EE578/EE978 Assignment 2 Wiener-Hopf Solution and LMS Algorithm December 30, 2020 Please submit electronic copies of your answers (scanned handwriting is perfectly acceptable) via MyPlace by Monday 8/2/2021. Your answers should contain appropriate plots of your results with any annotations and brief discussions or justifications of your answers. Q1. System Identification Via Wiener-Hopf. You are trying to identify an unknown system with impulse response c[n] ◦—• C(z) and trans- fer function C(z) = 1− 1 2 z−1+ 1 4 z−2 using the structure in Fig. 1. For the purpose of question Q1, the input signal x[n] ∈ N (0, 1) is uncorrelated. (a) For an adaptive filter of length L = 3, determine (by hand, using ensemble statistics) the covariance matrix R = E { xnx H n } and the cross-correlation vector P = E{xnd ∗[n]}. (b) Determine, by hand, the Wiener-Hopf solution wopt = R −1p, given the quantities in Q1.(a). (c) Determine the Wiener-Hopf solution for the case where your adaptive filter has a length of (i) L = 2, and (ii) L = 4, and briefly comment on the difference to the result in Q1.(b). (d) In Matlab, generate 1000 samples of the signal x[n]. From this signal, estimate the sample covariance matrix Rˆ and the sample cross-correlation vector pˆ, and determine the Wiener-Hopf solution wopt = Rˆ −1pˆ. Q2. System Identification Setup with Correlated Input. The setup is the same for for Q1 in Fig. 1, but the input is now x[n] = cos(npi). c[n] w[n] +x[n] e[n] d[n] y[n] − Fig. 1: Setup with an adaptive filter w[n]. (a) Even though x[n] is a deterministic signal, we can state an autocorrelation function. Show that rxx[τ ] = cos(τpi) expresses the autocorrelation of x[n]. (b) For an adaptive filter of length L = 3, what numerical values do you now obtain for the ensemble statistics R and p? (c) Does the Wiener-Hopf solution exist? If so, can you determine it? If not, can you find another solution that minimises the error e[n] appropriately? (d) Implement 1000 samples of the signal x[n] in Matlab, estimate the sample covariance matrix Rˆ and the sample cross-correlation vector pˆ for L = 3. Can you determine the Wiener-Hopf solution? How does this compare to Q1.(d) and Q2.(c)? Q3. LMS Algorithm. (a) Implement the LMS algorithm for the scenario of Fig. 1 in Matlab, and adapt the filter for L = 3 using 1000 samples for x[n] ∈ N (0, 1). State two performance metrics that allow to assess how well your implementation works. (b) Using your LMS implementation, now operate with x[n] = cos(npi). Check that your filter works, and in particular compare w[n] to c[n], or W (ejΩ) to C(ejΩ) for the adapted filter. What do you observe? How does this compare to the experimental Wiener-Hopf approach in Q2.(d)? S. Weiss, December 30, 2020 欢迎咨询51作业君
