Principles of Modern Radar - Volume 2 1891121537

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5.1 Introduction 149δ = M/N, the undersampling ratioρ = s/M, the normalized sparsityσ = the energy bound on the additive disturbance signal ef 0 = radar center frequency (Hz)λ = wavelength (meters)ω 0 = 2π f 0 , center frequency in (radians per second)c = speed of light (meters per second)d = array inter element spacing (meters)f = c/λ, frequency (Hz)k = 2π/λ, wavenumber (radians per meter)J = number of spatial channelsK = number of slow-time pulsesL = number of range gates in fast timeT = slow-time sampling period (seconds).5.1.4 AcronymsAcronyms used in this chapter include the following:AMP approximate message passingBP basis pursuitBPDN basis pursuit denoisingCoSaMP compressive sampling matching pursuitsCPI coherent processing intervalCS compressed sensingDFT discrete Fourier transformDLS data least squaresDWT discrete wavelet transformEM expectation maximizationFISTA fast iterative shrinkage-thresholding algorithmFOCUSS focal undetermined system solverIHT iterative hard thresholdingIRLS iterative reweighted least squaresISTA iterative shrinkage-thresholding algorithmLaMP lattice matching pursuitLASSO least absolute shrinkage and selection operatorMAP maximum a posterioriMIMO multiple input multiple outputML maximum likelihoodMM majorization minimizationMMSE minimum mean squared errorMP matching pursuitsMTI moving target indicationNESTA Nesterov’s algorithmOMP orthogonal matching pursuitsPSF point spread functionRIC restricted isometry constantRIP restricted isometry propertySAR synthetic aperture radar
150 CHAPTER 5 Radar Applications of Sparse ReconstructionSPSRSTAPSVDTVULAsubspace pursuitsparse reconstructionspace time adaptive processingsingular value decompositiontotal variationuniform linear array.5.2 CS THEORYThe origin of the name compressed sensing lies in a particular interpretation of CS algorithmsas an approach to signal compression. Many systems sample a signal of interest ata rate above the Nyquist sampling rate dictated by the signal’s bandwidth. This sampledsignal is then transformed to a basis where a few large coefficients contain most of thesignal’s energy. JPEG2000 is an excellent example of this sort of processing, relying on awavelet transformation. The signal can then be compressed by encoding only these largesignal coefficients and their locations.Since the signal can be encoded with just a few coefficients, it seems natural to askif the relatively large number of measurements is required in the first place. The originalsampling rate was dictated by Nyquist sampling theory to guarantee the preservation ofan arbitrary band-limited signal. However, perhaps one can use the knowledge that thesignal will be represented by only a few nonzero components in a known basis, such as awavelet transform, to reduce the required data acquisition. It turns out that, under certainconditions, a relatively small number of randomized or specially designed measurementsof the signal can be used to reconstruct this sparse representation. The key is that we do notneed to know which coefficients are nonzero; we require knowledge only of the basis ordictionary from which these elements or atoms are drawn. In fact, in the case of noiselesssignals, this reconstruction from a reduced data set will actually be perfect! Furthermore,we shall see that the reconstruction of the signal will be well-behaved both in the presenceof noise and when the signal is only approximately sparse. Because a reduced data set isbeing collected and compression is accomplished through the sampling procedure itself,this process is termed compressed sensing.This combination of randomized measurements with a sparse representation formsthe heart of CS. Indeed, CS combines measurement randomization with SR to provideperformance guarantees for solving ill-posed linear inverse problems [1,2]. We will explorethe implications and interpretation of this statement at length throughout this chapter. First,we will define the problem of interest and explore its relevance to radar.5.2.1 The Linear ModelMany radar signal processing problems can be represented with a linear measurementmodel. In particular, consider an unknown complex-valued signal of interest x true ∈ C N .We collect a set of measurements of the signal y ∈ C M using a forward model or measurementoperator A ∈ C M×N with additive noise e ∈ C M , that is,y = Ax true + e (5.1)Our fundamental goal will be to solve the inverse problem of determining the vector x truefrom the noisy measurements y. As we will see, even in the noise-free case e = 0, thisproblem is nontrivial with multiple feasible solutions.
