Post on 16-Oct-2021
Department of Electrical EngineeringUniversity 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 EngineeringUniversity of Arkansas
ELEG 3124 SYSTEMS AND SIGNALS
Ch. 1 Continuous-Time Signals
Dr. Jingxian Wu
wuj@uark.edu
OUTLINE
4
• Introduction: what are signals and systems?
• Signals
• Classifications
• Basic Signal Operations
• Elementary Signals
INTRODUCTION
• Examples of signals and systems (Electrical Systems)
– Voltage divider
• Input signal: x = 5V
• Output signal: y = Vout
• The system output is a fraction of the input (𝑦 =𝑅2
𝑅1+𝑅2𝑥)
– Multimeter
• Input: the voltage across the battery
• Output: the voltage reading on the LCD display
• The system measures the voltage across two points
– Radio or cell phone
• Input: electromagnetic signals
• Output: audio signals
• The system receives electromagnetic signals and convert them to
audio signal
Voltage divider
multimeter
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
– 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
EEG signal collection
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
9
• Introduction: what are signals and systems?
• Signals
• Classifications
• Basic Signal Operations
• Elementary Signals
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
– Signals changes with respect to time
10
audio signal
SIGNALS AND CLASSIFICATIONS
• Mathematical representation of signal:
– Signals can be represented as a function of time t
– Support of signal:
– E.g.
– E.g.
• and are two different signals!
– The mathematical representation of signal contains two components:
• The expression:
• The support:
– The support can be skipped if
– E.g.
)2sin()(1 tts =
),(ts21 ttt
21 ttt
+− t
)2sin()(2 tts = t0
)(1 ts )(2 ts
)(ts
21 ttt
+− t
)2sin()(1 tts =
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
13
• Introduction: what are signals and systems?
• Signals
• Classifications
• Basic Signal Operations
• Elementary Signals
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
• E.g. sinusoidal signal
• E.g. voice signal
• E.g. Rectangular pulse function
)4sin()( tts =
=otherwise,0
10,)(p
tAt
0 1 t
A
)(p t
14
Rectangular pulse function
• Discrete-time signal
– If the time t can only take discrete values, such as,
skTt = ,2,1,0 =k
then the signal is a discrete-time signal
– E.g. the monthly average precipitation at Fayetteville, AR (weather.com)
)()( skTsts =
– What is the value of s(t) at ?
• Discrete-time signals are undefined at !!!
• Usually represented as s(k)
ss kTtTk − )1(
month 1 =sT
skTt
12 , 2, 1, =k
SIGNALS: CONTINUOUS-TIME V.S. DISCRETE-TIME15
Monthly average precipitation
• 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
• Discrete-time, continuous-amplitude
– Example: samples of analog signal,
average monthly temperature
SIGNALS: ANALOG V.S. DIGITAL
10
2
3
0
21
10
23
0
21
16
Different types of signals
• Even v.s. odd
– x(t) is an even signal if:
• E.g.
– x(t) is an odd signal if:
• E.g.
– Some signals are neither even, nor odd
• E.g.
– Any signal can be decomposed as the sum of an even signal and odd
signal
• proof
SIGNALS: EVEN V.S. ODD
tetx =)(
)2cos()( ttx =
)2sin()( ttx =
0),2cos()( = tttx
)()( txtx −=
)()( txtx −=−
even odd
)()()( tytyty oe +=
17
SIGNALS: EVEN V.S. ODD
• Example
– Find the even and odd decomposition of the following signal
tetx =)(
• Example
– Find the even and odd decomposition of the following signal
SIGNALS: EVEN V.S. ODD
19
=otherwise0
0),4sin(2)(
tttx
• Periodic signal v.s. aperiodic signal
– An analog signal is periodic if
• There is a positive real value T such that
• It is defined for all possible values of t, (why?)
– Fundamental period : the smallest positive integer that satisfies
• E.g.
– So is a period of s(t), but it is not the fundamental period of
s(t)
SIGNALS: PERIODIC V.S. APERIODIC
)()( nTtsts +=
− t
0T
)()( 0nTtsts +=
01 2TT =
)()2()( 01 tsnTtsnTts =+=+
1T
20
0T
• Example
– Find the period of
– Amplitude: A
– Angular frequency:
– Initial phase:
– Period:
– Linear frequency:
)cos()( 0 += tAts − t
0
=0T
=0f
SIGNALS: PERIODIC V.S. APERIODIC
21
SIGNALS: PERIODIC V.S. APERIODIC
• Complex exponential signal
– Euler formula:
– Complex exponential signal
)sin()cos( xjxe jx +=
)sin()cos( 000 tjtetj
+=
– Complex exponential signal is periodic with period0
0
2
=
T
• Proof:
Complex exponential signal has same period as sinusoidal signal!
22
• 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.
• The period of the sum signal is
SIGNALS: PERIODIC V.S. APERIODIC
• The sum of two periodic signals
– x(t) has a period
– y(t) has a period
– Define z(t) = a x(t) + b y(t)
– Is z(t) periodic?
k
l
T
T=
2
1
2T
)()()( TtbyTtaxTtz +++=+
• In order to have x(t)=x(t+T), T must satisfy
• In order to have y(t)=y(t+T), T must satisfy
• Therefore, if
1kTT =
2lTT =
21 lTkTT ==)()()()()()( 21 tztbytaxlTtbykTtaxTtz =+=+++=+
1T
23
21 lTkTT ==
• Example
)3
sin()( ttx
= )9
2exp()( tjty
= )
9
2exp()( tjtz =
– Find the period of
– Is periodic? If periodic, what is the period?
– Is periodic? If periodic, what is the period?
– Is periodic? If periodic, what is the period?
)(),(),( tztytx
)(3)(2 tytx −
)()( tztx +
• Aperiodic signal: any signal that is not periodic.
)()( tzty
SIGNALS: PERIODIC V.S. APERIODIC
24
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:
– Signal power: the power of signal measured at R = 1 Ohm: )()( 2 txtp =
],[ ttt nn + )(tp
tnt
)( ntp
t
– Signal energy at:
ttpE nn )(
– Total energy
=→
n
nt
ttpE )(lim0
+
−= dttp )(
+
−= dttxE
2)(
– Review: integration over a signal gives the area under the signal.
Rtxtp /)()( 2=
25
Instantaneous power
SINGALS: ENERGY V.S. POWER
• Energy of signal x(t) over
−= dttxE
2)(
• Average power of signal x(t)
−→=
T
TTdttx
TP
2)(
2
1lim
– If then x(t) is called an energy signal.,0 E
],[ +−t
– If then x(t) is called a power signal.,0 P
• A signal can be an energy signal, or a power signal, or neither, but not both.
