In the simple case of univariate discrete data over a finite domain t ∈ {0, .., N − 1} (i.e., restricting to finite impulse response filters) we can simplify these operations to a discrete circular convolution.

Hamilton College

One critical aspect of the convolution operation is that it can be computed by
an element-wise product of the Fourier transforms of the two functions:

Fourier Transform of the Convolution Operation

Hamilton, W.L. Graph Representation Learning. *Synthesis Lectures on Artificial Intelligence and Machine Learning*. 2020, 14(3), 1–159. https://doi.org/10.2200/S01045ED1V01Y202009AIM046

Graph Representation Learning by William Hamilton

Discrete Circular Convolution

Fourier coefficient corresponding to the sequence (f(x0), f(x1), ..., f(xN−1)) is sk ∈ {s0..., sN−1}. In the equation shown below, we use the notation *N to emphasize that this is a circular convolution defined over the finite domain
{0, ..., N − 1}, but we will often omit this subscript for notational simplicity.

 Discrete Fourier Transform (DFT)

This property means that translating a signal and then convolving it by a filter
is equivalent to convolving the signal and then translating the result. 

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