Department of Electrical Engineering
University of Arkansas
ELEG 3124 SYSTEMS AND SIGNALS
Lecture Notes
Dr. Jingxian Wu
wuj@uark.edu
This work is licensed under:
Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)
OUTLINE
• Chapter 1: Continuous-Time Signals ………………………. 3
• Chapter 2: Continuous-Time Systems ……………………… 45
• Chapter 3: Fourier Series ……………………………………. 84
• Chapter 4: Fourier Transform ……………………………… 122
• Chapter 5: Laplace Transform ……………………………… 170
• Chapter 6: Discrete-time Signals and Systems ……………… 222
2
Department of Electrical Engineering
University of Arkansas
ELEG 3124 SYSTEMS AND SIGNALS
Ch. 1 Continuous-Time Signals
Dr. Jingxian Wu
wuj@uark.edu
OUTLINE
•
Introduction: what are signals and systems?
•
Signals
•
Classifications
•
Basic Signal Operations
•
Elementary Signals
4
INTRODUCTION
• Examples of signals and systems (Electrical Systems)
– Voltage divider
• Input signal: x = 5V
• Output signal: y = Vout
Voltage divider
𝑅2
𝑥)
1 +𝑅2
• The system output is a fraction of the input (𝑦 = 𝑅
– Multimeter
• Input: the voltage across the battery
• Output: the voltage reading on the LCD display
• The system measures the voltage across two points
multimeter
– Radio or cell phone
• Input: electromagnetic signals
• Output: audio signals
• The system receives electromagnetic signals and convert them to
audio signal
INTRODUCTION
• Examples of signals and systems (Biomedical Systems)
– Central nervous system (CNS)
• Input signal: a nerve at the finger tip senses the high
temperature, and sends a neural signal to the CNS
• Output signal: the CNS generates several output signals
to various muscles in the hand
• The system processes input neural signals, and generate
output neural signals based on the input
– Retina
• Input signal: light
• Output signal: neural signals
• Photosensitive cells called rods and cones in the retina convert
incident light energy into signals that are carried to the brain by the
optic nerve.
Retina
INTRODUCTION
• Examples of signals and systems (Biomedical Instrument)
– EEG (Electroencephalography) Sensors
• Input: brain signals
• Output: electrical signals
• Converts brain signal into electrical signals
EEG signal collection
– Magnetic Resonance Imaging (MRI)
• Input: when apply an oscillating magnetic field at a certain frequency,
the hydrogen atoms in the body will emit radio frequency signal,
which will be captured by the MRI machine
• Output: images of a certain part of the body
• Use strong magnetic fields and radio waves to form images of the
body.
MRI
INTRODUCTION
• Signals and Systems
– Even though the various signals and systems
could be quite different, they share some
common properties.
– In this course, we will study:
• How to represent signal and system?
• What are the properties of signals?
• What are the properties of systems?
• How to process signals with system?
– The theories can be applied to any general
signals and systems, be it electrical,
biomedical, mechanical, or economical, etc.
OUTLINE
•
Introduction: what are signals and systems?
•
Signals
•
Classifications
•
Basic Signal Operations
•
Elementary Signals
9
SIGNALS AND CLASSIFICATIONS
• What is signal?
– Physical quantities that carry information and changes with respect to time.
– E.g. voice, television picture, telegraph.
• Electrical signal
– Carry information with electrical parameters (e.g. voltage, current)
– All signals can be converted to electrical signals
• Speech → Microphone → Electrical Signal → Speaker → Speech
audio signal
– Signals changes with respect to time
10
SIGNALS AND CLASSIFICATIONS
• Mathematical representation of signal:
– Signals can be represented as a function of time t
s(t ),
– Support of signal: t1  t  t2
– E.g.
s1 (t ) = sin( 2t )
– E.g.
s2 (t ) = sin( 2t )
•
t1  t  t2
−   t  +
0t 
s1 (t ) and s2 (t ) are two different signals!
– The mathematical representation of signal contains two components:
• The expression: s(t )
t1  t  t2
• The support:
– The support can be skipped if −   t  +
s1 (t ) = sin( 2t )
– E.g.
11
SIGNALS AND CLASSIFICATIONS
• Classification of signals: signals can be classified as
–
–
–
–
–
–
–
Continuous-time signal v.s. discrete-time signal
Analog signal v.s. digital signal
Finite support v.s. infinite support
Even signal v.s. odd signal
Periodic signal v.s. Aperiodic signal
Power signal v.s. Energy signal
……
12
OUTLINE
•
Introduction: what are signals and systems?
•
Signals
•
Classifications
•
Basic Signal Operations
•
Elementary Signals
13
14
SIGNALS: CONTINUOUS-TIME V.S. DISCRETE-TIME
• Continuous-time signal
– If the signal is defined over continuous-time, then the signal is a
continuous-time signal
s(t ) = sin( 4t )
• E.g. sinusoidal signal
• E.g. voice signal
• E.g. Rectangular pulse function
p( t )
A
 A, 0  t  1
p( t ) = 
 0, otherwise
0
1
Rectangular pulse function
t
SIGNALS: CONTINUOUS-TIME V.S. DISCRETE-TIME
15
• Discrete-time signal
– If the time t can only take discrete values, such as,
t = kTs
k = 0,1,2,
then the signal s(t ) = s(kTs ) is a discrete-time signal
– E.g. the monthly average precipitation at Fayetteville, AR (weather.com)
Ts = 1 month
k = 1, 2, , 12
Monthly average precipitation
– What is the value of s(t) at (k −1)Ts  t  kTs ?
• Discrete-time signals are undefined at t  kTs !!!
• Usually represented as s(k)
16
SIGNALS: ANALOG V.S. DIGITAL
• Analog v.s. digital
– Continuous-time signal
• continuous-time, continuous amplitude→ analog signal
– Example: speech signal
• Continuous-time, discrete amplitude
– Example: traffic light
– Discrete-time signal
• Discrete-time, discrete-amplitude → digital signal
– Example: Telegraph, text, roll a dice
2
1
3
0
2
1
0
3
2
1
0
2
1
0
• Discrete-time, continuous-amplitude
– Example: samples of analog signal,
average monthly temperature
Different types of signals
17
SIGNALS: EVEN V.S. ODD
• Even v.s. odd
– x(t) is an even signal if: x(t ) = x( −t )
• E.g. x(t ) = cos(2t )
– x(t) is an odd signal if:
x( −t ) = − x(t )
• E.g. x(t ) = sin( 2t )
– Some signals are neither even, nor odd
x(t ) = cos(2t ), t  0
• E.g.
x (t ) = et
– Any signal can be decomposed as the sum of an even signal and odd
signal
y(t ) = ye (t ) + yo (t )
even
• proof
odd
SIGNALS: EVEN V.S. ODD
• Example
– Find the even and odd decomposition of the following signal
x (t ) = et
SIGNALS: EVEN V.S. ODD
• Example
– Find the even and odd decomposition of the following signal
t0
2 sin( 4t ),
x (t ) = 
0
otherwise

19
SIGNALS: PERIODIC V.S. APERIODIC
• Periodic signal v.s. aperiodic signal
– An analog signal is periodic if
• There is a positive real value T such that s(t ) = s(t + nT )
• It is defined for all possible values of t, −   t   (why?)
– Fundamental period T0 : the smallest positive integer T0 that satisfies
s (t ) = s (t + nT0 )
• E.g.
T1 = 2T0
s(t + nT1 ) = s(t + 2nT0 ) = s(t )
– So T1 is a period of s(t), but it is not the fundamental period of
s(t)
20
21
SIGNALS: PERIODIC V.S. APERIODIC
• Example
– Find the period of
–
–
–
–
–
s(t ) = Acos(0t +  )
Amplitude: A
Angular frequency:  0
Initial phase: 
Period: T0 =
Linear frequency: f 0 =
−t  
22
SIGNALS: PERIODIC V.S. APERIODIC
• Complex exponential signal
– Euler formula:
e jx = cos( x) + j sin( x)
– Complex exponential signal
e j0t = cos(0t ) + j sin( 0t )
– Complex exponential signal is periodic with period T0 =
• Proof:
Complex exponential signal has same period as sinusoidal signal!
2
0
SIGNALS: PERIODIC V.S. APERIODIC
• The sum of two periodic signals
– x(t) has a period T1
– y(t) has a period T2
– Define z(t) = a x(t) + b y(t)
– Is z(t) periodic?
z(t + T ) = ax(t + T ) + by(t + T )
• In order to have x(t)=x(t+T), T must satisfy T = kT1
• In order to have y(t)=y(t+T), T must satisfy T = lT
2
• Therefore, if T = kT = lT
1
2
z (t + T ) = ax(t + kT1 ) + by(t + lT2 ) = ax(t ) + by(t ) = z (t )
• The sum of two periodic signals is periodic if and only if the ratio of
the two periods can be expressed as a rational number.
T1 l
=
T2 k
• The period of the sum signal is
T = kT1 = lT2
23
24
SIGNALS: PERIODIC V.S. APERIODIC
• Example
x (t ) = sin(
–
–
–
–
•

