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

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452.5 Time-Frequency Analysis in NoiseThe discrete chirp signal in the presence of additive noise is given by:is the chirp phase, and v(n) are independent samples of acomplex white Gaussian noise process with zero mean and variance a 2 . The inputSNR is given by SNR = A²/σ² . From (2.40), the WVD kernel can be written as:In the expression above, the noise component isgiven by:The resultant noise power in the kernel isThere are only N/2 independent samples in the kernel (since it is conjugatesymmetric), so the variance of the noise at the output of the WVD is scaledby a factor of N/2 rather than N. Therefore, the SNR of the WVD output is:Equation 2.49 can be used to computea bound on the chirp estimation error. The performance of the WVD is comparedagainst the Cramér-Rao bound (CRB) on the IF estimation variance for a a complexsingle tone in additive white Gaussian noise. The CRB is achieved by the DiscreteFourier transform (DFT) of the single tone signal and is given by [45]:
46where for the DFT, the output SNR is N times the input SNR:The bound on the performance of the WVD is found by substituting the SNR fromEquation 2.49 in Equation 2.50. Furthermore, when compared to the DFT. theWVD has its frequency scaled by a factor of 2 (see the term2.40). This leads to a reduction by a factor of 4 in the variance of the frequencyestimate. Therefore, the variance of the WVD for a chirp signal is:At high input SNR (A 2 a²), this equation reduces to:which is the same as Equation 2.50. Since this equation is the variance of themaximum-likelihood estimate of the frequency of a stationary sinusoid in whiteGaussian noise [45], it can be said that the WVD is an optimal estimator of theIF for chirp signals at high SNR.2.6 Application of TF Analysis to IF EstimationThe estimate of the instantaneous frequency is provided by the peak of the WVDas a function of time [6]. For the WVD the peak is found from Equation 2.41 andoccurs when:It is evident from this relation that there is a linear relationship betweenthe timeand frequency coordinates (n and m, respectively). The slope of this line is givenby:
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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
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 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 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
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
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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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