulation within the target signature. An example 35

53 spectrogram showing this amplitude modulation is seen in Fig. 10, which is the spectrogram of a person walking towards the radar. Fig. 10: Spectrogram depicting amplitude modulations A number of models characterizing the micro-doppler signatures of different types of targets have been constructed [5-7]. However, these models tend to emphasize the micro-motion components of the scattering centers and generally assume the target to be composed of separable point scatterers [8-9]. Consequently, such models do not consider the impact of the combination of the scatterers that can result in a significant change to the measured phase as well as to the magnitude of the observed RCS. 36

54 Additionally, they don t capture aspects such as the limited angular range over which significant backscatter occurs for some scatterers. These multiple target scatterer combinations are the cause of the amplitude modulations seen in Fig Micro-Doppler of a Rotating Target To investigate the micro-doppler of interacting point scatters we first consider a target scene composed of a single point target rotating with a radius R, about a center O, located a distance D from the radar. In this demonstration example, the radius R is equal to the wavelength of the illuminating signal. This is illustrated in Fig. 11. It is further assumed that the rotating target (X) is an ideal point target Fig. 11: The two-scatterer point target scene considered. With radius R = λ and distance from radar D= 900 m 37

55 The expected phase return from this target scene is depicted in Fig. 12, and derived from the expected return of the target u 1 (t) = A 1 e j4πf o(r+d[cos ωt+sin ωt]) ( 24 ) c with phase term given by φ 1 = 4πf o (R + D[cos ωt + sin ωt]) c where ω is the rotation speed, and A is the amplitude of the return. The ( 25 ) amplitude, A, is not modeled to vary with time because the targets distance from the radar is large enough in comparison to the radius of rotation that the difference in amplitude due to difference in path length can be neglected. The simulation used to generate these results used a linear frequency modulated signal with a bandwidth of 4 GHz and a center frequency of 94 GHz in a noiseless environment. Fig. 12: The phase response of a single rotating target. 38

56 The phase response of the single rotating target can then be compared to a more complicated target scene. A center scatterer that has twice the amplitude of the rotating target at the position O is added and a new phase relationship can be calculated. The target return from the rotating target given in ( 24 ) can be added to the target return from a stationary target given by to produce the total target return at time t u 2 (t) = A 2 e j4πf or ( 26 ) c u(t) = A 1 e j4πf o c R + A 2 e j4πf o c (R+D[cos ωt+sin ωt]). ( 27 ) leading to u(t) = A 1 e jφ 1 + A 2 e jφ 2 ( 28 ) where φ 1 = 4πf o c (R + D[cos ωt + sin ωt]) and φ 2 = 4πf o R. Equation ( 28 ) can c then be expanded using Euler s formula producing u(t) = A 1 cos(φ 1 ) + A 2 cos(φ 2 ) + j[a 1 sin(φ 1 ) + A 2 sin (φ 2 )] ( 29 ) This equation can then be re-written further to fit the form u(t) = Ae jψ ( 30 ) where A = A A 2 2 is the amplitude of the return signal and ψ is the phase term of interest and can be calculated producing the final phase combined phase formula tan(ψ) = A 1 sin φ 1 + A 2 sin φ 2 A 1 cos φ 1 + A 2 cos φ 2 ( 31 ) Using this formula, the phase of the rotating target with a stationary target present in the center can be produced and subsequently compared to the single rotating target case. 39

57 Fig. 13: The phase response of a rotating target with a center scatterer of twice the amplitude present. Comparing Fig. 11 and Fig. 12 we see the shape of the plotted result is the same, but the extent of phase variation is less when the center target is present. The reason for this can be easily represented by a phasor diagram, see Fig. 14. [Q] [I] Fig. 14: Phasor diagram representing the target scene. The dashed line represents the response from stationary center scatterer. Its presence limits the summation of the two targets phases to +/- 30. This combination of the two phase centers in a target scene limit the phase response of the target as seen by the radar and can cause effects observable in micro-doppler processing that will be discussed in the proceeding section. 40

58 4.2. Processing and the Time-Frequency Relationship Taking the returns from the single rotating target and rotating target with center scatterer discussed in the previous section micro-doppler processing can be applied to produce spectrograms. The processing tool used is the STFT. Depending on the parameters used when performing a STFT different aspects of the resulting spectrogram are apparent. In the literature, it is typical to see STFT with a sliding window and large windows lengths. The use of a large window length allows for increased frequency resolution in the spectrogram, but gains this resolution in frequency by giving up resolution in the time domain. Using a sliding rectangular window with a window length of 40 pulses Fig. 15 shows the spectrogram of both the single rotating target and the rotating target with center scatterer. When comparing the two target scenes a previously unreported result is visible in the rotating target with center scatterer case. When the center scatterer is present, there is an amplitude modulation seen when comparing the return from the rotating target. This amplitude modulation is due to the combination of the phase of the two targets. The presence of the center scatterer does not only add a zero frequency component to the spectrogram that corresponds to a stationary target, but also the micro-doppler signature of the rotating target. By comparing the spectrogram results of the single rotating target and the rotating target with center scatterer using different length windows for the STFT more characteristic traits can be visualized (Fig. 16). The relationship between time and frequency in the visualization of a spectrogram and the subsequent analysis of the micro- 41

59 Doppler spectrum is paramount. Taking a closer look at Fig. 16 (b) it can be seen that as the window length used in the spectrogram generation is decreased the appearance of vertical frequency excursions is more visible. These vertical excursions in frequency occur when the two targets are destructively combine, which occurs when φ 1 = φ 2 For the given geometry this corresponds to when the difference between the distances of the point scatterers to the radar is λ/4 or 3λ/4. (a) (b) Fig. 15: (a) The micro-doppler spectrogram of a single rotating target using a STFT with sliding window length of 40 (b) The micro-doppler spectrogram of a rotating target and center scatterer of twice the amplitude using a STFT with sliding window length of

60 Continued Fig. 16: The spectrogram results using a STFT with different sliding window sized. (a) Single rotating target. (b) Rotating target with center scattering. 43

