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Unformatted text preview: x[n − 1]) .
a) Find the impulse response h[n] of the system.
b) Is it stable?
c) If the input to the system is
1,
x [n ] =
0, n ≥ 0 & n is even
otherwise what is the output y [n] of the system?
Recall:
−1 • Z{u[n]} = (1 − z −1 ) , z  > 1.
−1 • Z{an u[n]} = (1 − a z −1 ) , z  > a.
• Partial fraction expansion: Ak = (1 − dk z −1 )X (z )z=dk . Question 2 [35%]:
A continuous time signal xc (t) has the following Fourier transform: Figure 1: Pertaining to Q2
a) What is the Nyquist rate corresponding to the signal?
2 b) What is the lowest sampling frequency which can be applied to xc (t) without causing aliasing
distortion? Draw the DTFT of x[n] in this case.
c) Draw the blockdiagram of the reconstruction system that will recover xc (t) from the sampled
signal x[n] if the sampling rate is set to its lowest possible value. Sketch the frequency response
(i.e. Fourier transform) of the reconstruction ﬁlter that is used in this case. Question 3 [30%]:
Consider the following two systems: Figure 2: Pertaining to Q3
Write Y1 (z ) and Y2 (z ) as a function of X (z ). Are Y1 (z ) and Y2 (z ) equal? 3 University of Waterloo
Department of Electrical and Computer Engineering ECE 413 – Digital Signal Processing
Final Exam, Spring 2009
August 13, 2009, 9:0011:30 AM
Instructor: Dr. Oleg Michailovich Student’s name:
Student’s ID #: Instructions:
• This exam has 4 pages.
• No books and lecture notes are allowed on the exam. Please, turn oﬀ your cell
phones, PDAs, etc., and place your bags, backpacks, books, and notes under the table
or at the front of the room.
• Please...
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This document was uploaded on 02/28/2014 for the course ECE 413 at University of Waterloo, Waterloo.
 Fall '13
 OlegMichailovich
 Electrical Engineering, Digital Signal Processing, Signal Processing

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