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 xn
hk[n]
xk n k
k
yn T xk n k xk Tn k xk hk n
k
k
k
• From time invariance we arrive at convolution
yn
xk hn k xk hk
k
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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LTI System Example
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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
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2
1
0
-5
2
0
2
0
-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
xk hk
x[n]
k
k
xkhn k hkxn k hk xk
h[n]
y[n]
h[n]
x[n]
y[n]
• Convolution is distributive
xk h1 k h2 k xk h1 k xk 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]
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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
hk
k
– Let’s write the output of the system as
yn
k
k
hkxn k hk xn k
– If the input is bounded
x[n] Bx
– Then the output is bounded by
yn Bx
hk
k
– The output is bounded if the absolute sum is finite
• An LTI system is causal if and only if
hk 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 yn k b xn 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 yn k b xn 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:
xn ejn
• Let’s see what happens if we feed x[n] into an LTI system:
yn
hkxn k
k
j(n k )
h
k
e
k
jk jn
yn hk e
e H e j e jn
k
eigenfunction
eigenvalue
• The eigenvalue is called the frequency response of the system
hke
He
j
jk
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
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