61 Fig. 16 Continued (a) (b) 44

62 The presence of the vertical frequency excursions seen when a small STFT window is used, and the amplitude modulation seen when a larger STFT window is used, leads to a a greater understanding of characteristics in a micro-doppler signature. When a target scene consists of multiple scattering phase centers, unique characteristics of the phase center combinations can be seen visualy in a spectrogram. However, the length of the window used in the processing and other processing parameters can lead to different characteristics being visible. Therefore, it is extremely important to consider both the combination of phase centers in a target center, as well as the parameters used in post processing. By doing so more information can be derived from the spectrogram, and can potentially lead to improvements in target classification Experimental Validation In order to further examine the combination of rotating point scatterers an experiment to validate the findings was designed and conducted. This experiment consisted of using a VNA and a horn antenna to survey a set of metallic spheres on a turntable using frequencies from 1 GHz to 6 GHz (Fig. 17). Fig. 17: Experimental setup used observing a sphere rotating about another larger sphere. 45

63 The metalic spheres were set up in such a way that the larger sphere that was.095m in diameter was at the center of the turntable while the smaller of the spheres that was.067m in diameter was placed on the edge and therefore orbited the center sphere. The centers of the two spheres were.31m apart while the distance from the center scatterer and the horn antenna was 2.70 m. Given the sizes of the spheres the large sphere was expected to have a RCS of approximatly 3 times that of the rotating smaller sphere. The collection was performed by taking amplitude and phase measurements for trasnmissions between 1 GHz and 6 GHz in incriments of 5 MHz. This process was repeated over 360 of rotation angles in incriments of 1 degree. In order to achieve a large SNR and eliminate the stationary clutter seen in the results a background measurement taken and background subtraction was performed. Using the ampltude and frequency measurements collecetd at various frequencies an inverse fourier transform was performed in order to produce range profiles for each angle of rotation. After the range profiles were produced range gating was performed by only considering the range bins that contained a target response. The range bins of interest where then coherently summed to collapse the return into a single range bin. Taking the range bin with the target present a throsholding process was implemented to elimiate further and noise and sources of experimental error. Taking the resulting phase and amplitude measurements for the range bin with the target a STFT can be performed over the various aspect angles. The experimental results from the two sphere setup and an experimental setup consisting of just the smaller sphere for various lengths of STFT windows below in Fig. 19. The spectrograms produced using 46

64 the experimentaly collected data can confirm the same characteristics expeted using simulation work. This experimental work confirms the importance of scattering centers of a target and the effects of their combination. The experimental results can be compared to a simulation that uses the theoretical background (Fig. 18). The experimental results are slightlly shifted in time, but the meaurements agree with the analytically calculated results. The spectrograms produced using both the short and long window lengths show the presence of modulation due to the combination of the scatterers. (a) (b) Fig. 18: Comparison between (a) simulated results and (b) experimental measurements using processing consisting of both a window length of 5 and a window length of

65 Fig. 19: Experimental results showcasing the rotating target with center scatterer. 48

66 4.4. Electromagnetic Modeling and Micro-Doppler: Flat Plate In this section we will analyze the amplitude and phase response of a flat plate. The use of this simple target allows for the analysis of a more complex target than the point targets previously discussed, but is simple enough that the target returns can be calculated easily using electromagnetic calculations. The amplitude response seen from the flat plate has already be calculated by Ross in [107]. However, the phase return, crucial to the micro-doppler signature, has not been reported on (to the best of my knowledge). The flat plate is a relatively simple target, and is often simplified and modeled as a point target. CST Microwave Studio and analytic calculations were used to challenge this assumption. The target scene considered consist of a flat plate at an arbitrary angle θ to the radar (Fig. 20). We are interested in the amplitude and phase response as a function of θ. Fig. 20: Geometry of flat plate aspect angle analysis. 49

67 The theoretical RCS of a flat plate target, as a function of aspect angle, was reported by Ross [10]. Ross shows that the RCS magnitude of the flat plate over aspect angles -80 <θ<80 is given by the equation: σ H = 2b i sin 2ka sin φ ([cos 2ka sin φ + ] π sin φ ( 32 ) π 4ei2ka+i( 4 ) 2π (2ka) [ cos(φ) π + ei2ka i( 4 ) sinφ e i2ka 2 2π(2ka) 1 ( 2 1 sinφ + ei2ka sinφ 1 + sinφ π ei4ka+i( 2 ) 1 )] [1 2π(2ka) ] ) With magnitude σ H 2 and phase φ σh = tan 1 ( imag{σ H } ).Ross uses this real{σ H } equation to showcase the amplitude return of the flat plate over various aspect angles. However, he does not comment on the phase response. The amplitude and phase response calculated using this analytical formula can be seen below in Fig. 21 (a) and (b). In order to verify the results using the analytic calculations, a CST Microwave Studio simulation was performed. The CST simulations were run in a noiseless environment using LFM modulated signal with a center frequency of 10 GHz a bandwidth of 1 GHz, and a time bandwidth product of 20. The flat plate is an 8 cm square with a thickness of 1 mm. Results generated for positive aspect angle changes can be mirrored and considered accurate for negative aspect angles due to symmetry of target scene. The CST results and the theoretical calculations for both amplitude and phase over varying aspect angles are compared in Fig. 21 (a) and (b). From Fig. 21(a) it can be concluded that there is excellent agreement between theory and simulation. The flat plate 50

68 shows a significant RCS over a plus and minus 10 range from the broadside position and sidelobes are at a level consistent with an un-weighted top hat function. The phase response also shows good agreement between theory and simulation. The phase increases as the angle increases away from 0. In other words, for a flat plate rotating about an axis along its center line, there will be phase change as a function of rotation angle. This will impart a micro-doppler that will feature in observed spectrograms. (a) (b) Fig. 21: The amplitude (a) and phase (b) resposne of a flat plate as aspect angle changes comparing CST Microwave Studio results with theoretical calculations. 51

69 (a) (b) Fig. 22: The amplitude (a) and phase (b) resposne of a flat plate as aspect angle changes comparing noisy CST Microwave Studio results with theoretical calculations. Fig. 22 (a) shows that the sidelobes are now submerged into the noise and Fig. 22 (b) shows that outside of the plus and minus 10 range the phase shows violent fluctuations. Within the plus and minus 10 range where the echo return is significantly above the noise floor there is a clearly discernible increase in phase with rotation angle. This change in phase would in turn cause a Doppler frequency shift that would manifest itself in the micro-doppler signature. As the aspect angle changes the point on the flat plate that the response appears to come from (i.e., the phase center of the flat plate) moves from the center of the flat plate 52

