Adaptive Filters

Problems and Computer Projects

PROBLEMS

Problem V.1 (Weighted norms) Introduce the notation , where Σ is a Hermitian and non-negative definite matrix.

1. (a) Show that is a valid vector norm when Σ is positive-definite, i.e., verify that it satisfies all the properties of vector norms as described in Chapter B.
2. (b) When Σ is singular, which properties of vector norms are violated?

Problem V.2 (Positive-def initeness of the weighting matrix) Refer to expression (22.20) for Σ′ and assume Σ > 0.

1. (a) Verify that
   

   Conclude that Σ′ ≥ 0.

2. (b) Now let Σ′ = E Σ′. Clearly, Σ′ ≥ 0 in view of part (a). We wish to show that it is actually positive-definite. So assume, to the contrary, that there exists a nonzero constant vector x such that Σ′x = 0. Define
   

   Show from Σ′x = 0 that y = 0 almost surely. Argue that this is only possible if the regressor is non-random (i.e., a constant vector).

Problem V.3 (Special matrix) Consider a nonnegative-definite matrix F of the form = I – μA + μ²B, for some ...