The first animation and the second animation are actually Fourier transform analysis of the periodic rectangular wave in the time domain (approximating as a rectangular wave, not a rectangular square wave of strict meaning).
For the first graph, it focuses on one of the essences of transformation: superposition, each circle representing a harmonic component.
The second graph focuses on discrete spectrograms.
However, these two graphs actually only show the spectrum analysis of the periodic signal, corresponding to the discrete spectrum, and only analyze the very special time domain waveform. However, through these two animations, I must have a deeper impression on the Fourier changes!
Fourier transform is a very important algorithm in the field of digital signal processing. To understand the meaning of the Fourier transform algorithm, we must first understand the significance of the Fourier principle. The Fourier principle shows that any continuous measurement of timing or signal can be expressed as an infinite superposition of sinusoidal signals of different frequencies. The Fourier transform algorithm created according to this principle uses the directly measured raw signal to calculate the frequency, amplitude and phase of different sinusoidal signals in the signal in an accumulated manner.
Corresponding to the Fourier transform algorithm is the inverse Fourier transform algorithm. This inverse transformation is also essentially an accumulation process, so that a separately changed sine wave signal can be converted into a signal.
Therefore, it can be said that the Fourier transform converts the time domain signal that was originally difficult to process into a frequency domain signal (signal spectrum) that is easy to analyze, and some of the frequency domain signals can be processed and processed by some tools. Finally, these frequency domain signals can also be converted into time domain signals using an inverse Fourier transform.
From the perspective of modern mathematics, the Fourier transform is a special integral transform. It can represent a function that satisfies certain conditions as a linear combination or integral of a sinusoidal basis function. In different fields of research, Fourier transforms come in many different variants, such as continuous Fourier transforms and discrete Fourier transforms.
In the field of mathematics, although the initial Fourier analysis is used as a tool for analytical analysis of thermal processes, its method of thinking still has typical characteristics of reductionism and analyticism. An "arbitrary" function can be represented as a linear combination of sinusoidal functions by a certain decomposition, while a sine function is physically a well-studied and relatively simple function class:
1. The Fourier transform is a linear operator, and if given the appropriate norm, it is still a é…‰ operator;
2. The inverse transform of the Fourier transform is easy to find, and the form is very similar to the positive transform;
3. The sinusoidal basis function is the eigenfunction of the differential operation, so that the solution of the linear differential equation can be transformed into the solution of the algebraic equation with constant coefficients. The linear time-invariant convolution operation is a simple product operation, thus providing A simple means of calculating convolution;
4. The famous convolution theorem states that the Fourier transform can be transformed quickly using a digital computer (the algorithm is called the Fast Fourier Transform Algorithm (FFT)).
It is because of the above good properties that Fourier transform has a wide range of applications in the fields of physics, number theory, combinatorial mathematics, signal processing, probability, statistics, cryptography, acoustics, and optics.
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