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FALL 22 EC516 Problem Set 08

Due: Sunday November 13 (Before Midnight) on Blackboard Learn

Please Note that Problem Set 07 (previously posted) is also due on November 13

Please Note that Test 02 on Wednesday November 16 will cover up to Problem Set 08

Problem 8.1

(A) Let r" [n] = g[n] ∗ g[−n] where g[n] is a N-point real-valued signal. We call r" [n] the “autocorrelation” of g[n].

(a) Show that r" [n] = r" [ −n]

(b) Show that ∑./ g[n]g[n − m] =r" [m]

(c) Show that ∑./ g[n − k]g[n − m] = r" [m − k]

(B) In this part, we consider IIR all-pass filters. Assume z/  is a complex number and |z/ | < 1.

(a) If H(z) = 9(9) , show that ?H @eBC D ? = 1 for all 业 and draw a flowgraph for the IIR

filter with system function H(z) .

(b) If H(z) = 9(9) >:99(9); ,  show that ?H @eBC D ? = 1 for  all 业 . Also,  show that the difference equation for this second-order all-pass filter with a real impulse response has only real coefficients.

Problem 8.2

If the data g[n] is a real valued N-point signal and the Pade matching model for g[n] is:

H(z) = ><∑H(I)J;(F)GH9:H ; P < N

(a) Are the parameters G , a>, … , aQ  guaranteed to be real? Justify your answer.       (b) Is the impulse response ℎ[n] guaranteed to be real valued? Justify your answer.

Problem 8.3

Let g[n] = 6[n] + 26[n − 1] + 6[n − 2] − 0.56[n − 3] + 0.56[n − 4] − 0.256[n − 5]

(a) Determine the z-transform H> (z) of the parametric signal model ℎ > [n] of order P = 1 for the data g[n]. You may use MATLAB to aid your calculations.

(b) Determine the z-transform H[ (z) of the parametric signal model ℎ [ [n] of order P = 2 for the data g[n]. You may use MATLAB to aid your calculations.

(c) Calculate and display |H[  \e a | as a function of k . You may use MATLAB to do the calculations and the display.

(d) Calculate and display |G \e a | as a function of k . You may use MATLAB to do the calculations and the display.

(e) Comment on the similarities and differences between the plots you obtained in the previous two parts of this problem.

Problem 8.4

Please attempt Practice Test 2 that gets posted on Wednesday, November 9. It is advised that you give yourself 60 minutes for your attempt on Practice Test 2. The solution to Practice Test 2 will be presented during lecture on Monday, November 14.