Laplace Transform
Laplace Transform
Recall that the response of an LTI system to a complex
exponential
=
() =
The output is the same complex exponential with a
change of amplitude only.
Define the Laplace transform of a general signal () as
=
()

YANG Chunfeng
[email protected]
SHB726
Properties of System
LTI(Linear Time-Invariant) System
Represent signals as linear combinations of delayed
impulses
Convolution sum/convolution integral.
System:
A new signal(output) will be produced to respond to

Yang Chunfeng
[email protected]
SHB726
YANG Chunfeng
1
2012/11/5
Review(FS,CTFT,DTFT)
Relation between periodicity and continuity
Comparison between CTFT and DTFT(properties )
Discrete FT(DFT)
Self-exercise
YANG Chunfeng
2
2012/11/5
Fourier Series

YANG Chunfeng
SHB 726
[email protected]
YANG Chunfeng
1
2012.10
Review
Fourier Series representation of signals
Property of
Fouries Series
Exercise on Fouries Series and its properties
FS of Discrete-time signal
YANG Chunfeng
2
2012.10
Memory/Mem

IERG 2051 Tutorial 3
LU Lian
Oct. 4th, 2012
Outline
Convolution Algebra
Demo
Applications
Convolution Algebra
Convolution Algebra
Be careful here! We
have an exercise later
on.
Treat y(t)=x(t-D) as a new function
A Comprehensive Training
e(t)
1
-0.5
0

Tutorial note 11
YANG Chunfeng
Office hour: 10:00-11:00am Friday
SHB726
Outline
Z-transform
Inverse Z-transform
Common z-transform pair
Exercise
Property of Z-transform
Analysis of discrete system with Z-transform
Self-exercise
Z-transform
Z-trans

YANG Chunfeng
[email protected]
SHB726
Properties of System
LTI(Linear Time-Invariant) System
Represent signals as linear combinations of delayed
impulses
Convolution sum/convolution integral.
System:
A new signal(output) will be produced to respond to

Solutions for tutorial 1
I.
Signal Transformation
1. x(2t+2)
2. x(2(t+2)
3. x(1-3t)
II.
Even and Odd Function
1.
x(t ) e j ( t 1)
1
Evcfw_x(t ) (e j ( t 1) e j ( t 1) )
2
1
Odcfw_x(t ) (e j ( t 1) e j ( t 1) )
2
2. x(t) = 3t + 5
Evcfw_x(t) = 5
Odcfw_x(t)

x (t ), T :
ak e jkw t .
x (t ) =
k =
e jkw t
k
= cfw_, cos (w t ) , cos (w t ) , ., cos (kw t ) ,
sin (w t ) , sin (w t ) , ., sin (kw t ) .
e jkw t = (kw t ) + j (kw t ) .
w = , T
T

Yang Chunfeng
[email protected]
SHB726
YANG Chunfeng
1
2012/11/4
Review(FS,CTFT,DTFT)
Relation between periodicity and continuity
Comparison between CTFT and DTFT(properties )
Discrete FT(DFT)
Self-exercise
YANG Chunfeng
2
2012/11/4
Fourier Series

YANG Chunfeng
SHB 726
[email protected]
YANG Chunfeng
1
2012.10
Review
Fourier Series representation of signals
Property of
Fouries Series
Exercise on Fouries Series and its properties
FS of Discrete-time signal
YANG Chunfeng
2
2012.10
Memory/Mem

Basic Concepts about Signals and
Systems
Introduction to Signals
Signals are represented mathematically as functions of
one or more independent variables.
In simple terms, a signal is a mathematical function.
Examples
a speech signal: acoustic pressure

Fourier Series
Complex Exponentials
Continuous time:
Consider where is complex.
Response of an LTI system
=
() =
The output is the same complex exponential with a
change of amplitude only.
In mathematics, complex exponentials are eigenfunctions

IERG2051A
Tutorial 4
ZHANG, Yuqi
[email protected]
Outline
Fourier Series
Continuous Time Periodic Signals
Discrete Time Periodic Signals
In-class Assignment
Fourier Series - Continuous
Time
A signal is periodic with period :
The fundamental freq

