dependent signals. (a) Find expressions for the cross-spectrum Fyx(ω) and the spectra Fx(ω), Fs(ω), Fy(ω). Using these or otherwise find expressions for σ 2 x = var(xt), σ2 s = var(st), σ2 y = var(yt) and the variance signal to noise ratio V SNR = σ 2 s σ2 . (b) Simulate the system for t = 1, · · · , T = 1000. Use parameter values a = .8, b = 1, σ2 = 1 and θ = .7, σ2 ν = 1. Compute the VSNR. Show (on a single page) plots of yt, st, xt. (c) Using the simulated data (yt, xt) from (b) construct and display estimates of the cross-spectrum Fyx(ω) and the spectrum Fx(ω) 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. 1The use of stochastic signals as stimuli in experimental studies of open loop systems is common in practice e.g bio-engineering Question 2(20) Wiener Filter . Consider the AR(2)+WN 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 the signal and noise sequences are independent. st = 1 1−φ2z−2 t is AR(2) nt = νt is WN where t, νt are independent zero mean noise sequences with respective variances σ 2 , σ2 ν . (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 σ 2 = 1 and co = 1 + θ 2 , c1 = −θ for the two cases θ = ±.8. Show plots of the iterates in each case. (b) For the AR(2)+WN 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, λ = σ 2 σ2 ν (VSNR). (iii) ♦Hence using your mfile compute a representation of the Wiener filter as a forwards-backwards filter, in the case where φ = −.6, λ = 2. (c) ♦Simulate the AR(2)+WN system with: φ = −.6, λ = 2 for T = 100 time points. Compute the Wiener filter estimate sˆt and the error signal et and display them jointly on a single plot. Question 3(10) Kalman Filter . Consider the steady state Kalman filter (KF). Following the lectures it is given by ˆξ