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. 4 Follow the Homework Rules. 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). Show that σ 2 s = var(st) is given by σ2s = σ 2 u ∑∞ 0 h 2 r (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(βˆr) = √ var(βˆr), 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
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