The%20Convolution%20Integral

1
The Convolution Integral

• Convolution operation given symbol

y equals x convolved with h
2
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

3
Graphical Interpretation of Convolution Integral

• To correctly understand convolution it is often
  easier to think graphically

4
Graphical Interpretation of Convolution Integral

Take impulse response and reverse it in time
5
Graphical Interpretation of Convolution Integral

h(t-t)
t
t
Then shift it by time t
6
Graphical Interpretation of Convolution Integral

h(t-t)
x(t)
t
a
t
Overlay input function x(t) and integrate over
times where functions overlap - in this case
between a and t
7
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

8
Example

• Convolution of two gate pulses each of height 1

x1(t)
0 1 t
0 2 t
9
Example

x2(t)
x2(-t)
-2 0 2 t
Reverse function
10
Example

x1(t)
x2(-t)
-1 0 1 t
t
Reverse function, slide x2 over x1 and evaluate
integral
11
Example
x2(t-t)

x1(t)
0 1 t
t
Area of overlap is increasing linearly
12
Example

x2(t-t)
x1(t)
0 1 t
t-2
t
Area of overlap constant
13
Example

x1(t)
x2(t-t)
0 1 t
t
t-2
Area declining linearly - width of shaded area
1-(t-2)3-t
14
Example

x1(t)
x2(t-t)
0 1 t
t
After time t3 the convolution integral is zero
15
Example

0 1 2 3
t
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tint0 tfinal10 tstep.01
ttinttsteptfinal x5((tgt0)(tlt4)) subplot
(3,1,1), plot(t,x) axis(0 10 0
10) h3((tgt0)(tlt2)) subplot(3,1,2),plot(t,h)
axis(0 10 0 10) axis(0 10 0 5) t22tinttstep
2tfinal yconv(x,h)tstep subplot(3,1,3),plot(
t2,y) axis(0 10 0 40)
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18
Example 2

• Convolve the following functions

x1(t)
1.0
0 1 t
19
Example 2

Reversal
20
Example 2

x2(t-t)
0 t 1 t
-1
Shift reversed function
21
Example 2

22
Example 2

x1(t)
x2(t-t)
0 1 t t
-1
t-1
Overlay shift reversed function onto other
function and integrate overlapping section
23
Example 2

x1(t)x2(t)
0 1 2
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Example 3

25
Example 3

5
3
t
4
0
26
Example 3

5
t
Reverse h(t)
27
Example 3

5
t
t
4
Shift the reversed h(t) by t
28
Example 3

4
29
Example 3

30
Example 3

31
Example 3

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Example 3

33

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Commutativity of Convolution Operation

• The actions of flipping and shifting can be
  applied to EITHER function

35
Example 4

• Repeat example 3 by flipping and shifting x(t)
  rather than h(t)

0 t
36
Example 4

0 t
37
Example 4

0 t
t-4
38
Example 4

39
Example 4

Same result as before