ECSE-412, Winter 2014
Discrete-Time Signal Processing
Problem Set 1
Posted: Wednesday, January 15, 2014.
Due: Wednesday, January 22, 2014, 4h00pm.
Instructions:
Use the assignment box in Trottier.
Use THIS COVER PAGE with your assignment (mandatory).
Qu

Chapter 11
Multirate systems
284
CHAPTER 11. MULTIRATE SYSTEMS
11.1
285
Sampling rate modication
Problem overview:
Given the samples of continuous-time signal xc (t), that is
x[n] = xc (nTs ),
nZ
(11.1)
with sampling period Ts and corresponding sampling

Chapter 6
The discrete Fourier Transform and
its fast computation
115
CHAPTER 6. THE DISCRETE FOURIER TRANSFORM AND ITS FAST COMPUTATION116
Introduction:
Recall the denition of the DTFT:
X ( ) =
x[n]ejn ,
[, ]
(6.1)
n=
While the DTFT is useful from a t

Chapter 3: Discrete-time Fourier Transform
B. Champagne1
1 Department
of Electrical & Computer Engineering
McGill University
January 11, 2010
ECSE 412
3. The DTFT
Outline
3.1 The DTFT and its inverse
3.2 Convergence of the DTFT
3.3 Properties of the DTFT

ECSE 412
Discrete-Time Signal Processing
Tutorial # 2
Problem 1
Find the DTFT of the following sequences:
1. x[n] =
2. x[n] =
3. x[n] =
1n
2
u[n + 3]
1n
2
0
n = 0, 2, 4, . . .
otherwise
1 |n|
4
Problem 2
For each of the following pairs of signals, x[n] an

ECSE 412
Tutorial # 3
Siamak Youse, siamak.youse@mail.mcgill.ca
January 24-th, 2014
Problem 1
Prove the following properties
ZT
1. x[n n0 ] z n0 X (z )
ZT
2. n x[n] X (1 z ) ROC : |Rx
Problem 2
Find the Z-Transform of the following sequences
1. x[n] = cos

ECSE 412
Tutorial # 5
Siavash Rahimi, siavash.rahimi@mail.mcgill.ca
Feb 10, 2010
Problem 1
Find the N-point DFT of the sequence
x[n] = 4 + cos2
2n
N
,
n = 0, 1, . . . , N 1.
Problem 2
Given the two eight-point sequences x1 [n] and x2 [n] shown in the gure

ECSE 412
Discrete-Time Signal Processing
Tutorial # 4
Problem 1
Consider the LTI system described by the dierence equation
y [n] = 0.7y [n 2] + 0.15x[n] 0.15x[n 2].
1. Find the system function and plot the pole-zero diagram.
2. List the possible ROCs of t

Chapter 8:
Structures for the realization of
0 1 syste s
Benoit Champagne
Department o f Electrical and Computer Engineering
McGill University, Montreal, CANADA
O ctO_06
, ~"I
.
~/I
B. C hamoaane
Chaotar 1 - 0 1
Introduction
. . Consider a causal LTI sy

ECSE 412
Discrete-Time Signal Processing
Tutorial # 2
Siamak Youse
Email: siamak.youse@mail.mcgill.ca
17th January 2014
Problem 1
A discrete-time system can be
1. Static or dynamic
2. Linear or nonlinear
3. Time invariant or time varying
4. Causal or nonc

ECSE 412
Tutorial # 7
Siavash Rahimi, siavash.rahimi@mail.mcgill.ca
Feb. 28, 2011
Problem 1
The following system is used to process an analog signal with a discrete-time
system. Suppose that xa (t) is bandlimited with Xa (f ) = 0 for |f | > 5kHz as
shown

CHAPTER 6. THE DISCRETE FOURIER TRANSFORM AND ITS FAST COMPUTATION150
6.6
Fast Fourier transform (FFT)
Recap:
Let x[n] be discrete-time signal dened for 0 n N 1
The DFT:
N 1
kn
x [n ]W N ,
X [k ] =
n=0
j 2/N
WN
6.6.1
k = 0, 1, ., N 1
e
(6.71)
(6.72)
Dir

