[Digital Signal Processing] Bandpass Sampling Theorem and MATLAB Simulation

1. Bandpass sampling theorem

According to the Nyquist sampling theorem (low-pass sampling), the sampling frequency $f_{s}$ $f_{max}$ in the signal $f_{ma x}$ 2 times, it can ensure that the sampled digital signal can be restored to the original signal without distortion after being converted by DAC. However, if the frequency of the signal is distributed in a certain limited frequency band, and the highest frequency $f_{max}$ of the signal $B$ (band-pass signal), if it is still processed according to the low-pass sampling theorem at this time, a particularly high sampling rate is required. On the one hand, it will cause a huge amount of calculation for subsequent signal processing, and the real-time performance of digital signal processing cannot be guaranteed; On the one hand, the performance of the ADC device is limited, and the corresponding sampling frequency cannot be realized. Therefore, a sampling method suitable for band-pass signals is needed to meet the above requirements.

1.1 Content

Band-pass sampling theorem : Suppose a time-continuous analog signal $x (t)$ , which is band-limited at ( $f_{L}$ ， $f_{H}$ ), if the sampling frequency of the signal satisfies:
$f_{s} =\frac{2(f_{L}+f_{H})}{2m-1}=\frac{4f_{0}}{2m-1} ---------(1-1)$
$f_{s}≥=2(f_{H}-f_{L})=2B-----------(1-2)$
式中， $f_{0}=\frac{(f_{L}+f_{H})}{2}$ is the center frequency of the band-pass signal, $B=f_{H}-f_{L}$ is the bandwidth of the signal, $m = 1, 2, ...$ , take a positive integer that can satisfy the above two formulas.
Then use $f_{s}$ The signal sampling values obtained by sampling at equal intervals can be restored to the original signal without distortion.

1.2 Formula derivation

Second, MATLAB signal simulation

2.1 Signal simulation experiment

The 3KHz and 67KHz signals are sampled at a sampling frequency of 64KHz, what will happen to the frequency of the signal after sampling?
insert image description here
It can be found that the 67KHz signal is sampled at a sampling frequency of 64KHz, and the sampled signal waveform is consistent with the 3KHz signal waveform, indicating that the frequency spectrum of the sampled signal has been shifted, that is, 67-64=3.

2.2 MATLAB code

clc;
clear;
close all;

fs = 64000;     % 采样频率
f1 = 3000;
f2 = 67000;
N = 100;        % 数据长度
t = (0:N-1)/fs;

x1 = sin(2*pi*f1*t);
x2 = sin(2*pi*f2*t);

figure;
subplot(2,1,1);plot(t,x1);title('f1 = 3KHz');
subplot(2,1,2);plot(t,x2);title('f2 = 67KHz');

3. Summary

(1) The essence of sampling is to move the spectrum of the signal, and the most fundamental requirement is that the spectrum of the signal after sampling is not aliased.
(2) The low-pass sampling theorem requires the sampling frequency $f_{s}$ $f_{max}$ in the signal $f_{ma x}$ 2 times, and the sampling frequency of band-pass sampling has nothing to do with the highest frequency of the signal, but only with the bandwidth of the signal.
(3) The value of the sampling frequency in the band-pass sampling theorem is a discontinuous segmental interval, which is different from the minimum sampling frequency of the low-pass sampling signal.
(4) The minimum sampling frequency of band-pass sampling can be equal to twice the signal bandwidth, and it is often taken as four times or higher than the signal bandwidth in practical engineering applications.

reference

[1] Wang Po. Research and Implementation of Key Algorithms for PD Radar Signal Processing [D]. Nanjing University of Information Science and Technology, 2019. DOI: 10.27248/d.cnki.gnjqc.2019.000075. [
2] Bandpass Sampling Theorem in Engineering [School Zhiyong Series of Courses on Digital Signal Processing]
[3] Chen Boxiao, et al. Analysis and Design of Modern Radar Systems [M]. Xi'an: Xidian University Press, 2012.9.