In the process of analog/digital signal conversion, when the sampling frequency fs.max is greater than 2 times the highest frequency fmax in the signal (fs.max>2fmax), the digital signal after sampling completely retains the information in the original signal, In general practical applications, the sampling frequency is guaranteed to be 2.56 to 4 times the highest frequency of the signal; the sampling theorem is also known as the Nyquist theorem .
Before explaining the sampling theorem, it is necessary to understand the sampling tools "Fourier transform" and "unit impulse function".
Let's talk about the properties used by the Fourier transform: the product in the time domain is the convolution in the frequency domain.
Let’s talk about the properties of the unit impulse function:
Convolution with other functions
For details, see "Detection Technology Third Edition" P44
To sum up in human words: the convolution of the signal function and the unit impulse function in the time domain is to translate the signal function from the point as the origin to the intersection of the impulse function and the horizontal axis in the time domain coordinates. Similarly, the convolution in the frequency domain coordinates The product is also similar to image translation. previous image
The sampling process of the signal is to discretize the signal by multiplying the signal function and the unit response impulse function in the time domain. Then the sampled signal spectrum image is the result of the convolution of the two functions in the frequency domain coordinates. Therefore, to analyze the sampled signal spectrum, it is necessary to first find out the respective spectrums of the two functions, and perform convolution calculations. As shown below
As shown in the figure, if the sampling period is too small, the signal spectrum will overlap each other after each convolution and translation.
As shown in the figure, to eliminate this overlapping noise, fs>=2fm is required, that is, the sampling frequency needs to be greater than twice the highest frequency of the sampled signal.
The above is a graphic description of the signal sampling theorem.