70 radially outward towards the leading edge. Fig. 23 shows this apparent phase center change over the restricted angular interval plus and minus 10. Further it can be calculated that the point target has a radial acceleration change of μm/degree, a velocity change of 0.28 μm/degree, and an initial velocity of 356 μm/degree. This was calculated using a quadratic regression where the phase, φ. is dependent on the aspect angle in degrees, A degree, producing the relationship φ(a degree ) = aa degree 2 + va degree + v 0 ( 33 ) with acceleration a, velocity v, and initial velocity v 0. The change in position of the phase center of the flat plate with angle is caused by the role of the exposed edge of the side of the flat plate that has moved closest to the radar. The leading edge starts to contribute to the total echo response and pulls the phase center away from the physical center of the flat plate. Fig. 23: The phase resposne of a flat plate as aspect angle changes comparing theoretical calculations with a model using a single point scatterer. 53

71 This it is seen, again, that the micro-doppler signature of even simple targets contains subtle but significant effects that impact on the measured phase and resulting micro-doppler signatures. Any micro-doppler shift due to the change in RCS of the type examined above is an additional component that should be considered when modeling micro-doppler signatures. Further, this Doppler shift is characteristic of the target (or parts of a target) and therefore can also be used to aid target classification Combination of Target Scatterers and Angular Dependence In real-world targets there are complicated scattering effects as a result of the presence of multiple scattering centers that can change in position and amplitude depending on the viewing radar s aspect to the target. These principles can be combined to showcase the effects of multiple scattering centers and the complexity of the combination of phase centers that can lead to visible characteristics in the outputs of micro-doppler processing. The target scene considered is the combination of the rotating flat plate located at 900m and an ideal point scatterer that rotates around the flat plate starting at 902m. The spectrogram constructed using a STFT with a sliding window length of 5 is show in Fig. 24. The effects of the angular dependency are seen in Fig. 24 near the 2 second mark. At this point in time the flat plat is between the plus and minus 10 aspect angle where it has a large enough return power. Over the rest of the angular extent of the rotation the ideal point target dominates the return and its Doppler response is clearly modeled as the 54

72 expected sinusoid. Even though the majority of the response appears to be a traditional rotating target the areas that vary can lead to valuable information about the target scene. Fig. 24: The spectrogram produced from a rotating flat plate rotating concurrently with a point target at a distance of 2m from the center Over the angular extent where the flat plate response is above the threshold, between plus and minus 10, the combination of the angular dependent phase changes of the flat plate and the point target can be observed (Fig. 25). It can also be seen that at the point where the flat plate has high return it causes destructive combination between the return from the flat plate and the return from the ideal point scatterer causing large frequency excursions to be visible in the signature. The combination of angular extent, angular dependency, and interacting phase centers leads to a complex phase relationship between in what is considered a simple target scene. The differences of this target scene with the single rotating target can lead to 55

73 valuable information, including the presence of the angular dependent phase response of the flat plate which could potentially be overlooked. Fig. 25: A zoomed in look at of the aspect angles when the flat plate has an angular dependent response Micro-Doppler is a valuable tool providing much information about targets. The combination of even simple scatterers can produce subtle but significant features in the resulting micro-doppler signature. These features may be valuable for automated target recognition and could potentially lead to an increase of cognitive capabilities of a radar system. Such features can provide valuable information characteristic of a given target. This is a result of phase center position changes with aspect angle. The way in which the phase center position changes with angle for even simple targets, such as a flat plate, has 56

74 not been routinely reported in the literature but is essential to understand and model the micro-doppler response. 57

75 Chapter 5: HRR Micro-Doppler One approach to the target identification and classification problem is the use of high range resolution (HRR) profiles. High range resolution profiles provide an indication of the target scattering as a function of range. There have been good results for classification of targets by studying the scattering from objects at a variety of aspect angles and creating a library to use in distinguishing between the targets. However, the use of such a library can be cumbersome and require large amounts of data collection. Another approach to target characterization is the use of micro-doppler signature as an indicator of the underlying micro-motions of the target. It is common to observe the micro-doppler signature through the use of a time-frequency representation. The micro- Doppler signature is a useful tool used to identify characteristics of the target and has been used with some success for classification. The micro-doppler signature and HRR profiles both have capabilities to provide target signatures that are unique and can form the basis of radar automatic target classification or recognition systems. However, micro-doppler is usually considered with continuous wave (CW) or sampled CW waveforms that are not conducive to obtaining HRR profiles. As such, most analyses of the two techniques are independent. In this research, the signatures are considered together with the aim of providing a joint micro-doppler-hrr target characterization. 58

76 The move from narrowband to HRR micro-doppler, that permits extraction of both structural information through HRR and micro-motions through the Doppler signature, requires enhancement of the existing micro-doppler theory and development of new signal analysis methods. The existing literature, [10] [19], begins to approaching this with individual studies developing parts of the theory relevant to hypotheses being considered. However, a holistic theory that converts the narrowband theory of Chen [7], [43], [108] to the HRR case has yet to be proposed. Conversely, signal transformations that create a three dimensional data cube of range-time-frequency to help analyze and visualize the HRR micro-doppler signature have been developed in [10], [14], [17]. These papers show that by combining the HRR signal with micro-doppler analysis the complexity of the problem can increase. However, with appropriate processing tools this complexity can be mitigated and useful information extracted from the data. Here, we shall develop the theory of HRR micro-doppler from first principles. The theory development will be supported using simulations and experimental data. In addition, we shall summarize the steps necessary to calculate the range-time-frequency datacube and discuss as some of the limitations of the processing HRR Micro-Doppler Theory In [109], Chapter 2, Chen introduces micro-doppler for radar targets and derives a mathematical analysis using point-like scatterers that exhibit different types of micromotions. In his analysis Chen considers a continuous wave (CW) waveform that has no bandwidth, i.e. it is a pure tone, and considers the frequency response of a single point scatterer. More complicated targets are considered as collections of point scatterers. The 59

