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Li Ti I i t (LTI)Linear Time Invariant (LTI) SystemsSystems
Rui Wang, Assistant professorDept. of Information and Communication
T ji U i itTongji University
Email: [email protected]
OutlineOutline Discrete-time LTI system: The convolution y
Sum Continuous time LTI systems: The Continuous-time LTI systems: The
convolution integral Property of Linear Time-Invariant Systems Causal LTI Systems Described by Causal LTI Systems Described by
Differential and Difference Equations Singularity Functions
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2.1 Discrete-time LTI system: SThe convolution Sum
Using delta function to represent discrete Using delta function to represent discrete-time signal
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2.1 Discrete-time LTI system: SThe convolution Sum
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2.1 Discrete-time LTI system: SThe convolution Sum
Using delta function to represent discrete Using delta function to represent discrete-time signal
An example:
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2.1 Discrete-time LTI system: SThe convolution Sum
The response of a linear system to x[n] will The response of a linear system to x[n] will be the superposition of the scaled
f th t t h f thresponses of the system to each of these shifted delta functions.
The property of time invariance tells us that the responses to the time-shifted deltathat the responses to the time-shifted delta functions are simply time-shifted version of
thone another.
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2.1 Discrete-time LTI system: SThe convolution Sum
Let denote the response of the linear Let denote the response of the linear system to With
The output of the system The output of the system
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2.1 Discrete-time LTI system: SThe convolution Sum
An example: An example:
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2.1 Discrete-time LTI system: SThe convolution Sum
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2.1 Discrete-time LTI system: SThe convolution Sum
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2.1 Discrete-time LTI system: SThe convolution Sum
If linear system is time invariant we have If linear system is time invariant, we have
This simplify as This simplify as
This result is referred to as the convolution sum
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2.1 Discrete-time LTI system: SThe convolution Sum
Example 1: Consider an LTI system with Example 1: Consider an LTI system with impulse response h[n] and input x[n] as f ll D t i th t tfollows. Determine the output
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2.1 Discrete-time LTI system: SThe convolution Sum
Solution: Solution:
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2.1 Discrete-time LTI system: SThe convolution Sum
Solution: Solution:
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2.1 Discrete-time LTI system: SThe convolution Sum
Solution: Solution:
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2.1 Discrete-time LTI system: SThe convolution Sum
Solution: Solution:
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2.1 Discrete-time LTI system: SThe convolution Sum
Solution: Solution:
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2.1 Discrete-time LTI system: SThe convolution Sum
Example 2: Consider an LTI system with Example 2: Consider an LTI system with
Determine the output
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2.1 Discrete-time LTI system: SThe convolution Sum
Solution: Solution:
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2.1 Discrete-time LTI system: SThe convolution Sum
Solution: Solution:
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2.1 Discrete-time LTI system: SThe convolution Sum
Example 3: Consider an LTI system with Example 3: Consider an LTI system with
Determine the output
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2.1 Discrete-time LTI system: SThe convolution Sum
Solution: Solution: Interval 1: n<0, y[n] =0 Interval 2: 0≤n ≤4
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2.1 Discrete-time LTI system: SThe convolution Sum
Solution: Solution: Interval 3: for n>4, but n-6 ≤0, i.e., 4≤n ≤6
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2.1 Discrete-time LTI system: SThe convolution Sum
Solution: Solution: Interval 4: for n>6, but n-6 ≤4, i.e., 6≤n ≤10
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2.1 Discrete-time LTI system: SThe convolution Sum
Solution: Solution: Interval 4: for n-6>4, i.e., n>10i.e., 6≤n ≤10
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2.1 Discrete-time LTI system: SThe convolution Sum
Solution: Solution:
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2.2 Continuous-time LTI system: The convolution integral 2 2 1 Representing the continuous time 2.2.1 Representing the continuous-time
signals in terms of impulses
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2.2 Continuous-time LTI system: The convolution integral
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2.2 Continuous-time LTI system: The convolution integral If we define a pulse function as If we define a pulse function as
Since has unit amplitude we have Since has unit amplitude, we have
In above, for any value of t, only one term in , y , ythe summation on the right-hand side is non-zero.
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2.2 Continuous-time LTI system: The convolution integral As △0 the summation approaches an As △0, the summation approaches an
integral.
Check the unit step function:
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2.2 Continuous-time LTI system: The convolution integral 2 2 2 the unit impulse response and the 2.2.2 the unit impulse response and the
convolution integral As x(t) is approximated as the sum of shifted
version of the basic versions of the pulse signal
The response of the linear system will be
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2.2 Continuous-time LTI system: The convolution integral
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2.2 Continuous-time LTI system: The convolution integral
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2.2 Continuous-time LTI system: The convolution integral 2 2 2 the unit impulse response and the 2.2.2 the unit impulse response and the
convolution integral As , we have
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2.2 Continuous-time LTI system: The convolution integral 2 2 2 the unit impulse response and the 2.2.2 the unit impulse response and the
convolution integral When the linear system is time invariant, we
have
Define we have Define , we have
Referred to the above as the convolution integral.
