This module discusses convolution of continuous signals in the time and frequency domains.
Continuous Time Fourier Transform
The CTFT transforms a infinite-length continuous signal in the time domain into an infinite-length continuous signal in the frequency domain.
The convolution integral expresses the output of an LTI system based
on an input signal,
, and the system's impulse response,
convolution integral is expressed as
Convolution is such an important tool that it is represented
by the symbol , and can be written as
Convolution is commutative. For more information on the characteristics of the convolution
integral, read about the Properties of Convolution.
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).