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Principles of Modern Radar
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Published by SciTech Publishing, an
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Brief ContentsPreface xvPublisher A
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xContents3.6 Adaptive MIMO Radar 10
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xiiContents9.3 Adaptive Jammer Canc
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xivContents15.3 Multisensor Trackin
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xviPrefacehas always been the unlik
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Publisher AcknowledgmentsTechnical
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Editors and ContributorsVolume Edit
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xxiiEditors and ContributorsDr. Lis
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xxivEditors and ContributorsMr. Ara
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2 CHAPTER 1 Overview: Advanced Tech
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1.3 Radar and System Topologies 5FI
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1.4 Topics in Advanced Techniques 7
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1.6 References 15TABLE 1-1Summary o
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PART IWaveforms and SpectrumCHAPTER
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20 CHAPTER 2 Advanced Pulse Compres
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2.2 Stretch Processing 35is applied
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2.2 Stretch Processing 37TABLE 2-1
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2.5 Stepped Frequency Waveforms 61T
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2.5 Stepped Frequency Waveforms 63w
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2.5 Stepped Frequency Waveforms 69I
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2.9 References 812.7.4 SummaryWhile
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2.9 References 83[27] Taylor, Jr.,
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2.10 Problems 85shift across the pu
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88 CHAPTER 3 Optimal and Adaptive M
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3.2 Optimum MIMO Waveform Design fo
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200 CHAPTER 5 Radar Applications of
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Spotlight SyntheticAperture RadarCH
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6.1 Introduction 213• Image quali
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6.2 Mathematical Background 215andf
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6.2 Mathematical Background 2172π
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220 CHAPTER 6 Spotlight Synthetic A
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6.4 Sampling Requirements and Resol
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6.4 Sampling Requirements and Resol
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6.5 Image Reconstruction 239FIGURE
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6.6 Image Metrics 241The contrast r
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6.6 Image Metrics 243Along-track am
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6.7 Phase Error Effects 245TABLE 6-
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260 CHAPTER 7 Stripmap SAR3 m SAR O
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262 CHAPTER 7 Stripmap SAR• RMA i
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264 CHAPTER 7 Stripmap SARH 0 (ω)
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268 CHAPTER 7 Stripmap SARTABLE 7-1
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274 CHAPTER 7 Stripmap SARFIGURE 7-
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276 CHAPTER 7 Stripmap SAR−500Sce
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7.3 Doppler Beam Sharpening Extensi
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288 CHAPTER 7 Stripmap SARand in th
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7.4 Range-Doppler Algorithms 291to
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7.4 Range-Doppler Algorithms 293For
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296 CHAPTER 7 Stripmap SAR−150Col
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300 CHAPTER 7 Stripmap SAR33 cycles
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302 CHAPTER 7 Stripmap SARCrossrang
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304 CHAPTER 7 Stripmap SARfrom freq
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7.5 Range Migration Algorithm 307r,
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310 CHAPTER 7 Stripmap SARkx (rad/m
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316 CHAPTER 7 Stripmap SARThe diffe
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320 CHAPTER 7 Stripmap SARLet us re
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322 CHAPTER 7 Stripmap SARThe resul
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324 CHAPTER 7 Stripmap SARscene. Th
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326 CHAPTER 7 Stripmap SARdriven by
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328 CHAPTER 7 Stripmap SARTABLE 7-3
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330 CHAPTER 7 Stripmap SARFIGURE 7-
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332 CHAPTER 7 Stripmap SAR7.10 REFE
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334 CHAPTER 7 Stripmap SAR6. [MATLA
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Interferometric SAR andCoherent Exp
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8.1 Introduction 3398.1.1 Organizat
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8.1 Introduction 341Rsabnv rw x ,w
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8.2 Digital Terrain Models 343FIGUR
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8.3 Estimating Elevation Profiles U
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352 CHAPTER 8 Interferometric SAR a
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8.5 InSAR Processing Steps 365ξ ta
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8.5 InSAR Processing Steps 3670.1 0
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8.5 InSAR Processing Steps 373While
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8.6 Error Sources 375The second met
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8.6 Error Sources 377If sufficient
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Adaptive Digital BeamformingCHAPTER
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9.1 Introduction 403c k = clutter s
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408 CHAPTER 9 Adaptive Digital Beam
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9.2 Digital Beamforming Fundamental
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9.3 Adaptive Jammer Cancellation 42
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9.5 Wideband Cancellation 447FIGURE
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454 CHAPTER 10 Clutter Suppression
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10.2 Space-Time Signal Representati
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10.5 STAP Fundamentals 4810−5−1
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10.6 STAP Processing Architectures
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10.7 Other Considerations 4911 CPIM
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10.9 Summary 49310.7.3 Computationa
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10.10 References 495[12] Fenner, D.
