Nyquist theorem and Fourier transform

People always think things are running in the simplest way: for audio, people always think it's as low as possible.
If a tape recorder has a sampling rate of 8000Hz, the recorder can capture at most 4000Hz sound, and a sound with a maximum frequency of 100Hz, then the tape recorder sampling rate is at least 200Hz to record this sound perfectly.
The auditory range of the ear is 20hz~20000hz, so the sampling rate of the ear is 40000Hz, and the sound frequency range of the ear and the mechanical frequency range are different.
Popular explanation: A lamp light one off, it's frequency is 50Hz; the person sees the light, the eye opens immediately closes. In order to see the light every time, the human eye must open 100 times in the 1s clock, because the lamp 1s clock has 100 lights out state.

What if it violates the Nyquist theorem?
The low frequency recorder sees the world as if it is a low frequency, even if it is higher, it will be mistaken for low frequency.
I jump rope, 180 per minute, 3Hz, if the frequency of 1.3Hz for me to take pictures, up to record 78 of my state, at most think I jumped 39, that is, 0.65Hz. In other words, the limit frequency that the camera can see is 0.65Hz.
In fact, the camera doesn't think I jumped 39. But 3%1.3=0.4hz, 24.
In summer, the fan turns fast, and the human eye looks like the fan is turning backwards.
The frequency of electromagnetic movement in human eyes is different from the frequency of mechanical movement of human eyes.

Discrete Fourier is a special continuous Fourier
For n audio points, discrete Fourier always thinks its sample rate is N, playing time is 1s
The continuous Fourier sampling rate can be changed arbitrarily, and the discrete Fourier sampling rate is fixed. For n audio points, when the continuous Fourier sampling rate is less than n, the audio is distorted after the inverse transform, and when the continuous Fourier sampling rate equals N, the Fourier transform describes the audio signal perfectly, and when the continuous Fourier sampling rate is greater than n, the audio signal can also be perfectly described, but it is superfluous and unnecessary to exceed N.
The continuous Fourier form is more flexible, and discrete Fourier can be used to improve efficiency by using periodicity and division.
n Audio points need to be described with n sine waves

The time domain waveform has a positive negative, the offset position of the echo band
Audio processing: Timestretch,pitchshift,tempochange
Tempochange: Change the playback time, change the tone
Timestretch: Changes the playback time, does not change the tone
Pitchshift: Change tone, do not change playback time

Nyquist theorem and Fourier transform