Why Fourier transforms? – Limit Infinite

Fourier transforms have multiple uses. In practical applications, they are used to express a given function/vector in terms of the complex exponential basis. They are also useful as a mathematical tool to simplify computation of convolution of two functions/vectors.

Regarding their first use, Fourier transforms express an $N$ -dimensional vector in terms of complex exponential orthonormal basis $e^{i \frac{2\pi k}{N}}, k = 0,1,\cdots,N-1$ .

Expressing a vector in terms of a different basis gives the coefficients of the vector along each of the new basis’ components.

What is the significance of this particular basis? Why are the coefficients of a vector along the complex exponential basis’ components important?

One application is in identifying the frequency components of a function. Complex exponentials model the frequency components in a function. A complex exponential function of a continuous variable $e^{i \omega t}$ models is a function of frequency $\omega$ radians/sec. In signal processing applications, identifying the frequency components of a signal is a routine requirement.

Regarding the use of Fourier transforms as a mathematical tool, given two vectors $N$ -dimensional vectors, $\vec{x}$ and $\vec{y}$ , their convolution $\vec{x}*\vec{y}$ is defined as:

$(\vec{x}*\vec{y})_k = \sum\limits_{i=0}^{N-1} x_iy_{k-i}$

The convolution operation appears in many places. In signal processing applies, the output from a linear time-invariant (LTI) system of impulse response $\vec{h}$ given an input $\vec{x}$ is $\vec{h}*\vec{x}$ . In probability, convolution operation comes in computing the joint distribution (?) of two random variables. The Hilbert transform is defined as a convolution of a function with the function $\frac{1}{x}$ .

While convolution is a common operation, its computation is expensive, both in practical applications and in theoretical analysis. Fourier transforms come handy in computing convolutions because of the result: Fourier transform of a convolution of two functions(vectors) is equal to product of the Fourier transforms of the two functions(vectors). So, convolution of two functions(vectors) can be calculated by first computing the Fourier transforms of the two functions(vectors), multiplying them and then taking the inverse Fourier transform of the product.