26
SINGALS: ENERGY V.S. POWER
• Example 1: )exp()( tAtx −=
• Example 2:
dttxT
PT
=0
2)(
1
0t
• All periodic signals are power signal with average power:
)sin()( 0 += tAtx
• Example 3: tjejtx )1()( += 100 t
27
OUTLINE
28
• Introduction: what are signals and systems?
• Signals
• Classifications
• Basic Signal Operations
• Elementary Signals
OPERATIONS: SHIFTING
• Shifting operation
– : shift the signal x(t) to the right by )( 0ttx − 0t
– Why right?
Ax =)0( )()( 0ttxty −= Axttxty ==−= )0()()( 000
)()0( 0tyx =
29
Shifting to the right by two units
OPERATIONS: SHIFTING
• Example
o.w.
32
20
01
0
3
1
1
)(
−
+−
+
=t
t
t
t
t
tx
– Find )3( +tx
30
OPERATIONS: REFLECTION
• Reflection operation
– is obtained by reflecting x(t) w.r.t. the y-axis (t = 0))( tx −
31
-2 -1 1 2 3
-1
1
2
t
x(t)
-3 -2 -1 1
-1
1
2
t
x(-t)
Reflection
OPERATIONS: REFLECTION
• Example:
−+
=
o.w.
20
01
0
1
1
)( t
tt
tx
– Find x(3-t)
• The operations are always performed w.r.t. the time variable t directly!
32
OPERATIONS: TIME-SCALING
• Time-scaling operation
– is obtained by scaling the signal x(t) in time.
• , signal shrinks in time domain
• , signal expands in time domain
1a
1a
)(atx
33
-1 1
1
2
t
x(t)
-1.5 -1 -0.5 0.5 1 1.5
1
2
t
x(2t)
-2 -1 1 2
1
2
t
x(t/2)
Time scaling
OPERATIONS: TIME-SCALING
• Example:
o.w.
32
20
01
0
3
1
1
)(
−
+−
+
=t
t
t
t
t
tx
)( batx + 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)!
)63( −tx– Find
34
OUTLINE
35
• Signals
• Classifications
• Basic Signal Operations
• Elementary Signals
ELEMENTARY SIGNALS: UNIT STEP FUNCTION
• Unit step function
0
0
,0
,1)(
=t
ttu
−
=
otherwise0,22
,1
)(t
tp
• Example: rectangular pulse
Express as a function of u(t) )(tp
36
1
1
t
u(t)
t
u(t)
à /2
1/ Ã
- Ã /2
Unit step function
Rectangular pulse
ELEMENTARY SIGNALS: RAMP FUNCTION
• The Ramp function
)()( tuttr =
t
)(tr
0
– The Ramp function is obtained by integrating the unit step function u(t)
= −dttu
t
)(
37
Unit ramp function
ELEMENTARY SIGNALS: UNIT IMPULSE FUNCTION
• Unit impulse function (Dirac delta function)
=
=
=
− 0,0
0,1)(
0,0)(
)0(
t
tdtt
tt
t
)(t
– delta function can be viewed as the limit of the rectangular pulse
)(lim)(0
tpt Δ→
=
– Relationship between and u(t)
dt
tdut
)()( =
0 t
)(t
)()( tudttt
= −
38
t
u(t)
à /2
1/ Ã
- Ã /2
Unit impulse function
Rectangular pulse
ELEMENTARY SIGNALS: UNIT IMPULSE FUNCTION
• Sampling property
+
−=− )()()( 00 txdttttx
)()()()( 000 tttxtttx −=−
• Shifting property
– Proof:
39
ELEMENTARY SIGNALS: UNIT IMPULSE FUNCTION
• Scaling property
+=+
a
bt
abat
||
1)(
– Proof:
40
ELEMENTARY SIGNALS: UNIT IMPULSE FUNCTION
• Examples
=−+− dtttt )3()(4
2
2
=−+− dtttt )3()(1
2
2
=−−− dttt )42()1exp(3
2
41
ELEMENTARY SIGNALS: SAMPLING FUNCTION
• Sampling function
x
xxSa
sin)( =
– Sampling function can be viewed as scaled version of sinc(x)
)(sin
)(Sinc xsax
xx
==
42
t
sa(t)
-4 -3 -2 -1 1 2 3 4
1
t
sinc(t)
Sampling function
Sinc function
ELEMENTARY SIGNALS: COMPLEX EXPONENTIAL
• Complex exponential
– Is it periodic?
• Example:
– Use Matlab to plot the real part of
tjretx
)( 0)(+
=
)]4()2([)( )21( −−+= +− tutuetx tj
43
SUMMARY
• Signals and Classifications
– Mathematical representation
– Continuous-time v.s. discrete-time
– Analog v.s. digital
– Odd v.s. even
– Periodic v.s. aperiodic
– Power v.s. energy
• Basic Signal Operations
– Time shifting
– reflection
– Time scaling
• Elementary Signals
– Unit step, unit impulse, ramp, sampling function, complex exponential
44
),(ts21 ttt
Department of Electrical EngineeringUniversity 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.
continuous-time
System
)(tx )(tyDiscrete-time
System
)(nx )(ny
Continuous-time system discrete-time system
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
48
49
CLASSIFICATIONS: LINEAR AND NON-LINEAR
• Linear system
– Let be the response of a system to an input
– Let be the response of a system to an input
– The system is linear if the superposition principle is satisfied:
• 1. the response to is
• 2. the response to is
)(1 ty )(1 tx
)(2 ty )(2 tx
)()( 21 txtx + )()( 21 tyty +
)()( 21 txtx +
)(1 ty
Linear
System
)()( 21 tyty +
)(1 tx
• Non-linear system
– If the superposition principle is not satisfied, then the system is a
non-linear system
Linear system
50
CLASSIFICATIONS: LINEAR AND NON-LINEAR
• Example: check if the following systems are linear
– System 1:
– System 3: inductor. Input: i(t), output v(t)
)](exp[)( txty =
– System 2: charge a capacitor. Input: i(t), output v(t)
−=
t
diC
tv )(1
)(
dt
tdiLtv
)()( =
CLASSIFICATIONS: LINEAR AND NON-LINEAR
• Example
– System 4:
– System 5:
– System 6:
51
|)(|)( txty =
)()( 2 txty =
CLASSIFICATIONS: LINEAR V.S. NON-LINEAR
• Example:
– Amplitude Modulation:
• Is it linear?
