Fourier transform is an important algorithm in the field of digital signal processing. To know the meaning of Fourier transform algorithm, we must first understand the meaning of Fourier principle.

Fourier principle shows that any continuously measured time series or signal can be expressed as infinite superposition of sine wave signals with different frequencies. Based on this principle, the Fourier transform algorithm uses the directly measured original signal to calculate the frequency, amplitude and phase of different sine wave signals in this signal through accumulation.

Corresponding to the Fourier transform algorithm is the inverse Fourier transform algorithm. In essence, this inverse transformation is also an accumulation process, which makes the sine wave signal that changes independently be converted into a signal. Therefore, it can be said that Fourier transform is to transform the time domain signal that is difficult to process into the frequency domain signal (signal spectrum) that is easy to analyze, and these frequency domain signals can be processed and processed by some tools. Finally, these frequency domain signals can be converted into time domain signals by inverse Fourier transform.

From the point of view of modern mathematics, Fourier transform is a special integral transform. It can represent a function satisfying certain conditions as a linear combination or integral of sine basis functions.

In different research fields, Fourier transform has many different variants, such as continuous Fourier transform and discrete Fourier transform. In the field of mathematics, although Fourier analysis was originally used as an analytical tool for thermodynamic processes, its thinking method still has the characteristics of typical reductionism and analytical theory. The "arbitrary" function can be expressed as a linear combination of sine functions through a certain decomposition, and sine functions are relatively simple functions that have been fully studied in physics:

1) Fourier transform is a linear operator, and it is also a unitary operator if a proper norm is given;

2) The inverse transform of Fourier transform is easy to find, and the form is very similar to the forward transform;

3) Sine basis function is the intrinsic function of differential operation, which transforms the solution of linear differential equation into the solution of algebraic equation with constant coefficient. Linear convolution operation is a simple product operation, which provides a simple method to calculate convolution.

4) In the discrete Fourier physical system, frequency is an invariable property, so the response of the system to complex excitation can be obtained by combining its responses to sinusoidal signals with different frequencies; 5. The famous convolution theorem points out that Fourier transform can be transformed into complex transform and can be quickly calculated by digital computer (its algorithm is called FFT). Because of the above good properties, Fourier transform is widely used in physics, number theory, combinatorial mathematics, signal processing, probability, statistics, cryptography, acoustics, optics and other fields.

Reference link:

The Practical Significance of Fourier Series Expansion _ Baidu Library

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