Prepared by Dr Nidal Kamel 1 Chapter 2 Linear Time-Invariant Systems
Dec 21, 2015
Prepared by Dr Nidal Kamel 1
Chapter 2
Linear Time-Invariant Systems
Prepared by Dr Nidal Kamel 2
Introduction
Linearity and time invariance, play a fundamental role in
signal and systems analysis for two reasons:
Many physical processes can be modeled as linear time-invariant
(LTI) systems.
LTI systems can be analyzed intensively providing both insight into
their properties and a set of powerful tools form the core of signal
and system analysis.
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Introduction
LTI systems are amenable to analysis because it possess the
superposition property.
This fact will allow us to develop complete characterization
of any LTI system in term of its response to unit impulse.
Such representation is referred to as convolution sum in
DT systems and convolution integral in CT systems.
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DT Systems: The Convolution Sum Representation of DT Signals in term of impulses
1n 0,
1n ],1[]1[]1[
0n 0,
0n ],0[][]0[
-1n 0,
-1n ],1[]1[]1[
xnx
xnx
xnx
-k
knkx
nxnxnxnx
nxnxnxnx
-δ
...]3[]3[]2[]2[]1[]1[][]0[
]1[]1[]2[]2[]3[]3[...][
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DT Systems: The Convolution Sum
Unit Impulse Response and Convolution Sum of LTI
If h[n] is the output of the system to δ[n] and hk [n] is the output
to δ[n-k], then the output of linear system to x[n] is:
[ ] δ
... [ 3] [ 3] [ 2] [ 2] [ 1] [ 1]
[0] [ ] [1] [ 1] [2] [ 2] [3] [ 3] ...
k -
x n x k n-k
x n x n x n
x n x n x n x n
3 2 1
1 2 3
[ ] ... [ 3] [ ] [ 2] [ ] [ 1] [ ]
[0] [ ] [1] [ ] [2] [ ] [3] [ ] ..
k
k -
y n x h n x h n x h n
x h n x h n x h n x h n
x k h n
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DT Systems: The Convolution Sum Unit Impulse Response and Convolution Sum of LTI
1 1
[ ] [ 1] [ 1] [0] [ ] [1] [ 1]
[ ] [ 1] [ ] [0] [ ] [1] [ ]
x n x n x n x n
y n x h n x h n x h n
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DT Systems: The Convolution Sum Unit Impulse Response and Convolution Sum of LTI
The input-output relationship of a linear systems is given as:
If the linear system is also time-invariant, then,
Thus the output of LTI system is given
][][ knhnhk
-k
knhkxny ][
-k
knkxnx -δ][
[ ] [ ] [ ]k
y n x k h n k x n h n
8
DT Systems: The Convolution Sum Unit Impulse Response and Convolution Sum of LTI
Consider an LTI system with h[n] and x[n]
as shown in the figure. Find the system
output, y[n].
[ ] [ ] [ ]
[0] [ 0] [1] [ 1]
0.5 [ ] 2 [ 1]
k
y n x k h n k
x h n x h n
h n h n
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DT Systems: The Convolution Sum Unit Impulse Response and Convolution Sum of LTI
10
DT Systems: The Convolution Sum Unit Impulse Response and Convolution Sum of LTI
[ ] [ ] [ ]k
y n x k h n k
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DT Systems: The Convolution Sum Unit Impulse Response and Convolution Sum of LTI
1, 0 4[ ]
0,
, 0 6[ ]
0,
n
nx n
otherwise
nh n
otherwise
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DT Systems: The Convolution Sum Unit Impulse Response and Convolution Sum of LTI
1, 0 4[ ]
0,
, 0 6[ ]
0,
k
kx k
otherwise
kh k
otherwise
0 0
n nn k
k k
y n h n k[ ] [ ]
4 4
0 0
n k
k k
y n h n k[ ] [ ]
4 4
6 6
n k
k n k n
y n h n k[ ] [ ]
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CT Systems: The Convolution Integral Representation of CT Signals in Terms of Impulses
If we define
Since ΔδΔ(t) has unit amplitude, we have
otherwise 0,
t0 ,1
)(t
ˆ( ) ( )k
x t x k t k
14
CT Systems: The Convolution Integral
Convolution Integral Representation of LTI systems
The approximate representation of
x(t), is given as
Consequently, the response of
linear system is given as
k
ktkxtx )()(ˆ
)(ˆ)(ˆ thkxty
k
k
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CT Systems: The Convolution Integral Convolution Integral Representation of LTI systems
If the system is linear -time invariant (LTI), then
As Δ →0 the summation becomes integral
)()(
)()()(
thtx
dthxty
ˆˆ( ) ( )k
y t x k h t k
16
CT Systems: The Convolution Integral Convolution Integral Representation of LTI systems
Let x(t) be the input to LTI system with unit impulse response
h(t), where
)()(
0 ),()(
tuth
atuetx at
)(
)()()(
tuea
de
dthxty
at
ta
11
0
17
CT Systems: The Convolution Integral Convolution Integral Representation of LTI systems
Consider the convolution of the
following two signals:
otherwise 0,
20 ,)(
,0
0 ,1)(
Tttth
otherwise
Tttx
18
CT Systems: The Convolution Integral Convolution Integral Representation of LTI systems
There are three intervals:
2
2
2 2
( ) ( ) ( )
0, 0
1, 0
2
1( ) , 2
2
1 3, 2 3
2 2
0, 3
y t x h t d
t
t t T
y t Tt T T t T
t Tt T T t T
T
t
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Properties of Linear-Time Invariant System
The LTI systems are represented in terms of their unit impulse
responses.
