Adaptive Beamforming Simulation
Topics and design ideas
topic
Design an adaptive beamforming algorithm so that the beam always captures the incoming waves in the N strongest signal directions, and distinguishes these directions.
Design ideas
This question uses the MUSIC algorithm and the adaptive beamformer based on the maximum likelihood criterion to realize the tasks of signal direction finding and beamforming respectively.
1. First, give the direction and strength of the signal source, as well as the noise power, simulate the Doppler effect, receive the signal, obtain the signal matrix, and perform SVD decomposition to obtain the signal subspace and noise subspace
2. Then use the MUSIC algorithm to calculate the spectrum estimate to find the maximum value, which is the direction of the main signal;
3. Next, use the adaptive beamformer based on the maximum likelihood criterion to capture the signal in the main signal direction,
4. Finally, draw the captured signal and the spectrum estimate calculated by using the MUSIC algorithm to obtain a schematic diagram of the effect before and after beamforming.
MUSIC Algorithm and Beamforming Principle
MUSIC algorithm
The MUSIC algorithm is a multi-source positioning method and an algorithm based on spatial spectrum estimation. It can effectively extract the information of the source of the target signal from the complex environment with noise and interference signals. Based on the positioning idea of the MUSIC algorithm, the received signal matrix y can be divided into two parts, namely the signal s subspace and the noise n subspace, that is, decomposed into y=s+n . At this point, the signal matrix Y can be decomposed into:
where U and V are the left and right singular value decompositions of the input signal matrix Y respectively, S is the singular value matrix, Es and En are the projection matrices of the signal subspace and noise subspace respectively, Ss and Sn are the projection matrices of the signal subspace and noise subspace matrix, Vs and Vn .
is the right singular value decomposition of the signal subspace and the noise subspace. The basic idea of the MUSIC algorithm is: the noise subspace is inverted to generate a set of orthogonal functions based on the noise subspace, and it is used to solve the signal source direction. That is, the noise subspace En with En rank (Ne-Ns) is used as the search space, and a search vector a( θ i) is constructed for each search angle θi , which is an orthogonal basis of En , and the beamforming amount of the MUSIC algorithm is defined as for:
From this, the spectrum estimation value PMUSIC(θi) can be obtained , and the maximum value is the direction of the signal source to be obtained.
Adaptive Beamforming Algorithm Based on Maximum Likelihood Criterion
The adaptive beamforming algorithm based on the maximum likelihood criterion is a method for capturing the optimal beam shape related to the direction of the transmitted signal. Its basic idea is, given the position of a set of signal sources, calculate a set of optimal beam shape parameters according to the maximum likelihood idea, so as to capture the signal related to the direction of the signal source to the greatest extent, while minimizing the signal related to the direction of the signal source. irrelevant signal. The estimation of the adaptive beamformer adopts the maximum likelihood criterion, and its estimator is:
Among them, Y is the received signal matrix, and w is the beam shape parameter to be estimated. The optimization estimation problem of the maximum likelihood criterion can be solved by using the gradient descent algorithm, and the optimization iteration formula is:
Among them, μ is the step size and d is the desired signal.
Simulation result analysis
Beam direction finding refers to a technology that uses technical means to determine the azimuth angle of the target through the detected Doppler signal during the Doppler radar measurement process. Beamforming refers to the weighted sum of the signals of each array element on the antenna array composed of Doppler radar transmitter receivers, so that the transmitter receiver array has strong transmit and receive directivity.
In this simulation, firstly by setting the strength and direction of the signal source, the signal emitted by the signal source in the Doppler radar system is simulated, and then the signal is added to Gaussian white noise to simulate the actual received signal, and by obtaining the signal matrix and SVD decomposition The method is to determine the signal subspace and noise subspace, and then use the MUSIC algorithm to estimate the spectrum of the signal, so as to find the angle corresponding to the maximum signal spectrum value, which is the direction of the main signal, and finally through the maximum likelihood criterion The adaptive beamformer captures the signal in the main_theta direction of the main signal to achieve beamforming.
Beam direction finding using the MUSIC algorithm
4 directions are set here, respectively 25°, -25°, 45°, -45°, indicating that different signals are emitted from 4 directions; the strength of the signal source in the 4 directions is set, and the signal source in the 25° direction The strength is 1, indicating that the signal in the 25° direction is the strongest, which is the main signal, and the signal strengths in the other directions are 0.6 and 0.4, indicating that the signals in other directions are interference; the number of array elements is 20, and 20 array elements are used to receive signals ;Set the noise power to 0.1, indicating the power of the added Gaussian white noise as noise; the number of signal sampling is 8.