IERG 2051
TUTORIAL 1
YU YIDING
[email protected]
OUTLINE
Continuous-Time and Discrete-Time Signals
Energy and Power
Signal Transformation
Periodic and Aperiodic Signals
Even and Odd Signals
Unit Step and Unit Impulse Function
CONTINUOUS-TIME SIGN

IERG2051A Tutorial 3
Yongjun Zhao ([email protected])
Outline
Linear time invariant systems
Fourier Series
In-class assignment on Monday (06/02/2017)
Linear Time Invariant (LTI) System
Linear system:
Let 1 () be the response when the input is 1 ()
Let 2 () b

z-transform
z-transform
Recall that the response of an LTI system to a complex
exponential
[] =
[] =
=
[]
=
The output is the same complex exponential with a change
of amplitude only.
Define the z-transform of a general signal [] as
[]
=
=
[]
()

IERG 2051A Signals and Systems
Assignment 2, Due: Feb 13, 2017
Textbook:
Alan V. Oppenheim and Alan S. Willsky, Signals and Systems, Prentice Hall, second edition.
Chapter 2: LTI Systems
2.21 (d), 2.22(c), 2.28(f), 2.29(e), 2.40, 2.44

IERG 2051A Signals and Systems
Assignment 1, Due: Jan 26, 2017
Textbook:
Alan V. Oppenheim and Alan S. Willsky, Signals and Systems, Prentice Hall, second edition.
Chapter 1
1.21(c), 1.22(h), 1.27(c)(d), 1.28(e)

Fourier Transform
Intuition
Let the period go to infinity. Then a periodic
signal becomes aperiodic.
Start with Fourier Series
=
() 0
0
=
=
where 0 =
2
Continuous Time Fourier Transform
As ,
0
0 , which is the frequency of concern
Give a notat

Linear Time Invariant Systems
Discrete Time Signals
Discrete time signal as weighted sum of
shifted unit impulses:
E.g., ., 0, 0, 2, 1, 3, 4, 0, 0, 0,
can be viewed as the sum of
2 times , 0, 0, 1, 0, 0, 0, 0, 0, 0,
1 times , 0, 0, 0, 1, 0, 0, 0, 0, 0

IERG 2051 TUTORIAL 1
ZHANG Tao
Outline
Signal Transformation
Even and Odd Function
Periodic and Aperiodic Function
Complex Exponential Signal
Unit Step and Unit Impulse Function
Signal Transformation
Reflection
Continuous Signal
x(t) x(-t)
Discret

IEG2051 TUTORIAL 10
Zhang Tao
Outline
Z-transform definition
ROC (Region of Convergence)
Properties of pole-zero plot
Properties of z-transform
Z-transform
Z-transform
Where z is defined as
r is a real number
Why use z-transform?
Convergence issu

IERG2051A Exercise Solutions for Tutorial 2
1.
( ) [ ( )] ( )
ANS: The system is linear but time-varying.
Proof of linearity:
Consider two input signals x1(t) and x2(t), and the corresponding output signals y1(t) = T[x1(t)]
and y2(t) = T[x2(t)].
Denote th

IERG 2051A TUTORIAL 5
SHEN Siduo
Outline
Part 1: A DT periodic signal can be expressed as a linear
combination of DT periodic complex exponentials
1.
2.
3.
4.
5.
Discrete Time Complex Exponentials
Discrete Time Periodic Signals
DT Periodic Complex Expone

Tutorial 5 Some Examples
Solutions:
Key Points:
1. We can always write the complex exponentials in the form of trigonometric functions, i.e., cos
function or sin function according to the Euler s formula:
,
,
So, always start with the standard form which

IERG2051A
TUTORIAL 2
XIE Mengjie
Outline
Systems
System Properties
Signal Representation with Unit Impulse (t)/[n]
LTI System Characterization with Impulse Response
h(t)/h[n]
Convolution
Systems
A system T[.] takes an input signal x(t)/x[n] and prod