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2.2 Continuous-time LTI system: The convolution integral Example 1: Let x(t) be the input to an LTI
system with unit impulse response h(t), y p p ( ),where
determine the output y(t).
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2.2 Continuous-time LTI system: The convolution integral Solution:
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2.2 Continuous-time LTI system: The convolution integralWhen t<0, y(t) = 0
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2.2 Continuous-time LTI system: The convolution integralWhen t>0
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2.2 Continuous-time LTI system: The convolution integral Solution:
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2.2 Continuous-time LTI system: The convolution integral Example 2: Let x(t) be the input to an LTI
system with unit impulse response h(t), y p p ( ),where
1 0 t T 0 2t t T 1 0( )
0t T
x totherwise
0 2( )
0t t T
h totherwise
determine the output y(t).
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2.2 Continuous-time LTI system: The convolution integral Solution:
( ) ( ) ( ) ( ) ( )y t x t h t x h t d
( ) ( ) ( ) ( ) ( )y t x t h t x h t d
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2.2 Continuous-time LTI system: The convolution integral
21( )t
t d t 2
0( )
2y t d t
21( )t
y t d Tt T ( )2t T
y
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2.2 Continuous-time LTI system: The convolution integral
12 2 21( ) 2 ( )2
T
t Ty t d T t T
( ) 0y t
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2.2 Continuous-time LTI system: The convolution integral Example 3: Let x(t) be the input to an LTI
system with unit impulse response h(t), y p p ( ),where
determine the output y(t).
45
2.2 Continuous-time LTI system: The convolution integral Solution:
( ) ( ) ( ) ( ) ( )y t x t h t x h t d
( ) ( ) ( ) ( ) ( )y t x t h t x h t d
46
2.2 Continuous-time LTI system: The convolution integralWhen t-3<=0, i.e., t<=3
When t-3>0, i.e., t>3
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2 3 Properties of LTI Systems2.3 Properties of LTI Systems
Use convolution sum and convolution integral to obtain the output of discrete-g ptime and continuous-time systems, based on the unit impulse responsebased on the unit impulse response
An LIT system is completely An LIT system is completely characterized by its impulse response ( l f LTI t )(only for LTI system)
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2 3 Properties of LTI Systems
交换律
2.3 Properties of LTI Systems
2.3.1 The commutative property (交换律)
Proof: Proof:
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2 3 Properties of LTI Systems
律
2.3 Properties of LTI Systems
2.3.2 The distributive property (分配律) Definition:
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2 3 Properties of LTI Systems
律
2.3 Properties of LTI Systems
2.3.2 The distributive property (分配律) Definition:
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2 3 Properties of LTI Systems
结合律
2.3 Properties of LTI Systems
2.3.3 The associative property (结合律) Definition:
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2 3 Properties of LTI Systems
结合律
2.3 Properties of LTI Systems
2.3.3 The associative property (结合律) Definition:
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2 3 Properties of LTI Systems2.3 Properties of LTI Systems
2.3.4 LTI system with and without memory (有记忆和无记忆系统)y (有 无 ) Definition: the output only depends on value
of the input at the same timeof the input at the same time. We have
i e i.e.,
54
2 3 Properties of LTI Systems
性
2.3 Properties of LTI Systems
2.3.5 Inevitability of LTI system (可逆性)
Identical systemIdentical system
55
2 3 Properties of LTI Systems2.3 Properties of LTI Systems
Example: Delayed or advanced systemy y
W h We have
Can we find such that Can we find such that
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2 3 Properties of LTI Systems2.3 Properties of LTI Systems
Example: accumulator systemy
The inverse system is a first different ti ioperation, i.e.,
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2 3 Properties of LTI Systems
性
2.3 Properties of LTI Systems
2.3.6 Causality for LTI (因果性) LTI system should satisfyy y
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2 3 Properties of LTI Systems
性
2.3 Properties of LTI Systems
2.3.7 Stability for LTI (稳定性) LTI system should satisfy: absolutely y y y
summable and absolutely integrable
How to prove??
