The Convolution Integral Convolution operation given symbol ‘*’ “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 t h(t)
Graphical Interpretation of Convolution Integral h(-t) t h(t) Take impulse response and reverse it in time
Graphical Interpretation of Convolution Integral h(-t) h(t-t) t t Then shift it by time t
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
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 x2(t) x1(t)
Example x2(-t) x2(t) -2 0 2 t Reverse function
Example x2(-t) x1(t) -1 0 1 t t Reverse function, slide x2 over x1 and evaluate integral
Example x2(t-t) x1(t) 0 1 t t Area of overlap is increasing linearly
Example x2(t-t) x1(t) 0 1 t t-2 t Area of overlap constant
Example x1(t) x2(t-t) t t-2 Area declining linearly - width of shaded area = 1-(t-2)=3-t
Example x1(t) x2(t-t) 0 1 t t After time t=3 the convolution integral is zero
Example x1(t)*x2(t) 0 1 2 3 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 x2(t) x1(t) 1.0 0 1 t
Example 2 0 1 t x2(-t) -1 Reversal
Example 2 x2(t-t) -1 0 t 1 t Shift reversed function
Example 2 0 t 1 t x2(t-t) -1 Overlay shift reversed function onto other function and integrate overlapping section x1(t)
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
Example 2 x1(t)*x2(t) 0 1 2
Example 3
Example 3 5 3 t 4
Example 3 5 t Reverse h(t)
Shift the reversed h(t) by t Example 3 5 t t 4 Shift the reversed h(t) by t
Performing integral for 0<t<4 Example 3 5 t Performing integral for 0<t<4 4
Example 3
Performing integral for t>4 Example 3 5 t Performing integral for t>4 4
Example 3
Example 3
Commutativity of Convolution Operation The actions of flipping and shifting can be applied to EITHER function
Example 4 Repeat example 3 by flipping and shifting x(t) rather than h(t) 0 t
Example 4 0 t
Example 4 0 t t-4
Example 4
Example 4 Same result as before