文档介绍：INTRODUCTION
LTI systems possesses the superposition property.
Represent signals as binations of delayed impulses .
Convolution sum or convolution integral.
linear constant-coefficient difference or differential equations.
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1. Discrete-Time LTI: Convolution Sum
1) The Representation of Discrete-time Signals in Terms of Impulses ()
2) The Discrete-time Unit Impulse Response of LTI Systems ()
3) The Discrete-time Response of LTI Systems to any Input Signal: Convolution Sum
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1) The Representation of Discrete-time Signals in Terms of Impulses
If x[n]=u[n],then
Sifting Property of Unit Sample:
1. Discrete-Time LTI: Convolution Sum
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2) The Discrete-time Unit Impulse Response of LTI Systems
LTI
x[n]=[n]
y[n]=h[n]
Unit Impulse Response h[n] :response of the LTI system to the unit sample δ[n].
δ[n] → h[n]
Why do we need it?
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3) The Discrete-time Response of LTI Systems to any Input Signal: Convolution Sum
LTI
x[n]
y[n]=?
Solution:
Question:
[n] h[n]
[n-k] h[n-k]
x[k][n-k] x[k] h[n-k]
The response y[n] to x[n] is the weighted bination of delayed unit sample responses.
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Convolution Sum
So
Representing the convolution operation symbolically as: y[n] = x[n] * h[n]
--- Convolution Sum
That is, the unit impulse response --h[n] can fully characterize an LTI system.
Summary on calculating convolution sum
Time Inversal: h[k] h[-k]
Time Shift: h[-k] h[n-k]
Multiplication: x[k]h[n-k]
Summing:
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Example Consider a LTI system with unit sample response h[n] and input x[n], as illustrated in Figure (a). Calculate the convolution sum (convolution) of these two sequences graphically.
n
x[n]
0 1 2
n
h[n]
-2 0 2
(a)
1
2
2
k
x[k]
0 1 2
k
h[-k]
-2 0 2
(b)
2
2
1
7
k
x[k]
0 1 2
2
k
h[-k]
-2 0 2
2
1
n=0
k
h[-1-k]
-3 -2 0 1
2
1
n=-1
k
h[1-k]
-1 0 1 2 3
2
1
n=1
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Example Consider an input x[n] and a unit sample response h[n] given by
Determine and plot the output
Using