Principles of Modern Radar - Volume 2 1891121537

208 CHAPTER 5 Radar Applications of Sparse Reconstruction7. Generate a phase transition plot comparing the performance of BP, OMP, CoSaMP,and IHT for the noise free case. Select N = 256 for your simulations. Use Gaussianrandom entries for your A matrix. Be sure to normalize the columns of the A matrixand to generate a new random A matrix for each realization. For each algorithm, plotthe level curve representing the 50% probability of successful reconstruction. Hint:You will want to choose a reasonable numerical tolerance and count a realization as asuccess when the solution is correct to that tolerance. For BP, that is BP σ with σ = 0,you can use any of several available solvers found online. SPGL1 can be used, forexample.8. Obtain the MATLAB implementation of SPGL1 online. Generate a Gaussian A matrixand a sparse random vector. Attempt to reconstruct x true from the data y for a varietyof λ values. For each value of λ, solve the problem using SPGL1 with σ = ‖A ˆx − y‖ 2and verify that you obtain the same solution. Plot the Pareto curve represented by theseresults.9. Suppose that we wish to solve QP λ with a non-singular weighting matrix W in placeof the scalar weight λ, that is,ˆx = argminx‖Wx‖ 1 + ‖Ax − y‖ 2 2 (5.61)Demonstrate that we can use any algorithm which is capable of solving QP λ to obtainthe desired solution by solving a modified problem and then applying a simpletransformation to the output.