Signals & System
Lecture 7
Basis Functions & Fourier Series
Dr. Tahir Zaidi
Basis Functions & Fourier Series
3. Basis functions (3 lectures): Concept of basis
function. Fourier series representation of time
functions. Fourier transform and its properties.
Signals & System
Lecture 9
Fourier Transforms
Properties & Examples
Dr. Tahir Zaidi
Lecture 9: Fourier Transform
Properties and Examples
Basis functions (3 lectures): Concept of basis
function. Fourier series representation of time
functions. Fourier tran
Signals & System
Lecture 6
Linear Systems and Convolution
Dr. Tahir Zaidi
Lecture 6: Linear Systems and Convolution
Linear systems, Convolution: Impulse response, input
signals as continuum of impulses. Convolution,
discrete-time and continuous-time. LTI
Signals & System
Lecture 5
Linear Systems and Convolution
Dr. Tahir Zaidi
Lecture 5: Linear Systems and Convolution
2. Linear systems, Convolution (3 lectures): Impulse
response, input signals as continuum of
impulses. Convolution, discrete-time and
conti
Signals & System
Lecture 8
Fourier Series & Fourier Transforms
Dr. Tahir Zaidi
Lecture 8: Fourier Series and
Fourier Transform
Basis functions (3 lectures): Concept of basis
function. Fourier series representation of time
functions. Fourier transform and
Signals & System
Lecture 3
Signals Concepts & Properties
Dr. Tahir Zaidi
Lecture 3: Signals & Systems Concepts
Systems, signals, mathematical models. Continuoustime and discrete-time signals. Energy and power
signals. Linear systems. Examples for use
thro
Signals & System
Lecture 4
Linear Systems and Convolution
Dr. Tahir Zaidi
Lecture 4: Linear Systems and Convolution
Linear systems, Convolution: Impulse response,
input signals as continuum of impulses. Convolution,
discrete-time and continuous-time. LTI
Signals & System
Lecture 10
Sampling Discrete-Time Systems
Dr. Tahir Zaidi
Lecture 10: Sampling Discrete-Time Systems
Sampling & Discrete-time systems (2 lectures):
Sampling theorem, discrete Fourier transform
Specific objectives for today:
Sampling of a
Signals & System
Lecture 1
Introduction to Signals & Variables
Dr. Tahir Zaidi
Books/Resources
Essential
AV Oppenheim, AS Willsky: Signals and Systems, 2nd Ed
D Hanselman, B Littlefield Mastering Matlab 6: A
comprehensive tutorial and reference
Recommen
Signals & System
Lecture 2
Signals Concepts & Properties
Dr. Tahir Zaidi
Signals Concepts & Properties
(1) Systems, signals, mathematical models.
Continuous-time and discrete-time signals.
Energy and power signals. Linear systems.
Examples for use through
Re
2
Figure P203-1
x0) (i) Since the Fourier transform of stie exists, or n 1 must be in the R00.
cfw_i Fourier Therefore only one possible ROG exists, shown in Figure 320.3-1.
transform
of :1:(t)e "
converges
(ii) mm = D.
to: 10
Figure $20.34 A) = (s +
EE351: Spectrum Analysis and Discrete Time Systems
(Signals, Systems and Transforms)
Dr. Ha H. Nguyen
Associate Professor
Department of Electrical Engineering
University of Saskatchewan
August 2005
EE351Spectrum Analysis and Discrete Time Systems
Universi
Introduction to System
and Types of Systems
System
Systems processes input signals to produce output signals
How Are Signal & Systems Related
How to design a system to process a signal in particular
ways?
Design a system to restore or enhance a particu
Fourier Series
Representation of
Periodic Signals
Fourier Series Representation
LTI systems are based on representing signals as linear
combinations of shifted impulses
We now examine an alternative representation for signals
and LTI systems
We represe
Complex Exponential
and Sinusoidal Signals
Discrete Time
Exponential Signals
a)
b)
c)
d)
Real Exponential signal x[n] = Aean; A and a are Real
a > 0 and A >0; exponential rise
a < 0 and A >0; exponential decay
a > 0 and A <0;
a < 0 and A <0;
How will (c
Signal Transformations
Continuous and Discrete-Time Signals
Continuous-Time Signals
Most signals in the real world are
continuous time as the scale is
infinitesimally fine, for example
voltage, velocity
Denoted by x(t), where the time
interval may be b
Continuous Time Fourier
Transform (CTFT)
Continuous Time Fourier Transform
Example Square Wave
Synthesis and Analysis Equations
Synthesis and Analysis Equations
Synthesis and Analysis Equations
Synthesis and Analysis Equations
Example-Decaying Exponential
DT Convolution continued
DT Convolution Problem 1
DT Convolution Problem 2
Consider an input x[n] and a unit impulse response h[n]
given by:
Find
DT Convolution Problem 3
Consider an input x[n] and a unit impulse response h[n]
given by:
Find
Review Exercise
Periodicity
Draw the signal below and determine whether it is periodic
or not? If the signal is periodic, what is the fundamental
period?
x[n] =
4
[ 14 ]
Even and Odd Decomposition
Determine and sketch the even and odd parts of the sign
CT Convolution
CT Signals as Sum of Impulses
Consider the staircase approximation,
( ), as shown below:
, to a CT signal,
It may be seen that the CT signal can be approximately
expressed as a linear combination of delayed pulses.
CT Signals as Sum of Im
Teachers Introduction
Dr. Salman Abdul Ghafoor
[email protected]
BSc Electrical Engineering,
UET Peshawar
MSc Electronic Communications,
University of Nottingham, UK
PhD Fibre Optic Communication,
University of Southampton, UK
Introduction to Si
Linear Time-Invariant
(LTI) Systems DT
Convolution
Linear Time-Invariant (LTI) Systems
Systems that are linear and time-invriant
Focus of most of this course
LTI systems are of practical importance
A basic fact: If we know the response of an LTI syste
CT Convolution Problems
CT Convolution Problem 1
CT Convolution Problem 2
Evaluate y(t) = x(t) * h(t), where x(t) and h(t) are shown in
Figure below, (a) by an analytical technique, and (b) by a
graphical method.
Properties of Systems
System Properties
1.
2.
3.
4.
5.
6.
Memory
Invertible
Causal
Stability
Time Variance
Linearity
System Properties - Stability
A system is said to be bounded-input bounded-output stable
(BIBO stable or just stable) if the output signa
Properties of Signals
Signal Properties
Continuous Time Periodic signals: A periodic continuous-time signal
x(t) has the property that for a positive value of time T,
x(t) = x(t + T), for all value of t.
Then x(t) is periodic with time period T.
The funda
Complex Exponential
and Sinusoidal Signals
Continuous-Time
Motivation
These signals occur frequently in nature
They serve as basic building blocks from which we can
construct many other signals
Sinusoidal and periodic complex signals are used to
descr
Properties of Fourier
Series
Properties of CTFS
We have studied the following properties of CTFS:
1. Linearity
2. Conjugate Symmetry
3. Time Shift
Now we will study the remaining properties:
1. Time Reversal
2. Time Scaling
3. Multiplication
4. Parseval
Discrete Time (DT)
Fourier Series
DTFS for Periodic Signals
There are only N different signals in the set of discrete-time complex
exponential signals
Since the exponential sequences are distinct only over a range of N
successive values of k, the FS summa
Properties of LTI
Systems
Commutative Property
Hence the step response of an LTI system is the summation of its
impulse responses
Distributive Property
Associative Property
Associative Property
Memoryless and Identity LTI System
Invertible LTI System - DT