This module discusses convolution of discrete signals in the time and frequency domains.
The DTFT transforms an infinite-length discrete signal in the time domain into an finite-length (or -periodic) continuous signal in the frequency domain.
As mentioned above, the convolution sum provides a concise, mathematical way to express the output of an LTI system based on an arbitrary discrete-time input signal and the system's impulse response. The convolution sum is expressed as
Let and be two functions with convolution .. Let be the Fourier transform operator. Then
By applying the inverse Fourier transform , we can write:
The Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain (e.g., time domain) corresponds to point-wise multiplication in the other domain (e.g., frequency domain).