Problem 1. Let y=(x[n-2]+x[n])/2. (a) Derive the frequency response function in theory by doing a z transform to the time domain equation. Express the result in polar form. (b) What are h[n] and its Fourier transform? (c) Sketch the magnitude of the frequency response function. (d) Write a Matlab program to obtain the H(jw). Compare the theoretical and experimental H(jw) both in amplitude and in phase. (e) For an input function x[n]=2sin(2n/2)+3*cos(2n/8), predict its output based on theoretical H(jw). In Matlab, obtain output y[n] and compare Matlab output with your theoretical prediction.(f) What is the expected auto-correlation coefficient for y at lags -3 to 3, if x is a random sequence. Verify it in Matlab.
Problem 2. For y=x[n]-2x[n-1]+x[n-2], answer the same questions in problem 1.
Problem 3. Let x be a random sequence and have a Gaussian distribution with zero mean and unit standard deviation. Let y1=(x[n-4]+x[n-3]+x[n-2]+x[n-1]+x[n])/5 and y2=(x[n-4]+2x[n-3]+3x[n-2]+2x[n-1]+x[n])/9.a) Find the theoretical frequency transfer functions for the two systems/operations.b) Sketch the magnitudes of the above transfer functions as a function of normalized frequency and comment on the similarities and differences of the two.c) Sketch the expectations of the normalized auto-correlation functions for y1 and y2 for 10 lags (on both sides of zero lag). Please do this part by thinking about what the expectations should be. Comment on the differences and similarities of two autocorrelation functions. Give as many details as you can think of. d) Simulate the magnitudes of the two transfer functions in Matlab. Do the simulation by generating x, y1 and y2. Find their FFT’s and then transfer function. Please do multiple simulations and find the averages. e) Compare the matlab simulated results with the theoretical ones by overlaying them on top of each other. If they are not the same, explain why?f) Do part c) in Matlab. Do the Matlab simulations agree with your sketches? If they don’t, comment on your reasoning in your sketches and where your thought went wrong. Please do not “cheat” by doing the matlab part first.Bonus part (20 pts) g) In doing part c) for y1, work out the standard deviations at lag 6, 7, and 8? Verify this in Matlab.
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