Principles of Modern Radar - Volume 2 1891121537

824 Appendix A: Answers to Selected ProblemsChapter 52. M(A) = max |Am H A sin (π(m − n)M/N)n|=maxi ≠ ji ≠ j sin (π(m − n)/N)4. Consider a matrix A ∈ R M×N with condition number 1, e.g. a random subset ofa Fourier matrix. Consider a new matrix B ∈ R M×N+1 which is the concatenationof the matrix A with an additional copy of the last column of A. One can easilyargue that the condition number of B can be made arbitrarily close to 1 by choosingN sufficiently large for a fixed M. However, since the final two columns of B areidentical, [0 0 0 ...−11] T is in the null space of B, and thus R 2 (B) = 1.9. If W is nonsingular, we can define the (unknown) variable u = Wx and the modifiedforward-model matrix M = AW −1 , then solve the modified problemû = argmin ‖u‖ 1 + ‖Mu − y‖ 2 2 .uThe desired solution is then recovered by the linear transformation ˆx = W −1 û.Chapter 61. At low grazing angles, optical imagery provides a side-looking (or profile) view ofthe scene, while SAR presents a downward-looking view. Optical imagery gives adownward-looking at steep grazing angles–approaching normal incidence–while SARimagery transitions to a profile view. Because the scene content is projected into the imageplane, near-incident SAR imagery is often useless and these collection geometriesare avoided.3. Range resolution in the slant plane is always given by the familiar expression δR =c/2B. The image resolution in the ground plane is determined by the grazing angleδy = δR/ cos θ grz . The cross-range resolution is independent of grazing angle, and itdepends only on the wavelength and angle of integration: δx = λ/2θ int .7. Azimuth ambiguities will not be properly focused by the PFA. While the azimuthambiguities alias into the scene, the rate of change of phase (or range migration)does not correspond to any true location within the scene. Thus, PFA cannot properlycompensate for the ambiguous phase history.9. In the (k x ,k y ) plane the rectangular region of support is defined byandk x ∈ [−2k min cos θ grz tan(θ int /2), +2k min cos θ grz tan(θ int /2)]k y ∈ [2k min cos θ grz , 2k max cos(θ int /2) cos θ grz ],where k min = 2π f min /c and k max = 2π f max /c are the minimum and maximum wavenumberstransmitted by the radar.