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If f(x)and g(x)are integrable functions with fourier transforms f̂(ξ)and ĝ(ξ)respectively, then the fourier transform of the convolution is given by the product of the fourier transforms f̂(ξ)and ĝ(ξ)(under other conventions for the definition of the fourier transform a constant factor may appear). How does it do this magic? Fourier transform is a mathematical model that decomposes a function or signal into its constituent frequencies
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It helps to transform the signals between two different domains like transforming the frequency domain to the time domain. The fourier transform takes a curve and finds the the frequencise it is made of. If the laplace transform of a signal exists and if the roc includes the jω axis, then the fourier transform is equal to the laplace transform evaluated on the jω axis.
More specifically, the goal is for you to understand how it represents the inner workings of the fourier transform, an incredibly important tool for math, engineering, and most of science.
This inspired me to create an online tutorial dedicated to the fourier transform You can also check out my online tutorial on kalman filtering at www.kalmanfilter.net This tutorial focuses on the practical aspects of the fourier transform while also providing the necessary mathematical foundation.
