1
3
Examples of Simple Systems
ELEC 264 Winter 2013 (Week 3)
Continuous time (CT) and discrete time (DT) systems
Properties of Systems
Linear Time Invariant (LTI) Systems
To get some idea of typical systems (and their properties), consider the RC
elect
1
3
Impulse Response
ELEC 264 Winter 2013 (Week 5)
Signals and Systems I
input
n
More on Discrete Time Systems
Properties of LTI Systems
First order Difference Equation
T
LTI System
Unit Impulse
3.5
3
n
1
1, n 0
n
0, n 0
step response of y[n] =
1
3
Frequency analysis: why?
ELEC 264 - Winter 2013 (Week 6)
Fourier series provides an alternate way of representing data: instead of
representing the signal amplitude as a function of time, we represent the signal
by how much information is contained at
1
3
Examples of DT and CT Signals
ELEC 264 - Winter 2013 (Week 8)
Signals and Systems I
Unit impulse function
1, n 0
[n]
0, n 0
Unit step function
Midterm Review
1, t 0
u (t )
0, t 0
Unit ramp function
t , t 0
ramp(t ) tu (t )
0, t 0
Unit rectang
1
3
Properties of Fourier Series
ELEC 264 - Winter 2013 (Week 9)
Even and Odd Functions: if a function x(t) is even, x(t)=x(-t)
Signals and Systems I
Properties of Fourier Series
x(t ) x( t ) ak e jk 0t ak e jk0t a k e jk0t
Discrete time Fourier Series
1
3
Properties of CT Fourier Transform: Time Shifting
ELEC 264 - Winter 2013 (Week 11)
Signals and Systems I
FT
FT
j t
If x (t ) X ( j ) Then x (t t0 ) e X ( j )
0
Proof: Now replacing t by t-t0
x (t )
Properties of CT Fourier Transform
Applications of
1
3
CT Fourier Transform Representation of Aperiodic Signals
ELEC 264 - Winter 2013 (Week 10)
(An aperiodic signal is a periodic signal with infinite period.)
Let us consider continuous-time periodic square wave
Signals and Systems I
1,
x (t )
0,
Cont
1
3
Time and Frequency Shifting
ELEC 264 - Winter 2013 (Week 13)
FT
Signals and Systems I
FT
Properties of DT Fourier Transform
Important implications in DT because of periodicity
LTI Systems in Frequency Domain
Example
Magnitude of Fourier transform o
1
3
Reminder: Fourier Series Representation of DT Periodic Signals
ELEC 264 - Winter 2013 (Week 12)
Signals and Systems I
Discrete Time (DT) Fourier Transform
ak e jk0 n
Synthesis Equation
k N
Inverse Fourier Transform
Fourier Transform for Periodic S
ELEC 264
Formula Sheet
Final Exam
Continuous Domain
Discrete Domain
FS
FS
Fourier series: x(t) ak
Fourier series: x[n] ak
k=<N >
k=
1
Analysis equation: ak =
N
1
x(t)ejk0 t dt
Analysis equation: ak =
T T
FS
Linearity: x(t) + y(t) ak + bk
FS
Time shift: x(
1
3
Fourier Series of a CT Periodic Signal
ELEC 264 - Winter 2013 (Week 14)
Signals and Systems I
Consider a periodic signal,
x(t ) x(t T ) for t
Then, the fundamental frequency: o
Final Exam Review:
2
To
the fundamental period: minimum positive nonzero
ELEC 264
Sample Final Exam
Name:
ID:
Faculty of Engineering and Computer Science
ELEC 264 - Signals and Systems I
Sample Final Exam
Duration: 3 hours
Instructions:
Answer all questions in the exam booklet.
The use of any non-communicating calculator is p
1
3
What is ELEC 264?
ELEC 264 - Winter 2013 (Week 1)
Signals and Systems I
Because most systems are driven by signals, Electrical and Computer
Engineers study what is called signals and systems.
Go over Course Outline
Introduction to Signals and Syste
1
What is MATLAB?
