University of WaterlooDepartment of Electrical and Computer Engineering

ECE 413 – Digital Signal Processing

Midterm Exam, Spring 2012

June 8th, 2012, 6:30-7:50 PM

Instructor: Dr. Oleg Michailovich

Instructions:

• This exam has 2 pages.

• No books and lecture notes are allowed on the exam. Please, turn o↵ your cell phones,PDAs, etc., and place your bags, backpacks, books, and notes under the table or at the frontof the room.

• Please, place your WATCARD on the table, and fill out the exam attendance sheet whenprovided 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 yourassumptions 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 yourexam booklet, and remain seated in silence. The proctors will collect the exams, count them,and then announce you may leave.

1

Question 1 [40%]:

A function called autocorrelation for a real-valued, absolutely summable sequence x[n] is defined as

r

xx

[l] ,X

n

x[n] x[n� l]

Let X(z) be the z-transform of x[n].

a) Show that the z-transform of rxx

[l] is given by R

xx

(z) = X(z)X(z�1).

b) Let x[n] = a

n

u[n], |a| < 1. Determine R

xx

(z) and sketch its pole-zero plot and the ROC.

c) Determine the autocorrelation r

xx

[l] for the x[n] in (b) above.

Question 2 [25%]:

The signal x[n] = {1,�2, 3,�4, 0, 4,�3, 2,�1} (with 0 corresponding to n = 0, i.e. x[0] = 0) hasDTFT X(e|!). Without explicitly computing X(e|!), find the following quantities:

a) X(e| 0)

b) X(e| ⇡)

c)R

⇡

�⇡

X(e|!)d!

d)R

⇡

�⇡

|X(e|!)|2d!

Question 3 [35%]:

For the following input-output pairs, determine whether or not there is an LTI system producingy[n] when the input is x[n]. If such a system exists, determine its magnitude and phase responses;otherwise explain why such a system is not possible.

a) x[n] = sin⇡n/4⇡n

7! y[n] = sin⇡n/2⇡n

b) x[n] = (1/2)nu[n] 7! y[n] = (1/3)nu[n]

2

University of WaterlooDepartment of Electrical and Computer Engineering

ECE 413 – Digital Signal Processing

Final Exam, Spring 2012

August 7th, 2012, 7:30-10:00 PM

Instructor: Dr. Oleg Michailovich

Instructions:

• This exam has 3 pages.

• No books and lecture notes are allowed on the exam. Please, turn o↵ your cell phones,PDAs, etc., and place your bags, backpacks, books, and notes under the table or at the frontof the room.

• Please, place your WATCARD on the table, and fill out the exam attendance sheet whenprovided 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 yourassumptions 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 yourexam booklet, and remain seated in silence. The proctors will collect the exams, count them,and then announce you may leave.

1

Question 1 [20%]:

Determine all possible signals x[n] associated with the z-transform

X(z) =5z�1

(1� 2z�1)(3� z�1).

Question 2 [25%]:

Determine the coe�cients {h[n]} of a high-pass linear-phase FIR filter of length L = M + 1 = 4which has an antisymmetric impulse response, i.e. h[n] = �h[M � n], and a frequency responseH(e|!) that satisfies the condition

|H(ej⇡/4)| = 0.5 and |H(ej3⇡/4)| = 1.

Question 3 [35%]:

Consider the two systems shown in the figure below.

Ideal A/D Squaring Ideal D/A

Squaring Ideal D/AIdeal A/D

xc(t)

xc(t) x2c(t)

x[n] x2[n]

v[n]

y1(t)

y2(t)

Fs

Fs Fs

Fs

a) Find analytic relations between the signals y1(t) and xc(t), as well as between y2(t) and xc(t)for the case of Fs = 2B Hz (with B > 0) and

Xc(|F ) =

(1, |F | B

0, otherwise.

b) Determine y1(t) and y2(t) if xc(t) = cos(2⇡F0t) with F0 = 20 Hz and Fs = 50 Hz.

c) Repeat part (b) for Fs = 30 Hz.

2

Question 4 [20%]:

Define the N -point DFT of the Blackman window

w[n] = 0.42� 0.5 cos

✓2⇡n

N � 1

◆+ 0.08 cos

✓4⇡n

N � 1

◆, n = 0, 1, . . . , N � 1

in terms of a sampled version of the Dirichlet function

D(!) =sin(!N/2)

sin(!/2).

3