[ ] [ ] [ ] [ ] [ ]
( )
k
y n x k h n k x n h n
y t x h t d x t h t
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Properties of Linear-Time Invariant System Commutative Property
Convolution is a commutative operation.
k
x n h n h n x n h k x n k
x t h t h t x t h x t d
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Properties of Linear-Time Invariant System Distributive Property
Convolution is distributive over addition:
Parallel interconnected LTI systems
1 2 1 2
1 2 1 2
x n h n h n x n h n x n h n
x t h t h t x t h t x t h t
)()()(
)()()()(
)()()(
ththtx
thtxthtx
tytyty
21
21
21
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Properties of Linear-Time Invariant System Distributive Property
Find the convolution of the following two sequences:
We may use the distributive property
1
[ ] 22
[ ] [ ]
n
nx n u n u n
h n u n
1 2
1 2
1 2
[ ] [ ] [ ]
[ ] [ ] [ ] [ ]
y n x n x n h n
x n h n x n h n
= y n y n
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Properties of Linear-Time Invariant System Distributive Property
11 2
0 0
n nk
k k
y n x k
[ ] [ ]
2
0 0
2 0
2 0
= 2
[ ] [ ]
[ ]
n nk
k k
k
k k
y n x k for n
x k for n
1 2[ ] [ ] [ ]y n y n y n
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Properties of Linear-Time Invariant System Associative Property
Convolution is associative:
1 2 1 2
1 2 1 2
x n h n h n x n h n h n
x t h t h t x t h t h t
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Properties of Linear-Time Invariant System Systems with and without Memory
System is memoryless if
This is true if
The convolution sum reduces to
[ ] [ ]y n bx n
y t bx t
[ ] [ ] [ ] [ ]y n x n b n bx n
y t x t b t bx t
[ ] [ ]h n b n
h t b t
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Properties of Linear-Time Invariant System Invertibility of LTI Systems
The h(t) system is invertible if an
inverse system h1(t) exists.
The overall impulse response is
The same applies to DT system
1( ) ( ) ( )h t h t t
1[ ] [ ] [ ]h n h n n
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Properties of Linear-Time Invariant System Invertibility of LTI Systems
Consider an LTI with impulse response h[n] = u[n], find the
output of the system.
This system is invertible and its inverse has impulse response
n
k
k
kx
knukxny
][
][][][
1[ ] [ ] [ 1]h n n n
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Properties of Linear-Time Invariant System Invertibility of LTI Systems
The impulse response of inverse system is
We can verify this result by direct calculation:
1[ ] [ ] [ 1]h n n n
1[ ] [ ] [ ] [ ] [ 1
[ ]* [ ] - [ ]* [ -1]
[ ] - [ -1]
[ ]
h n h n u n n n
u n n u n n
u n u n
n
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Properties of Linear-Time Invariant System Causality of LTI Systems
LTI system is causal if
For causal LTI system, the convolution become
( ) 0
[ ] 0 0
h t for t < 0
h n for n .
t
n
k
thxty
knhkxny
)()()(
][][][
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Properties of Linear-Time Invariant System Causality of LTI Systems
Both the accumulator and its inverse are causal.
Causality for linear system is equivalent to the condition of
initial rest.
Initial rest: if the input to causal system is zero up to some point
in time, then the output must also be zero up to the same time.