59
Proof: Proof:
Example: time shift
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2 3 Properties of LTI Systems2.3 Properties of LTI Systems
2.3.8 The unit step response of LTI Use unit step response to describe the p p
system behavior
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2.4 Causal LTI system described ff & ffby differential & difference equs
An important class practical system can be described by usingy g Linear constant-coefficient differential
equationsequations Linear constant-coefficient difference
equationsequations
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2.4 Causal LTI system described ff & ffby differential & difference equs
Linear constant-coefficient differential equations
( ) ( )k kN Md y t d x t 0 0
( ) ( ),k kk kk k
d y t d x ta bdt dt
are constant, N is the order,k ka b The response to an input x(t) generally consists of
the sum of particular solution (特解) to the diff ti l ti & h l ti (其differential equation & a homogeneous solution (其次解)
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2.4 Causal LTI system described ff & ffby differential & difference equs
a homogeneous solution (nature response: 自然相应) l ti t th diff ti l ti ith i t应): a solution to the differential equation with input set to zero
Different choices of auxiliary conditions leads to Different choices of auxiliary conditions leads to different relationships between input and output
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2.4 Causal LTI system described ff & ffby differential & difference equs
Linear constant-coefficient difference equations
N M
0 0
( ) ( )N M
k kk k
a y n k b x n k
are constant, N is the order,k ka b The response to an input x[n] generally consists of
the sum of particular solution (特解) to the diff ti l ti & h l ti (其differential equation & a homogeneous solution (其次解)
65
2.4 Causal LTI system described ff & ffby differential & difference equs
a homogeneous solution (nature response: 自然相应) l ti t th diff ti l ti ith i t应): a solution to the differential equation with input set to zero
Re express difference equation as recursive Re-express difference equation as recursive equation
1( ) ( ) ( )M N
b k k
0 10
( ) ( ) ( )k kk k
y n b x n k a y n ka
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2.4 Causal LTI system described ff & ffby differential & difference equs
We see that
( )x n ( 1), ( 2), , ( )y y y N (0)y
(0)y ( 1), ( 2), , ( 1)y y y N (1)y
0 When N = 0, non-recursive equation
0n
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2.4 Causal LTI system described ff & ffNon recursive case:
by differential & difference equs Non-recursive case:
( ) ( )M
kby n x n k The unit impulse response is
0 0k a
Finite impulse response (FIR) system For recursive case: Infinite Impulse Response
(IIR) system
68
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2.4 Causal LTI system described ff & ffby differential & difference equs
Block diagram representation of the first-order systems described by differential and difference equationsequations
Addition
Multiplication by a coefficient
delay
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2.4 Causal LTI system described ff & ffby differential & difference equs
The following first-order difference equation can be described as
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2.4 Causal LTI system described ff & ffby differential & difference equs
1 M N 0 10
1( ) ( ) ( )k kk k
y n b x n k a y n ka
( )x n ( )w n0b
( ) ( )M
kw n b x n k D
1b
0( ) ( )k
kw n b x n k
D
2b
T Ⅰ
D1Mb
b
TypeⅠ
72
Mb
2.4 Causal LTI system described ff & ffby differential & difference equs
D
( )w n ( )y n01/ a
1
1( ) ( ) ( )N
kk
y n w n a y n ka
D
D 1a
a10 ka
2a
1Na
D
Na
73
2.4 Causal LTI system described ff & ffby differential & difference equs
1 M N 0 10
1( ) ( ) ( )k kk k
y n b x n k a y n ka
D
( )x n ( )y n0b01 / a
D
D
1b
2b
1a
2a
TypeⅡ
2
1Nb
2
1Na
DNbNa
74
2.4 Causal LTI system described ff & ffby differential & difference equs
For differential equation
Addition
Multiplication by a coefficient
differentiation
75
2.4 Causal LTI system described ff & ffby differential & difference equs
The following first-order differential equation can be described as
76
2.4 Causal LTI system described ff & ffby differential & difference equs
0 0
( ) ( )k kN N
k kk kk k
d y t d x ta bdt dt
0 0k k
( ) ( )( ) ( )N N
k N k k N ka y t b x t ( ) ( )0 0
( ) ( )k N k k N kk k
y
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2.4 Causal LTI system described ff & ffby differential & difference equs
11 N N 1
( ) ( )0 0
1( ) ( ) ( )N N
k N k k N kk kN
y t b x t a y ta
( )x t ( )w tNb ( )w t ( )y t1/ Na
1Nb
1Na
TypeⅠ2Nb
b
2Na
a
TypeⅠ1b
0b
1a
0a
78
0
2.4 Causal LTI system described ff & ffby differential & difference equs
11( ) ( ) ( )N N
t b t t
( ) ( )0 0
( ) ( ) ( )k N k k N kk kN
y t b x t a y ta
( )x t ( )y t1/ a b ( )x t ( )y t1/ Na Nb
b 1Na
a
1Nb
b
2Na
a
2Nb
b
Ⅱ
1a
a
1b
b
79
TypeⅡ0a 0b
2 5 Singularity functions2.5 Singularity functions
The delta function is the impulse response of the identity systemp y y
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2 5 Singularity functions2.5 Singularity functions
Unit doublet function : the derivative of the unit impulsep
also
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2 5 Singularity functions2.5 Singularity functions
We define as the k-th derivative of delta function
Property of unit doublet p y
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2 5 Singularity functions2.5 Singularity functions
For t = 0, yields
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2 5 Singularity functions2.5 Singularity functions
Unit step function:
The unit ramp function The unit ramp function
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2 5 Singularity functions2.5 Singularity functions
Similarly, we have
We have We have
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