3
t)
2
y (t ) = exp( j
t)
9
x(t ), y(t ), z (t )
2
z (t ) = exp( j t )
9
Find the period of
Is 2 x(t ) − 3 y(t )periodic? If periodic, what is the period?
Is x(t ) + z(t ) periodic? If periodic, what is the period?
Is y(t ) z(t ) periodic? If periodic, what is the period?
Aperiodic signal: any signal that is not periodic.
25
SINGALS: ENERGY V.S. POWER
• Signal energy
–
–
–
–
Assume x(t) represents voltage across a resistor with resistance R.
Current (Ohm’s law): i(t) = x(t)/R
Instantaneous power: p (t ) = x 2 (t ) / R
2
Signal power: the power of signal measured at R = 1 Ohm: p(t ) = x (t )
– Signal energy at: [tn , tn + t ]
En  p(tn )t
p(t )
p (t n )
– Total energy
E = lim
t →0
 p(t
n
+
)t =

+
2
n
−
t
p(t )dt
E =  x(t ) dt
tn
Instantaneous power
−
– Review: integration over a signal gives the area under the signal.
t
SINGALS: ENERGY V.S. POWER
t [−,+]
• Energy of signal x(t) over

E =  x(t ) dt
2
−
– If 0  E  , then x(t) is called an energy signal.
• Average power of signal x(t)
1
P = lim
T →  2T
– If 0  P  ,
•

T
−T
2
x (t ) dt
then x(t) is called a power signal.
A signal can be an energy signal, or a power signal, or neither, but not both.
26
27
SINGALS: ENERGY V.S. POWER
• Example 1:
•
•
Example 2:
•
Example 3:
x(t ) = A exp( −t )
t0
x(t ) = Asin( 0t +  )
x(t ) = (1 + j )e jt
0  t  10
All periodic signals are power signal with average power:
1
P=
T

T
0
2
x (t ) dt
OUTLINE
•
Introduction: what are signals and systems?
•
Signals
•
Classifications
•
Basic Signal Operations
•
Elementary Signals
28
29
OPERATIONS: SHIFTING
• Shifting operation
–
x(t − t0 ) : shift the signal x(t) to the right by t0
Shifting to the right by two units
– Why right?
x(0) = A
y(t ) = x(t − t0 )
x(0) = y (t0 )
y(t0 ) = x(t0 − t0 ) = x(0) = A
OPERATIONS: SHIFTING
• Example
– Find
 t +1 −1  t  0
 1
0t 2

x(t ) = 
− t + 3 2  t  3
 0
o.w.
x(t + 3)
30
31
OPERATIONS: REFLECTION
• Reflection operation
–
x(−t ) is obtained by reflecting x(t) w.r.t. the y-axis (t = 0)
x(-t)
x(t)
2
2
1
1
t
t
-2
-1
1
2
-3
3
-2
-1
1
-1
-1
Reflection
OPERATIONS: REFLECTION
• Example:
t + 1 − 1  t  0

x(t ) =  1 0  t  2
 0
o.w.

– Find x(3-t)
•
The operations are always performed w.r.t. the time variable t directly!
32
33
OPERATIONS: TIME-SCALING
• Time-scaling operation
–
x(at ) is obtained by scaling the signal x(t) in time.
• a  1 , signal shrinks in time domain
• a  1 , signal expands in time domain
x(2t)
x(t/2)
x(t)
2
2
1
1
2
1
t
t
-1
1
-1.5
-1
-0.5
0.5
Time scaling
1
1.5
t
-2
-1
1
2
OPERATIONS: TIME-SCALING
• Example:
– Find
 t +1 −1  t  0
 1
0t 2

x(t ) = 
− t + 3 2  t  3
 0
o.w.
x(3t − 6)
x(at + b) 1. scale the signal by a: y(t) = x(at)
2. left shift the signal by b/a: z(t) = y(t+b/a) = x(a(t+b/a))=x(at+b)
•
The operations are always performed w.r.t. the time variable t directly (be
careful about –t or at)!
34
OUTLINE
•
Signals
•
Classifications
•
Basic Signal Operations
•
Elementary Signals
35
36
ELEMENTARY SIGNALS: UNIT STEP FUNCTION
• Unit step function
u(t)
1, t  0
u (t ) = 
0, t  0
1
t
1
Unit step function
• Example: rectangular pulse
1



 , − t
p ( t ) =  
2
2

otherwise
 0,
Express p (t ) as a function of u(t)
u(t)
1/ Ã
t
- Ã /2
à /2
Rectangular pulse
37
ELEMENTARY SIGNALS: RAMP FUNCTION
• The Ramp function
r (t )
r (t ) = t  u(t )
0
t
Unit ramp function
– The Ramp function is obtained by integrating the unit step function u(t)

t
−
u (t )dt =
38
ELEMENTARY SIGNALS: UNIT IMPULSE FUNCTION
• Unit impulse function (Dirac delta function)
 (0) = 
 (t ) = 0, t  0

t
−
 (t )
1, t  0
0, t  0
 (t )dt = 
t
0
Unit impulse function
– delta function can be viewed as the limit of the rectangular pulse
 (t ) = lim pΔ (t )
→0
u(t)
1/ Ã
– Relationship between  (t ) and u(t)
t
- Ã /2

t
−
 (t )dt = u(t )
 (t ) =
du (t )
dt
à /2
Rectangular pulse
ELEMENTARY SIGNALS: UNIT IMPULSE FUNCTION
• Sampling property
x(t ) (t − t0 ) = x(t0 ) (t − t0 )
•
Shifting property

+
−
– Proof:
x(t ) (t − t0 )dt = x(t0 )
39
ELEMENTARY SIGNALS: UNIT IMPULSE FUNCTION
• Scaling property
1  b
 (at + b) =
 t + 
|a|  a
– Proof:
40
ELEMENTARY SIGNALS: UNIT IMPULSE FUNCTION
• Examples

4
(t + t 2 ) (t − 3)dt =
−2
1

−2

3
−2
(t + t 2 ) (t − 3)dt =
exp(t − 1) (2t − 4)dt =
41
42
ELEMENTARY SIGNALS: SAMPLING FUNCTION
sa(t)
• Sampling function
Sa ( x ) =
sin x
x
t
Sampling function
– Sampling function can be viewed as scaled version of sinc(x)
Sinc ( x) =
sin x
= sa (x)
x
sinc(t)
1
t
-4 -3 -2 -1
1 2 3 4
Sinc function
ELEMENTARY SIGNALS: COMPLEX EXPONENTIAL
• Complex exponential
x(t ) = e( r+ j0 )t
– Is it periodic?
• Example:
( −1+ j 2 ) t
– Use Matlab to plot the real part of x(t ) = e
[u(t + 2) − u(t − 4)]
43
44
SUMMARY
• Signals and Classifications
–
–
–
–
–
–
Mathematical representation s (t ),
Continuous-time v.s. discrete-time
Analog v.s. digital
Odd v.s. even
Periodic v.s. aperiodic
Power v.s. energy
t1  t  t2
• Basic Signal Operations
– Time shifting
– reflection
– Time scaling
• Elementary Signals
– Unit step, unit impulse, ramp, sampling function, complex exponential
Department of Electrical Engineering
University of Arkansas
ELEG 3124 SYSTEMS AND SIGNALS
Ch. 2 Continuous-Time Systems
Dr. Jingxian Wu
wuj@uark.edu
46
OUTLINE
• Classifications of continuous-time system
• Linear time-invariant system (LTI)
• Properties of LTI system
• System described by differential equations
47
CLASSIFICATIONS: SYSTEM DEFINITION
• What is system?
– A system is a process that transforms input signals into output signals
• Accept an input
• Process the input
• Send an output (also called: the response of the system to input)
– System examples:
• Radio: input: electrical signals from air, output: music
• Robot: input: electrical control signals, output: motion or action
• Continuous-time system
– A system in which continuous-time input signals are transformed to
continuous-time output signals
• Discrete-time system
– A system in which discrete-time input signals are transformed to discrete-time
output signals.
x(t )
continuous-time
System
Continuous-time system
y(t )
x(n)
Discrete-time
System
discrete-time system
y (n)
48
CLASSIFICATIONS: SYSTEM DEFINITION
• Classifications
–
–
–
–
–
–
Linear v.s. non-linear
Time-invariant v.s. time-varying
Dynamic v.s. static (memory v.s. memoryless)
Causal v.s. non-causal
Invertible v.s. non-invertible
Stable v.s. non-stable
49
CLASSIFICATIONS: LINEAR AND NON-LINEAR
• Linear system
– Let y1 (t )be the response of a system to an input
x1 (t )
– Let y (t )be the response of a system to an input
x2 (t )
2
– The system is linear if the superposition principle is satisfied:
• 1. the response to
is
x1 (t ) + x2 (t )
y1 (t ) + y2 (t )
• 2. the response to
is
x1 (t )
x1 (t ) + x2 (t )
y1 (t )
Linear
System
y1 (t ) + y2 (t )
Linear system
• Non-linear system
– If the superposition principle is not satisfied, then the system is a
non-linear system
50
CLASSIFICATIONS: LINEAR AND NON-LINEAR
• Example: check if the following systems are linear
– System 1:
y(t ) = exp[ x(t )]
– System 2: charge a capacitor. Input: i(t), output v(t)
v (t ) =
1 t
i ( )d
C −
– System 3: inductor. Input: i(t), output v(t)
v (t ) = L
di (t )
dt
51
CLASSIFICATIONS: LINEAR AND NON-LINEAR
• Example
– System 4:
– System 5:
y(t ) =| x(t ) |
– System 6:
y(t ) = x2 (t )
CLASSIFICATIONS: LINEAR V.S. NON-LINEAR
• Example:
– Amplitude Modulation:
• Is it linear?
Amplitude modulation
52
CLASSIFICATIONS: TIME-VARYING V.S. TIME-INVARIANT
• Time-invariant
– A system is time-invariant if a time shift in the input signal causes an
identical time shift in the output signal
x(t )
Time-invariant
System
y(t )
x(t − t0 )
Time-invariant system
• Examples
– y(t) = cos(x(t))
t
– y(t ) = 0 x(v)dv
Time-invariant
System
y(t − t0 )
53
54
CLASSIFICATIONS: MEMORY V.S. MEMORYLESS
• Memoryless system
– If the present value of the output depends only on the present value of
input, then the system is said to be memoryless (or instantaneous).
– Example: input x(t): the current passing through a resistor
output y(t): the voltage across the resistor
y(t ) = Rx(t )
– The output value at time t depends only on input value at time t.
• System with memory
– If the present value of the output depends on not only present value
of input, but also previous input values, then the system has
memory.
– Example: capacitor, current: x(t), output voltage: y(t)
1 t
y (t ) =  x ( ) d
C 0
– the output value at t depends on all input values before t
55
CLASSIFICATIONS: MEMORY V.S. MEMORYLESS
• Examples: determine if the systems has memory or not
–
N
y (t ) =  ai x(t − Ti )
i =0
–
y(t ) = sin( 2x2 (t ) +  ) x(t )
56
CLASSIFICATIONS: CAUSAL V.S. NON-CAUSAL
• Causal system
– A system is causal if the output
only on values of input
y (tdepends
0)
for
t  t0
• The output depends on only input from the past and present
– Example
y (t ) =
1
C
t
 x( )d
0
• Non-causal system
– A system is non-causal if the output depends on the input from the
future (prediction).
– Examples:
1 T /2
y
(
t
)
=
x( )d
y(t ) = x(t + a)

