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 t
h(t)

4. Graphical Interpretation of Convolution Integral
h(-t) t
h(t)
Take impulse response and reverse it in time

5. Graphical Interpretation of Convolution Integral
h(-t)
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 x2(t) x1(t)

9. Example
x2(-t)
x2(t) t
Reverse function

10. Example
x2(-t)
x1(t) t t
Reverse function, slide x2 over x1 and evaluate integral

11. Example
x2(t-t)
x1(t) t t
Area of overlap is increasing linearly

12. Example
x2(t-t)
x1(t) t
t-2 t
Area of overlap constant

13. Example x1(t) x2(t-t) t t-2 Area declining linearly -
width of shaded area = 1-(t-2)=3-t

14. Example
x1(t)
x2(t-t) t t
After time t=3 the convolution integral is zero

15. Example
x1(t)*x2(t) t

16. tint=0;
tfinal=10;
tstep=.01;
t=tint:tstep:tfinal;
x=5*((t>=0)&(t<=4));
subplot(3,1,1), plot(t,x)
axis([ ])
h=3*((t>=0)&(t<=2));
subplot(3,1,2),plot(t,h)
axis([ ])
axis([ ])
t2=2*tint:tstep:2*tfinal;
y=conv(x,h)*tstep;
subplot(3,1,3),plot(t2,y)
axis([ ])

18. Example 2
Convolve the following functions t
x2(t)
x1(t)
1.0 t

19. Example 2 t
x2(-t) -1
Reversal

20. Example 2
x2(t-t) -1
t t
Shift reversed function

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

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

23. Example 2
x1(t)*x2(t)

24. Example 3

25. Example 3 5 3 t 4

26. Example 3 5 t
Reverse h(t)

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

28. Performing integral for 0<t<4
Example 3 5 t
Performing integral for 0<t<4 4

29. Example 3

30. Performing integral for t>4
Example 3 5 t
Performing integral for t>4 4

31. Example 3

32. Example 3

34. 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) t

36. Example 4 t

37. Example 4 t
t-4

38. Example 4

39. Example 4
Same result as before