52
Amplitude modulation
53CLASSIFICATIONS: 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
Time-invariant
System
)( 0ttx −)(ty Time-invariant
System
)( 0tty −)(tx
• Examples
– y(t) = cos(x(t))
– =t
dvvxty0
)()(
Time-invariant system
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
)()( tRxty =
– 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)
=t
dxC
ty0
)(1
)(
– 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
i
ii Ttxaty0
)()(
– )())(2sin()( 2 txtxty +=
56
CLASSIFICATIONS: CAUSAL V.S. NON-CAUSAL
• Causal system
– A system is causal if the output depends only on values of input
for
• The output depends on only input from the past and present
– Example
)( 0ty
0tt
)()( atxty +=
• Non-causal system
– A system is non-causal if the output depends on the input from the future (prediction).
– Examples:
0a
– The output value at t depends on the input value at t + a (from future)
=t
dxC
ty0
)(1
)(
– All practical systems are causal.
−=2/
2/)(
1)(
T
Tdx
Tty
57
CLASSIFICATION: INVERTIBILITY
• Invertible
– A system is invertible if
• by observing the output, we can determine its input.
SystemInverse
System
)(tx )(ty )(tx
– Question: for a system, if two different inputs result in the same
output, is this system invertible?• Example
)(2)( txty =
)(cos)( txty =
– If two different inputs result in the same output, the system is non-
invertible
invertible system
58
CLASSIFICATION: STABILITY
• Bounded signal
– Definition: a signal x(t) is said to be bounded if
Btx |)(|
• 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.
21 |y(t)| |)(| BBtx
• Example: determine if the systems are BIBO stable
)(exp)( txty =
−=
t
dxty )()(
t
t
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
)(txi
System)(tyi
– Linear
Input:
Output: =
=+++=N
i
iiNN tyatyatyatyaty1
2211 )()()()()(
=
=+++=N
i
iiNN txatxatxatxatx1
2211 )()()()()(
– Time-invariant
Input: )()( 0ttxtx i −=
Output: )()( 0ttyty i −=
system
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)
• 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,
)()( ttx =System
)()( thty =
LTI system
62
LTI: CONVOLUTION
• Derivation
– Any signal can be approximated by the sum of a sequence of delta
functions
+
−=→
+
−−=−=
n
ntnxdtxtx )()(lim)()()(0
+
−=→
+
−=
n
nzdz )(lim)(0
t
x(t)
integration
63
LTI: CONVOLUTION
• Derivation
– Any signal can be approximated by the sum of a sequence of delta
functions
+
−=→
+
−−=−=
n
ntnxdtxtx )()(lim)()()(0
)(tSystem
)(th
– Time invariant
)( −ntSystem
)( −nth
– Linear
−+
−=
)()( ntnxn
System
−+
−=
)()( nthnxn
LTI system
64
LTI: CONVOLUTION
• Convolution
)(txSystem
+
−−= dthxty )()()(
– Definition: the convolution of two signals x(t) and h(t) is defined as
+
−−= dthxty )()()(
– The operation of convolution is usually denoted with the symbol
+
−−== dthxthtxty )()()()()(
)(txh(t)
)()( thtx
For LTI system, if we know input x(t) and impulse response h(t),
Then the output is )()( thtx
LTI system
LTI system
65
LTI: CONVOLUTION
• Examples
)()( ttx
)()( tutx
)()( 0tttx −
66
LTI: CONVOLUTION
• Examples
)()exp( tubt−)()exp( tuat−
?)( =ty
LTI system
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.
67
−=
t
diC
tv )(1
)(
68
LTI: CONVOLUTION PROPERTIES
• Commutativity
)()()()( txtytytx =
– Proof:
+
−−= dtyxtytx )()()()(
)(txh(t)
)()( thtx )(thx(t)
)()( txth ➔
commutativity
69
LTI: CONVOLUTION PROPERTIES
• Associativity
)()()()()()()()()( 212121 ththtxththtxththtx ==
– proof
)(tx)()( 21 thth
)(ty)(tx)(1 th )(1 th )(2 th
)(ty➔
)(1 ty
)(th
Associativity
70
LTI: CONVOLUTION PROPERTIES
• Distributivity
)()()()()()()( 1121 thtxthtxththtx +=+
– proof
)(tx
)(1 th
)(2 th
)(ty
+)(tx
)()( 21 thth +)(ty
➔
Distributivity
71
LTI: CONVOLUTION PROPERTIES
• Example
)(tx
)(1 th
)(3 th
)(ty
+
)(2 th
)(4 th
)()2exp()(1 tutth −= )()exp(2)(2 tutth −=)()3exp()(3 tutth −= )(4)(4 tth =
?)( =th
72
LTI: GRAPHICAL CONVOLUTION
• Graphical interpretation of convolution
– 1. Reflection )()( −= hg )())(()( 000 −=−−=− ththtg
– 3. Multiplication )()( 0 −thx
+
−−= dthxty )()()(
– 4. Integration +
−−= dthxty )()()( 00
)())(()( 000 −=−−=− ththtg
)(x )(h
t
x(t)
t
x(t)
t
x(t)
t
h(-t)
t
73
LTI: GRAPHICAL CONVOLUTION
• Example
)](2[)](2[)( 22 atpatpaty aa −=
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
)()( tKxty =
0for t
– The impulse response of Memoryless LTI system is
• Causal LTI system
– Review: output depends on only current input and past input.
– The impulse response of causal LTI system must satisfy:
)()( tKth =
0)( =th
– Why?
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
h(t) g(t))(tx )(ty )(tx
)()()()( txtgthtx =
– For invertible LTI systems with IR (impulse response) , there exists inverse system such that
)(th)(tg
)()()( tthtg =
– Example: find the inverse system of LTI system )()( 0ttth −=
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
+
−dtth )(
• Proof:
78
LTI PROPERTIES
• Examples
– Determine: causal or non-causal, memory or memoryless, stable or unstable
– 1.
– 2.
– 3.
)1()()3exp()()2exp()(1 −+−+−= ttuttuttth
)()2exp(3)(2 tutth −=
)5(5)(3 += tth
79
OUTLINE
• Classifications of continuous-time system
• Linear time-invariant system (LTI)
• Properties of LTI system
• System described by differential equations
• LTI system can be represented by differential equations
– Initial conditions:
– Notation: n-th derivative:
DIFFERENTIAL EQUATIONS
80
)()(')()()(')( )(
10
)(
10 txbtxbtxbtyatyatya M
M
N
N +++=+++
n
nn
dt
tydty
)()()( =
0
)(
=t
k
k
dt
tyd1,,0 −= Nk
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 initial current in the inductor.
The output of the system is the current across the inductor.
• Represent the system with a differential equation.