a0
−
T
/
2
T
– The output value at t depends on the input value at t + a (from future)
– All practical systems are causal.
57
CLASSIFICATION: INVERTIBILITY
• Invertible
– A system is invertible if
• by observing the output, we can determine its input.
x(t )
y(t )
System
Inverse
System
x(t )
invertible system
– Question: for a system, if two different inputs result in the same
output, is this system invertible?
• Example
y(t ) = 2 x(t )
y (t ) = cosx(t )
– If two different inputs result in the same output, the system is noninvertible
58
CLASSIFICATION: STABILITY
• Bounded signal
– Definition: a signal x(t) is said to be bounded if
| x(t ) | B  
t
• Bounded-input bounded-output (BIBO) stable system
– Definition: a system is BIBO stable if, for any bounded input x(t),
the response y(t) is also bounded.
t
| x(t ) | B1    | y(t) | B2  
• Example: determine if the systems are BIBO stable
y (t ) = expx(t )
t
y (t ) =  x( )d
−
59
OUTLINE
• Classifications of continuous-time system
• Linear time-invariant system (LTI)
• Properties of LTI system
• System described by differential equations
60
LTI: DEFINTION
• Linear time-invariant (LTI) system
– Definition: a system is said to be LTI if it’s linear and time-invariant
xi (t )
System
yi (t )
system
– Linear
Input:
N
x(t ) = a1 x1 (t ) + a2 x2 (t ) +  + a N x N (t ) =  ai xi (t )
i =1
N
Output: y (t ) = a1 y1 (t ) + a2 y2 (t ) +  + a N y N (t ) =  ai yi (t )
i =1
– Time-invariant
x(t ) = xi (t − t0 )
Input:
Output: y(t ) = yi (t − t0 )
61
LTI: IMPULSE RESPONSE
• Impulse response of LTI system
– Def: the output (response) of a system when the input is a unit impulse
function (delta function).
• Usually denoted as h(t)
x(t ) =  (t )
y(t ) = h(t )
System
LTI system
• For system with an arbitrary input x(t), we want to find
out the output y(t).
– Method 1: differential equations
– Methods 2: convolution integral
– Methods 3: Laplace transform, Fourier transform,
62
LTI: CONVOLUTION
• Derivation
– Any signal can be approximated by the sum of a sequence of delta
functions
+

+
−
z ( )d = lim
 →0
 z ( n ) 
n = −
+
x(t ) =  x( ) (t −  )d = lim
−
 →0
+
 x(n) (t − n)
n = −
x(t)
t
integration
63
LTI: CONVOLUTION
• Derivation
– Any signal can be approximated by the sum of a sequence of delta
functions
+
x(t ) =  x( ) (t −  )d = lim
−
 →0
 (t )
+
 x(n) (t − n)
n = −
h(t )
System
– Time invariant
 (t − n)
h(t − n)
System
– Linear
+
 x(n) (t − n)
n = −
+
 x(n)h(t − n)
System
LTI system
n = −
64
LTI: CONVOLUTION
• Convolution
+
x(t )
System
y(t ) =  x( )h(t −  )d
−
LTI system
– Definition: the convolution of two signals x(t) and h(t) is defined as
+
y(t ) =  x( )h(t −  )d
−
– The operation of convolution is usually denoted with the symbol 
+
y(t ) = x(t )  h(t ) =  x( )h(t −  )d
−
x(t )  h(t )
x(t )
h(t)
LTI system
For LTI system, if we know input x(t) and impulse response h(t),
Then the output is x(t )  h(t )
65
LTI: CONVOLUTION
• Examples
x(t )   (t )
x(t )  (t − t0 )
x(t )  u(t )
66
LTI: CONVOLUTION
• Examples
exp( −bt )u(t )
exp( −at )u(t )
LTI system
y(t ) = ?
67
LTI: CONVOLUTION
• Example
– Obtain the impulse response of a capacitor and use it to find the unit-step
response by using convolution. Assume the input is the current, and the
output is the voltage. Let C = 1F.
v (t ) =
1 t
i ( )d
C −
68
LTI: CONVOLUTION PROPERTIES
• Commutativity
x(t )  y(t ) = y(t )  x(t )
– Proof:
+
x(t )  y(t ) =  x( ) y(t −  )d
−
x(t )  h(t )
x(t )
h(t)
➔
commutativity
h(t )  x(t )
h(t )
x(t)
69
LTI: CONVOLUTION PROPERTIES
• Associativity
x(t )  h1 (t )  h2 (t ) = x(t )  h1 (t )  h2 (t ) = x(t )  h1 (t )  h2 (t )
– proof
h(t )
x(t )
h1 (t )
y1 (t )
h2 (t )
y(t )
Associativity
➔
x(t )
h1 (t )  h2 (t )
y(t )
70
LTI: CONVOLUTION PROPERTIES
• Distributivity
x(t )  h1 (t ) + h2 (t ) = x(t )  h1 (t ) + x(t )  h1 (t )
– proof
h1 (t )
x(t )
y(t )
+
➔
h2 (t )
Distributivity
x(t )
h1 (t ) + h2 (t )
y(t )
71
LTI: CONVOLUTION PROPERTIES
• Example
h1 (t )
h2 (t )
x(t )
+
h3 (t )
h1 (t ) = exp( −2t )u (t )
h3 (t ) = exp( −3t )u(t )
h(t ) = ?
y(t )
h4 (t )
h2 (t ) = 2 exp( −t )u (t )
h4 (t ) = 4 (t )
72
LTI: GRAPHICAL CONVOLUTION
• Graphical interpretation of convolution
x(t)
x(t)
+
y(t ) =  x( )h(t −  )d
−
t
t
x( )
h(-t)
t
x(t)
h( )
t
– 1. Reflection g ( ) = h(− )
t
– 3. Multiplication
x( )h(t0 − )
– 4. Integration
y(t0 ) =  x( )h(t0 −  )d
+
−
g ( − t0 ) = h(−( − t0 )) = h(t0 − )
73
LTI: GRAPHICAL CONVOLUTION
• Example
y(t ) = [2a  p2a (t )]  [2a  p2a (t − a)]
74
OUTLINE
• Classifications of continuous-time system
• Linear time-invariant system (LTI)
• Properties of LTI system
• System described by differential equations
75
LTI PROPERTIES
• Memoryless LTI system
– Review: present output only depends on present input
y(t ) = Kx(t )
– The impulse response of Memoryless LTI system is
h(t ) = K (t )
• Causal LTI system
– Review: output depends on only current input and past input.
– The impulse response of causal LTI system must satisfy:
h(t ) = 0
– Why?
for t  0
76
LTI PROPERTIES
• Invertible LTI Systems
– Review: a system is invertible iff (if and only if) there is an inverse system
that, when connected in cascade with the original system, yields an output
equal to original system input
x(t )
x(t )
y(t )
h(t)
g(t)
x(t )  h(t )  g (t ) = x(t )
– For invertible LTI systems with IR (impulse response) h(t ) , there
exists inverse system g (t ) such that
g (t )  h(t ) =  (t )
– Example: find the inverse system of LTI system h(t ) =  (t − t0 )
77
LTI PROPERTIES
• BIBO Stable LTI state
– Review: a system is BIBO stable iff every bounded input produces a
bounded output.
– LTI system: an LTI system is BIBO stable iff