77 use of a CW waveform eliminates the ability to separate these scatterers in range. This approach provides the background for micro-doppler: the target s radar signature is the summation of the micro-doppler responses of each scattering centre. When bandwidth is added to the signal, as happens in a pulse-doppler radar, two constraints are implied. First, the entirety of the target is contained in a single range bin. Second, the target remains in the range bin for the duration of the measurement of its signature. Essentially, the complex amplitudes of the range bin from each pulse may be treated as a sampled CW signal where the sampling frequency is the pulse repetition frequency. The case where the waveform allows for HRR, and the target can occupy multiple range bins and migrate between them during measurement, has yet to be fully considered. Fig. 26 Geometry of a radar target with rotating parts. A simple geometry to support derivation of the mathematical formulation of the micro-doppler effect is the observation of a point scatter that exhibits micro-motion as 60

78 well as bulk motion. As shown in Fig. 26, a stationary radar located at point O observes a point like scatterer, at point P, on a target. The points are defined in a fixed coordinate system (X,Y,Z). The target exhibits a bulk velocity v, defined in the fixed coordinates, and a rotation with respect to the radar that is described in terms of a local coordinate system (x,y,z) with its origin at the centre of rotation of the target, that is at point P 0 in the original fixed coordinate system. Over time, the position of the point P 0 will change due to the bulk velocity, v. The position of P relative to P 0 will also change due to the rotation of the vector r, that indicates the position of the point scatterer relative the centre of motion of the target in the local coordinates, by the rotation vector Ω = (ω x, ω y, ω z ) T. The summation of these two motions cause the change in the range to the point P, described by r(t), that leads to a progressive phase change in the received signal and ultimately the Doppler shift. Chen s signal is CW with carrier frequency, f c. The return from this signal assuming its interaction with a single point scatterer is. 2r(t) s(t) = ρ(x, y, z) exp (j2πf c ) = ρ(x, y, z) exp(jφ[r(t)]) c ( 34 ) Where ρ(x, y, z) is the reflectivity function of the point scatterer P described in the local coordinates (x, y, z), the phase of the baseband signal is represented by φ[r(t)] and r(t) represents the scalar range to the target as a function of time, t. The Doppler shift of the target is then given by the time derivative of the phase. f D = 1 dφ[r(t)] 2π dt ( 35 ) 61

79 Chen shows that this result can be calculated and the summation of the bulk velocity, v and the angular velocity vector, Ω, in the direction of the radar. f d = 2f c c [v + Ω r] n ( 36 ) Where n can be approximated as the unit vector along radar line of sight, given by vector OP in Fig. 26, when the range to the target is much greater than the distance the target moves during the observation. The results in ( 36 ) are very general and can be used to model a variety of motions. The use of a CW waveform limits the radar s ability to resolve targets in range. Here we extend Chen s analysis, to include waveform bandwidth, through consideration of the micro-doppler when a pulsed signal is used. The signal, s(t), can be described as s tx (t) = rect ( t τ 2 τ ) exp(jπf ct) q(t) ( 37 ) where q(t) is the modulation of the signal e.g. linear frequency modulation (LFM), also known as a chirp, f c is the carrier frequency of the signal, and τ is the pulse length. The rect( ) function is defined 1, x < 1 ( 38 ) rect(x) = { 2 0, x 1 2 Since the target is assumed an ideal point scatterer the received signal can be described as a time shifted version of the transmitted signal. Let the time delay to the target be represented by t d. This delay is dependent on the two way distance that the signal must travel to reach the point scatterer and return to the radar. If the target is moving, this distance changes for consecutive pulses. A variable to help characterize the 62

80 passing of time between pulses can be labelled as slow-time, and can be represented by, t s. Using slow time, the change in range of the target over consecutive pulses is given by r(t s ). The time delay, t d, to the scatterer can then be given as a function of the slow-time change in range. t d (t s ) = 2r(t s) c The return signal can then be expressed as s rx (t, t s ) = ρ(x, y, z)s t (t t d (t s )) This return is an approximation of the return signal that uses the stop-and-hop ( 39 ) ( 40 ) approximation described in [110] which makes the assumption that the target s motion is small compared to the speed of propagation and thus the movement of target while the pulse is in flight is negligible. By then combining ( 40 ) and ( 37 ) the expression for the return signal is given. s rx (t, t s ) = ρ(x, y, z)rect ( t t d(t s ) τ 2 ) exp(jπf τ c t)exp( jπf c t d (t s ))q(t ( 41 ) t d (t s )) The matched filter process [111], sometimes known as dechirping, of the received signal produces the final signal, s d (t, t s ), given by: s d (t, t s ) = s rx (t, t s ) s tx ( t) ( 42 ) Where the final signal is produced by the convolution denoted,of s rx and the complex conjugate of s tx. By substituting ( 37 ) and ( 41 ) into ( 42 ) the dechirped signal becomes s d (t, t s ) = ρ(x, y, z)tri ( t t d(t s ) t 0 τ 2 ) exp( jπf τ c (t d (t s ) t 0 ))Q (t ( 43 ) (t d (t s ) t 0 )) 63

81 where Q (t (t d (t s ) t 0 )) is the convolution output of the envelope of the applied modulation [20]. e.g. sin(x) x for a FM chirp and t 0 is the initial time of the generated pulse. The triangle function comes from the convolution of the rectangular envelope of the signals. The triangle function can be defined as 1 x, x < 1 ( 44 ) tri(x) = { 0, else The Q (x) function accounts for the convolution of the modulation of the transmit signal. If the modulation used was LFM Q (x) would be a sinc( ) function. However, this term could account for any modulation and would affect the outcome by augmenting the shape of the triangular function by Q (x). becomes: Following from ( 43 ), the phase,φ, of the return signal after matched filtering φ( t s ) = 2πf c (t d (t s ) t 0 ) ( 45 ) The micro-doppler frequency f D can now be found from the phase of the signal by taking its derivative, f D = 1 dφ(t s ) 2π dt s = 2f c dr(t s ) ( 46 ) c dt s Following Chen this expression for the Doppler frequency can be given by f D = 2f c c [v + Ω r] n ( 47 ) Looking at ( 47 ) the result shows that the Doppler frequency is independent of fast time, and only depends of the change in range over slow-time. However, considering the result of matched filtering further, we see the amplitude of the response in fast time can be described by 64

