Adaptive Filters

Problems and Computer Projects

PROBLEMS

Problem VI.1 (Similarity transformations) Let A and B be two similar matrices, i.e., B = T⁻¹AT for some square invertible matrix T. Show that A and B have the same eigenvalues.

Problem VI.2 (Nonnegativity of power spectra) Let r(k) = Eu(i)u*(i – k) and assume this auto-correlation sequence is exponentially bounded as in (26.2). Introduce the corresponding power spectrum (26.3). Let R_M be the Hermitian Toeplitz covariance matrix whose first column is col{r(0), …, r(M – 1)}. Pick any finite scalar a and define b = col{a, ae^−jw,…, ae^−j(M–1)}, where . Show that

Now take the limit as M → ∞ to conclude that S_u(e^jw) ≥ 0.

Problem VI.3 (Diagonal weighting matrix) Consider the stochastic version of the general transform domain LMS recursion stated in Alg. 26.1, namely,

Let denote the k–th diagonal entry of the covariance matrix . Argue that,

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