2.163J/6.455J/12.518Sonar, Radar and SeismicSignal ProcessingQuiz 2Issued: November 13, 2006Due: November 27, 20061Problem 1: System IdentificationPart 1) One problem with LTI models is that they are only incrementally linear aroundsome nominal operating point; hence, we must be able to find the effective impulse responsewhile operating the system. One way of doing this is to inject white noise and cross correlatewith the output as shown in the figure below. The input, x(t), is the nominal input. To thisH(f)Average[−T/2,T/2]+x(t)w(t)y(t)u(t) = w(t−T )oxCross correlation for Impulse responsewe add a small amount of white noise, w(t). The filter, H(f), is the incremental transferfunction which we seek. a) Show that the average of the output is the impulse responseh(To)b) Determine the mean and variance of the estimate of the transfer function assumming thetransfer function is smooth.c) Show that the maximum group delay through the system puts lower bounds upon thewindows one uses in either the direct or indirect methods (Your pick vis a vis method)Part 2) We consider a two dimensional version of this as in most systems there are multipleinputs and outputs. This is indicted in the figure below Describe a method (there are many)for estimating the transfer functions similar to descrbed above. One can put the inputwhite noises through any multichannel LTI system to output signals which are available foruse for cross correlation at the output. Indicate how one would determine the means andvariances. If you use a multistep process such as turning on a process wi(t) on and thenoff, describe how this influences your calculations of the mean and variance. For example,if both processes are on, then results are correlated, but perhaps with a lower variance. Ifon/off, then one has uncorrelated estimates whose variances add. Note that this part isa “thought” problem that is intended to probe your understanding of spectral estimation..2w (t)1w (t)2v (t)1v (t)2w (t)1v (t)1v (t)2v (t)2v (t)1w (t)2++++LTISystemx (t)x (t)12H11(f)H21(f)H12(f)H22(f)CrossCorrCorrCrossMultichannel system identificationProblem 2The data on the web page of the subject are 1000 sec in duration and sampled at 1000Hz. The numbers are formatted so any programming language can read them. Your task issimply to estimate the power density spectrum as well as the bias and variance. Please noteareas where your windowing may cause significant problems with bias as well as your variancecomputations. This means explicit statements about your window lengths, averaging extents,etc and not a philosophical discussion. Please submit a copy of your code. You may notuse MATLAB for your submitted solution, but obviously you can use MATLAB to checkyour work. (You may find minor differences between your work and MATLAB.) Please noteall significant spectral features such as peaks, their widths, sprectral slopes (log/log is oftenuseful). Also, there are some subtle aspects of this time series which may be useful to