82 s d (t, t s ) = ρ(x, y, z)tri ( t t d(t s ) t 0 τ 2 ) Q (t (t τ d (t s ) t 0 )) since the absolute of the complex exponent will be 1. The magnitude of s d (t, t s ) represents magnitude values for all (t, t S ) ( 48 ) coordinates creating a range-time surface. Conversely, the instantaneous frequency of s d (t, t s ) represents the micro-doppler frequency for each sample in slow time. These two functions can be combined to create a joint range-time-frequency space where range corresponds to fast time, t, time is represented by slow-time, t s, and the frequency is the Doppler shift, f D, caused by the motion of the point scatterer. This joint space consists of the range-time surface corresponding to s d (t, t s ) mapped into the three dimensional space by mapping the surface in the third dimension corresponding to f D. Υ((t, t s ), f D (t s )) = ( ρ(x, y, z)tri ( t t d(t s ) t 0 τ 2 τ ) Q (t (t d (t s ) t 0 )), 2f c [v + Ω c ( 49 ) r] n ) This function Υ is presented as the form of the ideal joint range-time-frequency (JRTF) data cube produced when a HRR waveform is used. The JRTF data cube can be related to Chen s narrowband formulation of the expected micro-doppler frequency shift by considering the boundary case as τ approaches infinity and becomes a continuous signal. As τ goes to infinity the argument of the triangle function, tri(x), changes according to ( 50 ). lim x = lim (t t d(t s ) t 0 τ τ τ τ 2 ) = 0 As x goes to zero the output of the triangle function becomes 1 as seen in ( 51 ). ( 50 ) 65

83 x, x < 1 ( 51 ) lim tri(x) = lim {1 = 1 x 0 x 0 0, else This shows that the as τ tends toward infinity the triangle function in ( 49 ) goes to 1. This causes the range-time surface portion of Υ to become dependent only on the modulation imparted on the signal represented by Q. For a pure tone CW signal Q is set to 1 and the expression for the JRTF data cube, Υ NB becomes Υ((t, t s ), f D (t s )) NB = ( ρ(x, y, z), 2f c c [v + Ω r] n ) ( 52 ) This expression for the narrowband case shows that dependence on fast time, t, is eliminated the range-time surface becomes a constant, ρ(x, y, z), and the expected micro- Doppler frequency remains unchanged compared with the HRR case Example Case: Pendulum Micro-Motion The micro-doppler shift for the special case of a swinging pendulum is considered in this section as an example of how HRR and micro-doppler may be combined into a HRR micro-doppler signature. The example comprises an analytical analysis, leading to a simulation, validated with experimental results measured in a laboratory. The radar target geometry is shown in Fig. 27. The target consists of an ideal point target P rotating about an anchor point P 0. The vector r is given by P 0 P in the local coordinate system that would be centered at point P 0 but is not depicted in the figure. The extent of the pendulum swing is depicted by the dashed lines and the maximum angular displacement from the center of rotation is given by θ. 66

84 Fig. 27 Geometry of swinging pendulum. A simulation showing the results of the high range resolution micro-doppler analysis of this geometry was performed through the use of a simple pulsed radar with no modulation. Both simulation and experimentation conducted in a laboratory environment used the following radar and geometrical parameters: Parameter Value PRF 1000 (Hz) Carrier Frequency 92 (GHz) L 1.7 (m) P (0,6,0) (m) P 0 θ max (0,6,1.7) (m) 13 (degrees) Table 2: Radar and geometrical parameters for simulation and experimental validation of pendulum. 67

85 The Doppler motion is given by ( 47 ). For this example, to simplify the result the anchor point P 0 will be assumed fixed corresponding to bulk velocity v = 0. Therefore, the target only exhibits the angular rotation corresponding to Ω. If the displacement angle of the pendulum is small, then the micro-motion of the pendulum is described by a sinusoid and the angular rotation of this geometry can be given by Ω(ω x, ω y, ω z ) = (θ max 0, sin(2πf p t), 0) T ( 53 ) where f p is the frequency of the pendulum s swing. If the pendulum is assumed to be a simple pendulum f p = 1 2π g L where L is the length of the pendulum and g is the acceleration of gravity [112] The results of the dechirped signal s d (t, t s ) at the first slow time sample are shown to have the triangular shape as expected, Fig. 28 (a). The triangular shape is clipped at the left side of the figure because the target is close to the radar and negative range is not shown. Observing the return over multiple slow time samples the target s position in time can be observed as well as the consistent triangular spreading in the range dimension, Fig. 28 (b). 68

86 Fig. 28 Fast and slow time response of a pendulum a Fast time representation of a single rectangular pulse. b Slow-time-fast-time representation.. Color scale of is db of signal intensity The target can be observed moving back and forth in range with a in sinusoidal nature a result of its micro-motion. The corresponding micro-doppler frequency signature is also sinusoidal as depicted in Fig. 29 (a). By combining the range-time surface and the frequency information of the target the JRTF signature can be presented according to ( 48 ) in Fig. 29 (b). The JRTF representation shows the range-time surface of the target scene and its amplitude using a db scale. The light yellow color represents the peak amplitude of the target response. The amplitude decays linearly, spreading frequency modulation to adjacent range bins. Through the use of this representation, both the range-time information and the Doppler information can be considered simultaneously. The height of the surface shows sinusoidal characteristics because the Doppler frequency of the target given the specified Ω is sinusoidal in the direction of the radar. For the experiment, a linear frequency modulation (LFM) was used over the pulse duration in order to increase the signal to noise ratio and provide a fine range resolution. 69

87 The addition of the LFM processing results in a window of the data in the range domain. The value of the convolved LFM is a sinc(x) function where sinc(x) is defined as sinc(x) = sin(x), which provides the input for Q in ( 49 ).This results in a higher peak x amplitude and lower sidelobes. However, the results of the experiment and the JRTF seen in Fig. 29 (c) representation shows that theoretical model is consistent with the experimental data. Fig. 29 Frequency signature of pendulum. a Micro-Doppler frequency signature. b Theoretical JRTF representation. Color scale in db of signal intensity. c Experimental JRTF representation of pendulum. Color scale in db of signal intensity. 70