• Find the output of the system with and
oI
0=oI 1=oI
DIFFERENTIAL EQUATION
• Zero-state response
– The output of the system when the initial conditions are zero
– Denoted as
• Zero-input response
– The output of the system when the input is zero
– Denoted as
• The actual output of the system
82
)()(')()()(')( )(
10
)(
10 txbtxbtxbtyatyatya M
M
N
N +++=+++
0
)(
=t
k
k
dt
tyd1,,0 −= Nk
)(tyzs
)(tyzi
)()()( tytyty zizs +=
DIFFERENTIAL EQUATION
• Example
– Find the zero-state output and zero-input response of the RL circuit in the
previous example.
83
Department of Electrical EngineeringUniversity 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
+
−=→
+
−−=−=
n
ntnxdtxtx )()(lim)()()(0
t
x(t)
Illustration of integration
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.
– 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
87
tfjtjee 00 2
=
2
00
=f
Department of Engineering Science
Sonoma State University
88
INTRODUCTION: REVIEW
• Complex exponential signal
)2sin()2cos(2 ftjfte ftj +=
– 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
f = 3 Hz
Sinusoidal at different frequencies
89
INTRODUCTION: ORTHONORMAL SIGNAL SET
• Definition: orthogonal signal set
– A set of signals, , are said to be orthogonal over an
interval (a, b) if ),(),(),( 210 ttt
kl
klCdttt
b
akl
=
= ,0
,)()( *
• Example:
– the signal set: are
orthogonal over the interval , where
tjk
k et 0)(
= ,2,1,0 =k],0[ 0T
0
0
2
T
=
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 can be decomposed as the sum of a set of complex exponential signals as
tjn
n
nectx 0)(
+
−=
=
• , Fourier series coefficients,2,1,0, =ncn
−=
0
0)(1
0T
tjn
n dtetxT
c
• derivation of :nc
0T
00
2
T
=
For a periodic signal, it can be either represented as s(t), or represented as
92
FOURIER SERIES
• Fourier series
tjn
n
nectx 0)(
+
−=
=
– 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:
,2,1,0, =ncn
nc
0n
• The periods of the n-th complex exponential function:
– The values of coefficients, , depend on x(t)
• Different x(t) will result in different
• There is a one-to-one relationship between x(t) and
n
TTn
0=
nc
)(ts ],,,,,[ 210,12 ccccc −−➔
nc
93
FOURIER SERIES
• Example
10
01
,
,)(
−
−
=t
t
K
Ktx
-3 -2 -1 1 2
t
x(t)
Rectangle pulses
FOURIER SERIES
• Amplitude and phase
– The Fourier series coefficients are usually complex numbers
– Amplitude line spectrum: amplitude as a function of
– Phase line spectrum: phase as a function of
94
nnn jbac +=
22
nnn bac +=
n
nn
a
btana=
0n
0n
nj
n ec
=
95
FOURIER SERIES: FREQUENCY DOMAIN
• Signal represented in frequency domain: line spectrum
– Each has its own frequency
– The signal is decomposed in frequency domain.
– is called the harmonic of signal s(t) at frequency
– Each signal has many frequency components.
• The power of the harmonics at different frequencies determines
how fast the signal can change.
nc
nc
amplitude phase
0n
0n
FOURIER SERIES: FREQUENCY DOMAIN
• Example: Piano Note
96
E5: 659.25 Hz
E6: 1318.51 Hz
B6: 1975.53 Hz
E7: 2637.02 Hz
E5
E6B6
E7
All graphs in this page are created by using the open-source software Audacity.
piano notes
One piano note
spectrum
97
FOURIER SERIES
• Example
– Find the Fourier series of )exp()( 0tjts =
FOURIER SERIES
• Example
– Find the Fourier series of
98
)cos()( 0 ++= tABts
)100sin(1)( tty +=
Time domain Amplitude spectrum Phase spectrum
99
FOURIER SERIES
• Example
– Find the Fourier series of
−
−−
=
2/2/,0
2/2/,
2/2/,0
)(
Tt
tK
tT
ts
5,1 == T
10,1 == T
15,1 == T
)(csinT
n
T
Kcn
=
t
x(t)
Time domain
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
Tdttx |)(|
– 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
0
0
2
=
T
– The Fourier series of the superposition of two signals is
+
−=
+=+
n
tjn
nn ekktyktxk 0)()()( 2121
+
−=
=
n
tjn
netx 0)(
)()()( 2121 nn kktyktxk +=+
– If
ntx =)(nty =)(
• then
+
−=
=
n
tjn
nety 0)(
103
PROPERTIES: EFFECTS OF SYMMETRY
• Symmetric signals
– A signal is even symmetry if:
– A signal is odd symmetry if:
– The existence of symmetries simplifies the computation of Fourier series
coefficients.
)()( txtx −=
)()( txtx −−=
-4 -3 -2 -1 1 2 3 4
t
x(t)
-5 -4 -3 -2 -1 1 2 3 4 5
t
x(t)
Even symmetric Odd symmetric
104
PROPERTIES: EFFECTS OF SYMMETRY
• Fourier series of even symmetry signals
– If a signal is even symmetry, then
( )+
−=
=n
n tnatx 0cos)( ( ) =2/
00
0
0
cos)(2 T
n dttntxT
a
• Fourier series of odd symmetry signals
– If a signal is odd symmetry, then
( )+
=
=1
0sin)(n
n tnbtx ( ) =2/
00
0
0
sin)(2 T
n dttntxT
b
105
PROPERTIES: EFFECTS OF SYMMETRY
• Example
−
−=
TtTAtT
A
TttT
AA
tx
2/,34
2/0,4
)(t
x(t)
Graph of x(t)
106
PROPERTIES: SHIFT IN TIME
• Shift in time
– If has Fourier series , then has Fourier series )(tx nc )( 0ttx −
00tjn
nec−
)(tx nc➔if , then )( 0ttx − ➔00tjn
nec−
– Proof:
107
PROPERTIES: PARSEVAL’S THEOREM
• Review: power of periodic signal
=T
dttxT
P0
2|)(|1
• Parseval’s theorem
+
−=
=m
m
T
dttxT
2
0
2 |||)(|1
)(txif ➔ n
then
– 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 )sin()( 0tAtx =
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
tjetx =)()(th
)(ty
)()()()()( txththtxty ==
dtxh+
−−= )()(
djhtj +
−−= )exp()()exp(
djhH +
−−= )exp()()(
• Transfer function
– For LTI system with complex exponential input, the output is
)exp()()( tjHty =
– It tells us the system response at different frequencies
PERIODIC INPUT
• Example:
– For a system with impulse response
find the transfer function
111
)()( 0ttth −=
112
PERIODIC INPUT:
• Example
– Find the transfer function of the system shown in figure.
