As mentioned above, the convolution integral provides an easy mathematical way to express the output of an LTI system based on an arbitrary signal, x (t), and the system's impulse response, h(t). The convolution integral is expressed as:
Convolution is such an important tool that it is represented by the symbol *, and can be written as
y (t) = x (t) * h (t)
By making a simple change of variables into the convolution integral, ז=t-ז, we can easily show that convolution is commutative:
x (t) * h (t) = h (t) * x (t)
Brief Overview of Derivation Steps:
1. An impulse input leads to an impulse response output.
2. A shifted impulse input leads to a shifted impulse response output. This is due to the time invariance of the system.
3. We now scale the impulse input to get a scaled impulse output. This is using the scalar multiplication property of linearity.
4. We can now "sum up" an infinite number of these scaled impulses to get a sum of an infinite number of scaled impulse responses. This is using the additivety attribute of linearity.
5. Now we recognize that this infinite sum is nothing more than an integral, so we convert both sides into integrals.
6. Recognizing that the input is the function f (t), we also recognize that the output is exactly the convolution integral.