Chapter 7:
Digital Processing of Analog
Signals
Benoit Champagne
champagne@ece.mcgill.ca
Department of Electrical and Computer Engineering
McGill University, Montreal, CANADA
October 5, 2005
Digital Processing of Analog Signals p. 1/2
Overview
I. Study of

Chapter 3: Discrete-time Fourier Transform
B. Champagne1
1 Department
of Electrical & Computer Engineering
McGill University
January 11, 2010
ECSE 412
3. The DTFT
Outline
3.1 The DTFT and its inverse
3.2 Convergence of the DTFT
3.3 Properties of the DTFT

ECSE-412, Winter 2011
Discrete-Time Signal Processing
Problem Set #1
Posted: Friday, January 14, 2011.
Due: Monday, January 24, 2011, 4h00pm.
Instructions:
Return your assignment in the assignment box reserved for this
course in Trottiers Building.
Plea

McGill University, Faculty of Engineering
Course ECSE-412: Discrete Time Signal Processing
Midterm Examination, Winter 2012
Date and time: Thursday, March 1, 2012, 10:05 - 11:25
Examiner: Prof. B. Champagne
Instructions: This is a closed book examination:

McGill University, Faculty of Engineering
Course ECSE-412: Discrete Time Signal Processing
Midterm Examination, Winter 2013
Date and time: Tuesday, Feb. 26, 2013, 10:05 - 11:25
Examiner: Prof. B. Champagne
Instructions: This is a closed book examination:

Chapter 2: Discrete-time signals
and systems
Benoit Champagne
Department of Electrical and Computer Engineering
McGill University, Montreal, CANADA
September 2006
B. Champagne
Chapter 1 p. 1/3
2.1 Discrete-time signals
B. Champagne
Chapter 1 p. 2/3
Signal

Chapter 1: Introduction
Benoit Champagne
Department of Electrical and Computer Engineering
McGill University, Montreal, CANADA
September 2006
B. Champagne
Chapter 1 p. 1/2
1.1 Signals and systems
B. Champagne
Chapter 1 p. 2/2
Signal
A physical quantity th

McGILL UNIVERSITY
Faculty of Engineering
FINAL EXAMINATION
Winter 2012
DISCRETE TIME SIGNAL PROCESSING
ECSE 412
Examiner: Prof. Beno Champagne
t
Signature:
Date: April 24, 2012
Co-Examiner: Prof. Fabrice Labeau
Signature:
Time: 9:00 to 12:00
INSTRUCTIONS:

Chapter 4
The z-transform (ZT)
Motivation:
While very useful, the DTFT has a limited range of applicability.
Example, the DTFT of simple signal like x[n] = 2n u[n] does not exist
One may view the ZT as a generalization of the DTFT that is applicable
to

Chapter 5
Z-domain analysis of LTI systems
5.1
The system function
LTI system (recap):
y [n] = x[n] h[n] =
x[k ]h[n k ]
(5.1)
k =
h[n] = Hcfw_ [n]
(5.2)
Response to arbitrary exponential:
Let x[n] = z n , n Z.
Corresponding output signal:
h[k ]z nk
Hcfw

Chapter 10
Quantization eects
Introduction:
Practical DSP systems use nite-precision (FP) number representations
and arithmetic.
The implementation in FP of a given LTI system H (z ) leads to deviations
in its theoretically predicted performance:
- Due

ECSE 412
Tutorial # 3
Siavash Rahimi, siavash.rahimi@mail.mcgill.ca
Jan 24, 2011
Problem 1
Prove the following properties
ZT
1. x[n n0 ] z n0 X (z )
ZT
2. n x[n] X (1 z ) ROC : |Rx
Problem 2
Find the Z-Transform of the following sequences
1. x[n] = cos(n0