Principles of Modern Radar - Volume 2 1891121537

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.