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This module discusses convolution of continuous signals in the time and frequency domains.
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, $x(t)$ , and the system's impulse response, $h(t)$ . The convolution integral is expressed as
Let $f$ and $g$ be two functions with convolution $f*g$ .. Let $F$ be the Fourier transform operator. Then
By applying the inverse Fourier transform ${F}^{-1}$ , 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).
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