+
−
• Proof:
h(t ) dt  
78
LTI PROPERTIES
• Examples
– Determine: causal or non-causal, memory or memoryless, stable or
unstable
– 1. h1 (t ) = t exp( −2t )u (t ) + exp(3t )u (−t ) +  (t − 1)
– 2. h2 (t ) = −3 exp( 2t )u (t )
– 3. h3 (t ) = 5 (t + 5)
79
OUTLINE
• Classifications of continuous-time system
• Linear time-invariant system (LTI)
• Properties of LTI system
• System described by differential equations
80
DIFFERENTIAL EQUATIONS
•
LTI system can be represented by differential equations
(N)
(M )
a
y
(
t
)
+
a
y
'
(
t
)
+

+
a
y
(
t
)
=
b
x
(
t
)
+
b
x
'
(
t
)
+

+
b
x
(t )
– Initial
conditions:
0
1
N
0
1
M
d k y (t )
dt k t =0
– Notation: n-th derivative:
d n y (t )
y (t ) =
dt n
(n)
k = 0,, N − 1
81
DIFFERENTIAL EQUATION
• Example:
– Consider a circuit with a resistor R = 1 Ohm and an inductor L = 1H, with
a voltage source v(t) = Bu(t), and is the initialIcurrent in the inductor.
The output of the system is the current across theoinductor.
• Represent the system with a differential equation.
• Find the output of the system with
and
Io = 0
Io = 1
82
DIFFERENTIAL EQUATION
a0 y(t ) + a1 y' (t ) +  + aN y ( N ) (t ) = b0 x(t ) + b1 x' (t ) +  + bM x ( M ) (t )
d k y (t )
dt k t =0
k = 0,, N − 1
• Zero-state response
– The output of the system when the initial conditions are zero
– Denoted as
• Zero-input response
yzs (t )
– The output of the system when the input is zero
– Denoted as
• The actual output of the system
yzi (t )
y (t ) = y zs (t ) + y zi (t )
83
DIFFERENTIAL EQUATION
• Example
– Find the zero-state output and zero-input response of the RL circuit in the
previous example.
Department of Electrical Engineering
University of Arkansas
ELEG 3124 SYSTEMS AND SIGNALS
Ch. 3 Fourier Series
Dr. Jingxian Wu
wuj@uark.edu
85
OUTLINE
• Introduction
• Fourier series
• Properties of Fourier series
• Systems with periodic inputs
86
INTRODUCTION: MOTIVATION
• Motivation of Fourier series
– Convolution is derived by decomposing the signal into the sum of a series
of delta functions
• Each delta function has its unique delay in time domain.
• Time domain decomposition
+
x(t ) =  x( ) (t −  )d = lim
−
 →0
+
 x(n) (t − n)
n = −
x(t)
t
Illustration of integration
87
INTRODUCTION: MOTIVATION
• Can we decompose the signal into the sum of other functions
– Such that the calculation can be simplified?
– Yes. We can decompose periodic signal as the sum of a sequence of
complex exponential signals ➔ Fourier series.
e
j0t
=e
j 2f 0t
f0 =
0
2
– Why complex exponential signal? (what makes complex exponential
signal so special?)
• 1. Each complex exponential signal has a unique frequency ➔
frequency decomposition
• 2. Complex exponential signals are periodic
88
INTRODUCTION: REVIEW
• Complex exponential signal
e j 2ft = cos(2ft) + j sin( 2ft)
– Complex exponential function has a one-to-one relationship with
sinusoidal functions.
– Each sinusoidal function has a unique frequency: f
• What is frequency?
– Frequency is a measure of how fast the signal can change within a
unit time.
• Higher frequency ➔ signal changes faster
f = 0 Hz
f = 1 Hz
Department of Engineering Science
Sonoma State University
Sinusoidal at different frequencies
f = 3 Hz
89
INTRODUCTION: ORTHONORMAL SIGNAL SET
• Definition: orthogonal signal set
, are
0 (t ),1 (t ), 2 (t ),
 said to be orthogonal over an
– A set of signals,
interval (a, b) if
C , l = k
a l (t ) (t )dt =  0, l  k
b
• Example:
*
k
k (t ) = e jk t
k = 0,1,2, are
– the signal set:
orthogonal over the interval [0, T0 ] , where
2
0 =
T0
0
90
OUTLINE
• Introduction
• Fourier series
• Properties of Fourier series
• Systems with periodic inputs
91
FOURIER SERIES
• Definition:
– For any periodic signal with fundamental period , it Tcan be decomposed
as the sum of a set of complex exponential signals as 0
+
c e
x (t ) =
n = −
jn 0t
n
0 =
• cn , n = 0,1,2, , Fourier series coefficients
cn =
1
T0
• derivation of cn :

T0 
x (t )e − jn0t dt
2
T0
92
FOURIER SERIES
• Fourier series
x (t ) =
+
c e
n = −
jn 0t
n
– The periodic signal is decomposed into the weighted summation of
a set of orthogonal complex exponential functions.
– The frequency of the n-th complex exponential function: n 0
• The periods of the n-th complex exponential function: Tn =
T0
n
– The values of coefficients, cn , n = 0,1,2, , depend on x(t)
• Different x(t) will result in different c n
• There is a one-to-one relationship between x(t) and cn
s(t )
➔
[, c−2 , c−1,c0 , c1 , c2 ,]
For a periodic signal, it can be either represented as s(t), or
represented as cn
93
FOURIER SERIES
• Example
x(t)
− K , − 1  t  0
x(t ) = 
 K, 0  t  1
t
-3
-2
-1
1
Rectangle pulses
2
94
FOURIER SERIES
• Amplitude and phase
– The Fourier series coefficients are usually complex numbers
j
cn = an + jbn = cn e n
– Amplitude line spectrum: amplitude as a function of
cn = an2 + bn2
– Phase line spectrum: phase as a function of
bn
 n = a tan
an
n 0
n 0
95
FOURIER SERIES: FREQUENCY DOMAIN
• Signal represented in frequency domain: line spectrum
–
–
–
–
phase
amplitude
Each c n has its own frequency n 0
The signal is decomposed in frequency domain.
c n is called the harmonic of signal s(t) at frequency n 0
Each signal has many frequency components.
• The power of the harmonics at different frequencies determines
how fast the signal can change.
96
FOURIER SERIES: FREQUENCY DOMAIN
• Example: Piano Note
piano notes
One piano note
E5
E6 B6
E7
E5: 659.25 Hz
E6: 1318.51 Hz
B6: 1975.53 Hz
E7: 2637.02 Hz
spectrum
All graphs in this page are created by using the open-source software Audacity.
97
FOURIER SERIES
• Example
– Find the Fourier series of
s(t ) = exp( j 0t )
98
FOURIER SERIES
• Example
– Find the Fourier series of
s(t ) = B + Acos(0t +  )
y(t ) = 1 + sin( 100t )
Time domain
Amplitude spectrum
Phase spectrum
99
FOURIER SERIES
• Example
– Find the Fourier series of
− T / 2  t  − / 2
 0,

s (t ) =  K ,
 0,

− / 2  t   / 2
 /2t T /2
 = 1, T = 5
x(t)
 = 1, T = 10
t
Time domain
cn =
K
n
sin c( )
T
T
 = 1, T = 15
100
FOURIER SERIES: DIRICHLET CONDITIONS
• Can any periodic signal be decomposed into Fourier series?
– Only signals satisfy Dirichlet conditions have Fourier series
• Dirichlet conditions
– 1. x(t) is absolutely integrable within one period

T 
| x(t ) | dt  
– 2. x(t) has only a finite number of maxima and minima.
– 3. The number of discontinuities in x(t) must be finite.
101
OUTLINE
• Introduction
• Fourier series
• Properties of Fourier series
• Systems with periodic inputs
102
PROPERTIES: LINEARITY
• Linearity
– Two periodic signals with the same period
x (t ) =
+
 e
n = −
jn 0t
y (t ) =
n
T0 =
+
2
0
jn t

e
 n
0
n = −
– The Fourier series of the superposition of two signals is
k1 x (t ) + k2 y (t ) =
– If
x(t ) =  n
+
jn0t
(
k

+
k

)
e
 1 n 2 n
n = −
y(t ) = n
• then
k1x(t ) + k2 y(t ) = (k1 n + k2 n )
103
PROPERTIES: EFFECTS OF SYMMETRY
• Symmetric signals
– A signal is even symmetry if:
x(t ) = x(−t )
– A signal is odd symmetry if:
x(t ) = − x(−t )
– The existence of symmetries simplifies the computation of Fourier series
coefficients.
x(t)
x(t)
t
-5 -4 -3 -2 -1
1
2
3
t
-4 -3
-2 -1
1
2
Even symmetric
3
4
Odd symmetric
4
5
104
PROPERTIES: EFFECTS OF SYMMETRY
• Fourier series of even symmetry signals
– If a signal is even symmetry, then
x (t ) =
+
a
n = −
n
cos(n 0t )
2
an =
T0

T0 / 2
0
x (t ) cos(n0t )dt
• Fourier series of odd symmetry signals
– If a signal is odd symmetry, then
+
x (t ) =  bn sin (n0t )
n =1
2
bn =
T0

T0 / 2
0
x (t ) sin (n 0t )dt
105
PROPERTIES: EFFECTS OF SYMMETRY
• Example
x(t)
4A