88 Visualizing HRR Micro-Doppler Signatures Before we turn our attention from theory to human motion, we first consider how the results might be evaluated when considering more complex targets. The theory assumed that a target could be deconstructed into its individual point scatterers and then the JRTF data cube, Υ((t, t s ), f D (t s )), could be evaluated for the motion of each scatterer separately. However, this result is the analytical formulation of the JRTF datacube and cannot be realized for a complicated target. However, it can be estimated. The JRTF signature can be estimated using conventional time-frequency analysis tools. Here we use the short-time-fourier-transform (STFT). The STFT does not exhibit cross terms and is therefore a common technique used in observing micro-doppler signatures [113]. With a narrowband system the time-frequency representation that does not provide range information. However, with a HRR system targets that would be unresolvable with a narrowband waveform can now be separated. The samples in fast time provides range information about the target and we would like a data transform that retains this information. This transform can be achieved by taking the STFT over slow time for each of the fast time/range samples. This process creates a time-frequency representation for each range bin in the target scene Fig. 30. The time-frequency representations (TFR) for each range bin can then be combined to estimate a datacube that contains the JRTF signature. Through the use of the STFT the JRTF signature can be estimated for complex data scenes. This technique permits further analysis of experimental results for targets consisting of multiple scattering centers. 71

89 Range Bin or Fast Time Slow Time Pulse-Doppler Radar Hardware Joint Range-Time-Frequency Representation Data Cube TFR TFR Pulse Index or Slow Time TFR TFR One TFR per range bin Frequency Range / Fast Time Fig. 30: Calculation of the Joint-Range-Time-Frequency data cube using a TFR HRR micro-doppler of a walking human target Following the simple example of a pendulum signature given above, we now turn our attention to a more complicated target: a walking person. To demonstrate the effect of HRR waveforms on the micro-doppler signature of such a target, a simulation and an experiment of a walking person was used. Both the simulation and the experiment used the radar and geometrical parameters listed in Table 3. PRF (Hz) Carrier Frequency 92.5 (GHz) Bandwidth 1 (GHz) Distance to Target 7 (m) Velocity ~1 m/s Table 3 Radar and geometrical parameters for simulation and experimental walking pedestrian. The human gait simulation was originally developed by Chen and was based on a human gait estimation model by Van Dorp [109], [114]. The simulation decomposes the human motion into 16 different parts corresponding to segements of each limb as well as 72

90 the torso and head. It then uses a perfect electric conductor (PEC) model of ellipsoid to model the RCS of each segment. The model is a basic representation of the target scattering and does not consider complex scattering principles such as multipath; multibounce; and target occlusions. Even with these omissions, however, the simulation provides a well-structured approximation of human motion by including the expected return from the main contributors, as investigated and concluded by Van Dorp, to the micro-doppler signature such as the torso, arm, and leg motion. The simulation results are presented and provide insight on how the HRR micro-doppler processing can be viewed, but is not presented as an ideal model therefore, experimental verification of the methods is also provided. The geometry used in the simulation consists of a single person walking towards the radar 7 meters away. A high range resolution range profile was simulated for each pulse. The series of HRR profiles is shown in Fig. 31 (a) where the color scale represents the uncalibrated power of the backscatter signal in db. The range profiles show human target return spread over 16 range bins. The general decrease in range over slow-time shows that the motion of the person is towards the radar. The dominant response was attributed to the person s torso. The scattering observed originating in front of and behind the main torso line; was attributed to the limbs of the person. In addition to the range profile the traditional micro-doppler signature was generated by coherently summing range bins containing the target response and taking the STFT [115] Fig. 31 (b). 73

91 Fig. 31: Simulation and experimental results of person walking toward radar. a Simulated range profile of person walking toward radar. The color scale is in db normalized intensity. b Simulated micro-doppler signature of person walking toward radar. The color scale is in db intensity. c Experimental Micro-Doppler signature of person walking toward radar. The color scale is in db signal intensity. The micro-doppler signature shows the different limb motion of the person without the use of a HRR waveform. The motion of the torso can be seen in high intensity line(s) in the middle of the signature with an approximately sinusoidal shape. The limb motion is also visible in the lower intensity lines that surround the torso response. While these lines are clearly periodic in nature, their shape no longer a pure sinusoid. 74

92 For the HRR case the STFT was applied on the each of the 16 range bins that the target was present in during the measurement. The results of the STFT show the time-frequency relationship in each range bin. With the increased complexity of the target s radar signature, it is no longer practical to identify a surface in the range-time-frequency datacube upon which the signature can be observed. Instead, time-frequency cuts at each range bin are shown in Fig. 32 where range bin 1 corresponds to the first range bin containing a target response and it is closest to the radar s location. The color scale shows the intensity of the plots in db. Fig. 32 Spectrogram of each range bin contain scattering of simulated walking person. The color scale is in db intensity and common to all sub-plots. 75

93 In the figure it can be seen that bin in the first 2000 pluses the frequency content in the range bins is very similar. This is due to the presence of large RCS scatterers whose sidelobes are present in adjacent range bins. This spreading of the scatterer in range was predicted by the tri( ) and Q functions of ( 49 ).We also see that the JRTF representation allows for the observation of the target s migration in range simultaneously with the micro-doppler frequency information. The equivalent narrowband micro-doppler signature for the experimental results can be seen in Fig. 31 (c) while the HRR signature that separates the frequency response for each range bin is shown in Fig. 33. Fig. 33 Experimental spectrogram of each range bin contain scattering of walking person. The color scale is in db intensity and common to all sub-plots. 76

94 The results of the simulation are consistent with the experimental data. Similar to the simulation range bins the spreading effect of a high amplitude scatter continues to appear in the experimental data. This spreading effect of the frequency information in the range dimension does present challenges for localization of micro-motion scatterers in the range-time-frequency data cube. From the figure, it can be seen that the experimental target scatterer migrated through range bins more slowly than the simulated target. This was due to the person be measured walking more slowly than the velocity used in the simulation. This difference in range migration was easily observed using the JRTF datacube, but would be harder to observe using narrowband micro-doppler analysis. It can also be observed that the density of scatterers in the experimental case is higher than that of the simulation. This difference was attributed to the small number of point scatterers used in the simulation to represent the complex scattering of a person. Despite this discrepancy, the shape and structure of the time-frequency signature is the same Experimental Processing Approach When implementing HRR micro-doppler in an experimental setting, consideration should be given to noise and sidelobes caused by having a finite length pulsed signal. The sidelobes cause spreading of the frequency components in range. In order to mitigate this spreading a thresholding technique is proposed. Thresholding was implemented using an ordered statistic (OS) CFAR technique. The OS CFAR procedure selects a single value from a distribution X (k), k {1,2,, N} where X (1) X (2) X (N) and uses it to estimate the average clutter power [116]. 77

