ECE 413 Digital Signal Processing Midterm Exam, Spring Instructions:

Transcription

1 University of Waterloo Department of Electrical and Computer Engineering ECE 4 Digital Signal Processing Midterm Exam, Spring 00 June 0th, 00, 5:0-6:50 PM Instructor: Dr. Oleg Michailovich Student s name: Student s ID #: Instructions: This exam has 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, place your WATCARD on the table, and fill out the exam attendance sheet when provided by the proctor after the exam starts. Question marks are listed by the question. Please, do not separate the pages, and indicate your Student ID at the top of every page. Be neat. Poor presentation will be penalized. No questions will be answered during the exam. If there is an ambiguity, state your assumptions and proceed. No student can leave the exam room in the first 45 minutes or the last 0 minutes. If you finish before the end of the exam and wish to leave, remain seated and raise your hand. A proctor will pick up the exam from you, at which point you may leave. When the proctors announce the end of the exam, put down your pens/pencils, close your exam booklet, and remain seated in silence. The proctors will collect the exams, count them, and then announce you may leave.

2 Question [5%]: Consider the difference equation y[n] 5 6 y[n ] + 6 y[n ] = x[n ]. () What are the impulse response, frequency response, and step response (i.e. a response to x[n] = u[n]) for the causal LTI system satisfying this difference equation. Reminder: Question [0%]: Z a n u[n] (when z > a ); az N n=0 r n = rn r. () Let x[n] be a causal stable sequence with z-transform X(z). The complex cepstrum ˆx[n] is defined as the inverse transform of the logarithm of X(z), i.e. ˆX(z) =logx(z) where the ROC of ˆX(z) includestheunitcircle. Determine the complex cepstrum for the sequence. Z ˆx[n], () x[n] =δ[n]+a δ[n N], where a <. (4) Reminder: log( + x) = ( ) n= n+ xn n. (5) Question [5%]: Consider a standard system (shown below) consisting of a C/D converter, a discrete-time system, and a D/C converter. Suppose that the Fourier transform X c (jω) ofx c (t) obeysx c (jω) =0for Ω π 000, and that the discrete-time system is a squarer, i.e. y[n] =x [n]. What is the largest value of T such that y c (t) =x c(t)? x c (t) x[n] y[n] y C/D y = x c (t) D/C

3 Solution to the Midterm exam, Spring 00 Solution Taking Fourier Transform of the given difference equation, Y (e jω ) 5 6 e jω Y (e jω )+ 6 e jω Y (e jω )= e jω Y (e jω ) Therefore the frequency response is H(e jω )= Y (ejω ) X(e jω ) = e jω 5 6 e jω + 6 e jω To find the impulse response, we express H(e jω )as H(e jω )= e jω e jω e jω = [ e jω e jω ] e jω e jω = e jω e jω Therefore h[n] = n n u[n] u[n]

4 Let s[n] be the step response. Then s[n] = = = = k= n k= k= k=0 h[k]u[n k] h[k] n k k u[k] u[k] n k k u[k] u[k] Now clearly s[n] = 0 for n<0. For n 0, s[n] = (/)n+ / =+(/) n (/) n (/)n+ / Therefore s[n] = + n n u[n] Solution x[n] =δ[n] aδ[n N], a < Taking the z-transform, X(z) =+az N Therefore the z-transform of the complex cepstrum ˆx[n] is given by ˆX(z) = log X(z) = log( + az N )= ( ) n+ (az N ) n n n= Therefore ˆx[n] = ( ) k+ a k δ[n kn] k k=

5 Solution Since y[n] =x [n], therefore Y (e jω )= π X(ejω ) X(e jω ). Therefore Y (e jω ) will occupy twice the frequency band that X(e jω ) does if no aliasing occurs. Now, if Y (e jω ) = 0for π < ω < π, then X(e jω ) = 0for π/ < ω < π/, and X(e jω )=0forπ/ ω π. Now it is given that X c (jω) =0forΩ π000. Using ω = ΩT, π T.π(000) = T 4000

6 University of Waterloo Department of Electrical and Computer Engineering ECE 4 Digital Signal Processing Final Exam, Spring 00 August, 00, 4:00-6:0 PM Instructor: Dr.OlegMichailovich Student s name: Student s ID #: Instructions: This exam has pages. No books and lecture notes are allowed on the exam. Please,turnoffyourcell phones, PDAs, etc., and place your bags, backpacks, books, and notes under the table or at the front of the room. Please, place your WATCARD on the table, and fill out the exam attendance sheet when provided by the proctor after the exam starts. Question marks are listed by the question. Please, do not separate the pages, and indicate your Student ID at the top of every page. Be neat. Poor presentation will be penalized. No questions will be answered during the exam. Ifthereisanambiguity,state your assumptions and proceed. No student can leave the exam room in the first 45 minutes or the last 0 minutes. If you finish before the end of the exam and wish to leave, remain seated and raise your hand. A proctor will pick up the exam from you, at which point you may leave. When the proctors announce the end of the exam, put down your pens/pencils, close your exam booklet, and remain seated in silence. The proctors will collect the exams, count them, and then announce you may leave.

7 Problem (0%) Consider the following sampling system: v c (t) Sampling at Hz v[n] w[n] Shannon reconstruction w c (t) The continuous time signal v c (t) isgivenby sin πt, t = 0 πt v c (t) = sinc(t) =, t =0. a) Sketch the Fourier transform V c (jω) of the continuous-time signal v c (t). b) Sketch the discrete-time Fourier transform (DTFT) V (e jω )ofthesampledsignalv[n]. c) The signal w[n] isobtainedfromv[n] by interpolation according to v[n/], if (n) =0 w[n] = 0, otherwise. Sketch the DTFT W (e jω )ofw[n]. d) The signal w[n] ispassedthroughanideal(shannon)interpolatortoresultin t nt w c (t) = w[n]sinc T n= with T =/6. Find an expression for w c (t) andsketchitsfouriertransformw c (jω). e) Are there values of T for which w c (t) =v c (t)? Explain your answer. Problem (0%) You are given two finite-length signals x [n], n =0,,...,N andx [n], n =0,,...,N, and your task is to perform linear convolution of these signals. Describe a way to compute the convolution if you are only allowed to use DFT/DFT and a frontal zero-padding of the form 0, for n =0,,...,L y [n] = x [n L], for n = L, L +,...,N + L

8 and y [n] = 0, for n =0,,...,L x [n L], for n = L, L +,...,N + L for some value of L. Problem (5%) Let x[n] beareal-valued signal of length N = L,whereL is an integer. Another signal y[n] of length M = N/ isdefinedfromx[n] accordingtoy[n] =x[n]+jx[n +],0 n<m. Let X[k] (with0 k<n)andy [k] (with0 k<m) denote the DFT coefficients of x[n] and y[n], respectively. a) For all k =0,,...,M, find an expression for Y [k] intermsofx[k]. b) Find an expression for X[0] and X[M] intermsofy [0]. Problem 4 (5%) When an input to an LTI system is x[n] =5u[n], the output is y[n] =[(/) n +( /4) n ] u[n]. a) Determine the system function H(z) of the system. Plot the poles and zeros of H(z), and indicate the ROC. b) Determine the impulse response h[n] ofthesystemforallvaluesofn. c) Write the difference equation that characterizes the system.

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