University of Toronto
Department of Electrical and Computer Engineering
ECE216 SIGNALS AND SYSTEMS
Information Sheet
Spring 2015
Lecturers
Professor Raymond Kwong (course coordinator)
Oce GB343; email [email protected]
Section LEC0102:
Mon 11-12n
ECE 216 Signals and Systems
Stark C. Draper
Course Notes
Version 3.11
c 2017
Copyright
Stark C. Draper
All rights reserved
University of Toronto
ECE 216 Signals & Systems
Remarks, feedback, and versions
These notes are in development in winter term 2017.
ECE 216 Signals and Systems
Stark C. Draper
Course Notes
Version 3.06
c 2017
Copyright
Stark C. Draper
All rights reserved
University of Toronto
ECE 216 Signals & Systems
Remarks, feedback, and versions
These notes are in development in winter term 2017.
ECE216
Signals and Systems
University of Toronto
Spring 2016
Problem Set #11 Solutions
Problem 11.1
1
Problem 11.2
2
Problem 11.3
3
Problem 11.4
(a)
(b)
(c)
4
Problem 11.5
(a)
(b)
(c)
(d)
5
Problem 11.6
C=4
6
Problem 11.7
(a)
7
(b)
(c)
Problem 11.8
(a)
8
ECE216
Signals and Systems
University of Toronto
Spring 2016
Problem Set #11
Problem 11.1
Use the convolution integral to determine the convolution of two exponential signals with
the same exponent, i.e.
y(t) = [eat u(t)] [eat u(t)].
Problem 11.2
A linear
ECE216
Signals and Systems
University of Toronto
Spring 2016
Problem Set #9
Problem 9.1
Compute using the graphical method (flip and shift, see pages 110-114 in course notes) the
convolution between the following signal and impulse response
1, 0 n 4
x[n]
ECE216
Signals and Systems
University of Toronto
Spring 2016
Problem Set #10
Problem 10.1
Consider the 4-point moving average filter
3
y[n] =
1X
x[n k].
4 k=0
Recall that |H(ej )| 0 and < H(ej ) are real-valued functions satisfying
j )
H(ej ) = |H(ej )|ej
University of Toronto
Edward S. Rogers Department of Electrical and Computer Engineering
ECE216: SIGNALS AND SYSTEMS
Project 1: Convolution and Fourier Series
Topic
This project will give you practice using the Matlab program to explore the concepts of c
ECE216
Signals and Systems
University of Toronto
Spring 2016
Problem Set #7 Solutions
Problem 7.1
1
2
c)
h[n] = 2[n] 3[n 1] + 2[n 2]
3
2
1
0
1
2
3
4
0
2
4
6
8
10
Figure 3: Impulse response h[n] for n cfw_0, . . . , 10
Problem 7.2
Problem 7.3
4
H ]?RA ! %\
ECE216
Signals and Systems
University of Toronto
Spring 2016
Problem Set #1
Problem 1.1 (Complex Numbers)
Some parts of the following problem use Eulers relation:
ejt = cos t + j sin t
and its application to give
1 jt
(e ejt )
2j
1
cos t = (ejt + ejt )
2
ECE216
Signals and Systems
University of Toronto
Spring 2016
Problem Set #5 Solutions
Problem 5.1
1
2
3
Problem 5.2
4
5
Problem 5.3
6
7
8
Problem 5.4
(d) N0 = 5.
(e)
4
2
1X
y[n]ej 5 kn
5 n=0
2
2
1
=
1 + ej 5 k(1) + ej 5 k(1)4
5
2
8
1
=
1 + ej 5 k + ej 5 k
ECE216
Signals and Systems
University of Toronto
Spring 2016
Problem Set #8
Problem 8.1 (Impulse response)
Given an LTI discrete time system, suppose the output due to an input x[n] = [n] is given
as Fig. 1.
(a) Find the output due to an input x[n] = [n 1
ECE216
Signals and Systems
University of Toronto
Spring 2016
Problem Set #6 Solutions
Problem 6.1
Problem 6.2
(a) For all k cfw_0, . . . , N0 1
hk , k i =
=
=
N
0 1
X
n=0
N
0 1
X
n=0
N
0 1
X
2
2
jN
kn j N
kn
e
0
e
0
e(0)n
1
n=0
= N0
(b) For periodic x[n]
ECE216
Signals and Systems
University of Toronto
Spring 2016
Problem Set #4 Solutions
Problem 4.1
1. A. The fundamental period is LCM (2/2.5, 2/3.25, 1/9) = 8.
2. D. Using trig identity we have
1
x(t) = (cos(2(1 + )t) + cos(2(1 )t).
2
Fundamental periods
ECE216
Signals and Systems
University of Toronto
Spring 2016
Problem Set #7
Problem 7.1
A linear time-invariant system is described by the difference equation
y[n] = 2x[n] 3x[n 1] + 2x[n 2].
(a) When the input to this system is
0
n + 1
x[n] =
5n
1
n < 0,
ECE216
Signals and Systems
University of Toronto
Spring 2016
Problem Set #6
Problem 6.1 (Discrete-Time Fourier Basis)
In this question, we are concerned with the N0 -periodic, discrete-time Fourier basis. We
define this basis here as
1 j 2 kn
k [n] = e N0
ECE216
Signals and Systems
University of Toronto
Spring 2016
Problem Set #5
Problem 5.1 (DTFS Analysis)
Use the definition of the DTFS coefficients to determine the DTFS representation of the
following signals.
n + cos 10
n +1
(a) x[n] = 2 sin 14
19
19
P
ECE216
Signals and Systems
University of Toronto
Spring 2016
Problem Set #3
Problem 3.1 (Differentiation and the CTFS)
Let x(t) be a periodic signal with fundamental period T0 . Let y(t) be defined as y(t) = dx(t)
.
dt
This signal also has period T0 . Let
ECE216
Signals and Systems
University of Toronto
Spring 2016
Problem Set #4
Problem 4.1 (Multiple Choice)
1. The fundamental frequency of cos(2.5t) + sin(3.25t) + cos(18t) is:
(A)
(B)
(C)
1
.
8
1
.
4
1
.
2
(D) 1
2. Let x(t) = cos(2t) cos(2 2 t). Which of
ECE216
Signals and Systems
University of Toronto
Spring 2016
Problem Set #3 Solutions
Problem 3.1
1
Problem 3.2
Problem 3.3
1.
2
2. Let T1 , 1 , and bk be the fundamental period, fundamental frequency, and CTFS coefficients of y(t). Let s = t. By the peri
ECE216
Signals and Systems
University of Toronto
Spring 2016
Problem Set #2 Solutions
Problem 2.1 (Fundamental Frequency)
(a) 0 = 30 radians/s, f0 =
0
2
= 15 Hz.
(b) Using trig identity
sin(2t)2 =
1 cos(4t)
,
2
hence 0 = 4 radians/s, f0 = 2 Hz.
(c) Let x(
ECE216
Signals and Systems
University of Toronto
Spring 2015
Problem Set #2
Problem 2.1 (Fundamental Frequency)
Find the fundamental frequency for each of the following periodic functions.
(a) f (t) = sin(30t)
(b) f (t) = (sin(2t)2
(c) f (t) = sin(10t) +