95 The clutter power can then be used to determine the threshold level such that values below the threshold level are removed from the signature. The OS CFAR technique, assumes a Rayleigh clutter distribution and provides a probability of false alarm given by P fa = k ( N (k 1)! (T + N k)! ( 54 ) ) k (T + N)! where T is a scaling factor selected in order to achieve the desired probability of false alarm [116]. OS CFAR can be applied to the JRTF signature by using the frequency values for each time-range coordinate in the JRTF datacube as X (k). This is process is depicted in Fig. 34. Fig. 34: OS CFAR of the JRTF signature. These frequency values can be sorted by amplitude so that X (k) meets the constraint of X (1) X (2) X (N) where N is the length of the FFT used when creating the JRTF signature Fig

96 Fig. 35: Sorted frequency values for OS CFAR. Results in this section set N = 256. Given the X (k) distribution, a single value from the distribution is selected in order to estimate the average clutter power. Using k = 64 = 1 N, the value of X 4 ( 1 can be used to estimate the clutter power. This value N) 4 of k is a valid selection because the PRF was chosen such that the maximum frequency observed from the target was less 3/8*PRF. This guarantees X 1 ( to be a measure of the N) 4 noise level. Using X 1 ( N), values of X (k) X 1 ( T are determined to be clutter and set to N) 4 4 zero. In order to achieve a P fa = the scalar T = 8. After the amplitude threshold has been applied a local maxima peak search was implemented for the time-frequency slice at each range bin. The search operated on a single column of the slice at a time and identified the local maxima for each time step. This process extracts the location in frequency of any local maxima in each time step. This processes is then repeated for every time step in the time-frequency slice. Pseudo code of the described process is given: 79

97 Alg. 1: Local Max Extraction Pseudo Code Threshold Value = Calculated through ordered statistic CFAR For: All Values In Data Cube If: Data Cube Value < Threshold Value Data Cube Value = 0 End End For: Each Range Bin For: Each Pulse For: Each Frequency If: Data Cube Value Is Local Maxima Local Max Extraction Value = 1 Else Local Max Extraction Value = 0 End End End End By extracting the local maxima in the described manner, the result produces the location in time, range and frequency of all the target scatterers. This information can then be used in multiple ways to extract new information from the data and portray the three dimensional data of the joint range-time-frequency data cube in a two dimensional format. To generate results using this method a walking person is considered as a target. A walking person has been considered frequently in µd research as a topic of interest[45], [78]. However, most µd research focuses on a person walking towards or away from the radar. In this paper we also consider the geometry of a person walking perpendicular to the line of sight of the radar. The geometry is shown in Fig. 36 where the radar is considered to be located at the origin with its boresight direction along the y- axis. This geometry benefits greatly from µd-hrr processing trajectory the micromotions of the target, such as the swinging of the arms and legs, are primarily perpendicular to the radar and thus tend to exhibit smaller µd signatures. 80

98 We have chosen to research this scenario because of its applicability to automotive radar. When the radar is mounted on the front of the car a pedestrian crossing the road ahead is moving perpendicular to the LOS. Many automotive radars operate at the 77 GHz frequency[117] where it is possible to have a large bandwidth, to obtain fine range resolution, but still have a small fractional bandwidth allowing the narrow band radar assumption to be made for Doppler processing. The experiments in this research were conducted at W-band, using a 92.5 GHz center frequency, which is a reasonable approximation to the 77 GHz currently used by automotive systems. Fig. 36: The geometry considered showing a pedestrian walking perpendicular to the radar line of sight which is located at the origin looking down the y axis. Using the local maxima extracted data the number of distinct frequency components in each range bin can be determined. By summing the number of local maxima present in each range bin for each time step, the number of distinct frequency components present in a range bin over time can be determined. Performing this summation for all range bins and all time steps allows for the creation of a new range map depicting the number of distinct frequency components seen in Fig

99 Fig. 37: Number of frequency components (indicated by image intensity) as a function of range and time. Being able to extract the number of discrete frequency components in each range bin has potential benefits. It would be expected that the number of discrete frequency components would relate to the number of scatterers in each range bin. This estimation of number of scatterers in each range bin at each time interval could potentially be used as a dynamic model of the target scene in which no assumption about the number of scatterers present is required. The local maxima extraction can also be used to create a hybrid two dimensional image that allows for partial visualization of the three dimensional data cube. Taking the local maxima extraction data in its binary form, i.e. one where a local maxima was detected and zero elsewhere, it is possible to set the data for each range bin to a different color in the image when plotting. This technique allows for the visualization of each range bin local maxima extraction simultaneously and can be seen in Fig

100 Fig. 38: Joint Range-Time-Frequency Cube visualized using colors to represent range bin. Using colors to represent range can have benefits and can allow for the visualization of aspects of a target scene that may otherwise not be apparent. One application of this is when a complex target scene is considered. In a complex target scene, especially when observing people, targets can move in and out of range bins rapidly and multiple complex scattering signatures can overlap making it difficult to distinguish what frequency components belong to which target. By separating the targets in range using HRR and then visualizing the data using color to represent the range it gives insight as to what frequency components belong to which complex scattering target. This topic is further discussed and represented by experimental data and observations of multiple people walking in close proximity in section Walking People Experiments The experimental setup that was used consists of a radar with a center frequency of 92.5 GHz operating with a bandwidth of 1 GHz. The walking targets were set up to be 83

101 walking within 10 meters from the radar in order to achieve a high signal to noise ratio. The experiments were conducted in a lab environment so in order to isolate the desired target scene range gating was used to remove the back wall. The Experimental setup and radar parameters are depicted below in Fig. 39 and Table 4. The bandwidth of the signal was chosen to be 1 GHz. Using this bandwidth and pulse compression a 3dB range resolution of δr 3dB c 2B =.15 m. This range resolution is considered to be a high range resolution when observing human motion because the human target signature is larger than the.15m range resolution resulting in the target signature of the person to be contained in multiple range bins. Since the stride length of a typical person is around.7 m [118] the.15 m range resolution provides separation within the target signature and can allow for the separation of the motion of the limbs from the motion of the torso. 4m 8m C D 1m 1m A B Fig. 39: Experimental setup where A, B, C, D are paths people walk in the four experiments Radar Parameter Value Units Bandwidth 1 GHz Center Frequency 92.5 GHz PRF 3 KHz Pulse Length 800 ns Table 4: Radar parameters used in experiment 84

