Space/time/frequency methods in adaptive radar - New Jersey ...

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47Consequently, when the slope S, is measured from the WVD, the chirp rate can befound from the relation:As can be seen from Equations 2.42 and 2.44, both the STFT and the Gaborexpansion reach maxima along the line w Using the frequency and timeand the slope S, measured from theSTFT and Gabor expansion respectively, the chirp rate p, can be calculated usingEquation 2.55.The instantaneous frequency of a simulated linear-FM signal (chirp) wasestimated by each of the TF analysis techniques considered. For each method, theinput signal was a linear FM signal of 256 points sampled at 1000 Hz. The chirp ratewas 8.84 Hz/sec. The SNR of the input signal was varied from 15 dB to —15 dB in1 dB steps and 250 trials were averaged for each SNR level. The data was analyzedas follows: for the WVD a K=64 point square window, for the STFT a 64 pointhamming window, for the Gabor expansion, a 32.x16 TF grid that was oversampledby 4, and the scalogram (CWT) was analyzed with a window of 32 points at thecoarsest scale. For each transform, and in order to minimize frequency quantizationerrors, the peak response at each time sample was found from an interpolation alongthe frequency axis. After the peaks for each time sample were determined the chirprate was found from the least-squares linear fit through these maxima. The errorbetween this line and the actual IF of the input was found and is plotted for eachTF analysis technique in Figure 2.23. The CRB for N 64 is plotted on the figurealso. It can be seen that the WVD approaches this bound at high SNR. The STFTis closer to the bound than the Gabor expansion and the scalogram. All methodsexhibit a threshold. This figure confirms the optimality of the WVD [7] and therelative advantages of the STFT over the Gabor expansion [11].
48Figure 2.23 Input SNR vs 1/mse of CRB, WVD, STFT, Gabor expansion, andCWT2.7 Estimation of Target ParametersIn this section we apply the TF analysis techniques to the SAR data for the purposeof extracting the target motion parameters. The SAR data was provided by theNaval Air Warfare Center, Warminster PA and was collected using their P-3 SARsystem at X-band. The data consisted of a boat target of opportunity moving inparallel with the SAR platform within an open ocean background. The data wasprocessed using each of the TFD techniques and the boat speed was estimated foreach case. However, the absolute accuracy of the estimates could not be assessedsince the true boat velocity was not available. The SAR data has been analyzedafter range compression, but prior to azimuth compression. The SAR signal phasereturned from a single point target is given by the functional relation:
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Copyright Warning & RestrictionsThe
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ABSTRACTSPACE/TIME/FREQUENCY METHOD
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SPACE/TIME/FREQUENCY METHODS IN ADA
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APPROVAL PAGESpace/Time/Frequency M
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ACKNOWLEDGMENTI would like to begin
Page 12 and 13: ChapterPage3.2.1 Non-Adaptive Beamf
Page 14 and 15: FigurePage2.22 Scalogram of a linea
Page 16 and 17: CHAPTER 1INTRODUCTIONVarious space,
Page 18 and 19: 3the target. In this work we study
Page 20 and 21: 5support consists of a population o
Page 22 and 23: 7processing. This is explained by t
Page 24 and 25: 92.1 An Overview of Synthetic Apert
Page 26 and 27: 11Figure 2.2 SAR cross-range Dopple
Page 28 and 29: 13Through the expression for the on
Page 30 and 31: 15Illumination pattern onEarth's su
Page 32 and 33: 17Figure 2.6 Time-frequency descrip
Page 34 and 35: 19Figure 2.7 Computation of the sho
Page 36 and 37: 21Figure 2.9 Spectrogram of a multi
Page 38 and 39: 23where 9/ is the Hilbert transform
Page 40 and 41: Figure 2.11 Gabor Transform of a si
Page 42 and 43: 272.3.3 Continuous Wavelet Transfor
Page 44 and 45: 29Figure 2.13 Time-Frequency resolu
Page 46 and 47: Figure 2.14 Scalogram of a sine wav
Page 48 and 49: Figure 2.16 Regions of influence co
Page 50 and 51: Figure 2.17 WVD transform of a sine
Page 52 and 53: 37Figure 2.18 Example of WVD cross
Page 54 and 55: 39(a) Contour plot of the WVD of a
Page 56 and 57: 41(a) Contour plot of the STFT of a
Page 58 and 59: 43(a) Contour plot of the Gabor exp
Page 60 and 61: 452.5 Time-Frequency Analysis in No
Page 64 and 65: 49where:Doppler phase shift due to
Page 66 and 67: 51(a) Contour plotFigure 2.24 SAR i
Page 68 and 69: 53(a) Contour plotFigure 2.26 SAR i
Page 70 and 71: 55calculated in each case. In this
Page 72 and 73: 57(a) Moving target with synthetic
Page 74 and 75: Figure 2.31 SAR with 2 moving targe
Page 76 and 77: 61Figure 3.1 Space-time adaptive ar
Page 78 and 79: 63Figure 3.3 Airborne radar basic g
Page 80 and 81: subspace is formed using the remain
Page 82 and 83: 67where k is a gain constant and R
Page 84 and 85: 69The scalar beamformed noise estim
Page 86 and 87: 71are the desired signal, interfere
Page 88 and 89: CHAPTER 4REDUCED-RANK PROCESSINGThe
Page 90 and 91: 75Figure 4.2 Reduced-rank GSCThis m
Page 92 and 93: 77This relation establishes an equi
Page 94 and 95: 79The conditioned SNR is a random v
Page 96 and 97: 81The performance of the GSC proces
Page 98 and 99: 83For large interference eigenvalue
Page 100 and 101: 85the steering vector s, and the ve
Page 102 and 103: 87(a) CSNR vs Rank order for other
Page 104 and 105: 89Figure 5.4 CSNR vs Rank order for
Page 106 and 107: 91Figure 5.6 Theoretical and simula
Page 108 and 109: 93Figure 5.8 CSNR vs CNRThe signal
Page 110 and 111: 950.80.80.6 0.6~(5'0.4IC!J0.40.20.2
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97the spatial domain. The target is
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99(a) Training outside the target r
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101(a) Training around the target r
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103Figure 5.15 Angle scan with the
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CHAPTER 6CONCLUSIONSThis work has b
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107also presented. The Sample Matri
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APPENDIX ADISCRETE COSINE TRANSFORM
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111This matrix is formed using the
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REFERENCES1. Leon Cohen. Time-frequ
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11523. Franz Hlawatsch and G. Faye
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11750. Daniel F. Marshall. A two st
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INDEXanalytic signal, 21array imper
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