Some considerations of system sampling frequency

In digital signal processing, it is required that the signals to be processed are discrete and quantified. But in the real world, such as voltage, temperature and so on is a continuous quantity, also is commonly said analog signal. Therefore, before the digital signal processing, these continuous signals need to be converted into digital signals. This conversion is done by the ADC. The conversion of analog signal to digital signal, actually contains two parts of the work, one is discrete, and the second is quantization.

The problem of continuous signal discretization is first seen. What transformation does a continuous signal undergo to become a discrete signal? How do you ensure that this transformation does not lose the information that a continuous signal carries? The discretization of the continuous signal is equivalent to the multiplication of the continuous signal and the impulse string, so that the corresponding continuous signal is preserved and the continuous signal is discretized only where the impulse string has value. What about the interval of the impulse string without losing the information carried by the continuous signal? The answer lies in the Nyquist sampling theorem: The signal bandwidth with a sampling frequency greater than or equal to twice times. Because the spectrum of the impulse string is still an impulse string , and the interval is sampling frequency fs. It is known from convolution theorem that the multiplication of time domain is equivalent to that of frequency domain, and if the Nyquist theorem is satisfied, there will be no spectral aliasing in the frequency domain, and the information carried by the continuous signal is not lost. In other words, in the case of satisfying the Nyquist sampling theorem, it is theoretically possible to reconstruct the original contact signal completely by means of discrete signals. That is to say, from the point of view of discretization, the sampling frequency only satisfies the Nyquist theorem.

Then look at the problem of quantification. Since digital signals have values only at certain points, they are specifically related to the number of digits quantified. But a continuous signal can have a value at any point. This problem arises when a value of a continuous signal is between two numeric values. For example, with 8-bit quantization, the continuous signal range is 0 to 1. At this time the minimum number of digital signal can be expressed in units of 0.00390625, if the continuous signal at a certain point in the value of 0.004, after quantization can only be expressed as 0.00390625, also is 00000001. In this way, the quantization process produces an error, which is called quantization error, or quantization noise. The research shows that the quantization noise power is (LSB) 2/12, wherein the LSB is the lowest bit represented by the value, in just the example, is 0.00390625. Accordingly, the power spectrum for quantization noise is (LSB) 2/(12*FS). In the actual sampling process, we certainly want to minimize the impact of noise. There are two options: one is to increase the number of sample bits to reduce the value of the LSB, and the other is to increase the sampling frequency. This means that in order to reduce the quantization error, the sampling frequency is expected to be as large as possible. However, in practice, the sampling frequency can not be infinite, because the higher the sampling rate means that the higher the data rate, the storage and processing of the system has brought a great burden. In order to solve this contradiction, the Oversampling method is often used, and the sampling frequency is more than the Nyquist frequency, which is generally 2-4 times. This can effectively reduce the impact of quantization noise. A multi-rate method is used in the process to reduce the sampling rate during the actual processing . About multi-rate processing, discussed later.

In summary, in practice, the selection of sampling frequency should follow the following principles: The most important is the Nyquist sampling theorem, followed by appropriate oversampling.

Some considerations of system sampling frequency