Initial rest = no input no output
][][ nunh
[ ] [ ] [ 1]h n n n
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Properties of Linear-Time Invariant System Stability of LTI Systems
A system is stable if BIBO
Consider input to LTI that is bounded in magnitude:
The magnitude of the output is:
[ ]x n B for all n.
[ ]k
y n h k x n k
[ ]
[ ]
k
k
y n h k x n k
B h k for all n
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Properties of Linear-Time Invariant System Stability of LTI Systems
Thus, DT-LTI system is stable if
CT-LTI, the system is stable if
[ ]k
h k
( )h d
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Properties of Linear-Time Invariant System Stability of LTI Systems
Find whether pure time shift system is stable or not.
thus the system is stable.
Find whether accumulator system, h[n] = u[n] is stable or not.
thus system is unstable.
0
0
[ ] 1
( ) ( 1
n n
h n n n
h d t d
0
0
[ ] [ ]
( )
n n
u n u n
u d d
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Properties of Linear-Time Invariant System Unit step Response of an LTI
step response is obtained when u[n] is applied at the input
of the system
Thus h[n] can be recovered from s[n] using the relation
[ ] [ ] [ 1] [ ] [ ] [ 1]n u n u n h n s n s n
k
n
k
khknukh
nunhnhnuns
][][][
][][][][][
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Properties of Linear-Time Invariant System Unit step Response of an LTI
For the CT system, the unit step response is given as:
In analogy to the DT part,
t
dhdtuh
tuththtuts
)()()(
)()()()()(
dt
tdsth
dt
tdut
)()(
)()(
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Causal LTI Systems Described by Differential
and Difference Equations
Important class of CT systems has input-output relationships in
form of linear constant-coefficient differential equations.
A general Nth-order equation is given by
In case of N = 0, we have
0 0
( ) ( )k kN M
k kk kk k
d y t d x ta b
dt dt
00
1 ( )( )
kM
k kk
d x ty t b
a dt
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Causal LTI Systems Described by Differential
and Difference Equations
auxiliary conditions are required for determination of input-
output relationship.
Different auxiliary conditions result in different input-output
relationships.
In practical systems we use auxiliary condition of initial rest.
Under condition of initial rest, the system is causal and LTI.
1
0 0
0 1
( ) ( )( ) ... 0
N
N
dy t d y ty t
dt dt
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Causal LTI Systems Described by Differential
and Difference Equations
DT systems have input-output relationships in form of linear
constant-coefficients difference equations:
This form is called the recursive form.
0 0
0 10
[ ] [ ]
1[ ] [ ] [ ]
N M
k k
k k
M N
k k
k k
a y n k b x n k
y n b x n k a y n ka
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Causal LTI Systems Described by Differential
and Difference Equations
When N = 0, we have the following form for input-output:
This often called nonrecursive equation.
By direct computation, this system has a finite impulse
response (FIR) of the form:
0 0
[ ] [ ]M
k
k
by n x n k
a
0
, 0[ ]
0,
nbn M
ah n
otherwise
0
[ ] [ ] [ ]M
k
y n h k x n k
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Causal LTI Systems Described by Differential
and Difference Equations
Consider the difference equation with x[n] = bδ[n]. Find y[n].
Suppose we impose condition of initial rest .
Condition of initial rest implies x[n] = 0 for n < 0, and y[n] =
0 for n < 0.
1[ ] [ 1] [ ]
2
1[ ] [ ] [ 1]
2
y n y n x n
y n x n y n
41
Causal LTI Systems Described by Differential
and Difference Equations
Starting with this initial condition, we may solve for
successive values of y[n] for n ≥ 0 as follows:
System impulse response is
2
1[0] [0] [ 1] ,
2
1 1[1] [1] [0] ,
2 2
1 1[2] [2] [1]
2 2
1 1[ ] [ ] [ 1]
2 2
n
y x y b
y x y b
y x y b
y n x n y n b
1
[ ] [ ]2
n
h n u n
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Summary
In this chapter we developed important representations
for LTI systems.
In discrete time we derived the representation of signals
as weighted sums of shifted impulse.
Next, we used this representation to derive the
convolution sum representation for LTI discrete systems.
In continuous time we derived representation of signals as
weighted integral of shifted unit impulse.
Later, this representation is used to derive the convolution
integral for LTI continuous systems.
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Summary
Moreover, the convolution sum and integral provided us
with a means of analyzing the properties of LTI systems,
including causality and stability.
Important class of continuous time systems described by
linear constant-coefficients differential equations, is
discussed.
Discrete time systems represented by linear constant-
coefficients difference equations, are also discussed.