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The Convolution Integral
• Convolution operation given symbol ‘*’
dthxthtxty )(*)(
“y” equals “x” convolved with “h”
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
Graphical Interpretation of Convolution Integral
• To correctly understand convolution it is often easier to think graphically
h(
Graphical Interpretation of Convolution Integral
h(
h(-
Take impulse response and reverse it in time
Graphical Interpretation of Convolution Integral
h(-
Then shift it by time t
h(t-
t
Graphical Interpretation of Convolution Integral
Overlay input function x(t) and integrate over times where functions overlap - in this case between a and t
h(t-
t a
x(
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
Example
• Convolution of two gate pulses each of height 1
0 1
x1(
0 2
x2(
dtxxxxy 2121 *
Example
-2 0 2
x2(x2(-
Reverse function
Example
-1 0 1
x1(x2(-
Reverse function, slide x2 over x1 and evaluate integral
t
Example
0 1
x1(x2(t-
t
tdtxxy
tt
0
21 1*
10for
Area of overlap is increasing linearly
Example
0 1
x1(x2(t-
t)(
pulsesmaller of area
1*
21for
1
21
x
xxy
t
Area of overlap constant
t-2
1
0
1
0111 d
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
Example
0 1
x1(
x2(t-
t
0*
3for
21
xxy
t
After time t=3 the convolution integral is zero
Example
0 1 2 3
x1(t)*x2(t)
tint=0; tfinal=10; tstep=.01; t=tint:tstep:tfinal; x=5*((t>=0)&(t<=4));subplot(3,1,1), plot(t,x)axis([0 10 0 10])h=3*((t>=0)&(t<=2));subplot(3,1,2),plot(t,h)
axis([0 10 0 10])axis([0 10 0 5])t2=2*tint:tstep:2*tfinal;y=conv(x,h)*tstep;subplot(3,1,3),plot(t2,y)axis([0 10 0 40])
Example 2
• Convolve the following functions
0 1 t
1.0
x1(t)
0 1 t
x2(t)
Example 2
0 1
x2
-1
Reversal
Example 2
0 t 1
x2t
-1
Shift reversed function
Example 2
0 t 1
x2t
-1
Overlay shift reversed function onto other function and integrate overlapping section
x1
tdtxx
tt
0
21 1*
10for
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
Example 2
x1(t)*x2(t)
0 1 2
Example 3
3.height of pulse second 4 a ish input whican todue
system thisofoutput thecompute 5)(
is system LTIan of response impulse Given the2 u(t)eth t
Example 3
5 )(5)( 2 tueth t
t0 4
3
)(tx
Example 3
5
)( h
Reverse h(
Example 3
5
)( th
Shift the reversed h(by t
t 4
Example 3
5
)( th
Performing integral for 0<t<4
t
t
t dety0
253)(Output
4
Example 3
t
tt
tt
tt
ety
ee
deedety
2
0
22
0
22
0
2
15.7)(
2
115
1515)(
Example 3
5
)( th
Performing integral for t>4
t 4
4
0
215)( dety t
Example 3
15.72
115
1515)(
824
0
22
4
0
224
0
2
eeee
deedety
tt
tt
Example 3
415.7
4015.7
00
)(82
2
tee
te
t
tyt
t
Commutativity of Convolution Operation
• The actions of flipping and shifting can be applied to EITHER function
)(*)(
)(*)(
txthdtxh
dthxthtx
Example 4
• Repeat example 3 by flipping and shifting x(t) rather than h(t)
0 t
tt
dedety
t
0
2
0
2 1553)(
40for
Example 4
0 t
t
tt
ety
edety
t
2
02
0
2
15.7)(
5.715)(
40for
Example 4
0 t
t
t
dety
t
4
215)(
4for
t-4
Example 4
ttt
t
t
t
t
eeeety
edety
t
28242
4
2
4
2
15.75.7)(
2
11515)(
4for
Example 4
415.7
4015.7
00
)(82
2
tee
te
t
tyt
t
Same result as before
