ELEC 9741: Assignment 1, 2023
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ELEC 9741: Assignment 1, 2023
Instructions
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due in Moodle, Friday June 23, 4pm |
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Signed School Cover Sheet attached |
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TYPED PDF only - not handwritten. |
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Follow the Homework Rules. |
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Computer output |
: no commentary ÷ no marks. |
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Analytical results |
: no working ÷ no marks. |
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◆ means you can use Matlab; else not. |
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No Copying No Discussion except from lectures . No web searching |
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Q1 (14) 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. mu(2)). Show that ms(2) = war(st ) is given by
ms(2) = mu(2) 0(o)hr(2)
(ii) Suppose the impulse response is
hr = ryr . r = 0. 1. 2. . . . where y = e_ 1/τ
(iia) Explain what are the stability restrictions on a if any.
(iib) Prove that the maximum of hr occurs at the integer closest to a . Find the value of that maximum.
(iic) Derive a closed form formula for ms(2) .
(b) Noise Model.
Consider the stationary process
yt = a + ryt _ 1 + ct _ 9ct _2 . t = 1. 2. . . .
where ct is a Gaussian white noise sequence of zero mean and variance m2 .
(i) Explain what are the stability/stationarity con- straints on r. 9?
(ii) Derive closed form expressions for the mean and acs of yt .
Q2(14) (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. m2 ) independent of the ut sequence. Also hr = 0. r > mo + 1.
The variance signal to noise ratio (vsnr) is defined by
war(st ) ms(2)
war(nt ) m2
With mo = 40. a = 12. T = 500. wsnr = 1. m2 = 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 ms(2) from the formula in Q1 on the histogram. The value of ms(2) 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 a = 12. T = 400. wsnr = 1 simulate the system once and compute the penalised least squares estimator of 8 for a grid of m. A 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 A. 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 8’s2 . Comment on the results.
Q3 (8). ◆ Statistical Graphics.
The graphics/plots you display in Q1, Q2, Q4 will earn up to 8 marks.
Q3(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 that Vo > 3.
(b) ◆ Using your mfile simulate an AR(3) with roots (o8. o9e土jθ. 9 = π/3) for T=200. List the true pa-rameter 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?
2023-06-27