Principles of Modern Radar - Volume 2 1891121537

178 CHAPTER 5 Radar Applications of Sparse Reconstruction110.50.500 2 400 2 4110.50.500 2 400 2 4110.50.500 2 400 2 4FIGURE 5-7 An example application of OMP to a simple radar problem. The signal of interestis a set of three-point scatterers. The collected data represent the returns from a single pulsewith 500 MHz of bandwidth. Each row depicts a single iteration of the OMP algorithm. Thepanes on the left depict the backprojected residual A H ( y − A ˆx) at each iteration. The rightpanes depict the signal estimate ˆx with a curve and the true signal x true with crosses.obtain a RIP-based guarantee of the same form as others that we have seen. In particular,if R 4s (A) ≤ 0.1 [39], then∥ x true − ˆx ∥ 2≤ 20 [∥∥ x true − x true∥2+ s −1/2 ∥∥ x true − x true∥1+ σ ] (5.37)sIn addition, CoSaMP will converge, assuming exact arithmetic, in at most 6(s + 1) iterations.This guaranteed convergence and numerical simplicity come at a price. In particularthe requirement on the RIC is more stringent and the error bound is looser. A comparisonof IHT, CoSaMP, and BP for noise-free signals can be found in [94]. A detailed numericalinvestigation along with parameter tuning strategies comparing CoSaMP and SP to iterativethresholding algorithms can be found in [95]. As we shall see one advantage of thesegreedy approaches is that they can be easily extended to cases where signal knowledgebeyond simple sparsity is available.s5.3.5 Bayesian ApproachesWe have already mentioned the interpretation of QP λ as the MAP estimate under a Laplacianprior. Several efforts have been made to apply other priors to the vector x true to achievesparse solutions. Many of these approaches seek minimum mean square error (MMSE)estimates of the signal.5.3.5.1 Averaging SolutionsThe first example along these lines is a randomized version of OMP described in [96].Instead of always selecting the atom having the largest correlation with the remainingresidual, this algorithm selects the next atom at random with a distribution whose