3
MATLAB Desktop contd
MATLAB is a commercial "Matrix Laboratory" package which operates as an
interactive programming environment.
MATLAB is well adapted to numerical experiments.
MATLAB program and script files (m-files) always have fi
1
3
Example
ELEC 264 - Winter 2013 (Week 4)
Signals and Systems I
Impulse response
LTI Systems and Convolution
Differential and difference systems
The discrete signal x[n]
is decomposed into the following
additive components
x[-4][n+4] +
x[-3][n+3] +
Chapter 4
Continuous Time Fourier Transforms
Periodic signals represented by Fourier Series of frequencies of
fundamental and harmonics.
Non-periodic signals are represented by Fourier integrals. Frequencies are
continuous.
Representation of continuous
Midterm Exam 1
ELEC264
Winter 2009
Question 1:
Consider a discrete-time system which has input of x[n] and output of y[n] = x 2 [n] .
a) Is this system linear? Why?
System is not linear as shown below:
Consider x1 [ n ] y1 [ n] = x12 [ n]
2
Also consider
ELEC264/4-W
Midterm Exam II
Signals and Systems I
Instructions for the Examination
Answer all 4 Questions.
Questions have equal weight.
Parts a and b of each question have equal weight.
Show all the intermediate steps of your solution.
Make reasonable ass
ELEC264
Midterm Exam III
Signals and Systems I
Question 1:
Evaluate the Fourier Transform of the continuous time signal:
d
x (t ) = t cfw_te 2t sin(t )u (t )
dt
Solution 1:
FT
te 2t u (t )
1
(2 + j )2
e jt e jt 2t
1
1
1
FT
te 2t cfw_sin(t )u (t ) =
te
Chapter 4
Continuous Time Fourier Transforms
Periodic signals represented by Fourier Series of frequencies of
fundamental and harmonics.
Non-periodic signals are represented by Fourier integrals. Frequencies are
continuous.
Representation of continuous
Continuous-Time & Discrete-Time Signals
Continuous-Time:
x(t)
x(t)
0
0
t
(a)
Discrete-Time:
x[n]
0
(b)
n
Signal Energy & Power:
Signal x(t) and x[n] - associated with energy & power:
Energy E & Power P:
t2
Continuous -time :E = t x( t )2 dt
1
t
P=
1 2
2
Chapter 2: Linear Time-Invariant (LTI) Systems
LTI systems support superposition. Hence,
If x(t) or x[n] is expressed as a linear combination of a set of basic
functions, the output will be the summation of individual responses.
A general signal may be e
Chapter 3
Fourier Series Representation of Periodic Signals
If an arbitrary signal x(t) or x[n] is expressed as a linear combination of
some basic signals, the response of an LTI system becomes the sum of the
individual responses of those basic signals.
S
Chapter 5
Discrete Time Fourier Transform
DT Fourier Transform of Aperiodic Signals:
Consider a general aperiodic signal x[n]
given as:
x[ n ] = 0, outside
x[n]
- N1 n N2.
. . .
Now consider a periodic signal x^[n]
of which x[n] is its one period as shown
ELEC264
Assignment 1 Signals and Systems I
Study Chapter 1 and solve following assignment.
Question 1) Find the even and odd components of the following signals:
a) x (t ) = e 2t cos t
b) x[n] = (1) n 1
Question 2) For each of the following signals, deter
ELEC264
Assignment 3 Signals and Systems I
Study sections 4.1 to 4.6 of Chapter 4.
Use the defining equation for Fourier Transform to evaluate the frequency-domain
representations of the following three signals:
Question 1)
x(t ) = e 2t u (t 3)
m
Question
ELEC264
Assignment4SignalsandSystemsI
Study sections 5.1 to 5.7 of Chapter 5.
Question 1) Evaluate the frequency-domain representations of the following signals
a) Use the defining equation for Fourier Transform.
x[n]
1
n
1
b)
x[n ] isadiscreteperiodicsig