” ELEC 9741程序 写作、 辅导Java，PythonELEC 9741: Assignment 1, 2021Instructions1 due in Moodle, Wednesday June 30, 4pm2 Signed School Cover Sheet attached3 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 isthe input signal and hr, r = 0, is the impulse re-sponse.(i) Suppose the input is a white noise sequence i.e.iid(0, 2u).(ii) Suppose the impulse response ishr = rr, r = 0, 1, 2, where = e?1/(iia) Explain what are the stability restrictionson if any.(iib) Prove that the maximum of hr occurs at theinteger closest to . Find the value of thatmaximum.(iic) Derive a closed form formula for 2s .(b) Noise Model.Consider the stationary processYt = a+ Yt?2 + t ? t?2, t = 1, 2, where t is a Gaussian white noise sequence of zeromean and variance 2.(i) Explain what are the stability/stationarity con-straints on , (ii) Derive closed form expressions for the mean andacs 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 measuredin noiseyt = st + nt t = 1, , Twhere nt are iid(0, 2) Independent of the ut sequence.Also hr = 0, r mo + 1.The variance signal to noise ratio (vsnr) is defined byvsnr =var(st)var(nt)=2s2With mo = 45, = 15, T = 500, vsnr = 1, 2 = 1,repeatedly simulate the system for R = 100 repeats.(i) For each repeat compute the sample variance ofst. Display the R sample variances in a histogramand mark the true value 2s from the formula in Q1 onthe histogram. The Value of 2s from Q1 is not quitethe correct value to use here; why? But it should bevery close; why? Comment on the histogram.(b) ? Parameter Estimation.Write an m-file to compute the penalized least squaresestimator and its standard errors1(i) With = 15, T = 400, vsnr = 1 simulate thesystem once and compute the penalised least squaresestimator of for a grid ofm, values. Compute anddisplay the BIC for this grid.(ii) Derive a formula for the variance of the penalizedleast squares estimator.(iii) Find the values of ,m that minimize BIC and ontop of the true FIR, plot the corresponding estimatedFIR together with 95% confidence curves based onthe standard errors of the estimated s2. Commenton the results.Q3 (5). ? Statistical Graphics.The graphics/plots you display in Q1, Q2 will earn up to 5marks.1se(?r) =var(?r), r = 1, ,m2we ignore the biasQ3(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) timeseries driven by a zero mean Gaussian white noise ofunit variance.Your mfile should accept as input, three real roots orone 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) onefor each of the above cases. List the two sets of pa-rameters used. In each case ensure that o 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 forthe 3 parameters, the Noise variance as well as stan-dard errors for the parameters.Are the estimates within 2 standard errors of the truevalues?(c) Using your mfile simulate new data (T=100) fromthe same model (ii) compute BIC4 and find its mini-mizing order p?. Show a single plot of BIC togetherwith its two components.Give the parameter estimates corresponding to p? andtheir standard errors.Also do a statistical model diagnosis using just the acsof the residuals. What Conclusions do you draw aboutthe quality of the estimated parameters and model or-der?3write your own mfile; dont use any matlab command for any regres-sion related computations4using your own mfile; Not matlabs BIC command请加QQ：99515681 或邮箱：99515681@qq.com WX：codehelp
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