Since convolution is commutative (), we have also
That and are transform pairs is expressed by writing or .
In words, convolution in the time domain is multiplication in the frequency domain.
Taking the -transform of both sides of Eq. (2.2.1) and applying the convolution theorem gives
where is the -transform of the filter impulse response. Thus the-transform of the filter output is the -transform of the input times the -transform of the impulse response.
The general difference equation for an LTI filter appears as
Taking the -transform of both sides, denoting the transform by gives
using linearity and the shift theorem. Replacing by , by , and solving for , which equals the transfer function , yields