A
−
t, 0  t  T / 2

T
x(t ) = 
4A
 t − 3 A, T / 2  t  T
T
t
Graph of x(t)
106
PROPERTIES: SHIFT IN TIME
• Shift in time
– If
x(t ) has Fourier series
,cthen
n
x(t −has
t0 ) Fourier series
cne− jn0t0
if x(t ) ➔ cn , then x(t − t0 ) ➔
– Proof:
cne− jn0t0
107
PROPERTIES: PARSEVAL’S THEOREM
• Review: power of periodic signal
1 T
P =  | x(t ) |2 dt
T 0
• Parseval’s theorem
if x(t ) ➔  n
then 1 T | x(t ) |2 dt =
T

0
+
2
|

|
 m
m = −
– Proof:
The power of signal can be computed in frequency domain!
108
PROPERTIES: PARSEVAL’S THEOREM
• Example
– Use Parseval’s theorem find the power of
x(t ) = Asin( 0t )
109
OUTLINE
• Introduction
• Fourier series
• Properties of Fourier series
• Systems with periodic inputs
110
PERIODIC INPUTS: COMPLEX EXPONENTIAL INPUT
• LTI system with complex exponential input
x(t ) = e jt
h(t )
y(t )
y(t ) = x(t )  h(t ) = h(t )  x(t )
+
=  h( ) x(t −  )d
−
+
= exp( jt )  h( ) exp( − j )d
−
• Transfer function
+
H () =  h( ) exp( − j )d
−
– For LTI system with complex exponential input, the output is
y(t ) = H () exp( jt )
– It tells us the system response at different frequencies
111
PERIODIC INPUT
• Example:
– For a system with impulse response
find the transfer function
h(t ) =  (t − t0 )
112
PERIODIC INPUT:
• Example
– Find the transfer function of the system shown in figure.
RL circuit
113
PERIODIC INPUTS
• Example
– Find the transfer function of the system shown in figure
RC circuit
114
PERIODIC INPUTS: TRANSFER FUNCTION
• Transfer function
– For system described by differential equations
n
py
i =0
i
m
(i )
(t ) =  qi x ( i ) (t )
i =0
m
H () =
 q ( j)
i
i
i =0
n
 p ( j)
i =0
i
i
115
PERIODIC INPUTS
• LTI system with periodic inputs
– Periodic inputs:
x (t ) =
+
c
n = −
e
linear:
+
jn0t
c e
n = −
n
h(t )
e jn0t H (n0 )
+
jn0t
n
exp( jn0t )
2
0 =
T
h(t )
jn0t
c
e
 n H ( n 0 )
n = −
+
x(t )
h(t )
jn0t
c
e
 n H ( n 0 )
n = −
For system with periodic inputs, the output is the weighted
sum of the transfer function.
116
PERIODIC INPUTS
• Procedures:
– To find the output of LTI system with periodic input
• 1. Find the Fourier series coefficients of periodic input x(t).
1
n =
T

T
0
x (t )e − jn0t dt
 0 = 2f 0 =
• 2. Find the transfer function of LTI system H ()
• 3. The output of the system is
y (t ) =
+
jn 0t
c
e
 n H (n0 )
n = −
2
T
period of x(t)
117
PERIODIC INPUTS
• Example
– Find the response of the system when the input is
x(t ) = 4 cos(t ) − 2 cos(2t )
RL Circuit
118
PERIODIC INPUTS
• Example
– Find the response of the system when the input is shown in figure.
x(t)
t
-3
RC circuit
-2
-1
1
Square pulses
2
119
PERIODIC INPUTS: GIBBS PHENOMENON
• The Gibbs Phenomenon
– Most Fourier series has infinite number of elements→ unlimited
bandwidth
x (t ) =
+
jn 0t
c
e
n
n = −
• What if we truncate the infinite series to finite number of elements?
x N (t ) =
+N
jn 0t
c
e
n
– The truncated signal,n = − N
signal x(t)
, is an approximation of the original
xN (t )
120
PERIODIC INPUTS: GIBBS PHENOMENON
x(t)
t
-3
-2
-1
1
2
 2K 1
, n odd,

cn =  j n
 0,
n even.
x N (t ) =
+N
jn 0t
c
e
n
n=− N
Square pulses
x3 (t )
x5 (t )
x19 (t )
121
FOURIER SERIES
• Analogy: Optical Prism
– Each color is an Electromagnetic wave with a different frequency
Optical prism
Department of Electrical Engineering
University of Arkansas
ELEG 3124 SYSTEMS AND SIGNALS
Ch. 4 Fourier Transform
Dr. Jingxian Wu
wuj@uark.edu
123
OUTLINE
• Introduction
• Fourier Transform
• Properties of Fourier Transform
• Applications of Fourier Transform
124
INTRODUCTION: MOTIVATION
• Motivation:
– Fourier series: periodic signals can be decomposed as the summation of
orthogonal complex exponential signals
+
 cn exp  jn0t 
1 T
cn =  x (t ) exp  jn0t dt
n = −
T 0
• each harmonic contains a unique frequency: n/T
x (t ) =
x(t)
t
Time domain
• time domain ➔ frequency domain
Frequency domain
(T =  )
How about aperiodic signals
?
125
INTRODUCTION: TRANSFER FUNCTION
• System transfer function
e
e jt H ()
jt
h(t )
H ( ) =  h(t ) exp jt dt
+
−
• System with periodic inputs
e
+
jn0t
c e
n = −
h(t )
+
jn0t
n
e jn0t H (n0 )
h(t )
jn0t
c
e
 n H ( n 0 )
n = −
+
x(t )
h(t )
jn0t
c
e
 n H ( n 0 )
n = −
126
OUTLINE
• Introduction
• Fourier Transform
• Properties of Fourier Transform
• Applications of Fourier Transform
127
FOURIER TRANSFORM
• Fourier Transform
+
X ( ) =  x(t )e − jt dt
−
– given x(t), we can find its Fourier transform X ( )
• Inverse Fourier Transform
x (t ) =
1
2

+
−
X ( )e jt d
– given X ( ) , we can find the time domain signal x(t)
– signal is decomposed into the “weighted summation” of complex
exponential functions. (integration is the extreme case of
summation)
x(t ) ➔ X ( )
128
FOURIER TRANSFORM
• Example
– Find the Fourier transform of x(t ) = rect (t /  )
x(t)
x(t)
t
t
129
FOURIER TRANSFORM
• Example
– Find the Fourier transform of
x(t ) = exp( −a | t |)
a0
130
FOURIER TRANSFORM
• Example
– Find the Fourier transform of
x(t ) = exp( −at )u(t )
a0
131
FOURIER TRANSFORM
• Example
– Find the Fourier transform of x(t ) =  (t − a)
132
FOURIER TRANSFORM: TABLE
133
FOURIER TRANSFORM
• The existence of Fourier transform
– Not all signals have Fourier transform
– If a signal have Fourier transform, it must satisfy the following two
conditions
• 1. x(t) is absolutely integrable
+
 | x(t ) | dt  
−
• 2. x(t) is well behaved
– The signal has finite number of discontinuities, minima,
and maxima within any finite interval of time.
• Example
–
x(t ) = exp(t )u(t )
134
OUTLINE
• Introduction
• Fourier Transform
• Properties of Fourier Transform
• Applications of Fourier Transform
135
PROPERTIES: LINEARITY
• Linearity
– If
x1 (t )  X 1 ( )
– then
• Example
x2 (t )  X 2 ( )
ax1 (t ) + bx2 (t )  aX 1 ( ) + bX 2 ( )
– Find the Fourier transform of x(t ) = 2rect (t /  ) + 3 exp( −2t )u(t ) + 4 (t )
136
PROPERTY: TIME-SHIFT
• Time shift
– If
– Then
x(t )  X ( )
x(t − t0 )  X () exp[ − jt0 ]
phase shift
• Review: complex number
c =| c | e j =| c | cos( ) + j | c | sin(  ) = a + jb
a =| c | cos
b =| c | sin 
| c |= a 2 + b 2
 = a tan( b / a)
– Phase shift of a complex number c by  0 : c exp( j0 ) =| c | exp j ( + 0 )
time shift in time domain ➔ frequency shift in frequency domain
137
PROPERTY: TIME SHIFT
• Example:
– Find the Fourier transform of
x(t ) = rect t − 2
138
PROPERTY: TIME SCALING
• Time scaling
– If
x(t )  X ( )
– Then
1  
x(at ) 
X 
|a|  a 
• Example
– Let X ( ) = rect( − 1) / 2 , find the Fourier transform of x(−2t + 4)
139
PROPERTY: SYMMETRY
• Symmetry
– If x(t )  X ( ) , and x(t ) is a real-valued time signal
– Then
X (−) = X * ()
140
PROPERTY: DIFFERENTIATION
• Differentiation
– If
x(t )  X ( )
– Then
dx(t )
 jX ( )
dt
• Example
– Let
d n x(t )
n
(
)

j

X ( )
n
dt
X ( ) = rect( − 1) / 2 , find the Fourier transform of
dx(t )
dt
141
PROPERTY: DIFFERENTIATION
• Example
– Find the Fourier transform of
(Hint:
d 1

