Applied Mathematics/General Fourier Transform

Fourier Transform

The Fourier transform relates the function's time domain, shown in red, to the function's frequency domain, shown in blue. The component frequencies, spread across the frequency spectrum, are represented as peaks in the frequency domain.

Fourier Transform is to transform the function which has certain kinds of variables, such as time or spatial coordinate, $f(t)$ for example, to the function which has variable of frequency.

Definition

$\hat{f}(\xi )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)e^{-i2\pi t\xi }dt$...(1)

This integral above is referred to as Fourier integral, while $\hat{f}(\xi )$ is called Fourier transform of $f(t)$. $t$ denotes "time". $\xi$ denotes "frequency".

On the other hand, Inverse Fourier transform is defined as follows:

$f(t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hat{f}(\xi )e^{i2\pi t\xi }d\xi$ ...(2)

In the textbooks of universities, the Fourier transform is usually introduced with the variable Angular frequency $\omega$. In other word, $\xi \to \omega =2\pi \xi$ is substituted to (1) and (2) in the books. In that case, the Fourier transform is written in two different ways.

1.

$\hat{f}(\omega )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)e^{-i\omega t}dt$

$f(t)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hat{f}(t)e^{i\omega t}d\omega$

2.

$\hat{f}(\omega )=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)e^{-i\omega t}dt$

$f(t)=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hat{f}(t)e^{i\omega t}d\omega$