Transcript Document

Linear Time-Invariant Systems
Quote of the Day
The longer mathematics lives the more abstract –
and therefore, possibly also the more practical – it
becomes.
Eric Temple Bell
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall
Inc.
Linear-Time Invariant System
• Special importance for their mathematical tractability
• Most signal processing applications involve LTI systems
• LTI system can be completely characterized by their impulse
response
[n-k]
T{.}
• Represent any input xn 
hk[n]

 xk n  k 
k  


 

yn  T   xk n  k    xk Tn  k    xk hk n
k  
k  
 k  
• From time invariance we arrive at convolution
yn 

 xk hn  k  xk  hk
k  
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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LTI System Example
1
1
LTI
0.5
0
-5
0
0.5
0
-5
5
2
LTI
0
LTI
0
5
0
5
LTI
0
0
5
Copyright (C) 2005 Güner Arslan
1
0
-5
4
5
1
0
-5
0
1
0
-5
2
5
1
0
-5
2
5
2
1
0
-5
2
0
2
0
-5
5
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Convolution Demo
Joy of Convolution Demo from John Hopkins University
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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Properties of LTI Systems
• Convolution is commutative
xk   hk  
x[n]


k  
k  
 xkhn  k   hkxn  k  hk  xk
h[n]
y[n]
h[n]
x[n]
y[n]
• Convolution is distributive
xk  h1 k  h2 k  xk  h1 k  xk  h2 k
h1[n]
x[n]
+
y[n]
x[n]
h1[n]+ h2[n]
y[n]
h2[n]
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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Properties of LTI Systems
• Cascade connection of LTI systems
x[n]
h1[n]
h2[n]
y[n]
x[n]
h2[n]
h1[n]
y[n]
x[n]
Copyright (C) 2005 Güner Arslan
h1[n]h2[n]
y[n]
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Stable and Causal LTI Systems
• An LTI system is (BIBO) stable if and only if
– Impulse response is absolute summable

 hk   
k  
– Let’s write the output of the system as
yn 


k  
k  
 hkxn  k   hk xn  k
– If the input is bounded
x[n]  Bx
– Then the output is bounded by
yn  Bx

 hk 
k  
– The output is bounded if the absolute sum is finite
• An LTI system is causal if and only if
hk  0 for k  0
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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Linear Constant-Coefficient Difference Equations
• An important class of LTI systems of the form
N
M
 a yn  k   b xn  k
k 0
k
k 0
k
• The output is not uniquely specified for a given input
– The initial conditions are required
– Linearity, time invariance, and causality depend on the initial
conditions
– If initial conditions are assumed to be zero system is linear, time
invariant, and causal
• Example
– Moving Average
y[n]  x[n]  x[n  1]  x[n  2]  x[n  3]
– Difference Equation Representation
0
3
 a yn  k   b xn  k
k 0
k
Copyright (C) 2005 Güner Arslan
k 0
k
where ak  bk  1
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Eigenfunctions of LTI Systems
• Complex exponentials are eigenfunctions of LTI systems:
xn  ejn
• Let’s see what happens if we feed x[n] into an LTI system:
yn 

 hkxn  k  
k  

j(n k )


h
k
e

k  
 
 
 jk  jn
yn    hk e
e  H e j e jn
 k  

eigenfunction
eigenvalue
• The eigenvalue is called the frequency response of the system
    hke
He
j

 jk
k  
• H(ej) is a complex function of frequency
– Specifies amplitude and phase change of the input
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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Eigenfunction Demo
LTI System Demo
From FernÜniversität, Hagen, Germany
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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