)
sgn(
t
)
=

(
t
)

dt  2

x(t ) = sgn( t )
142
PROPERTY: CONVOLUTION
• Convolution
– If
x(t )  X ( ),
– Then
h(t )  H ( )
x(t )  h(t )  X ( ) H ( )
x(t )  h(t )
x(t )
h(t )
time domain
X ( )
H ( )
X ( ) H ( )
frequency domain
143
PROPERTY: CONVOLUTION
• Example
– An LTI system has impulse response
h(t ) = exp (− at )u (t )
If the input is
x(t ) = (a − b) exp (− bt )u (t ) + (c − a) exp( −ct )u (t )
Find the output
(a  0, b  0, c  0)
144
PROPERTY: MULTIPLICATION
• Multiplication
– If
x(t )  X ( ) ,
– Then
x(t )m(t ) 
m(t )  M ()
1
X ( )  M ( )
2
145
PROPERTY: DUALITY
• Duality
– If
– Then
g (t )  G( )
G(t )  2g (− )
146
PROPERTY: DUALITY
• Example
– Find the Fourier transform of
(recall:
  )
rect (t /  )   sinc 

 2 
t
h(t ) = Sa 
2
147
PROPERTY: DUALITY
• Example
– Find the Fourier transform of
– Find the Fourier transform of
x(t ) = 1
x(t ) = e j0t
148
PROPERTY: SUMMARY
149
PROPERTY: EXAMPLES
• Examples
– 1. Find the Fourier transform of
x(t ) = cos(0t )
– 2. Find the Fourier transform of
x(t ) = u(t )
1
sgn( t ) + 1
2
2
j
u (t ) =
sgn( t ) 
150
PROPERTY: EXAMPLES
• Examples
– 3. A LTI system with impulse response
Find the output when input is
– 4. If
(Hint:
h(t ) = exp− at u (t )
x(t ) = u(t )
x(t )  X ( ) , find the Fourier transform of

t
−
x( )d = x(t )  u (t ) )

t
−
x( )d
151
PROPERTY: EXAMPLES
• Example
– 5. (Modulation) If
, ) m(t ) = cos( t )
x(t )  X (
0
Find the Fourier transform of
x(t )m(t )
– 6. If
X ( ) =
1 , find x(t)
a + j
152
PROPERTY: DIFFERENTIATION IN FREQ. DOMAIN
• Differentiation in frequency domain
– If:
– Then:
x(t )  X ( )
d n X ( )
(− jt) x(t ) =
d n
n
PROPERTY: DIFFERENTIATION IN FREQ. DOMAIN
• Example
– Find the Fourier transform of
t exp( −at )u(t ),
a0
153
154
PROPERTY: FREQUENCY SHIFT
• Frequency shift
– If:
– Then:
• Example
– If
x(t )  X ( )
x(t ) exp( j0t )  X ( − 0 )
X ( ) = rect( − 1) / 2 , find the Fourier transform x(t ) exp( − j 2t )
155
PROPERTY: PARSAVAL’S THEOREM
• Review: signal energy
+
E =  | x(t ) |2 dt
−
• Parsaval’s theorem
+
1
|
x
(
t
)
|
dt
=
−
2
2

+
−
| X ( ) |2 d
156
PROPERTY: PARSAVAL’S THEOREM
• Example:
– Find the energy of the signal
x(t ) = exp( −2t )u(t )
157
PROPERTY: PERIODIC SIGNAL
• Fourier transform of periodic signal
– Periodic signal can be written as Fourier series
x(t ) =
+
c
n = −
n
exp  jn0t 
– Perform Fourier transform on both sides
X ( ) = 2
+
 c  ( − n )
n = −
n
0
158
OUTLINE
• Introduction
• Fourier Transform
• Properties of Fourier Transform
• Applications of Fourier Transform
159
APPLICATIONS: FILTERING
• Filtering
– Filtering is the process by which the essential and useful part of a signal is
separated from undesirable components.
• Passing a signal through a filter (system).
• At the output of the filter, some undesired part of the signal (e.g. noise)
is removed.
– Based on the convolution property, we can design filter that only allow
signal within a certain frequency range to pass through.
x(t )  h(t )
x(t )
X ( )
h(t )
H ( )
filter
filter
time domain
X ( ) H ( )
frequency domain
160
APPLICATIONS: FILTERING
• Classifications of filters
Passband
Stop
band
Low pass filter
Stop Passband Stop
band
band
Band pass filter
Stop
band
Passband
High pass filter
Passband
Stop
Passband
band
Band stop (Notch) filter
161
APPLICATION: FILTERING
• A filtering example
– A demo of a notch filter
X ( )
H ( )
Corrupted sound
Filter
X ( ) H ( )
Filtered sound
162
APPLICATIONS: FILTERING
• Example
– Find out the frequency response of the RC circuit
– What kind of filters it is?
RC circuit
163
APPLICATION: SAMPLING THEOREM
• Sampling theorem: time domain
– Sampling: convert the continuous-time signal to discrete-time signal.
x(t )
p (t ) =
+
  (t − nT )
n = −
sampling period
xs (t ) = x(t ) p(t )
Sampled signal
164
APPLICATION: SAMPLING THEOREM
• Sampling theorem: frequency domain
– Fourier transform of the impulse train
• impulse train is periodic
+
1
p(t ) =   (t − nTs ) =
Ts
n = −
Fourier series
+
jns t
1

e

n = −
2
s =
Ts
• Find Fourier transform on both sides
2
P( ) =
Ts
+
 ( − n )
n = −
s
• Time domain multiplication ➔ Frequency domain convolution
x (t ) p (t ) 
1
X ( )  P( )
2
1
x(t ) p(t ) 
Ts
+
 X ( − n )
n = −
s
165
APPLICATION: SAMPLING THEOREM
• Sampling theorem: frequency domain
– Sampling in time domain ➔ Repetition in frequency domain
Time domain
Frequency domain
166
APPLICATION: SAMPLING THEOREM
• Sampling theorem
– If the sampling rate is twice of the bandwidth, then the original signal can
be perfectly reconstructed from the samples.
s  2B
s  2B
s = 2B
s  2B
Frequency domain
167
APPLICATION: AMPLITUDE MODULATION
• What is modulation?
– The process by which some characteristic of a carrier wave is
varied in accordance with an information-bearing signal
Information
bearing signal
modulation
Modulated signal
Carrier wave
• Three signals:
– Information bearing signal (modulating signal)
• Usually at low frequency (baseband)
• E.g. speech signal: 20Hz – 20KHz
– Carrier wave
• Usually a high frequency sinusoidal (passband)
• E.g. AM radio station (1050KHz) FM radio station
(100.1MHz), 2.4GHz, etc.
– Modulated signal: passband signal
168
APPLICATION: AMPLITUDE MODULATION
• Amplitude Modulation (AM)
s(t ) = Ac m(t ) cos(2f ct )
– A direct product between message signal and carrier signal
m(t )
s(t )
Mixer
Ac cos(2f ct )
Local
Oscillator
Amplitude modulation
169
APPLICATION: AMPLITUDE MODULATION
• Amplitude Modulation (AM)
S( f ) =
Ac
M ( f − f c ) + M ( f + f c )
2
Amplitude modulation
Department of Electrical Engineering
University of Arkansas
ELEG 3124 SYSTEMS AND SIGNALS
Ch. 5 Laplace Transform
Dr. Jingxian Wu
wuj@uark.edu
171
OUTLINE
• Introduction
• Laplace Transform
• Properties of Laplace Transform
• Inverse Laplace Transform
• Applications of Laplace Transform
172
INTRODUCTION
• Why Laplace transform?
– Frequency domain analysis with Fourier transform is extremely useful for
the studies of signals and LTI system.
• Convolution in time domain ➔ Multiplication in frequency domain.
– Problem: many signals do not have Fourier transform
x(t ) = exp( at )u(t ), a  0
x(t ) = tu(t )
– Laplace transform can solve this problem
• It exists for most common signals.
• Follow similar property to Fourier transform
• It doesn’t have any physical meaning; just a mathematical tool
to facilitate analysis.
– Fourier transform gives us the frequency domain
representation of signal.
173
OUTLINE
• Introduction
• Laplace Transform
• Properties of Laplace Transform
• Inverse Lapalace Transform
• Applications of Fourier Transform
LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM
• Bilateral Laplace transform (two-sided Laplace transform)
+
X B ( s) =  x(t ) exp( − st )dt ,
−
s =  + j
– s =  + j is a complex variable
– s is often called the complex frequency
– Notations:
X B ( s ) = L[ x(t )]
x (t )  X B ( s )
•
Time domain v.s. S-domain
– x(t ) : a function of time t → x(t) is called the time domain signal
– X B (s ) : a function of s → X B (s ) is called the s-domain signal
– S-domain is also called as the complex frequency domain
174
175
LAPLACE TRANSFORM
• Time domain v.s. s-domain
– x(t ) : a function of time t → x(t) is called the time domain signal
– X B (s ) : a function of s → X B (s ) is called the s-domain signal
• S-domain is also called the complex frequency domain
– By converting the time domain signal into the s-domain, we can usually
greatly simplify the analysis of the LTI system.
– S-domain system analysis:
• 1. Convert the time domain signals to the s-domain with the Laplace
transform
• 2. Perform system analysis in the s-domain
• 3. Convert the s-domain results back to the time-domain
LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM
• Example
– Find the Bilateral Laplace transform of
x(t ) = exp( −at )u(t )
• Region of Convergence (ROC)
– The range of s that the Laplace transform of a signal converges.
– The Laplace transform always contains two components
• The mathematical expression of Laplace transform
• ROC.
176
177
LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM
• Example
– Find the Laplace transform of
x(t ) = − exp( −at )u(−t )
LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM
• Example
– Find the Laplace transform of
x(t ) = 3 exp( −2t )u(t ) + 4 exp(t )u(−t )
178
179
LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM
• Unilateral Laplace transform (one-sided Laplace transform)
+
X ( s) =  − x(t ) exp( − st )dt
0
– 0− :The value of x(t) at t = 0 is considered.
– Useful when we dealing with causal signals or causal systems.
• Causal signal: x(t) = 0, t < 0.
• Causal system: h(t) = 0, t < 0.
– We are going to simply call unilateral Laplace transform as
Laplace transform.
180
LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM
• Example: find the unilateral Laplace transform of the following
signals.
– 1.
x(t ) = A
– 2. x(t ) =  (t )
181
LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM
• Example
– 3.
x(t ) = exp( j 2t )
– 4.
x(t ) = cos(2t )
– 5.
x(t ) = sin( 2t )
LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM
182
183
OUTLINE
• Introduction
• Laplace Transform
• Properties of Laplace Transform
• Inverse Lapalace Transform
• Applications of Fourier Transform
184
PROPERTIES: LINEARITY
• Linearity
x (t )  X ( s)
x1 (t )  X 1 ( s)
2
2
– If
ax1 (t ) + bx2 (t )  aX 1 ( s) + bX 2 ( s)
– Then
The ROC is the intersection between the two original signals
• Example
– Find the Laplace transfrom of
A + B exp( −bt )u(t )
185
PROPERTIES: TIME SHIFTING
• Time shifting
t0  0
x(t )  X (s)and
– If
x(t − t0 )u(t − t0 )  X (s) exp( −st0 )
– Then
The ROC remain unchanged
186
PROPERTIES: SHIFTING IN THE s DOMAIN
• Shifting in the s domain
– If
– Then
Re(s)  
x(t )  X (s)
y(t ) = x(t ) exp( s0t )  X (s − s0 )
Re( s)   + Re( s0 )
• Example
– Find the Laplace transform of
x(t ) = A exp( −at ) cos(0t )u(t )
187
PROPERTIES: TIME SCALING
• Time scaling
– If
– Then
x(t )  X (s)
1 s
x(at )  X  
a a
• Example
– Find the Laplace transform of
Re{s}   1
Re{s}  a 1
x(t ) = u(at )
188
PROPERTIES: DIFFERENTIATION IN TIME DOMAIN
• Differentiation in time domain
– If
– Then
g (t )  G(s)
dg (t )
 sG ( s ) − g (0 − )
dt
d 2 g (t )
2
−
−

