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###### 日期：2021-07-01 11:05

ELEC 9741: Assignment 1, 2021

Instructions

1 due in Moodle, Wednesday June 30, 4pm

2 Signed School Cover Sheet attached

3 TYPED PDF only - no microsoft word docs.

5 Computeroutput : no commentary? no marks.

6 Analyticalresults : no working? no marks.

7 means you can use Matlab; else not.

8 No Copyingexcept from lectures ; No Discussion.

Q1 (15) Theory

(a) Impulse Response.

Consider the LTI system st = (h ? u)t where ut is

the input signal and hr, r = 0, ・ ・ ・ is the impulse re-

sponse.

(i) Suppose the input is a white noise sequence i.e.

iid(0, σ2u).

(ii) Suppose the impulse response is

hr = rβ

r, r = 0, 1, 2, ・ ・ ・ where β = e?1/τ

(iia) Explain what are the stability restrictions

on τ if any.

(iib) Prove that the maximum of hr occurs at the

integer closest to τ . Find the value of that

maximum.

(iic) Derive a closed form formula for σ2s .

(b) Noise Model.

Consider the stationary process

Yt = a+ φYt?2 + t ? θt?2, t = 1, 2, ・ ・ ・

where t is a Gaussian white noise sequence of zero

mean and variance σ2.

(i) Explain what are the stability/stationarity con-

straints on φ, θ

(ii) Derive closed form expressions for the mean and

acs of Yt.

Q2(15) (Impulse Response Estimation)

(a) Simulation.

Write an mfile to simulate an FIR version of the sys-

tem described in Q1(a) when the output is measured

in noise

yt = st + nt t = 1, ・ ・ ・ , T

where nt are iid(0, σ2) independent of the ut sequence.

Also hr = 0, r ≥ mo + 1.

The variance signal to noise ratio (vsnr) is defined by

vsnr =

var(st)

var(nt)

=

σ2s

σ2

With mo = 45, τ = 15, T = 500, vsnr = 1, σ2 = 1,

repeatedly simulate the system for R = 100 repeats.

(i) For each repeat compute the sample variance of

st. Display the R sample variances in a histogram

and mark the true value σ2s from the formula in Q1 on

the histogram. The value of σ2s from Q1 is not quite

the correct value to use here; why? But it should be

very close; why? Comment on the histogram.

(b) ? Parameter Estimation.

Write an m-file to compute the penalized least squares

estimator and its standard errors1

(i) With τ = 15, T = 400, vsnr = 1 simulate the

system once and compute the penalised least squares

estimator of β for a grid ofm,λ values. Compute and

display the BIC for this grid.

(ii) Derive a formula for the variance of the penalized

least squares estimator.

(iii) Find the values of λ,m that minimize BIC and on

top of the true FIR, plot the corresponding estimated

FIR together with 95% confidence curves based on

the standard errors of the estimated β’s2. Comment

on the results.

Q3 (5). ? Statistical Graphics.

The graphics/plots you display in Q1, Q2 will earn up to 5

marks.

1se(β?r) =

var(β?r), r = 1, ・ ・ ・ ,m

2we ignore the bias

Q3(15) (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 that γo ≥ 3.

(b) ? Using your mfile simulate an AR(3) with roots

(.9,.7,.5) for T=200. List the true parameter values.

Using least squares regression3 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 BIC4 and find its mini-

mizing order p?. Show a single plot of BIC together

with its two components.

Give the parameter estimates corresponding to p? 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?

3write your own mfile; don’t use any matlab command for any regres-

sion related computations

4using your own mfile; not matlab’s BIC command