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

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17Figure 2.6 Time-frequency description of a frequency-shift keyed signal, x(t) =e327rf“, t k < t < tk+1 -Within each time interval [t k , tk+1], only a singlefrequency ƒk is presentThe time localization of the spectral components are described by the instantaneousfrequency and the group delay for a group of signals. The instantaneousfrequency represents the frequency as a function of time, ƒ ƒs (t) and assumesthere exists a single frequency component at each time t. This assumption doesnot apply in the case of the signal s(t) ej2rf i t e327f2t, containing two frequencycomponents (ƒ1 and f2 ) at all times. For the group delay, the assumption is that agiven frequency is concentrated around a single instance of time.If the signal can be described by a surƒace spanning the time-frequency planeinstead of a curve on this plane, the restrictions associated with the instantaneousfrequency and the group delay can be minimized. This concept of a surface spanningthe plane corresponds to a joint function Ts(t, ƒ) of time and frequency and is knowas the time-frequency representation of the signal.
182.3.1 Short-Time Fourier TransformThe short-time Fourier transform (STFT) has played a large role in the history ofdigital signal processing [30, 31, 32]. It is defined aswhere s(t) is the signal being analyzed and g (t) is a time-shifted windowing function.The squared magnitude of the STFT is known as the spectrogram. A simple interpretationof the STFT is a "sliding-window Fourier transform". The analysis of thefrequency content of s at time t o may be found by sliding the window function, g (t)across the signal until it is centered at t o and then taking the Fourier transform ofthe signal-window product. Each constant-time slice, $(t, ƒ), of the STFT representsthe frequency content of the signal around time t o . Many functions have beenused successfully as STFT windowing functions: Gaussian, Hamming, Hanning, andsquare/rectangular functions. The computation of the time slice at time to is shownin Figure 2.7. The spectrogram of s(t) = sin(2π ƒt) using a hamming window isshown in Figure 2.8.The STFT can also be expressed in the terms of the Fourier transform of thesignal and windowing functions in the formThe STFT of the multicomponent signalcomputed using a hamming window is shown in Figure 2.9. As is shown, the threesignal components are centered at the points (T 1 , ƒ ƒ1), (T2 , ƒ2), and (T3 , ƒ3).
Page 1 and 2: Copyright Warning & RestrictionsThe
Page 3 and 4: ABSTRACTSPACE/TIME/FREQUENCY METHOD
Page 5 and 6: SPACE/TIME/FREQUENCY METHODS IN ADA
Page 7: APPROVAL PAGESpace/Time/Frequency M
Page 10 and 11: 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 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 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
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67where k is a gain constant and R
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69The scalar beamformed noise estim
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71are the desired signal, interfere
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CHAPTER 4REDUCED-RANK PROCESSINGThe
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75Figure 4.2 Reduced-rank GSCThis m
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77This relation establishes an equi
Page 94 and 95:
79The conditioned SNR is a random v
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81The performance of the GSC proces
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83For large interference eigenvalue
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85the steering vector s, and the ve
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87(a) CSNR vs Rank order for other
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89Figure 5.4 CSNR vs Rank order for
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91Figure 5.6 Theoretical and simula
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93Figure 5.8 CSNR vs CNRThe signal
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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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