Principles of Modern Radar - Volume 2 1891121537

166 CHAPTER 5 Radar Applications of Sparse Reconstructionfunction appropriately. This view was explored for the ambiguity function, along witha deterministic approach to constructing waveforms that yield low mutual coherence forthe resulting A, in [56]. In a nutshell, the thumbtack ambiguity functions that are knownto be desirable for radar systems [57] are also beneficial for CS applications. In [12],the authors use mutual coherence as a surrogate for RIP when designing waveforms formultistatic SAR imaging. As one might expect, noise waveforms provide good results inboth scenarios. 20The ambiguity function characterizes the response of a matched filter to the radardata. At the same time, the ambiguity function determines the mutual coherence of theforward operator A, which provides insights into the efficacy of SR and CS. Thus, CSdoes not escape the limitations imposed by the ambiguity function and the associatedmatched filter. Note that virtually all SR algorithms include application of the matchedfilter A H repeatedly in their implementations. Indeed, SR algorithms leverage knowledgeof the ambiguity function to approximately deconvolve it from the reconstructed signal.Put another way, SR can yield signal estimates that lack the sidelobe structure typical of amatched filtering result, but the extent to which this process will be successful is informedby the ambiguity function.5.3 SR ALGORITHMSIn the previous section, we surveyed much of the underlying theory of CS. As we haveseen, CS combines randomized measurements with SR algorithms to obtain performanceguarantees for signal reconstruction. In this section, we will review several examples ofSR algorithms and their associated CS performance guarantees. It is worth mentioning thatthese algorithms can and are used in situations when the sufficient conditions associatedwith CS are not satisfied. In spite of this failure to satisfy these conditions, the resultingreconstructions are often desirable.One issue is that the traditional CS theory provides error bounds on reconstructingx true . In many radar problems, the signal x true represents fine sampling of a parameterspace, such as the set of image voxels or the angle-Doppler plane. In these scenarios,producing a reconstruction whose nonzero elements are slightly shifted in the vector ˆxmay be perfectly acceptable to a practitioner, as this would correspond to a small error inestimating the relevant parameter. However, the traditional error definitions of CS wouldsuggest that this reconstruction is extremely poor.To give a concrete example, suppose that our signal of interest is a single tone in noise.The vector x true represents the discrete Fourier transform (DFT) of the signal sampled ona fine grid, and A is simply a DFT matrix. If the true signal is zero except for a singleentry in the first position equal to 1 and ˆx σ contains a single 1 in the second position,then we have almost perfectly reconstructed the signal. The model order is correct with asingle sinusoid, and the frequency has been estimated to an accuracy equal to the samplingdensity of our frequency grid. Yet the CS measure of error ∥ ∥x true − ˆx σ∥∥2 would be larger20 We note that it is not possible to improve the Kruskal rank of a matrix by left-multiplication; see [58].This motivates attempts to change the waveform or collection geometry in radar problems to improvesparse reconstruction performance rather than simply considering linear transformations of the collecteddata.