University of Waterloo

Department of Electrical and Computer Engineering

ECE 413 – Digital Signal ProcessingMidterm Exam, Spring 2010

June 10th, 2010, 5:30-6:50 PM

Instructor: Dr. Oleg Michailovich

Student’s name:

Student’s ID #:

Instructions:

• This exam has 2 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 10 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.

1

Question 1 [35%]:

Consider the difference equation

y[n]− 5

6y[n− 1] +

1

6y[n− 2] =

1

3x[n− 1]. (1)

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:

1

1− a z−1

Z−1

−→ anu[n] (when |z| > |a|);N−1

n=0

rn =1− rN

1− r. (2)

Question 2 [30%]:

Let x[n] be a causal stable sequence with z-transform X(z). The complex cepstrum x[n] is definedas the inverse transform of the logarithm of X(z), i.e.

X(z) = logX(z)Z←→ x[n], (3)

where the ROC of X(z) includes the unit circle.

Determine the complex cepstrum for the sequence.

x[n] = δ[n] + a δ[n−N ], where |a| < 1. (4)

Reminder:

log(1 + x) =∞

n=1

(−1)n+1 x

n

n. (5)

Question 3 [35%]:

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 Xc(jΩ) of xc(t) obeys Xc(jΩ) = 0 for

|Ω| ≥ 2π 1000, and that the discrete-time system is a squarer, i.e. y[n] = x2[n]. What is the largest

value of T such that yc(t) = x2c(t)?

xc(t)C/D

x[n]y = x2 y[n]

D/Cyc(t)

2

Solution to the Midterm exam, Spring 2010

Solution 1

Taking Fourier Transform of the given difference equation,

Y (ejω)− 5

6e−jω

Y (ejω) +1

6e−j2ω

Y (ejω) =1

3e−jω

Y (ejω)

Therefore the frequency response is

H(ejω) =Y (ejω)

X(ejω)

=13e−jω

1− 56e−jω + 1

6e−j2ω

To find the impulse response, we express H(ejω) as

H(ejω) =13e−jω

1− 1

3e−jω

1− 1

2e−jω

=2[

1− 1

3e−jω

−

1− 1

2e−jω

]

1− 13e−jω

1− 1

2e−jω

=2

1− 12e−jω

− 2

1− 13e−jω

Therefore

h[n] = 2

1

2

n

u[n]− 2

1

3

n

u[n]

1

Let s[n] be the step response. Then

s[n] =∞

k=−∞

h[k]u[n− k]

=n

k=−∞

h[k]

=n

k=−∞

2

1

2

k

u[k]− 2

1

3

k

u[k]

=n

k=0

2

1

2

k

u[k]− 2

1

3

k

u[k]

Now clearly s[n] = 0 for n < 0. For n ≥ 0,

s[n] = 21− (1/2)n+1

1− 1/2− 2

1− (1/3)n+1

1− 1/3

= 1 + (1/3)n − 2(1/2)n

Therefore

s[n] =

1 +

1

3

n

− 2

1

2

nu[n]

Solution 2

x[n] = δ[n]− aδ[n−N ], |a| < 1

Taking the z-transform,X(z) = 1 + az

−N

Therefore the z-transform of the complex cepstrum x[n] is given by

X(z) = log X(z) = log(1 + az−N) =

∞

n=1

(−1)n+1 (az−N)n

n

Therefore

x[n] =∞

k=1

(−1)k+1

ka

kδ[n− kN ]

2

Solution 3

Since y[n] = x2[n], therefore

Y (ejω) =1

2πX(ejω) ∗X(ejω).

Therefore Y (ejω) will occupy twice the frequency band that X(ejω) does ifno aliasing occurs.

Now, if Y (ejω) = 0 for −π < ω < π, then X(ejω) = 0 for −π/2 < ω <

π/2, and X(ejω) = 0 for π/2 ≤ |ω| ≤ π.Now it is given that Xc(jΩ) = 0 for Ω ≥ 2π1000. Using ω = ΩT ,

π

2≥ T.2π(1000)

=⇒ T ≤ 1

4000

3

University of WaterlooDepartment of Electrical and Computer Engineering

ECE 413 – Digital Signal ProcessingFinal Exam, Spring 2010

August 13, 2010, 4:00-6:30 PM

Instructor: Dr. Oleg Michailovich

Student’s name:

Student’s ID #:

Instructions:

• This exam has 3 pages.

• No books and lecture notes are allowed on the exam. Please, turn off your cellphones, PDAs, etc., and place your bags, backpacks, books, and notes under the tableor at the front of the room.

• Please, place your WATCARD on the table, and fill out the exam attendance sheetwhen 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 everypage.

• Be neat. Poor presentation will be penalized.

• No questions will be answered during the exam. If there is an ambiguity, stateyour assumptions and proceed.

• No student can leave the exam room in the first 45 minutes or the last 10minutes.

• If you finish before the end of the exam and wish to leave, remain seated and raiseyour 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, closeyour exam booklet, and remain seated in silence. The proctors will collect the exams,count them, and then announce you may leave.

1

Problem 1 (30%)

Consider the following sampling system:

3Sampling at 2 Hz Shannonreconstruction

vc(t) v[n] w[n] wc(t)

The continuous time signal vc(t) is given by

vc(t) = sinc(t) =

sinπtπt , t = 0

1, t = 0.

a) Sketch the Fourier transform Vc(Ω) of the continuous-time signal vc(t).

b) Sketch the discrete-time Fourier transform (DTFT) V (eω) of the sampled signal v[n].

c) The signal w[n] is obtained from v[n] by interpolation according to

w[n] =

v[n/3], if (n)3 = 0

0, otherwise.

Sketch the DTFT W (eω) of w[n].

d) The signal w[n] is passed through an ideal (Shannon) interpolator to result in

wc(t) =∞

n=−∞w[n] sinc

t− nT

T

with T = 1/6. Find an expression for wc(t) and sketch its Fourier transform Wc(Ω).

e) Are there values of T for which wc(t) = vc(t)? Explain your answer.

Problem 2 (20%)

You are given two finite-length signals x1[n], n = 0, 1, . . . , N1−1 and x2[n], n = 0, 1, . . . , N2−1, and your task is to perform linear convolution of these signals. Describe a way to computethe convolution if you are only allowed to use DFT/DFT−1 and a “frontal” zero-padding ofthe form

y1[n] =

0, for n = 0, 1, . . . , L− 1

x1[n− L], for n = L,L+ 1, . . . , N1 + L− 1

2

and

y2[n] =

0, for n = 0, 1, . . . , L− 1

x2[n− L], for n = L,L+ 1, . . . , N2 + L− 1

for some value of L.

Problem 3 (25%)

Let x[n] be a real-valued signal of length N = 2L, where L is an integer. Another signal y[n]of length M = N/2 is defined from x[n] according to y[n] = x[2n] + x[2n+ 1], 0 ≤ n < M .Let X[k] (with 0 ≤ k < N) and Y [k] (with 0 ≤ k < M) denote the DFT coefficients of x[n]and y[n], respectively.

a) For all k = 0, 1, . . . ,M − 1, find an expression for Y [k] in terms of X[k].

b) Find an expression for X[0] and X[M ] in terms of Y [0].

Problem 4 (25%)

When an input to an LTI system is x[n] = 5u[n], the output is y[n] = [2(1/2)n + 3(−3/4)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] of the system for all values of n.

c) Write the difference equation that characterizes the system.

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