RL circuit
PERIODIC INPUTS
• Example
– Find the transfer function of the system shown in figure
113
RC circuit
114
PERIODIC INPUTS: TRANSFER FUNCTION
• Transfer function
– For system described by differential equations
= =
=n
i
m
i
i
i
i
i txqtyp0 0
)()( )()(
=
=
=n
i
i
i
m
i
i
i
jp
jq
H
0
0
)(
)(
)(
115
PERIODIC INPUTS
• LTI system with periodic inputs
– Periodic inputs:
tjne 0
)(th)( 0
0
nHetjn
+
−=
=n
n tjnctx )exp()( 0
linear: tjn
n
nec 0+
−=
)(th
)( 00
+
−=
nHectjn
n
n
)(tx)(th
)( 00
+
−=
nHectjn
n
n
For system with periodic inputs, the output is the weighted
sum of the transfer function.
T
20 =
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).
−
=T
tjn
n dtetxT 0
0)(1
• 2. Find the transfer function of LTI system
Tf
22 00 ==
period of x(t)
• 3. The output of the system is
)()( 00 =
+
−=
nHectytjn
n
n
)(H
117
PERIODIC INPUTS
• Example
– Find the response of the system when the input is
)2cos(2)cos(4)( tttx −=
RL Circuit
118
PERIODIC INPUTS
• Example
– Find the response of the system when the input is shown in figure.
-3 -2 -1 1 2
t
x(t)
RC circuitSquare pulses
PERIODIC INPUTS: GIBBS PHENOMENON
• The Gibbs Phenomenon
– Most Fourier series has infinite number of elements→ unlimited
bandwidth
• What if we truncate the infinite series to finite number of elements?
– The truncated signal, , is an approximation of the original
signal x(t)
119
tjn
n
nectx 0)(
+
−=
=
tjnN
Nn
nN ectx 0)(
+
−=
=
)(txN
PERIODIC INPUTS: GIBBS PHENOMENON
120
=
even. 0,
odd, ,12
n
nnj
K
cn tjn
N
Nn
nN ectx 0)(
+
−=
=
)(3 tx )(5 tx )(19 tx
-3 -2 -1 1 2
t
x(t)
Square pulses
FOURIER SERIES
• Analogy: Optical Prism
– Each color is an Electromagnetic wave with a different frequency
121
Optical prism
Department of Electrical EngineeringUniversity 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
tjnctxn
n 0exp)( +
−=
=
• each harmonic contains a unique frequency: n/T
=T
n dttjntxT
c0
0exp)(1
How about aperiodic signals ?
( )=T• time domain ➔ frequency domain
t
x(t)
Time domain Frequency domain
125
INTRODUCTION: TRANSFER FUNCTION
• System transfer function
• System with periodic inputs
tje
)(th)( He tj
+
−= dttjthH exp)()(
tjne 0
)(th)( 0
0 nHe
tjn
tjn
n
nec 0+
−= )(th)( 0
0 nHec
tjn
n
n+
−=
)(tx)(th
)( 00
nHectjn
n
n+
−=
126
OUTLINE
• Introduction
• Fourier Transform
• Properties of Fourier Transform
• Applications of Fourier Transform
127
FOURIER TRANSFORM
• Inverse Fourier Transform
• Fourier Transform
– given x(t), we can find its Fourier transform
– given , 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)
+
−
−= dtetxX tj )()(
+
−=
deXtx tj)(2
1)(
)(X
)(X
➔)(tx )(X
128
FOURIER TRANSFORM
• Example
– Find the Fourier transform of )/()( trecttx =
t
x(t)
t
x(t)
129
FOURIER TRANSFORM
• Example
– Find the Fourier transform of |)|exp()( tatx −= 0a
130
FOURIER TRANSFORM
• Example
– Find the Fourier transform of 0a)()exp()( tuattx −=
131
FOURIER TRANSFORM
• Example
– Find the Fourier transform of )()( attx −=
132
FOURIER TRANSFORM: TABLE
133
FOURIER TRANSFORM
+
−dttx |)(|
)()exp()( tuttx =
• Example
–
• 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
• 2. x(t) is well behaved
– The signal has finite number of discontinuities, minima,
and maxima within any finite interval of time.
134
OUTLINE
• Introduction
• Fourier Transform
• Properties of Fourier Transform
• Applications of Fourier Transform
135
PROPERTIES: LINEARITY
• Linearity
– If
– then
)()( 11 Xtx )()( 22 Xtx
)()()()( 2121 bXaXtbxtax ++
• Example
– Find the Fourier transform of )(4)()2exp(3)/(2)( ttuttrecttx +−+=
136
PROPERTY: TIME-SHIFT
• Time shift
– If
– Then)()( Xtx
]exp[)()( 00 tjXttx −−
• Review: complex number
jbacjcecc j +=+== )sin(||)cos(||||
cos|| ca = sin|| cb =
22|| bac += )/tan( aba=
phase shift
time shift in time domain ➔ frequency shift in frequency domain
– Phase shift of a complex number c by : 0 )(exp||)exp( 00 += jcjc
137
PROPERTY: TIME SHIFT
• Example:
– Find the Fourier transform of 2)( −= trecttx
138
PROPERTY: TIME SCALING
• Time scaling
– If
– Then
• Example
– Let , find the Fourier transform of
)()( Xtx
aX
aatx
||
1)(
( ) 2/1)( −= rectX )42( +− tx
139
PROPERTY: SYMMETRY
• Symmetry
– If , and is a real-valued time signal
– Then
)()( Xtx )(tx
)()( * XX =−
140
PROPERTY: DIFFERENTIATION
• Differentiation
– If
– Then
)()( Xtx
)()(
Xjdt
tdx ( ) )(
)( Xj
dt
txd n
n
n
• Example
– Let , find the Fourier transform of ( ) 2/1)( −= rectXdt
tdx )(
141
PROPERTY: DIFFERENTIATION
• Example
– Find the Fourier transform of
(Hint: )
)sgn()( ttx =
)()sgn(2
1tt
dt
d=
142
PROPERTY: CONVOLUTION
• Convolution
– If ,
– Then
)()( Xtx )()( Hth
)()()()( HXthtx
)(tx
)(th)()( thtx )(X
)(H)()( HX
time domain frequency domain
143
PROPERTY: CONVOLUTION
• Example
– An LTI system has impulse response
If the input is
Find the output
( ) )(exp)( tuatth −=
( ) )()exp()()(exp)()( tuctactubtbatx −−+−−=
)0,0,0( cba
144
PROPERTY: MULTIPLICATION
• Multiplication
– If ,
– Then
)()( Xtx )()( Mtm
)()(2
1)()(
MXtmtx
145
PROPERTY: DUALITY
