ELEC 9741: Assignment 1, 2024
Instructions
1due in Moodle, Friday June 21, 4pm
2Signed School Cover Sheet attached
3TYPED PDF only - not handwritten.
4Follow the Homework Rules.
5
Computer
output
: no commentary⇒no marks.
6
Analytical
results
: no working⇒no marks.
7♦means you can use Matlab; else not.
8
No Copying
except from lectures
•
No Discussion
No web searching
9No use of Large Language Models
Q1(14) Theory
(a) Impulse Response.
Consider the LTI systems
t
= (h∗u)
t
whereu
t
is
the input signal andh
r
,r= 0,···is the impulse re-
sponse.
(i) Suppose the input is an AR(2) sequence
u
t
=φu
t−2
+η
t
whereη
t
is an iid(0,σ
2
) sequence.
Show thatvar(u
t
) =σ
2
u
=
σ
2
1−φ
2
.
Suppose the impulse response is FIR
[h
r
,r= 1,2,3] =A[1,−2α,1]andh
r
= 0,r >3.
Show thats
t
is an ARMA sequence and find its
parameters.
(ii) What are the stability/stationarity restrictions on
the parameters ofs
t
?
Assumings
t
is stationary, derive a closed form
formula forσ
2
s
in terms ofφ,A,α,σ
2
.
(b) Noise Model.
Consider the stationary process
Y
t
=a+φY
t−3
+
t
,t= 1,2,···
where
t
is a Gaussian white noise sequence of zero
mean and varianceσ
2
.
(i) Derive the stability/stationarity constraints onφ.
(ii) Derive closed form expressions for the mean and
acs ofY
t
.
Q2(14) (Impulse Response Estimation)
(a)♦Simulation.
Write an mfile to simulate the system described in
Q1(a) when the output is measured in noise
y
t
=s
t
+n
t
t= 1,···,T
wheren
t
are iid(0,σ
2
)independent of theu
t
sequence.
The variance signal to noise ratio (vsnr) is defined by
vsnr=
var(s
t
)
var(n
t
)
=
σ
2
s
σ
2
(i) For each of the two sets of values
φ
α
vsnr
=
.8.8
1 1
.5 1
show plots (on a single page) ofu
t
,s
t
,y
t
(T= 250).
Also show plots of their histograms.
Compute the sample/empirical variances and com-
pare them with their theoretical values.
(ii) Also display (on a single page):
A Bode plot of the corresponding transfer func-
tion. Describe the effect of the system on the
input?
(b)♦Least Squares System identification.
Using your mfile generate data from the model for
φ,α,vsnr = -.8,.1,.5 andT=1000.
(i) Find the FIR least squares estimate of the im-
pulse response for the 4 values of m = 2, 4, 8, 16
and display (on a single page) 4 separate plots
each showing the results on top of the true im-
pulse response.
(ii) Also display (on a single page) the correspond-
ing 4 separate Bode plots each on top of the true
Bode plot. Comment on the performance of the
IR and TF estimators.
(iii) By repeating the simulation B = 50 times con-
struct histograms of each of the FIR parameters
for m = 4. Comment on the performance of
these estimators.
Q3(8).♦Statistical Graphics.
The graphics/plots you display in Q1, Q2, Q4 will earn up
to 8 marks.
Q4(14) (Noise Modeling)
Do not use any specialised matlab commands such as zp2tf,
arima, aic, bic etc.
(a)♦Write an mfile to simulate a stationary AR(3) time
series driven by a zero mean Gaussian white noise of
unit variance.
Your mfile should accept as input, three real roots or
one real root and a complex root; all non-zero.
It should produce the AR parameters & variance di-
rectly as well as the simulated values as output.
Show two simulations (T=200) (on a single page) one
for each of the above cases. List the two sets of pa-
rameters used. In each case ensure thatγ
o
≥3.
(b)♦Using your mfile simulate an AR(3) with roots
(.9,.1,.6) for T=200. List the true parameter values.
Using least squares regression
1
produce estimates for
the 3 parameters, the noise variance as well as stan-
dard errors for the parameters.
Are the estimates within 2 standard errors of the true
values?
(c)♦Using your mfile simulate new data (T=100) from
the same model (ii) compute BIC
2
and find its mini-
mizing orderp
∗
. Show a single plot of BIC together
with its two components.
Give the parameter estimates corresponding top
∗
and
their standard errors.
Also do a statistical model diagnosis using just the acs
of the residuals. What conclusions do you draw about
the quality of the estimated parameters and model or-
der?
1
write your own mfile; don’t use any matlab command for any regres-
sion related computations
2
using your own mfile; not matlab’s BIC command