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10.11 Problems 497the PSD in Figure
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11.2 Colored Space-Time Exploration
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11.2 Colored Space-Time Exploration
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506 CHAPTER 11 Space-Time Coding fo
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11.8 References 525Cost reduction o
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11.9 Problems 52711.9.2 Equivalence
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530 CHAPTER 12 Electronic Protectio
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12.2 Electronic Attack 535TABLE 12-
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12.2 Electronic Attack 541variation
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12.5 Antenna-Based EP 557are adjust
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12.6 Transmitter-Based EP 561polari
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12.11 Summary 583TABLE 12-4Summary
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12.14 Problems 585[9] Stimson, G.W.
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Introduction to RadarPolarimetryCHA
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13.1 Introduction 591and precipitat
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13.1 Introduction 593S: target scat
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13.2 Polarization 597After substitu
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13.2 Polarization 599zFIGURE 13-4Po
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13.3 Scattering Matrix 601left-hand
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604 CHAPTER 13 Introduction to Rada
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13.4 Radar Applications of Polarime
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13.4 Radar Applications of Polarime
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13.5 Measurement of the Scattering
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13.5 Measurement of the Scattering
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13.8 References 623• Lee, Jong-Se
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13.8 References 625[32] Kraus, J.D.
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13.9 Problems 627with the result in
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Automatic Target RecognitionCHAPTER
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14.2 Unified Framework for ATR 633P
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14.3 Metrics and Performance Predic
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14.3 Metrics and Performance Predic
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14.4 Synthetic Aperture Radar 639TA
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14.4 Synthetic Aperture Radar 641Te
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14.4 Synthetic Aperture Radar 64314
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14.4 Synthetic Aperture Radar 645Un
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14.4 Synthetic Aperture Radar 649se
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14.4 Synthetic Aperture Radar 651th
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14.5 Inverse Synthetic Aperture Rad
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14.5 Inverse Synthetic Aperture Rad
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14.6 Passive Radar ATR 657radar sys
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14.7 High-Resolution Range Profiles
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14.9 Further Reading 661take a targ
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14.10 References 663[19] Q. H. Pham
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14.10 References 665[56] M. Martore
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14.10 References 667[92] E. Libby,
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Multitarget, MultisensorTrackingCHA
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15.1 Review Of Tracking Concepts 67
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15.1 Review Of Tracking Concepts 67
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678 CHAPTER 15 Multitarget, Multise
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15.2 Multitarget Tracking 683TABLE
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15.2 Multitarget Tracking 685remove
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15.5 Further Reading 69515.4 SUMMAR
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15.6 References 697[11] Blair, W.D.
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15.7 Problems 6992. Common metrics
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⎡⎤3 0 0 0 0 00 3 0 0 0 0P 2 =0
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Human Detection With Radar:Dismount
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environment that may mask the gener
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D s % = Boulic-Thalmann model param
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16.2 Characterizing the Human Radar
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714 CHAPTER 16 Human Detection With
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16.4 Technical Challenges in Human
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16.4 Technical Challenges in Human
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16.5 Exploiting Knowledge for Detec
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16.7 Further Reading 729enable not
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16.8 References 731[13] Broggi, A.,
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16.8 References 733[53] G.E. Smith,
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16.8 References 735Conference on So
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16.9 Problems 737FIGURE 16-13Pendul
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740 CHAPTER 17 Advanced Processing
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17.3 Direct Signal and Multipath/Cl
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17.3 Direct Signal and Multipath/Cl
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17.4 Signal Processing Techniques f
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17.5 2D-CCF Sidelobe Control 787TAB
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17.6 Multichannel Processing for De
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17.10 References 815While far from
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17.10 References 817[20] Chetty, K.
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17.11 Problems 819[52] Gao, Z., Tao
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17.11 Problems 8215. Using the sign
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824 Appendix A: Answers to Selected
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ResolutionResolution withDimension
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830 IndexASAR. See Advanced SAR (AS
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832 IndexCross-polarization (XPOL),
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834 IndexExciter-based EP (cont.)fr
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836 IndexInterferencecolored noise,
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838 IndexMoving target indication (
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840 IndexPOI. See Probability of in
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842 IndexSaturated (constant amplit
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844 IndexStripmap SAR (cont.)remote
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846 IndexWWald sequential test, com
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