Principles of Modern Radar - Volume 2 1891121537

5.9 Problems 207[187] Devaney, A.J., “Super-Resolution Processing of Multistatic Data Using Time Reversal andMusic,” 2000. Available at http://www.ece.neu.edu/faculty/devaney.[188] Luke, D.R. and Devaney, A.J., “Identifying Scattering Obstacles by the Construction of NonscatteringWaves,” SIAM Journal on Applied Mathematics, vol. 68, no. 1, pp. 271–291, 2007.[189] Cheney, M., “The Linear Sampling Method and the Music Algorithm,” Inverse Problems,vol. 17, no. 4, p. 591, 2001.[190] Paige, C.C. and Strakos, Z., “Unifying Least Squares, Data Least Squares, and Total LeastSquares,” in Total Least Squares and Errors-in-Variables Modeling, Ed. S. v. Huffel andP. Lemmerling, Springer, 2002.[191] Wang, Y., Yang, J., Yin, W., and Zhang, Y., “A New Alternating Minimization Algorithmfor Total Variation Image Reconstruction,” SIAM Journal of Imaging Sciences, vol. 1,no. 3, pp. 248–272, 2008.[192] Romberg, J., “Compressive Sensing by Random Convolution,” SIAM Journal on ImagingSciences, vol. 2, no. 4, pp. 1098–1128, 2009.[193] Ender, J.H., “On Compressive Sensing Applied to Radar,” Signal Processing, vol. 90,no. 5, pp. 1402–1414, 2010. Special Section on Statistical Signal & Array Processing.5.9 PROBLEMS1. Generate several small matrices in MATLAB and numerically compute their mutualcoherence and restricted isometry constants. Notice how rapidly the problem of estimatingthe RIC becomes intractable as you increase M, N, and s.2. Assume that A represents oversampling of a one-dimensional discrete Fourier transform.Find an analytical expression for the mutual coherence of this dictionary. Verifythe result in MATLAB. (Hint: Utilize the expression for the Dirichlet kernel.)3. We noted in the text that M(A) ≤ R s (A) ≤ (s − 1)M(A). Prove these bounds.Hint: ‖x‖ 1 ≤ √ s ‖x‖ 2 if x is s sparse. Also, notice that ‖Ax‖ 2 2 = x H A H Ax =‖x‖ 2 2 + ∑ i≠ j x i ∗x j G ij , where G = A H A.4. The condition number of a matrix is given by the ratio of the largest singular value to thesmallest singular value. An identity matrix has condition number 1, and the conditionnumber will approach infinity as a matrix becomes closer to being rank deficient. Onemight be tempted to conclude that a good condition number indicates that a matrix Ahas a good RIP constant. Provide a counterexample to this claim. Verify your counterexample in MATLAB.5. We have alluded to the idea that mutual coherence is linked to the radar ambiguityfunction. In particular, the values encoded in the Gramian matrix A H A are samples ofthe radar ambiguity function. Demonstrate that this relationship holds.6. Implement the following algorithms in MATLAB or your language of choice. Testthe algorithm using a variety of problem sizes. Try generating the A matrix from aGaussian distribution, Rademacher distribution (±1), and as a subset of a DFT matrix.(a) OMP(b) CoSaMP(c) FISTA(d) IHT