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8/3/2019 Convolution 4327
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The Convolution Integral
Convolution operation given symbol *
dthxthtxty )(*)(
y equals x convolved with h
8/3/2019 Convolution 4327
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The Convolution Integral
The time domain output of an LTI system is
equal to the convolution of the impulse
response of the system with the input signal
Much simpler relationship between
frequency domain input and output
First look at graphical interpretation of
convolution integral
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Graphical Interpretation of
Convolution Integral
To correctly understand convolution it is
often easier to think graphically
h(
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Graphical Interpretation of
Convolution Integral
h(
h(-
Take impulse response and reverse it in time
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Graphical Interpretation of
Convolution Integral
h(-
Then shift it by time t
h(t-
t
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Graphical Interpretation of
Convolution Integral
Overlay input functionx(t) and integrate over times
where functions overlap - in this case between a and t
h(t-
ta
x(
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Graphical Interpretation of the
Convolution Integral
Convolving two functions involves
flipping or reversing one function in time
sliding this reversed or flipped function over
the other and
integrating between the times when BOTH
functions overlap
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Example
Convolution of two gate pulses each of
height 1
0 1
x1(
0 2
x2(
dtxxxxy2121
*
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Example
-2 0 2
x2(x2(-
Reverse function
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Example
-1 0 1
x1(x2(-
Reverse function, slidex2overx1 and evaluate integralt
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Example
0 1
x1(x2(t-
t
tdtxxy
tt
0
21 1*
10for
Area of overlap is increasing linearly
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Example
0 1
x1(
x2(t-
t)(
pulsesmallerofarea
1*
21for
1
21
x
xxy
t
Area of overlap constant
t-2
1
0
1
0111 d
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Example
0 1
x1(
x2(t-
t
txxy
t
3*
32for
21
Area declining linearly -
width of shaded area = 1-(t-2)=3-t
t-2 ttdt
t
32111
1
2
2
2
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Example
0 1
x1(
x2(t-
t
0*
3for
21
xxy
t
After time t=3 the convolution integral is zero
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Example
0 1 2 3
x1(t)*x2(t)
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tint=0;
tfinal=10;
tstep=.01;
t=tint:tstep:tfinal;x=5*((t>=0)&(t=0)&(t
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Example 2
Convolve the following functions
0 1 t
1.0
x1(t)
0 1 t
x2(t)
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Example 2
0 1
x2
-1
Reversal
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Example 2
0 t 1
x2t
-1
Shift reversed function
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Example 2
0 t 1
x2t
-1
Overlay shift reversed function ontoother function and integrate overlapping
section
x1
tdtxx
t
t
0
21 1*
10for
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Example 2
0 1 t
x2t
-1
Overlay shift reversed function onto
other function and integrate overlapping
section
x1
tdtxx
t
t
21*
21for
1
1
21
t-1
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Example 2
x1(t)*x2(t)
0 1 2
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Example 3
3.heightofpulsesecond4aishinput whicantodue
systemthisofoutputthecompute5)(
issystemLTIanofresponseimpulseGiven the
2 u(t)etht
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Example 3
5 )(5)(2
tuetht
t0 4
3
)(tx
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Example 3
5
)( h
Reverse h(
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Example 3
5
)( th
Shift the reversed h(by t
t 4
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Example 3
5
)( th
Performing integral for 0
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Example 3
t
t
t
t
t
t
t
ety
ee
deedety
20
22
0
22
0
2
15.7)(
2
115
1515)(
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Example 3
5
)( th
Performing integral for t>4
t 4
4
0
215)( dety t
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Example 3
15.72
115
1515)(
82
4
0
22
4
0
22
4
0
2
eeee
deedety
tt
tt
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Example 3
415.7
4015.7
00
)(
82
2
tee
te
t
ty
t
t
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Commutativity of Convolution
Operation
The actions of flipping and shifting can be
applied to EITHER function
)(*)(
)(*)(
txthdtxh
dthxthtx
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Example 4
Repeat example 3 by flipping and shifting
x(t) rather than h(t)
0 t
tt
dedety
t
0
2
0
21553)(
40for
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Example 4
0 t
t
tt
ety
edety
t
2
02
0
2
15.7)(
5.715)(
40for
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Example 4
0 t
t
t
dety
t
4
215)(
4for
t-4
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Example 4
ttt
t
t
t
t
eeeety
edety
t
28242
4
2
4
2
15.75.7)(
2
11515)(
4for
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Example 4
415.7
4015.7
00
)(
82
2
tee
te
t
ty
t
t
Same result as before