s
G
(
s
)
−
sg
(
0
)
−
g
'
(
0
)
2
dt
d n g (t )
n
n −1
−
( n−2)
−
( n −1)
−

s
G
(
s
)
−
s
g
(
0
)
−

−
sg
(
0
)
−
g
(
0
)
n
dt
• Example
2
– Find the Laplace transform of g (t ) = sin t  u(t ),
g (0 − ) = 0
189
PROPERTIES: DIFFERENTIATION IN TIME DOMAIN
• Example
– Use Laplace transform to solve the differential equation
y' ' (t ) + 3 y' (t ) + 2 y(t ) = 0,
y(0− ) = 3
y ' (0 − ) = 1
190
PROPERTIES: DIFFERENTIATION IN S DOMAIN
• Differentiation in s domain
– If
– Then
x(t )  X (s)
d n X (s)
(−t ) x(t ) 
ds n
n
• Example
– Find the Laplace transform of
t n u (t )
191
PROPERTIES: CONVOLUTION
• Convolution
x(t )  X (s)
h(t )  H ( s)
– If
– Then
x(t )  h(t )  X (s) H (s)
The ROC of X ( s) H ( s) is the intersection of the ROCs of X(s) and
H(s)
192
PROPERTIES: INTEGRATION IN TIME DOMAIN
• Integration in time domain
– If
– Then
x(t )  X (s)
t
1
0 x( )d  s X (s)
• Example
– Find the Laplace transform of r(t ) = tu(t )
193
PROPERTIES: CONVOLUTION
• Example
– Find the convolution
t −a
t −a
rect 

rect



 2a 
 2a 
194
PROPERTIES: CONVOLUTION
• Example
– For a LTI system, the input is
system is
x(t ) = exp( −2t ), uand
(t )the output of the
y (t ) = exp( −t ) + exp( −2t ) − exp( −3t )u(t )
Find the impulse response of the system
195
PROPERTIES: CONVOLUTION
• Example
– Find the Laplace transform of the impulse response of the LTI system
described by the following differential equation
2 y' ' (t ) − 3 y' (t ) + y(t ) = 3x' (t ) + x(t )
(n)
(n)
assume the system was initially relaxed ( y (0) = x (0) = 0 )
196
PROPERTIES: MODULATION
• Modulation
– If
– Then
x(t )  X (s)
1
x(t ) cos(0t )  X ( s + j0 ) + X ( s − j0 )
2
j
x(t ) sin( 0t )  X ( s + j0 ) − X ( s − j0 )
2
197
PROPERTIES: MODULATION
• Example
– Find the Laplace transform of
x(t ) = exp( −at ) sin( 0t )u(t )
198
PROPERTIES: INITIAL VALUE THEOREM
• Initial value theorem
– If the signal x(t ) is infinitely differentiable on an interval around
then
x(0 + ) = lim sX ( s)
s →
x(0+ )
s =  must be in ROC
– The behavior of x(t) for small t is determined by the behavior of
X(s) for large s.
199
PROPERTIES: INITIAL VALUE THEOREM
• Example
– The Laplace transform of x(t) is
Find the value of
+
x(0 )
cs + d
X ( s) =
( s − a)( s − b)
200
PROPERTIES: FINAL VALUE THEOREM
• Final value theorem
– If
– Then:
x(t )  X (s)
lim x(t )  lim sX (s)
t →
s →0
s = 0 must be in ROC
• Example
– The input x(t ) = Au(t ) is applied to a system with transfer
c
function
, find the value of lim y(t )
H ( s) =
s ( s + b) + c
t →
201
PROPERTIES
202
OUTLINE
• Introduction
• Laplace Transform
• Properties of Laplace Transform
• Inverse Lapalace Transform
• Applications of Fourier Transform
203
INVERSE LAPLACE TRANSFORM
• Inverse Laplace transform
1  + j
x(t ) =
X ( s) exp( st )ds


−
j

2j
– Evaluation of the above integral requires the use of contour
integration in the complex plan ➔ difficult.
• Inverse Laplace transform: special case
– In many cases, the Laplace transform can be expressed as a
rational function of s
bm s m + bm−1s m−1 +  + b1s + b0
X (s) =
an s n + an −1s n −1 +  + a1s + a0
– Procedure of Inverse Laplace Transform
• 1. Partial fraction expansion of X(s)
• 2. Find the inverse Laplace transform through Laplace
transform table.
204
INVERSE LAPLACE TRANSFORM
• Review: Partial Fraction Expansion with non-repeated linear
factors
X (s) =
A = (s − a1 ) X (s) s =a
1
A
B
C
+
+
s − a1 s − a2 s − a3
B = (s − a2 ) X (s) s =a
2
C = (s − a3 ) X (s) s =a
• Example
– Find the inverse Laplace transform of
X (s) =
3
2s + 1
s 3 + 3s 2 − 4 s
205
INVERSE LAPLACE TRANSFORM
• Example
– Find the Inverse Laplace transform of
2s 2
X ( s) = 2
s + 3s + 2
• If the numerator polynomial has order higher than or equal to the order
of denominator polynomial, we need to rearrange it such that the
denominator polynomial has a higher order.
206
INVERSE LAPLACE TRANSFORM
• Partial Fraction Expansion with repeated linear factors
X (s) =
1
A2
A1
B
=
+
+
( s − a ) 2 ( s − b) (s − a )2 s − a s − b
A2 = (s − a ) X (s)
2
s =a
A1 =

d
(s − a )2 X (s)
ds

B = (s − b )X ( s) s =b
s =a
207
INVERSE LAPLACE TRANSFORM
• High-order repeated linear factors
AN
A1
A2
B
X ( s) =
=
+
++
+
N
2
N
( s − a ) ( s − b) s − a ( s − a )
( s − a)
s −b
1