102 The experiment which was conducted indoors in a laboratory environment. Because of this, the experimental setup was constrained by the laboratory size and the observed targets were close to the radar. This resulted in the targets to be considered in the near field of the radar which is defined as r < 2D2 λ where D is the largets dimension of the target and λ is the wavelength. For the set parameters and assuming a target height of 1.8m the range required to be considered in the far-field would be 200 m. The target being in the near-field results in variation of the illuminating field across the target, and variation in the RCS across the target [119]. This is caused by the fact that distance to the extremities of the target cannot be considered to be the same as the distance to the center of the target. For a target that is placed in the center of the radar beam pattern at a distance r, from the radar the difference in range between the center point and the edge of the largest dimension is given by δ r = r 2 ( D 2 ) 2 r ( 55 ) The δ r for the experimental geometry is provided in Fig. 40 assuming a 1.8 m tall person observed from a distance of 5m from the radar. In Fig. 40 it can be seen that the distance from the top of the target to the radar is further from than the distance to the middle of the target by.08m. Considering the bandwidth of 1 GHz this δ r would cause spreading of the signature in range that is approximately equal to half of the radar s range resolution. 85

103 Fig. 40:Near-field geometry of walking person. The first experiment that was conducted was the conventional µd scenario of a person walking directly towards the radar represented in Fig. 39 by path A. The spectrogram, the extraction of discrete frequency components and the two dimensional visualization of the joint range-time-frequency data cube are pictured in Fig. 41. The spectrogram, pictured in Fig. 41 (a), shows the bi-pedal motion that has been depicted in many µd research papers see [80] and references there in. In this spectrogram the offset of the bulk velocity can be seen. The cyclical motion that is caused by the motion of the limbs can also be observed, all in accordance with what was expected due to previous gait analysis. It should be noted that some stationary lab clutter was observed by the radar. The high amplitude zero frequency component can be attributed to this clutter. This lab clutter remains present in all of the experiments, and will not be addressed in further analysis. 86

104 (a) (b) (c) Fig. 41: Radar results of single person walking towards radar. (a) Depicts the traditional spectrogram. (b) Depicts the number of frequency components in each range bin. (c) Depicts the two dimensional portrayal of the joint range-time-frequency data cube. The distinct frequency component extraction figure pictured in Fig. 41 (b) shows the number of scatterers in each range bin as the person walks toward the radar. In this figure it is clear that the target is approaching the radar with its bulk velocity causing the linear slope of the data. In addition to the moving person, the stationary clutter seen in (a) 87

105 is also detected in (b). The clutter is represented by the horizontal lines that experience no range migration. Their color indicates there is only one clutter scatterer at each range. Physically, it is possible that there may be more, however, since they do not exhibit micro-motion they cannot be separated and therefore a combined response is observed. There are also visible areas where more distinct frequency components are present in each range bin. If the positions of the areas of higher density of scatterers in (b) are compared to the frequency content of the spectrogram in (a), it can be seen that there are scatterers moving at high velocities at these points. The point in a person s stride where the highest velocity is seen in the direction of the radar corresponds to when the limbs are near the torso. Therefore, it can be said that when the limbs are close to the torso there are high velocities observed as well as a higher density of scatterers in the range bins. The two dimensional view of the joint range-time-frequency cube depicted in Fig. 41 (c) shows the progression of the target in range as it approaches the radar by transitioning from the red colored range bins towards the blue colored range bins. This transition showcases the range bin migration, and depicts both the range bin migration of the target as seen by the radar as well as the micro-motions. The second experimental setup that was observed was a single person walking in the direction perpendicular to the radar represented by path C in Fig. 39. For this geometry a smaller frequency response is expected because less of the motion is in the direction of the radar. The results of this experiment can be seen in Fig. 42. The spectrogram from this experiment (a) show that, as expected, the frequency response is lower observing this geometry than when the person was walking directly towards the 88

106 radar. Note there is no strong DC line present since for this target geometry it was possible to range gate out the clutter scatterers. The discrete frequency component extraction depicted in Fig. 42 (b) is different than that seen in Fig. 41 (b). When considering the person walking perpendicular to the radar the range migration of the target is not as great as when the person is walking towards the radar. This difference is seen in Fig. 42 (b) by the target signature being almost horizontal in range, but slightly increasing as the target moves away from the boresight line. The cyclical motion of the walking person seen in Fig. 41 (b) is still observable in this figure. A repetition of high density of scatterers can be seen near time step one thousand, three thousand, six thousand, and eight thousand. Looking at the two dimensional visualization seen in Fig. 42 (c) it is clear that there is less range variation (fewer colors are present) than the case where the person was walking toward the radar. This is due to the reduced range migration due to bulk target motion for the geometry. This reduced bulk range bin migration makes it is easier to see the change in color in the representation due to the transition of the target s limbs through the range bins. This effect is consistent with the results of Fig. 42 (b) which shows that the number of frequency components in each range bin is changing as the person progresses. 89

107 (a) (b) (c) Fig. 42: Radar results of single person walking perpendicular to the radar. (a) Depicts the traditional spectrogram. (b) Depicts the extracted number of frequency components in each range bin. (c) Depicts the two dimensional portrayal of the joint range-time-frequency data cube. The third experimental set up that has been considered is when two targets walk towards the radar starting 1 meter apart represented in Fig. 39 by path A and B. The results of this experiment can be seen in Fig. 43. Due to the presence of multiple people in the target scene, the spectrogram pictured in Fig. 43. (a) is more complex since it consists of their superposition of their responses. It can be noted that the frequency contributions from each person are present, but not readily separable by eye. 90

108 (a) (b) (c) Fig. 43: Radar results of two people walking towards radar. (a) Depicts the traditional spectrogram. (b) Depicts the extracted number of frequency components in each range bin. (c) Depicts the two dimensional portrayal of the joint range-time-frequency data cube. However, looking at the two dimensional visualization of the joint range-timefrequency data cube in Fig. 43 (c) it is possible to determine which frequency components correspond to which person. By looking at Fig. 43 (c) is can be seen that greater frequency content towards the start of the collection, around time step 1000, is represented in a yellow color, while there is another frequency content represented by the 91