• Duality
– If
– Then
)()( Gtg
)(2)( − gtG
146
PROPERTY: DUALITY
• Example
– Find the Fourier transform of
(recall: )
=
2)(
tSath
2sinc )/(rect t
147
PROPERTY: DUALITY
• Example
– Find the Fourier transform of 1)( =tx
tjetx 0)(
=– Find the Fourier transform of
148
PROPERTY: SUMMARY
149
PROPERTY: EXAMPLES
• Examples
– 1. Find the Fourier transform of )cos()( 0ttx =
– 2. Find the Fourier transform of )()( tutx =
1)sgn(2
1)( += ttu
jt
2)sgn(
150
PROPERTY: EXAMPLES
• Examples
– 3. A LTI system with impulse response
Find the output when input is )(exp)( tuatth −=
)()( tutx =
– 4. If , find the Fourier transform of
(Hint: )
)()( Xtx −
t
dx )(
)()()( tutxdxt
= −
151
PROPERTY: EXAMPLES
• Example
– 5. (Modulation) If ,
Find the Fourier transform of )()( Xtx )cos()( 0ttm =
)()( tmtx
– 6. If , find x(t)
ja
X+
=1
)(
152
PROPERTY: DIFFERENTIATION IN FREQ. DOMAIN
• Differentiation in frequency domain
– If:
– Then:
)()( Xtx
n
nn
d
Xdtxjt
)()()( =−
PROPERTY: DIFFERENTIATION IN FREQ. DOMAIN153
),()exp( tuatt − 0a
• Example
– Find the Fourier transform of
154
PROPERTY: FREQUENCY SHIFT
• Frequency shift
– If:
– Then:
)()( Xtx
)()exp()( 00 − Xtjtx
• Example
– If , find the Fourier transform ( ) 2/1)( −= rectX )2exp()( tjtx −
155
PROPERTY: PARSAVAL’S THEOREM
• Review: signal energy
+
−= dttxE 2|)(|
• Parsaval’s theorem
+
−
+
−=
dXdttx 22 |)(|
2
1|)(|
156
PROPERTY: PARSAVAL’S THEOREM
• Example:
– Find the energy of the signal )()2exp()( tuttx −=
157
PROPERTY: PERIODIC SIGNAL
• Fourier transform of periodic signal
– Periodic signal can be written as Fourier series
tjnctxn
n 0exp)( +
−=
=
– Perform Fourier transform on both sides
)(2)( 0 ncXn
n −= +
−=
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.
)(tx
)(th)()( thtx )(X
)(H)()( HX
time domain frequency domain
filter filter
160
APPLICATIONS: FILTERING
• Classifications of filters
Low pass filter
Band pass filterBand stop (Notch) filter
PassbandStop
band PassbandStop
band
High pass filter
Passband Stop
band
Stop
band
Stop
bandPassband Passband
161
APPLICATION: FILTERING
• A filtering example
– A demo of a notch filter
)(X
)(H
)()( HX
Corrupted sound Filter 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.
+
−=
−=n
nTttp )()(
sampling period
)()()( tptxtxs =
)(tx
Sampled signal
164
APPLICATION: SAMPLING THEOREM
• Sampling theorem: frequency domain
– Fourier transform of the impulse train
• impulse train is periodic
+
−=
+
−=
=−=n
tjn
sn
sse
TnTttp
11
)()(
• Find Fourier transform on both sides
+
−=
−=n
s
s
nT
P )(2
)(
• Time domain multiplication ➔ Frequency domain convolution
)()(2
1)()(
PXtptx
+
−=
−n
s
s
nXT
tptx )(1
)()(
s
sT
2=
Fourier series
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.
Bs 2
Bs 2
Bs 2=
Bs 2
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
modulationInformation
bearing signal
Carrier wave
Modulated signal
• 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)
)2cos()()( tftmAts cc =
– A direct product between message signal and carrier signal
Mixer
Local
Oscillator
)(tm
)2cos( tfA cc
)(ts
Amplitude modulation
169
APPLICATION: AMPLITUDE MODULATION
• Amplitude Modulation (AM)
)()(2
)( ccc ffMffM
AfS ++−=
Amplitude modulation
Department of Electrical EngineeringUniversity 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
0),()exp()( = atuattx )()( ttutx =
– 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
174LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM
• Bilateral Laplace transform (two-sided Laplace transform)
,)exp()()( +
−−= dtsttxsX B
– is a complex variable
– s is often called the complex frequency
– Notations:
– : a function of time t → x(t) is called the time domain signal
– a function of s → is called the s-domain signal
– S-domain is also called as the complex frequency domain
js +=
)()( sXtx B
js +=
)]([)( txLsX B =
)(tx
:)(sX B)(sX B
• Time domain v.s. S-domain
LAPLACE TRANSFORM
• Time domain v.s. s-domain
– : a function of time t → x(t) is called the time domain signal
– a function of 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
175
)(tx:)(sX B
)(sX B
176
• Example
– Find the Bilateral Laplace transform of )()exp()( tuattx −=
• 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.
LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM
177
• Example
– Find the Laplace transform of )()exp()( tuattx −−−=
LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM
178
• Example
– Find the Laplace transform of )()exp(4)()2exp(3)( tuttuttx −+−=
LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM
179LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM
• Unilateral Laplace transform (one-sided Laplace transform)
+
−−=
0)exp()()( dtsttxsX
– :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.
−0
+
−−=
0)exp()()( dtsttxsX
180
• Example: find the unilateral Laplace transform of the following
signals.
– 1. Atx =)(
– 2. )()( ttx =
LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM
181
• Example
– 3. )2exp()( tjtx =
– 4.
)2sin()( ttx =– 5.
