EECE 340 Exam, Fall 2007

AMERICAN UNIVERSITY OF BEIRUT Department of Electrical and Computer Engineering EECE440 Signals and Systems Lecturer: Prof. Fadi N Karameh Sample problems 3 PROBLEM 1 (12%) Consider a DT LTI system with the impulse response h[n] = �1 2 �n u[n] Determine, using DTFTs, the response to each of the following inputs: (i) x[n] = � 3 4 �n u[n] (ii) x[n] = (n + 1) � 1 4 �n u[n] (iii) x[n] = (−1)nu[n] PROBLEM 2 Consider a system consisting of the cascade of two DT LTI systems with the frequency responses H1(ejΩ) = 2 − e−jΩ 1 + 1 2e−jΩ and H2(ejΩ) = 1 1 − 1 2e−jΩ + 1 4e−j2Ω a) Find the difference equation describing the overall system. b) Determine the impulse response of the overall system. PROBLEM 3 y1[n] and y2[n] are two real 8-pt sequences. You are given only the first five points of their DFTs as ˜Y1[k] = [2 1 0 1 2] ˜Y2[k] = [0 1 2 0 0] b) Find the DTFT ˜ M[k] for k = 1 and k = 4 where m[n] is m[n] = y1[n].y2[n] Hint: Exploit knowledge of DFT properties for real sequences. 1 PROBLEM 4 Find the DTFT of the following: a) x[n] = � 1 2 �n u[n + 2] b) x[n] = (n − 2)(u[n + 4] − u[n − 5]) PROBLEM 5 Find the signal x[n] which has the following DTFT a) X(ejΩ) = j sin(4Ω) − 2 b) � e−j2Ω sin(15Ω/2) sin(Ω/2) � ∗ sin(7Ω/2) sin(Ω/2) . PROBLEM 6 (10 %) Consider the analog-to-discrete-to-analog system shown in figure 1. The CT signal xa(t) is sampled at a frequency of Fs = 2000 Hz (Ts = 0.5 msec). The resulting impulse train is then converted to a discrete time sequence xd[n]. The Lowpass DT filter Hd(ejΩ) is subsequently used to filter xd[n] giving yd[n]. Finally, a CT version of the output ya(t) is created, using an ideal DT-to-CT converter (at the same sampling frequency Fs = 2000 Hz). Note : Hd(ejΩ), which is obviously periodic, is shown for only one period. Figure 1: Problem 6 a) For the CTFT of xa(t) given by Xa(w) in the figure with B = 2000π rad/sec, sketch the Xd(ejΩ), the DTFT of the DT sequence xd[n]. b) Sketch Ya(w), the CTFT of the CT signal ya(t). Again, assume that we are using the same frequency as that of sampling Fs = 2000 Hz. 2 c) (BONUS 5%) Describe, using Matlab code, how would you use Matlab to obtain X(w), the CTFT of a CT signal x(t). Assume that I sample x(t) at 200 Hz and I have 1 second recording of it. Clearly describe the various input and output vectors you obtain, and label the coordinates of your axes. Hint: you have done this twice in the course so far. 3