1
d N −k
N
(
)
Ak =
s
−
a
X ( s)
N −k
( N − k )! ds
B = (s − b )X ( s) s =b

k = 1,, N
s =a
208
OUTLINE
• Introduction
• Laplace Transform
• Properties of Laplace Transform
• Inverse Lapalace Transform
• Applications of Laplace Transform
209
APPLICATION: LTI SYSTEM REPRESENTATION
• LTI system
– System equation: a differential equation describes the input output
relationship of the system.
y ( N ) (t ) + aN −1 y ( N −1) (t ) +  + a1 y (1) (t ) + a0 y(t ) = bM x( M ) (t ) +  + b1 x (1) (t ) + b0 x(t )
N −1
y
(N)
M
(t ) +  an y (t ) =  bm x ( m ) (t )
(n)
n =0
m =0
– S-domain representation
 N N −1 n 
M
m
 s +  an s Y ( s) =   bm s  X ( s)
n =0


 m =0

– Transfer function
M
Y (s)
H (s) =
=
X (s)
b
m =0
m
sm
N −1
s +  an s n
N
n =0
210
APPLICATION: LTI SYSTEM REPRESENTATION
• Simulation diagram (first canonical form)
Simulation diagram
211
APPLICATION: LTI SYSTEM REPRESENTATION
• Example
– Show the first canonical realization of the system with transfer function
s 2 − 3s + 2
H (S ) = 3
s + 6s 2 + 11s + 6
212
APPLICATION: COMBINATIONS OF SYSTEMS
• Combination of systems
– Cascade of systems
H ( S ) = H1 ( s ) H 2 ( s )
– Parallel systems
H ( S ) = H1 ( s ) + H 2 ( s )
213
APPLICATION: LTI SYSTEM REPRESENTATION
• Example
– Represent the system to the cascade of subsystems.
s 2 − 3s + 2
H (S ) = 3
s + 6s 2 + 11s + 6
214
APPLICATION: LTI SYSTEM REPRESENTATION
• Example:
– Find the transfer function of the system
LTI system
215
APPLICATION: LTI SYSTEM REPRESENTATION
• Poles and zeros
H (s) =
– Zeros:
– Poles:
( s − z M )( s − z M −1 )  ( s − z1 )
( s − p N )( s − p N −1 )  ( s − p1 )
z1 , z 2 ,, z M
p1 , p2 ,, pN
216
APPLICATION: STABILITY
• Review: BIBO Stable
– Bounded input always leads to bounded output

+
−
| h(t ) | dt  
• The positions of poles of H(s) in the s-domain
determine if a system is BIBO stable.
H (s) =
AN
A1
A2
+
+

+
s − s1 ( s − s2 ) m
s − sN
– Simple poles: the order of the pole is 1, e.g. s1
sN
– Multiple-order poles: the poles with higher order. E.g. s 2
217
APPLICATION: STABILITY
• Case 1: simple poles in the left half plane
1
1
=
(s −  k )2 + k2 ( s −  k + jk )( s −  k − jk )
p1 =  k − j k
hk (t ) =

+
−
1
k
k  0
p2 =  k + jk
exp( k t ) sin( k t )u (t )
hk (t ) dt =
Impulse response
• If all the poles of the system are on the left half plane,
then the system is stable.
218
APPLICATION: STABILITY
• Case 2: Simple poles on the right half plane
1
1
=
(s −  k )2 + k2 ( s −  k + jk )( s −  k − jk )
p1 =  k + jk
hk (t ) =
1
k
k  0
p2 =  k − jk
exp( k t ) sin( k t )u (t )
Impulse response
• If at least one pole of the system is on the right half
plane, then the system is unstable.
219
APPLICATION: STABILITY
• Case 3: Simple poles on the imaginary axis
1
1
=
(s −  k )2 + k2 ( s −  k + jk )( s −  k − jk )
hk (t ) =
1
k
k = 0
sin( k t )u (t )
• If the pole of the system is on the imaginary axis, it’s
unstable.
220
APPLICATION: STABILITY
• Case 4: multiple-order poles in the left half plane
1 m
k  0
stable
hk (t ) =
t exp( k t ) sin( k t )u (t )
k
• Case 5: multiple-order poles in the right half plane
1 m
hk (t ) =
t exp( k t ) sin( k t )u (t )
unstable
k  0
k
• Case 6: multiple-order poles on the imaginary axis
hk (t ) =
1
k
t m sin( k t )u (t )
k  0
unstable
k  0
221
APPLICATION: STABILITY
• Example:
– Check the stability of the following system.
H ( s) =
3s + 2
s 2 + 6 s + 13
Department of Electrical Engineering
University of Arkansas
ELEG 3124 Signals & Systems
Ch. 6 Discrete-Time System
Dr. Jingxian Wu
wuj@uark.edu
223
OUTLINE
• Discrete-time signals
• Discrete-time systems
• Z-transform
224
SIGNAL
• Discrete-time signal
– The time takes discrete values
n
x(n) = cos 
4
1
n
x ( n) = exp  
2
4
225
SIGNAL: CLASSIFICATION
• Energy signal v.s. Power signal
– Energy:
E = lim
N →
N

x ( n)
2
n=− N
– Power:
N
1
2
P = lim
x ( n)

N → 2 N + 1
n=− N
– Energy signal:
E
– Power signal:
P
226
SIGNAL: CLASSIFICATION
• Periodic signal v.s. aperiodic signal
– Periodic signal
x(n) = x(n + N )
• The smallest value of N that satisfies this relation is the fundamental
periods.
– Is
periodic?
cos(n)
cos(n) is periodic if
– Example:
cos(3n)
cos(n)
3
cos( n)
4
2 k

is integer for integer k.
227
SIGNAL: ELEMENTARY SIGNAL
• Unit impulse function
1, n = 0,
 ( n) = 
0, n  0.
• Unit step function
0, n  0,
u ( n) = 
1, n  0.
• Relation between unit impulse function and unit step
function
 (n) = u(n) − u(n − 1)
u ( n) =
n
  (k )
k = −
228
SIGNAL: ELEMENTARY SIGNAL
• Exponential function
x(n) = exp(n)
• Complex exponential function
x(n) = exp( j0n) = cos(0n) + j sin( 0n)
229
OUTLINE
• Discrete-time signals
• Discrete-time systems
• Z-transform
230
SYSTEM: IMPULSE RESPONSE
• Impulse response of LTI system
 (n)
– The response of the system when the input is
x(n) =  (n)
y(n) = h(n)
System
LTI system
• System response for arbitrary input
– Any signal can be decomposed as the sum of time-shifted impulses
x ( n) =
– Time invariant
 (n − k )
+
 x(k ) (n − k )
k = −
h( n − k )
System
LTI system
– Linear
+
+
 x(k ) (n − k )
k = −
 x(k )h(n − k )
System
k = −
LTI system
231
SYSTEM: CONVOLUTION SUM
• Convolution sum
– The convolution sum of two signals x(n) and
x ( n)  h( n) =
h(n) is
+
 x(k )h(n − k )
k = −
• Response of LTI system
– The output of a LTI system is the convolution sum of the input and
the impulse response of the system.
x(n)  h(n)
x(n)
h(n)
LTI system
232
SYSTEM: CONVOLUTION SUM
• Example
– 1.
x(n)   (n − m)
– 2.
x(n) =  nu(n),
x(n)  h(n) =
h(n) =  nu(n)
233
SYSTEM: CONVOLUTION SUM
• Example:
– Let
x(n) = [1,3,−1,−2]
sequences, find
two
h(n) = [1,2,0,−1be,1],
x(n)  h(n)
234
STSTEM: COMBINATION OF SYSTEMS
• Combination of systems
➔
Two systems in series
+
➔
Two systems in parallel
235
SYSTEM: DIFFERENCE EQUATION REPRESENTATION
• Difference equation representation of system
N
a
k =0
M
k
y (n − k ) =  bk x(n − k )
k =0
236
OUTLINE
• Discrete-time signals
• Discrete-time systems
• Z-transform
237
Z-TRANSFORM
• Bilateral Z-transform
X ( z) =
+
−n
x
(
n
)
z

n = −
• Unilateral Z-transform
+
X ( z ) =  x(n)z − n
n =0
• Z-transform:
– Ease of analysis
– Doesn’t have any physical meaning (the frequency domain
representation of discrete-time signal can be obtained through
discrete-time Fourier transform)
– Counterpart for continuous-time systems: Laplace transform.
238
Z-TRANSFORM
• Example: find Z-transforms
– 1.
x(n) =  (n)
n
– 2. x(n) =  1  u (n)
2
239
Z-TRANSFORM
• Example
– 3.
n
1
x(n) = −  u (− n − 1)
2
• Region of convergence (ROC)
Region of convergence
240
Z-TRANSFORM: CONVERGENCE
• Convergence of causal signal
x(n) =  nu(n)
• Convergence of anti-causal signal
x(n) =  nu(−n −1)
241
Z-TRANSFORM: TIME SHIFTING PROPERTY
• Time Shifting
– Let x (n ) be a causal sequence with the Z-transform
– Then
Z x(n + n0 ) = z X ( z ) − z
n0
Z x(n − n0 ) = z
− n0
X ( z) + z
n0
− n0
n0 −1
 x ( m) z
−m
m =0
−1
 x(m) z
m = − n0
−m
X (z )
242
Z-TRANSFORM: LTI SYSTEM
• LTI System
– Difference equation representation
N
a
k =0
M
k
y (n − k ) =  bk x(n − k )
k =0
– Z-domain representation
N
M
−k 
−k 
a
z
Y
(
z
)
=
b
z
 k 
 k  X ( z )
 k =0

 k =0

– Transfer function
M
−k 
b
z
 k 
Y ( z )  k =0
H ( z) =
= N
X ( z) 
−k 
a
z
 k 
 k =0

243
Z-TRANSFORM: LTI SYSTEM
• Example
– Find the transfer function of the system described by the following
difference equation
1
y (n) − 2 y (n − 1) + 2 y (n − 2) = x(n) + x(n − 1)
2
244
Z-TRANSFORM: STABILITY
• Stability
z
H ( z) =
z−a
h( n) = a n u ( n)
– A LTI system is BIBO stable is all the poles are within the unit
circle (|a| < 1)
– A LTI system is unstable is at least one pole is on or outside of the
unit circult ( | a | 1 )