)2cos()( ttx =
LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM
LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM182
183
OUTLINE
• Introduction
• Laplace Transform
• Properties of Laplace Transform
• Inverse Lapalace Transform
• Applications of Fourier Transform
184
PROPERTIES: LINEARITY
• Linearity
– If
– Then
The ROC is the intersection between the two original signals
)()( 11 sXtx )()( 22 sXtx
)()()()( 2121 sbXsaXtbxtax ++
• Example
– Find the Laplace transfrom of )()exp( tubtBA −+
185
PROPERTIES: TIME SHIFTING
• Time shifting
– If and
– Then
The ROC remain unchanged
)()( sXtx
)exp()()()( 000 stsXttuttx −−−
00 t
186
PROPERTIES: SHIFTING IN THE s DOMAIN
• Shifting in the s domain
– If
– Then )()exp()()( 00 ssXtstxty −=
• Example
– Find the Laplace transform of )()cos()exp()( 0 tutatAtx −=
)Re(s)()( sXtx
)Re()Re( 0ss +
187
PROPERTIES: TIME SCALING
• Time scaling
– If
– Then )()( sXtx
1}Re{ as
a
sX
aatx
1)(
1}Re{ s
• Example
– Find the Laplace transform of )()( atutx =
188
PROPERTIES: DIFFERENTIATION IN TIME DOMAIN
• Differentiation in time domain
– If
– Then)()( sGtg
)0()()( −− gssG
dt
tdg
• Example
– Find the Laplace transform of ),(sin)( 2 tuttg =
)0()0()0()()( )1()2(1 −−−−−− −−−− nnnn
n
n
gsggssGsdt
tgd
0)0( =−g
)0(')0()()( 2
2
2−− −− gsgsGs
dt
tgd
189
PROPERTIES: DIFFERENTIATION IN TIME DOMAIN
• Example
– Use Laplace transform to solve the differential equation
,0)(2)('3)('' =++ tytyty 3)0( =−y 1)0(' =−y
190
PROPERTIES: DIFFERENTIATION IN S DOMAIN
• Differentiation in s domain
– If
– Then
)()( sXtx
n
nn
ds
sXdtxt
)()()( −
• Example
– Find the Laplace transform of )(tut n
191
PROPERTIES: CONVOLUTION
• Convolution
– If
– Then
The ROC of is the intersection of the ROCs of X(s) and
H(s)
)()( sXtx )()( sHth
)()()()( sHsXthtx
)()( sHsX
192
PROPERTIES: INTEGRATION IN TIME DOMAIN
• Integration in time domain
– If
– Then
)()( sXtx
)(1
)(0
sXs
dxt
• Example
– Find the Laplace transform of )()( ttutr =
193
PROPERTIES: CONVOLUTION
• Example
– Find the convolution
−
−
a
atrect
a
atrect
22
194
PROPERTIES: CONVOLUTION
• Example
– For a LTI system, the input is , and the output of the
system is )()2exp()( tuttx −=
)()3exp()2exp()exp()( tutttty −−−+−=
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
)()('3)()('3)(''2 txtxtytyty +=+−
assume the system was initially relaxed ( )0)0()0( )()( == nn xy
196
PROPERTIES: MODULATION
• Modulation
– If
– Then
)()( sXtx )()( sXtx
)()(2
1)cos()( 000 jsXjsXttx −++
)()(2
)sin()( 000 jsXjsXj
ttx −−+
197
PROPERTIES: MODULATION
• Example
– Find the Laplace transform of )()sin()exp()( 0 tutattx −=
198
PROPERTIES: INITIAL VALUE THEOREM
• Initial value theorem
– If the signal is infinitely differentiable on an interval around
then
)(tx )0( +x
)(lim)0( ssXxs →
+ =
– The behavior of x(t) for small t is determined by the behavior of X(s) for large s.
=s must be in ROC
199
PROPERTIES: INITIAL VALUE THEOREM
• Example
– The Laplace transform of x(t) is
Find the value of ))(()(
bsas
dcssX
−−
+=
)0( +x
200
PROPERTIES: FINAL VALUE THEOREM
• Final value theorem
– If
– Then: )()( sXtx
)(lim)(lim0
ssXtxst →→
• Example
– The input is applied to a system with transfer
function , find the value of
0=s must be in ROC
)()( tAutx =
cbss
csH
++=
)()(
)(lim tyt →
PROPERTIES
201
202
OUTLINE
• Introduction
• Laplace Transform
• Properties of Laplace Transform
• Inverse Lapalace Transform
• Applications of Fourier Transform
203
INVERSE LAPLACE TRANSFORM
• Inverse Laplace transform
01
1
1
01
1
1)(asasasa
bsbsbsbsX
n
n
n
n
m
mmm
++++
++++=
−
−
−
−
– 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
– Procedure of Inverse Laplace Transform
• 1. Partial fraction expansion of X(s)
• 2. Find the inverse Laplace transform through Laplace
transform table.
+
−=
j
jdsstsX
jtx
)exp()(
2
1)(
204
INVERSE LAPLACE TRANSFORM
• Review: Partial Fraction Expansion with non-repeated linear
factors
321
)(as
C
as
B
as
AsX
−+
−+
−=
1
)()( 1 assXasA
=−=
2
)()( 2 assXasB
=−=
3
)()( 3 assXasC
=−=
• Example
– Find the inverse Laplace transform of sss
ssX
43
12)(
23 −+
+=
205
INVERSE LAPLACE TRANSFORM
• Example
– Find the Inverse Laplace transform of
23
2)(
2
2
++=
ss
ssX
• 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
( ) bs
B
as
A
as
A
bsassX
−+
−+
−=
−−= 1
2
2
2 )()(
1)(
( )as
sXasA=
−= )(2
2 ( ) as
sXasds
dA
=
−= )(2
1( )
bssXbsB
=−= )(
207
INVERSE LAPLACE TRANSFORM
• High-order repeated linear factors
bs
B
as
A
as
A
as
A
bsassX
N
N
N −+
−++
−+
−=
−−=
)()()()(
1)(
2
21
( )bs
sXbsB=
−= )(
( ) as
N
kN
kN
k sXasds
d
kNA
=
−
−
−−
= )()!(
1Nk ,,1=
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.
)()()()()()()( 0
)1(
1
)(
0
)1(
1
)1(
1
)( txbtxbtxbtyatyatyaty M
M
N
N
N +++=++++ −
−
=
−
=
=+M
m
m
m
N
n
n
n
N txbtyaty0
)(1
0
)()( )()()(
– S-domain representation
)()(0
1
0
sXsbsYsasM
m
m
m
N
n
n
n
N
=
+
=
−
=
– Transfer function
−
=
=
+
==1
0
0
)(
)()(
N
n
n
n
N
M
m
m
m
sas
sb
sX
sYsH
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
6116
23)(
23
2
+++
+−=
sss
ssSH
212
APPLICATION: COMBINATIONS OF SYSTEMS
• Combination of systems
– Cascade of systems
– Parallel systems
)()()( 21 sHsHSH =
)()()( 21 sHsHSH +=
213
APPLICATION: LTI SYSTEM REPRESENTATION
• Example
– Represent the system to the cascade of subsystems.
