Adaptive Filters

Problems and Computer Projects

PROBLEMS

Problem XI.1 (Indefinite quadratic cost) Consider the quadratic cost function (44.1) and choose Π = I, H = I, W = (0 ⊕ –2), y = col{α, β}.

1. (a) What is the signature of the coefficient matrix Π + H*W H?
2. (b) Verify that the gradient vector of J(w) evaluates to zero at , and that . Show that is the only vector where the gradient of J(w) is zero.
3. (c) Are there any other choices for w at which J(w) evaluates to 2|β|²?
4. (d) Plot the contour curves of J(w).

Problem XI.2 (Negative-definite coefficient) Refer to (45.19).

1. (a) Conclude from (45.20) that the matrices and Γ_i are congruent.
2. (b) Use the existence condition (45.15) to conclude that Δ < 0.

Problem XI.3 (Robust formulation with weighted noise energy) Refer to the discussion in Sec. 45.1. Assume again that we are given measurements d(i) = UiW° + v(i) and that we wish to estimate s(i) = U_iW° in order to satisfy the criterion

for some α > 0. Repeat the arguments of that section to show that ...