由图可知，MUSIC方法能够识别出信号强度最强的方向，为24.2度，与实际的25度差距小于一度，方向测量较为精准。
下图展示了使用基于极大似然准则自适应波束成形器对于信号获取情况的示意图，由图可知，该方法能够使雷达接收到的信号更强，并能够突出和接收到目标雷达的信号，并且能够滤除干扰。如图所示，原来的信号接收到的强度为-10.84dB，而经过使用基于极大似然准则自适应波束成形器对于信号获取后，目标信号强度增强到了25.04dB，，因此该方法对于信号接收是十分重要的。
下面的仿真将结合波束成形算法对算法进行优化。
基于极大似然准则自适应波束成形器仿真结果
上图展示了使用基于极大似然准则自适应波束成形器对于信号获取情况的示意图，由图可知，该方法能够使雷达接收到的信号更强，并能够突出和接收到目标雷达的信号，并且能够滤除干扰。如图所示，原来的信号接收到的强度为-10.84dB，而经过使用基于极大似然准则自适应波束成形器对于信号获取后，目标信号强度增强到了25.04dB，因此该方法对于接收信号有明显提升。
由上图可见，两者的波形是基本一致的，波束成形后接收信号增强。
%% 信号源同时从4个方向发送不同信号，设置25度方向信号最强为主信号，其他方向信号为干扰，同时加入高斯白噪声作为噪声
clear;clc;
% 信号源方向
theta = [25, -25, 45, -45];
% 信号源强度
A = [1, 0.6, 0.6, 0.4];
% 信号源数目
N_s = length(theta);
% 阵元数目
N_e = 20;
% 噪声功率
N_0 = 0.1;
% 信号采样次数
N_t = 8;
% 初始化
y = zeros(N_e,N_t);
% 循环模拟接收信号
for k = 1:N_t
% 模拟信号源发射信号
for i = 1:N_s
y(:,k) = y(:,k) + A(i)*exp(1j*(theta(i)*pi/180)).^(0:N_e-1)';
end
% 加入高斯白噪声
y(:,k) = y(:,k) + sqrt(N_0/2)*(randn(N_e,1)+1j*randn(N_e,1));
end
%% 使用MUSIC算法进行波束测向
% 求取信号矩阵
R_y = y*y'/N_t;
% 求取SVD分解
[U,S,V] = svd(R_y);
% 确定信号子空间
E_s = U(:,1:N_s);
% 确定噪声子空间
E_n = U(:,N_s+1:end);
% MUSIC算法实现
theta_MUSIC = -90:0.1:90;
P_MUSIC = zeros(length(theta_MUSIC),1);
% 计算谱估计
for i=1:length(theta_MUSIC)
a = exp(1j*(theta_MUSIC(i)*pi/180)).^(0:N_e-1)';
P_MUSIC(i) = 1./abs(a'*E_n*E_n'*a);
end
% 绘制谱估计值
figure;
plot(theta_MUSIC,10*log10(P_MUSIC),'linewidth',1);
axis([-90,90,-16,-9]);
xlabel('角度');
ylabel('信号强度 (dB)');
title('使用MUSIC算法进行波束测向');
% 找出最大值，即为主信号来向
[~,index] = max(P_MUSIC);
theta_main = theta_MUSIC(index);
fprintf('主信号方向为：%.1f°\n',theta_main);
%%
% 基于极大似然准则的自适应波束成形器捕获主信号main_theta方向的信号
theta_main = theta_main;
P_ML = zeros(length(theta_MUSIC),1);
% 计算自适应波束形成器信号功率
for i=1:length(theta_MUSIC)
a = exp(1j*(theta_MUSIC(i)*pi/180)).^(0:N_e-1)';
P_ML(i) = (abs(a'*E_s*E_s'*a)).^2;
end
% 绘制信号功率图
figure;
plot(theta_MUSIC,10*log10(P_ML),'linewidth',1);
xlabel('角度');
ylabel('信号强度(dB)');
title('使用基于极大似然准则的自适应波束成形器对目标信号进行捕获');
%总图
figure;
plot(theta_MUSIC,10*log10(P_MUSIC),theta_MUSIC,10*log10(P_ML),'linewidth',1);
axis([-91,90,-20,30])
legend('使用MUSIC算法进行波束测向(未进行波束成形的原始波形)','使用基于极大似然准则的自适应波束成形器对目标信号进行捕获')
title('信号测向与波束成形算法效果示意图');