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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


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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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h(

h(-

Take impulse response and reverse it in time


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h(-

Then shift it by time t

h(t-

t


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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


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