Principles of Modern Radar - Volume 2 1891121537

3.10 Problems 1179. A constrained optimum MIMO approach can be developed based on recognizingthat the N-dimensional eigenspectrum of the generally positive definite compositechannel kernel H ′ H (or E { H ′ H } for the stochastic case) forms a continuum for whichsome number k of eigenfunctions (and corresponding eigenvalues) retain matchingproperties.a. Assume that k orthonormal eigenfunctions of H ′ H, denoted by u 1 ,...u k , withassociated eigenvalues λ 1 ≥ λ 2 ≥ ... ≥ λ k > 0, are available and have bettermatching properties than, say, a nominal nonadaptive quiescent waveform s q .Derive an expression for the waveform s p that resides in the matched subspacespanned by the k best eigenvectors. The resulting waveform can be viewed as atype of constrained optimization in which the properties of the nominal waveform(e.g., good range sidelobes) are traded for better SNR (see, e.g., [36]).b. Show that in the limit as k → N, the matched subspace waveform s p → s q . (Hint:The eigenfunctions of a positive definite (Hermitian) matrix form a complete basis.)