Department of Electrical, Computer, and Systems Engineering
ECSE 4530: Digital Signal Processing, Fall 2021
数字信号处理代写 1.(10 points) Write a Matlab function whose syntax is x=ar（a，sig，L） that produces a length L realization of an autoregressive···
Show all work for full credit! 数字信号处理代写
1.(10 points) Write a Matlab function whose syntax is x=ar（a，sig，L） that produces a length L realization of an autoregressive process with parameters a and σ such that
x[n]+ a[1]x[n −1]+ a[2]x[n −2]+…+ a[M]x[n − M] = v[n]
where v[n] is a white Gaussian noise process with variance σ2 . Hand in a plot of a realization you created with coeffificients a[1] = −0.9, a[2] = 0.7, and zero otherwise. σ = 0.5, and L = 200. Note: you should assume the process begins with x(−1) = ··· = x(−M) = 0, but these zeros should not be included in the output.
2.(10 points) Write a Matlab function whose syntax is r=autocorr（x，N） that returns an estimate of the N-lag autocorrelation vector, defifined as
(Note that (1) is the defifinition of autocorrelation when x[n] is complex. (1) reduces to the defifinition we have in class if x[n] is real. )
What is the estimate of r returned by your algorithm when the input is the vector x you created above and N = 3? What about when you use the same AR parameters with an L = 10000 realization?
数字信号处理代写
3.(10 points) Compute the normalized correlation coeffificients ρ(1) = 
for an AR process with the same parameters as in problem 1. You should be able to get explicit, closed-form solutions by using the Yule-Walker equations. Then, obtain the value of r (0) by using the equation for the variance of the white-noise process in terms of the r (j). Next, use the code in problem 1 to generate x[n] and use the code in problem 2 to compute the autocorrelations of x[n]. Then compute the normalized correlation coeffificients. Do your exact answers above agree with the estimates you obtained numerically?
4.(10 points) Write a Matlab function whose syntax is 
that computes the parameters a and σ for a given realization x as if it were an AR process of order M. That is, your function should implement the Yule-Walker equations to estimate a and the equation for the variance of the white-noise process to estimate σ. Do NOT use the MATLAB command “aryule.” What is the output of your function when M = 2 and the input is a L = 200 realization of an AR process using the same parameters as in problem 1? What about the results using a L = 10000 realization? What happens if you use M = 5 instead of M = 2? Discuss whether you get the results that you would expect.
5.(20 points) Suppose we put an AR process x[n] with the same parameters as in problem 1 through a communications channel with transfer function
Thus, the output u(n) of the channel is related to the input x(n) by u[n]−0.5u[n −1] = x[n]. (2) 数字信号处理代写
At the output of the channel, the signal is corrupted by a white noise process w(n) with variance 0.1. The total channel output is therefore y[n] = u[n]+w[n]. We assume that the two white-noise processes v(n) and w(n) are uncorrelated, and that all the signals are real. Compute a 3-tap FIR Wiener fifilter that operates on the received signal y(n) to produce an estimate of x(n) that is optimal in the mean-square sense. Show your work! Note that this problem requires both some paper-and-pencil analysis and some MATLAB (MATLAB can be used to solve the linear equations you obtain).
(Hint 1: fifirst fifind the equivalent AR model of x[n]. u[n] can be viewed as the output of applying a COMBINED fifilter to a white Gaussian noise process. Then you could fifind the statistical properties of u[n]. Hint 2: if you need to compute E(x[n]u[n−k]), you could multiple both sides of (2) by u[n−k] and then take expectations. )
更多代写：EE代写 网上考试作弊 写作辅导 论文proposal 做assignment 圣达菲学院代写
