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pset9000

Course: ECE 2025, Fall 2008
School: Georgia Tech
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GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 2025 Fall 1999 Problem Set #9 Assigned: 22 October 99 Due Date: 29 October 99 (F RIDAY ) Quiz #2 will be given on Monday, October 25-th in your regular class time. Review Sessions are planned for Saturday 11-1pm and Sunday at 7 pm in the ECE Auditorium. Reading: In DSP First, Chapters 7 and 8 on Z-Transforms and IIR Filters. = The...

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GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 2025 Fall 1999 Problem Set #9 Assigned: 22 October 99 Due Date: 29 October 99 (F RIDAY ) Quiz #2 will be given on Monday, October 25-th in your regular class time. Review Sessions are planned for Saturday 11-1pm and Sunday at 7 pm in the ECE Auditorium. Reading: In DSP First, Chapters 7 and 8 on Z-Transforms and IIR Filters. = The two (2) STARRED problems will have to be turned in for grading. The other problems are from EE2200, Spring 1999, Problem Set #7. They are excellent exercises on the current material. You will see more of these type of problems on Quiz #3. Next week a solution will be posted. Some similar problems solutions can be found on the CD-ROM and in old homeworks, especially the &quot;unstarred&quot; problems. PROBLEM 9.1*: A linear time-invariant system has system function H(z) = (1 - z-1 ) 1 z-1 = - . (1 + 0.5z-1 ) (1 + 0.5z-1 ) (1 + 0.5z-1 ) (a) Determine the difference equation that is satisfied by the output y[n] and the input x[n] of this system. (b) Plot the poles and zeros of H(z) in the complex z-plane. ^ (c) Determine the frequency response H(e j ). Use freqz( ) in MATLAB to evaluate the frequency ^ ^ response of this system for - . Plot |H(e j )| and H(e j ) in a two part plot and hand it in ^ with your solution to this problem. (d) Determine h[n], the <a href="/keyword/impulse-response/" >impulse response</a> of this system. (We did this in lecture 15 on Friday, October 22). Plot the values of this sequence for -3 n 10. (e) The input to this system is x[n] = 3 + 7[n] + 10 cos(n/2). Use superposition to determine the output of the system y[n] corresponding to the above input x[n]. Give an equation for y[n] that is valid for all n. (Note: This is an easy problem if you approach it correctly!) PROBLEM 9.2*: An inverse filter is an LTI system which, when cascaded with another LTI system, &quot;undoes&quot; the effects of the other LTI system. You saw an approximate inverse filter in one of the Laboratory experiments. The reason that you were only able to do an approximation then was that we had not yet studied IIR systems. x[n] w[n] y[n] H1 (z) H2 (z) (a) Suppose that H1 (z) describes a given LTI system. Determine H2 (z) so that the output is y[n] = x[n]; i.e., so that the second system compensates exactly for the effects of the first system. (b) If H1 (z) represents an FIR system, what can you say about the second system; i.e., is it FIR or IIR? Explain. (c) Suppose that the first system is defined by the difference equation 9 w[n] = k=0 k x[n - k] where 0 &lt; &lt; 1. Show that H1 (z) can be expressed the following ratio of polynomials in the variable z-1 : H1 (z) = 1 - 10 z-10 . 1 - z-1 Plot the poles and zeros of H1 (z) in the complex z-plane. Hint: You will need to find the values of z that satisfy the equation 1 - 10 z-10 = 0. This is done just as in the case of the moving average filter. You should find that the zeros are not on the unit circle, but on a circle of a different radius. (d) Now, for the system H1 (z) of part (c), determine the inverse system function H2 (z) and plot its poles and zeros in the complex z-plane. What happens to the poles and zeros of H1 (z) and H2 (z) when we form the product H1 (z)H2 (z)? (e) From H2 (z) obtained in part (d), determine the difference equation that relates the output y[n] to w[n], the input to the second system. (f) The system of part (e) could be implemented in Matlab by the statement yy=filter(b,a,ww); What should b and a be in order to implement the inverse system for the example of part (c)? PROBLEM 9.3: For each of the difference equations below, determine the poles and zeros of the corresponding system function, H(z). 1 S1 : S3 : S6 : S7 : y[n] = 0.4y[n - 1] + x[n] + x[n - 1] y[n] = -0.75y[n - 1] + x[n] - x[n - 1] y[n] = x[n] - x[n - 1] + x[n - 2] - x[n - 3] y[n] = x[n] + 1 x[n - 1] - 3 x[n - 2] 4 4 1 These systems are a subset of those in the following two problems. PROBLEM 9.4: <a href="/keyword/impulse-response/" >impulse response</a> : J 1 1 0.5 0 -1 -5 1 0.5 0 -0.5 -5 1 1.5 0 1 0.5 -1 -5 0 5 10 TIME INDEX (n) 15 0 -5 0 5 10 TIME INDEX (n) 15 0 5 10 15 -1 -5 0 5 10 15 0 5 10 15 -5 0 5 10 15 <a href="/keyword/impulse-response/" >impulse response</a> : K 0 <a href="/keyword/impulse-response/" >impulse response</a> : L 1 0 <a href="/keyword/impulse-response/" >impulse response</a> : M <a href="/keyword/impulse-response/" >impulse response</a> : N <a href="/keyword/impulse-response/" >impulse response</a> : O For each of the impulse-response plots (J, K, L, M, N, O), determine which one of the following systems2 (specified by either an H(z) or a difference equation) matches the <a href="/keyword/impulse-response/" >impulse response</a> . In addition, derive a formula for the <a href="/keyword/impulse-response/" >impulse response</a> , h[n], for S1 and S4 . S1 : S2 : S3 : S4 : S5 : S6 : S7 : S8 : y[n] = 0.4y[n - 1] + x[n] + x[n - 1] H(z) = 1 + z-1 1 - 0.7...

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