The Evaluation of the Convolution Integral

Consider the system described by the differential equation:

which has an impulse response given  by

We will use convolution to find the zero input response of this system to the input given by

The next section reiterates the development of the page deriving the convolution integral.  If you feel you know that material, you can skip ahead to the mechanics of using the convolution integral.

Convolution as sum of impulse responses

For continuity with the page deriving the convolution integral we can approximate the input by a series of impulses...
plot  the response of the system to each of these impulses...
and plot the response as the sum of the individual responses  (also shown is the exact solution).

Convolution Integral

Likewise the convolution can be considered from the point of view of the convolution integral.

Click on picture below to see animation.

The Mechanics of Using the Convolution Integral

To find the output of the system with impulse response

to the input

we will use the convolution integral

Because the input function has three distinct regions t<0, 0<t<1 and 1<t, we will need to split up the integral into three parts.

Part 1:  t<0

For t<0 the argument of the impulse function (t-t) is always negative.  Since h(t-t)=0 for (t-t)<0, the result of the integral is zero for t<0. 

This situation is depicted graphically below (t=-0.2):

 So the result for the first part of our solution is

Part 2:  0<t<1

For 0<t<1 we need to evaluate the integral only from t=0 to t=t, since f(t)=0 when t<0, and h(t-t)=0 when (t-t)<0 (or, equivalently t<t).  So the integral becomes, in effect: 

This situation is depicted graphically below (t=0.5):

We can now evaluate the integral of the solid black line:

Thus, the result for the second part of the solution  is

 

Part 3:  1<t

For 1<t we need to evaluate the integral only from t=0 to t=1, since f(t)=0 when t<0 and when t>1.  So the integral becomes, in effect: 

This situation is depicted graphically below (t=1.2):

We can now evaluate the integral:

Thus, the result for the third part of the solution is:

 

The complete answer. 

We can get the results for all time by combining the solutions from the three parts.

This result is shown below.  Click on the image to see an animation of the convolution operation.

The convolution integral is a completely general method for finding the output of a linear system for any input.  The integral is often difficult to evaluate, but this page gives one example of how this can be accomplished for a relatively simple system.

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Comments or Questions?

Erik Cheever Engineering Department Swarthmore College