Convolution: A Systems Approach


These notes are taken from notes by Searl Dunn.
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Convolution is a powerful tool for determining the output of a system to any input.  The Convolution Theorem is developed here in a completely mathematical way.  On this page we will derive the convolution theorem: If the input to a system is x(t), and the impulse response of that system is h(t), then we can determine the output of the system, y(t), from the integral:

\[y\left( t \right) = \int_0^t {h\left( {t - \lambda } \right)f\left( \lambda \right)d\lambda } \]

Start with a linear time-invariant (LTI) system (in box).  There is an input to the system, x(t), and an output from the system, y(t).

Now consider the same system with the input as an impulse, δ(t).  The output is then, by definition, the impulse response, h(t).

If we use a delayed impulse into the system, the output is just the impulse response with the same delay.

Because of linearity, if we scale the input by any factor, the output will be scaled by the same factor.  In particular, we can scale the input (and hence the output) by the factor x(λ)dλ.

If we integrate the input, the output is also integrated.  In this case we'll take the upper limit of the integration to be t+; but it could just as well have been positive infinity.  We could also take the lower lower limit to be negative infinity.

By the sifting property of the impulse function we know that (if λ>0, t>0):

\[\int_0^t {\delta \left( {t - \lambda } \right)x\left( \lambda \right)d\lambda = x\left( t \right)} \]

Therefore the input to the system is x(t)

If the input is x(t), then the output must be y(t) (by definition, from the first diagram). 

Equating the outputs of the last two diagrams leads directly to the convolution theorem:

\[y\left( t \right) = \int_0^t {h\left( {t - \lambda } \right)f\left( \lambda \right)d\lambda } \]