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

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49where:Doppler phase shift due to relative motion between platform and target2π/λis the wavenumberrange to target= SAR platform velocity= target along-track (azimuth) velocityIf the derivative of Ψ(t) is taken with respect to t we obtain a relation between therate of change of the phase (angular frequency) and the SAR-target relative velocity.If the SAR pulse repetition interval is T, , then t nTr . From Equation 2.57, itcan be seen that the relation between frequency and time is linear. The slope of thefrequency curve is given by:The relative velocity v, v p, vt, can then be found from the relation:S, is determined from the outputs of each of the transforms. Finally, the relationbetween the slope of the time-frequency characteristic at the output of each TFanalysis technique and the relative velocity is given by:The relative velocity of the boat was estimated from the SAR data using eachtransform and Equation 2.61. The SNR was coarsely estimated from the SAR data
50to be 15.4 dB. In the case of the WVD, the window was set to K=64 points. Thetarget's relative velocity was estimated to be 7.1 meters/sec. The contour plot forthe output of the WVD is shown in Figure 2.24. In the case of the STFT, a 64 pointhamming window was utilized. The target's relative velocity was estimated to be7.2 meters/sec. The contour plot for the output in the case of the STFT is shownin Figure 2.25. For the case of the Gabor expansion, the target's relative velocitywas estimated to be 6.5 meters/sec using a 32x16 TF grid oversampled by 4. Thecontour plot for the output in the case of the Gabor expansion is shown in Figure 2.26.Finally, for the case of the scalogram (CWT), the target's relative was estimated tobe 7.4 meters/sec. The contour plot for the scalogram is shown in Figure 2.27. Ascan be seen from the estimated relative velocities, the numerical results from thefour techniques are very similar, however, the representation of the data is not. Inour scenario, the contour plot of the WVD provides the representation with the mostsharply defined chirp characteristic. This is partially due to the particular windowsize that was used in the transforms and the specific transformed signal. The chirpcharacteristics of the moving target for the STFT, the Gabor expansion, and thescalogram are not as obvious but are also apparent. In our simulations, a gaussianfunction was used with the CWT. If a chirp had been used in some form with theCWT as a basis function, the results for the CWT may have improved.2.7.1 Another Chirp in the SignalA second chirp was added to the SAR signal containing the moving target at asignal level that was 5 dB higher that the original signal. This level was calculatedby comparing the energy contained in the added signal and the original SAR image.After transforming the combination of this new signal into a TF representation, amask was created that nulled the areas of the transform domain above a certainlevel for each particular transform. The relative velocity of the remaining chirp was
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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
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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 62 and 63: 47Consequently, when the slope S, i
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
Page 112 and 113: 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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