Looking back at Fourier algorithm from the application of signal analysis

Fourier transform is a very important algorithm in the field of digital signal processing. But what is the significance of this algorithm?

To understand the significance of the Fourier transform algorithm, we must first understand the significance of the Fourier principle. The Fourier principle states that any continuously measured time series or signal can be represented as an infinite superposition of sine wave signals of different frequencies. The Fourier transform algorithm created based on this principle uses the original signal directly measured to calculate the frequency, amplitude and phase of different sine wave signals in the signal in an accumulative manner.

The inverse Fourier transform algorithm corresponds to the Fourier transform algorithm. The inverse transform is essentially an accumulation process, which can convert the individually changed sine wave signals into one signal.

Therefore, it can be said that Fourier transform converts the originally difficult-to-process time domain signals into easy-to-analyze frequency domain signals (signal spectrum), and these frequency domain signals can be processed and processed using some tools. Finally, the inverse Fourier transform can be used to convert these frequency domain signals into time domain signals.

From the perspective of modern mathematics, Fourier transform is a special kind of integral transform. It can express a function that meets 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.

Fourier transform belongs to the content of harmonic analysis. The word "analysis" can be interpreted as in-depth research. From a literal point of view, the word "analysis" actually means "analyze in detail". It achieves an in-depth understanding and research of complex functions by "analyzing in detail" the functions. From a philosophical point of view, "analysisism" and "reductionism" are to achieve the purpose of enhancing the understanding of the essence of things through appropriate analysis of the internal parts of things. For example, modern atomic theory attempts to analyze the origin of all matter in the world into atoms, but there are only hundreds of atoms. Compared with the infinite richness of the material world, this kind of analysis and classification undoubtedly provides a good means for understanding the various properties of things.

The same is true in the field of mathematics. Although Fourier analysis was originally used as a tool for analytical analysis of thermal processes, its thinking method still has typical reductionist and analytical characteristics. "Any" function can be expressed as a linear combination of sine functions through certain decompositions, and sine functions are a well-studied and relatively simple function class in physics. This idea is very similar to the atomic theory in chemistry! The amazing thing is that modern mathematics has discovered that Fourier transform has very good properties, making it so easy to use and useful that people have to marvel at the magic of creation:

1. The Fourier transform is a linear operator. If given an appropriate norm, it is also a unitary operator;

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

3. Sine basis functions are eigenfunctions of differential operations, so that the solution of linear differential equations can be transformed into the solution of algebraic equations with constant coefficients. In a linear time-invariant physical system, frequency is an invariant property, so the response of the system to complex excitations can be obtained by combining its responses to sinusoidal signals of different frequencies;

4. The famous convolution theorem states that Fourier transform can transform complex convolution operations into simple product operations, thus providing a simple means of calculating convolution;

5. The discrete form of the Fourier transform can be calculated quickly using a digital computer (the algorithm is called the Fast Fourier Transform (FFT) algorithm).

It is precisely because of the above-mentioned good properties that Fourier transform has a wide range of applications in physics, number theory, combinatorial mathematics, signal processing, probability, statistics, cryptography, acoustics, optics and other fields.