At that time, there were two famous mathematicians who reviewed this paper, namely Joseph Louis Lagrange (1736- 18 13) and Laplacian (1749- 1827). When Laplace and other examiners voted to publish this paper, Lagrange firmly opposed it. In recent 50 years, Lagrange insisted that Fourier method could not represent angular signals, such as discontinuous slopes in square waves. The French scientific society succumbed to Lagrange's prestige and rejected Fourier's work. Fortunately, Fourier has other things to do. He took part in the political movement. After Napoleon's expedition to Egypt, the French Revolution was put to the guillotine, and he has been escaping. This paper was not published until 15 after the death of Lagrange. Who is right? Lagrange is right: sine curves cannot be combined into angular signals. But we can use sine curve to represent it very closely, so there is no energy difference between the two representations. Based on this, Fourier is right. Why use sine curve instead of original curve? For example, we can also use square waves or triangular waves instead. There are countless ways to decompose signals, but the purpose of decomposing signals is to process the original signals more simply. It will be simpler to represent the original signal with sine and cosine, because sine and cosine have a property that the original signal does not have: sine fidelity. After a sinusoidal signal is input, the output is still sinusoidal, only the amplitude and phase may change, but the frequency and waveform are still the same. Only sine curve has this property, which is why we don't use square wave or triangular wave to express it. 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. Any function can be expressed as a linear combination of sine functions through a certain decomposition, and sine functions are relatively simple function classes that have been fully studied in physics: 1. Fourier transform is a linear operator, and it is still 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 inherent function of differential operation, so that the solution of linear differential equation is transformed 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.