ELEC 9741 2021 Mid-term Exam
Instructions
1 due in Moodle, Friday July 9, 4pm,
2 Signed School Cover Sheet attached
3 TYPED only - not handwritten.
4 Follow the course homework guide.
5 Computer output: no discussion⇒ no marks.
6 Analytical results: no working⇒ no marks.
7 ♦ icon means you can use Matlab; else not.
8 No Copyingexcept from lectures ; No Discussion.
Question 1(15) Spectral Estimation .
Consider a FIR open loop system with output measured
in noise
yt = st + t where st = (a+ bz−1)xt
Also t is a white noise; mean = 0, variance = σ2 .
Suppose the input xt is an AR(1) process 1 xt = 1(1−gz−1)νt
νt is a white noise; mean = 0,variance = σ2ν .
Also t and νt are independent 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
σ2x = var(xt), σ
2
s = var(st), σ
2
y = var(yt)
and the variance signal to noise ratio V SNR = σ
2
s
σ2
.
(b) Simulate the system for t = 1, · · · , T = 600.
Use parameter values a = .8, b = 1, σ2 = 1 and
g = .7, σ2ν = 1. Compute the VSNR.
Show plots of yt, st, xt. Also show histograms of st
and t. To what extent is the signal ’hidden’ in the
noise’?
(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 func-
tion. Show plots of the true transfer function and
the transfer function estimator for lag window values
M = 10, 20, 30, 40. Using the transfer function esti-
mator construct an estimator of the impulse response.
Compare the respective estimators to the correspond-
ing true quantities.
1The use of stochastic signals as stimuli in experimental studies of open
loop systems is common in practice e.g bio-engineering
Question 2(15) Wiener Filter .
Consider the MA(1)+AR(1) signal extraction problem
yt = st + nt, t = 1, 2, · · ·
where yt is the observed sequence, and the signal st and
noise sequence nt are independent.
st = (1− θz−1)t is MA(1)
nt =
1
1−φz−1)νt is AR(1)
where t, νt are independent zero mean noise sequences with
respective variances σ2 , σ
2
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 an MA(2) process. Show
plots of the iterates for each parameter.
(b) For the MA(1) + AR(1) problem above:
(i) Derive a formula for the Wiener filter as a ratio-
nal filter.
(ii) Derive a formula for the denominator MA co-
variances in terms of: φ, λ = σ
2

σ2ν
(VSNR).
(iii) ♦Hence using your mfile compute a representa-
tion of the Wiener filter as a forwards-backwards
filter, in the case where φ = −.6, λ = 2.
(c) ♦Simulate the MA(1)+AR(1) system with:
φ = −.2, θ = .9, σ2ν = 1, λ = 2 for T = 100.
Compute the Wiener filter estimate sˆt and the error
signal et and display them jointly on a single plot.
Question 3(12) Kalman Filter .
Consider the steady state Kalman filter (KF) for a noisy
unstable AR(1) model with scalar state ξt
ξt+1 = φξt + wt and yt = ξt + vt and |φ| > 1.
To simplify discussion we introduce non-dimensional
quantities:
correlation ρ = N√
QR
; VSNR α = 12
√
Q√
R
;
scaled state error variance Po = P√
QR
.
(a)Assuming ρ = 0 show that:
(a) Po = β +
√
β2 + 1 where β = α+ φ
2−1
4α .
(b) Hence show that Po ≥
√
φ2 − 1.
(c) Show that the steady state KF is an AR(1) with pa-
rameter φ∗ = φ2αPo+1 . Show that |φ∗| < 1.
(b) ♦Simulate the system with φ = 1.1; α = 1; T = 100
and implement the steady state Kalman filter. Plot ξt, ξˆt
overlaid and comment on the results.
Question 4(8) Graphical Display .
Marks for displays given in Q1,Q2,Q3.

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