6116
23)(
23
2
+++
+−=
sss
ssSH
214
APPLICATION: LTI SYSTEM REPRESENTATION
• Example:
– Find the transfer function of the system
LTI system
215
APPLICATION: LTI SYSTEM REPRESENTATION
• Poles and zeros
)())((
)())(()(
11
11
pspsps
zszszssH
NN
MM
−−−
−−−=
−
−
– Zeros:
– Poles:
Mzzz ,,, 21
Nppp ,,, 21
216
APPLICATION: STABILITY
• Review: BIBO Stable
– Bounded input always leads to bounded output
+
−dtth |)(|
• The positions of poles of H(s) in the s-domain
determine if a system is BIBO stable.
N
N
m ss
A
ss
A
ss
AsH
−++
−+
−=
)()(
2
2
1
1
– Simple poles: the order of the pole is 1, e.g.
– Multiple-order poles: the poles with higher order. E.g.
1s Ns
2s
217
APPLICATION: STABILITY
• Case 1: simple poles in the left half plane
kk jp −=1
0k( ) 22
1
kks +−
)()sin()exp(1
)( tuttth kk
k
k
=
))((
1
kkkk jsjs −−+−=
kk jp +=2
=+
−dtthk )(
• If all the poles of the system are on the left half plane,
then the system is stable.
Impulse response
218
APPLICATION: STABILITY
• Case 2: Simple poles on the right half plane
kk jp +=1
0k( ) 22
1
kks +− ))((
1
kkkk jsjs −−+−=
kk jp −=2
)()sin()exp(1
)( tuttth kk
k
k
=
• If at least one pole of the system is on the right half
plane, then the system is unstable.
Impulse response
219
APPLICATION: STABILITY
• Case 3: Simple poles on the imaginary axis
)()sin(1
)( tutth k
k
k
=
0=k( ) 22
1
kks +− ))((
1
kkkk jsjs −−+−=
• 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
)()sin()exp(1
)( tutttth kk
m
k
k
= 0k stable
• Case 5: multiple-order poles in the right half plane
)()sin()exp(1
)( tutttth kk
m
k
k
= 0k
0k 0k
unstable
• Case 6: multiple-order poles on the imaginary axis
)()sin(1
)( tuttth k
m
k
k
= unstable
221
APPLICATION: STABILITY
• Example:
– Check the stability of the following system.
136
23)(
2 ++
+=
ss
ssH
Department of Electrical EngineeringUniversity 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
=
4cos)(
nnx
=
4exp
2
1)(
nnx
225
SIGNAL: CLASSIFICATION
• Energy signal v.s. Power signal
– Energy:
−=
→=
N
NnN
nxE2
)(lim
– Power:
−=
→ +=
N
NnN
nxN
P2
)(12
1lim
– Energy signal: E
– Power signal: P
226
SIGNAL: CLASSIFICATION
• Periodic signal v.s. aperiodic signal
– Periodic signal
• The smallest value of N that satisfies this relation is the fundamental
periods.
– Is periodic?
)()( Nnxnx +=
– Example: )3cos( n
)cos( n
)cos( n
)4
3cos( n
)cos( n is periodic if is integer for integer k.
k2
227
SIGNAL: ELEMENTARY SIGNAL
• Unit impulse function
==
.0
,0
,0
,1)(
n
nn
• Unit step function
=
.0
,0
,1
,0)(
n
nnu
• Relation between unit impulse function and unit step function
)1()()( −−= nunun
−=
=n
k
knu )()(
228
SIGNAL: ELEMENTARY SIGNAL
• Exponential function
)exp()( nnx =
• Complex exponential function
)sin()cos()exp()( 000 njnnjnx +==
229
OUTLINE
• Discrete-time signals
• Discrete-time systems
• Z-transform
230
SYSTEM: IMPULSE RESPONSE
• Impulse response of LTI system
– The response of the system when the input is )(n
)()( nnx =System
)()( nhny =
• System response for arbitrary input
– Any signal can be decomposed as the sum of time-shifted impulses
)()()( knkxnxk
−= +
−=
)( kn−System
)( knh −– Time invariant
– Linear
)()( knkxk
−+
−=
System
)()( knhkxk
−+
−=
LTI system
LTI system
LTI system
231
SYSTEM: CONVOLUTION SUM
• Convolution sum
– The convolution sum of two signals and is )(nx )(nh
)()()()( knhkxnhnxk
−= +
−=
• Response of LTI system
– The output of a LTI system is the convolution sum of the input and
the impulse response of the system.
)(nx)(nh
)()( nhnx
LTI system
232
SYSTEM: CONVOLUTION SUM
• Example
– 1. )()( mnnx −
– 2. ),()( nunx n= )()( nunh n=
= )()( nhnx
233
SYSTEM: CONVOLUTION SUM
• Example:
– Let be two
sequences, find
],1,1,0,2,1[)( −=nh]2,1,3,1[)( −−=nx
)()( nhnx
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
==
−=−M
k
k
N
k
k knxbknya00
)()(
236
OUTLINE
• Discrete-time signals
• Discrete-time systems
• Z-transform
237
Z-TRANSFORM
• Bilateral Z-transform
n
n
znxzX −+
−=
= )()(
• Unilateral Z-transform
n
n
znxzX −+
=
=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. )()( nnx =
– 2. )(2
1)( nunx
n
=
239
Z-TRANSFORM
• Example
– 3. )1(
2
1)( −−
−= nunx
n
• Region of convergence (ROC)
Region of convergence
240
Z-TRANSFORM: CONVERGENCE
• Convergence of causal signal
)()( nunx n=
• Convergence of anti-causal signal
)1()( −−= nunx n
Z-TRANSFORM: TIME SHIFTING PROPERTY
• Time Shifting
– Let be a causal sequence with the Z-transform
– Then
241
)(nx )(zX
−
=
−−=+1
0
0
0
00 )()()(n
m
mnnzmxzzXznnxZ
−
−=
−−−+=−
1
0
0
00 )()()(nm
mnnzmxzzXznnxZ
242
Z-TRANSFORM: LTI SYSTEM
• LTI System
– Difference equation representation
= =
−=−N
k
M
k
kk knxbknya0 0
)()(
– Z-domain representation
)()(00
zXzbzYzaM
k
k
k
N
k
k
k
=
=
−
=
−
– Transfer function
==
=
−
=
−
N
k
k
k
M
k
k
k
za
zb
zX
zYzH
0
0
)(
)()(
243
Z-TRANSFORM: LTI SYSTEM
• Example
– Find the transfer function of the system described by the following
difference equation
)1(2
1)()2(2)1(2)( −+=−+−− nxnxnynyny
244
Z-TRANSFORM: STABILITY
• Stability
az
zzH
−=)( )()( nuanh 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 ( )1|| a