# UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE Fall 2010 Linear Systems Fundamentals

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1 UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE Fall 2010 Linear Systems Fundamentals FINAL EXAM WITH SOLUTIONS (YOURS!) You are allowed one 2-sided sheet of notes. No books, no other notes, no calculators. PRINT YOUR NAME Signature Student ID Number Problem Weight Score 1 20 pts 2 20 pts 3 20 pts 4 20 pts 5 20 pts Total 100 pts Please do not begin until told. Write your name on all pages. Show your work. Tables from the textbook are at the back of the exam. Good luck! 1

2 Problem 1 Consider the discrete-time system where the input signal x[n] and output signal y[n] are related by: y[n] = k= x[k]g[n 2k] where g[n] =u[n] u[n 4]. (a) Sketch g[n] precisely. (b) Determine and sketch precisely the output y 1 [n] when the input is x 1 [n] =δ[n 1]. 2

3 Problem 1 (cont.) (c) Determine and sketch precisely the output y 2 [n] when the input is x 2 [n] =δ[n 2]. (d) Is the system linear? Is it time-invariant? Justify your answers. 3

4 Problem 2 (a) Let x(t) = cos(100πt). Determine and sketch precisely X(jω), the Fourier Transform of x(t). (b) Let s(t) be the periodic unit impulse train s(t) = δ(t kt) k= with period T = Determine the sampling rate, i.e., the fundamental frequency ω s of the signal s(t). Then determine and sketch precisely S(jω), the Fourier Transform of s(t). 4

5 Problem 2 (cont.) (c) Let x s (t) =x(t)s(t). Determine and sketch precisely X s (jω), the Fourier Transform of x s (t). (d) Consider an ideal low-pass filter with cut-off frequency ω c = ω s 2 and gain T, that is, a filter with frequency response { T if ω ω s H(jω) = 2 0 if ω > ω s 2. Let y(t) be the output produced when the input to this filter is x s (t). Determine and sketch precisely Y (jω), the Fourier Transform of y(t). 5

6 Problem 2 (cont.) (e) Finally, determine y(t), the output signal from part (d). Has the filter correctly reconstructed the original signal x(t) from the sampled signal? If not, determine whether a low-pass filter with a different cut-off frequency and gain can be used to reconstruct x(t) fromy(t). 6

7 Problem 3 (a) Consider the discrete-time signal x[n] =δ[n +3] δ[n 3]. Determine X(e jω ), the discrete-time Fourier Transform (DTFT) of the signal x[n]. (b) Is X(e jω ) real, pure imaginary, or neither? (c) Sketch precisely one period of X(e jω ), the magnitude of X(e jω ), in the range [ π, π]. 7

8 Problem 3 (cont.) (d) Determine the discrete-time Fourier Transform (DTFT) Y (e jω )ofthe signal y[n] =1+cos( π 4 n + π). (e) Sketch precisely one period of Y (e jω ), the magnitude of Y (e jω ), in the range [ π, π]. 8

9 Problem 4 Consider a continuous-time LTI system where the input signal x(t) andoutput signal y(t) are related by the linear constant-coefficient differential equation d 2 y(t) dt 2 + dy(t) dt 2y(t) = dx(t) dt + x(t). (a) Determine the algebraic part of the system transfer function H(s). (b) Identify all poles and zeros associated with H(s) and sketch precisely the corresponding pole-zero plot. 9

10 Problem 4 (cont.) (c) Give precise descriptions of all possible regions of convergence (ROCs) that can be associated with the pole-zero pattern you found in part (b). (d) For each ROC you identified in part (c), indicate whether the associated system is causal and/or stable. Use one row in the table below for each of the ROCs you identified and put an X in a column to indicate that the ROC satisfies the corresponding property. Ignore any extra rows in the table. Justify your answers. ROC Causal Stable 10

11 Problem 4 (cont.) (e) You should have identified at least one causal system in part (d). For each such system, determine the corresponding impulse response h(t). 11

12 Problem 4 (cont.) (f) For the system whose ROC contains the jω-axis, determine H(jω), the magnitude of the frequency response. Calculate the value of H(jω) at ω = 0 and the asymptotic value as ω. Justify your answers. 12

13 Problem 5 Let x(t) be a signal for which the Fourier Transform satisfies 1+ ω 10π if 10π ω 0 X(jω) = 1 ω if 0 <ω 10π 10π 0 if ω > 10π. (a) Suppose that the Fourier Transform Y (jω) of another signal y(t) hasthe property that Y (jω)=2x(j(ω 20π)). Find a signal c(t) such that y(t) =x(t)c(t). 13

14 Problem 5 (cont.) (b) Let z(t) be the signal obtained by modulating the signal y(t) from part (a) according to z(t) =y(t)cos(10πt). Determine and sketch precisely Z(jω), the Fourier Transform of z(t). (c) Now consider the signal w(t) obtained by modulating the signal z(t) from part (c) according to w(t) =z(t)cos(10πt). Determine and sketch precisely W (jω), the Fourier Transform of w(t). 14

15 Problem 5 (cont.) (d) Can y(t) be recovered from w(t) by means of a low-pass, high-pass, or bandpass filter? If not, explain why. If so, determine and sketch precisely the frequency response of the appropriate reconstruction filter. (e) Can x(t) be recovered from w(t) by means of a low-pass, high-pass, or bandpass filter? If not, explain why. If so, determine and sketch precisely the frequency response of the filter. 15

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