Fourier transform is a mathematical tool to convert time domain signals into frequency domain signals, which is widely used in signal processing, image processing and other fields. The following is a simple deduction of Fourier transform:

For the continuous time periodic function f(x), it can be expressed by the series expansion of trigonometric function:

Where an and bn are coefficients, which can be obtained by the function) f(x). This expansion is called trigonometric series expansion.

We can also write trigonometric functions in complex exponential form:

Substituting it into the above formula, we get:

These include:

Cn is the Fourier coefficient of f(x). On the contrary, Fourier series expansion can be written as:

In this way, we get the Fourier transform.