Digital Signal Processing and Digital Filters
Practice Sheet 6
1) Let A (ejω) = 1 + e−jω a and B (ejω) = b. We try to approximate a filter given by
The weight function is W (ω) = 1, ∀ω ∈ [0, π]. We choose ω ∈ {0, 4/π, π}.
(a) Derive the numerical expressions of U , V and d such that
(b) Compute the real values a,b that minimize ∑k|EE(ωk)|2.
2) In Optimal IIR filter design, we aim to approximate a target filter D (ω) by adjusting the parameters of a filter H . Typically, the estimation is performed by minimising one of the two following equations
(a) Given that both equations involve nonlinear operations due to weighting functions WS (ω) , WE (ω), explain what is the challenge in optimising ES (ω).
(b) The optimisation is performed at frequency samples Explain why we can estimate the filter when K is only half of the number of unknowns, which is K = 2/M+N+1.
(c) Explain what happens if K is very large, e.g. K = 100 · 2/M+N+1. Select one from the options below, and explain the choice.
i. The estimation would fail.
ii. The algorithm would estimate D (ω) with a very high degree of accuracy
iii. The algorithm works, but the accuracy is limited.
Explain what happens ifK < 2/M+N+1. Select one from the options below, and explain
the choice.
i. The algorithm returns an error.
ii. The algorithm returns multiple solutions.
iii. The algorithm returns one correct solution.
(e) Let WE (ω) = cos2(p (ω − ω0)). Find p, ω0 such that EE(k 20/π) = 0, k = 1, . . . , 20, for any H (ejω) and D (ω).