ELEC 9741 Electrical Engineer Spectral, Wiener, Kalman Filter Assignment

Assignment Task

Instructions

vt is a white noise; mean = 0, variance = σv2

Also, et and vt are independent signals.

(a) Find expressions for the cross-spectrum Fyx(w) and the spectra Fx(w), Fs(w), Fy(w).

Using these or otherwise find expressions for

σ²_x=var(xt), σ²_s=var(st), σ²_y​=var(yt) and the variance signal to noise ratio VSNR=σs2/σe2

(b) Simulate the system for t = 1,..., T = 600. Use parameter values a = .8, b = 1, σ²_e=1 g = .7, σ²_v=1. Compute the VSNR.

Show plots of yt, st, tx, et, and amplitude histograms. To what extent is the signal 'hidden' in the noise?

(c) Using the simulated data (yt, xt,) from (b) construct (by writing your own files) and display estimates of the cross-spectrum Fx(w) and hence the transfer function. Show plots (overlaid & on a single page) of the true transfer function and the transfer function estimator for lag window values M=10,20,30,40 Compare the estimators to the true transfer function and true input spectrum. NB. Just showing results is not enough - you have to explain how you got them.

Question 2(17) Wiener Filter

Consider the ARMA(1,1) signal extraction problem yt=st+nt,t=1,2,.... where yt is the observed sequence, st is a signal of interest, nt is a noise and they are independent. st=1(1−ϕz−1)1−θz−1 is ARMA(1,1) and nt=vt where et, vt are independent zero mean noise sequences with respective variances σ²_e,σ²_v

(a) Write an mfile to implement Wilson's algorithm for general m. Check your program by using it to do a spectral factorisation of the MA(1) process with σ²=1 and c₀=1+θ²,c1=−θ when θ=±.9. Show plots of the iterates in each case.

(b) For the ARMA(1,1) problem above:

(i) Derive a formula for the Wiener filter as a rational filter.

(ii) Derive a formula for the denominator MA covariances in terms of: ϕ,θ,λ=σ²_e/σ²_v(VSNR)

(iii) Using your mfile compute a representation of the Wiener filter as a forwards-backwards filter, in the case where ϕ=.7, θ=.6, λ=1.5

(c) Simulate the ARMA(1,1) system with: ϕ=.7, θ=.6, λ=1.5 for T=120 time points. Compute the Wiener filter estimate st and the error signal et and display them jointly on a single plot.

Question 3(16) Kalman Filter

Consider the steady-state Kalman filter (KF). Following the lectures it is given by εt + 1 = Fεt + k_et and yt =c^tεt + et where et is a zero mean white noise sequence of variance V. Also k=g/V and the steady state (gain) equations for g, V, P are given in the lecture notes.

We consider state estimation of the random walk (RW), scalar state, model εt + 1 = εt + wt and yt = εt + vt and where ε₀ = 0

(ai) Find a formula for the RW variance as a function of time. Hence show the RW is not stationary.

(aii) Show the conditions for a steady state Kalman filter are satisfied in this case. Can there be a steady-state Kalman filter even though the RW is not stationary?

(b) Simulate the system with Q=2, λ=1, T=100. Implement the steady-state Kalman filter. Show appropriate plots (no more than four) to illustrate how well (or poorly) it works.

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