An important property of convolution that is very useful in its calculation is the commutative property.
This property can be proved easily as follows:
Now that the basic idea of convolution has been introduced, we can do a few examples of the convolution integral.
Consider an LTI system with an exponential impulse response, h(t), as shown below.
If an input comprising of two time shifted delta functions as shown in the figure is input to this sytem, determine the output.
Method A: The impulse response h(t) means that if a d(t) is input, then h(t) is output. Now, if d(t+1) is input, then h(t+1) will be output, and if d(t-1) is input then h(t-1) will be output, since the system is time-invariant.
When d(t+1)+d(t-1) is input, the output will be h(t+1)+h(t-1). This is drawn below.
Method B:
A system with a rectangular impulse response, h(t), as shown below, is given a step input, x(t)=u(t). Calculate the output y(t).
The output, y(t), can be calculated using the convolution integral:
The product of the two functions x(T) and h(t-T) in variable T is shown graphically below. Obviously, when t<0, the overlap of x(T) and h(t-T) is zero and y(t)=0. When t>=0 the overlap is from 0 to t. Both the functions have amplitude =1. The area of the product is:
When t>1, then the entire h(t-T) is in the area of overlap as shown below, and the output is:
Review the interactive example given in Lec#2A to see how the above calculation is performed graphically.
In addition to the commutative property given above, there are two other important properties of convolution.
Convolution is associative
x(t)*[h 1 (t) * h 2 (t)] = [x(t)*h 1 (t) ] * h 2 (t)Convolution is distributive
x(t)*[h 1 (t) + h 2 (t)] = x(t)*h 1 (t) + x(t) * h 2 (t)
© Suresh Devasahayam