多重信号分类MUSIC算法
1. MUSIC算法原理
MUSIC算法，叫做多信号分类算法 (Multiple Signal Classification)，是一种基于特征结构的高分辨率DOA算法。该算法利用了信号子空间和噪声子空间正交性的特点，构造噪声空间然后通过谱峰搜索来检测信号的波达方向。需要注意的是，该算法有一个前提，即各个入射信号之间互不相关，这样才能保证入射信号的协方差矩阵是满秩的。
然后就可以用 P ( ω ) = 1 a H ( ω ) U ^ N U ^ N H a ( ω ) \boldsymbol P(\omega)=\frac{1}{\boldsymbol a^{\mathrm{H}}(\omega) \boldsymbol {\hat U_N} \boldsymbol {\hat U^{\mathrm{H}}_N} \boldsymbol a(\omega)} P(ω)=aH(ω)U^NU^NHa(ω)1进行谱峰搜索，寻找r个入射信号的来波方向。
2. MATLAB仿真代码
% Code For Music Algorithm % Author:痒羊羊 % Date：2020/10/28 clc; clear all; close all; %% -------------------------initialization------------------------- f = 500; % frequency c = 1500; % speed sound lambda = c/f; % wavelength d = lambda/2; % array element spacing M = 10; % number of array elements N = 100; % number of snapshot K = 6; % number of sources doa_phi = [-30, 0, 20, 40, 60, 75]; % direction of arrivals %% generate signal dd = (0:M-1)'*d; % distance between array elements and reference element A = exp(-1i*2*pi*dd*sind(doa_phi)/lambda); % manifold array, M*K S = sqrt(2)\(randn(K,N)+1i*randn(K,N)); % array of random signal, K*N X = A*S; % received data without noise, M*N X = awgn(X,10,'measured'); % received data with SNR 10dB %% calculate the covariance matrix of received data and do eigenvalue decomposition Rxx = X*X'/N; % covariance matrix [U,V] = eig(Rxx); % eigenvalue decomposition V = diag(V); % vectorize eigenvalue matrix [V,idx] = sort(V,'descend'); % sort the eigenvalues in descending order U = U(:,idx); % reset the eigenvector P = sum(V); % power of received data P_cum = cumsum(V); % cumsum of V %% define the noise space J = find(P_cum/P>=0.95); % or the coefficient is 0.9 J = J(1); % number of principal component Un = U(:,J+1:end); %% music for doa; seek the peek theta = -90:0.1:90; % steer theta doa_a = exp(-1i*2*pi*dd*sind(theta)/lambda); % manifold array for seeking peak music = abs(diag(1./(doa_a'*(Un*Un')*doa_a))); % the result of each theta music = 10*log10(music/max(music)); % normalize the result and convert it to dB %% plot figure; plot(theta, music, 'linewidth', 2); title('Music Algorithm For Doa', 'fontsize', 16); xlabel('Theta(°)', 'fontsize', 16); ylabel('Spatial Spectrum(dB)', 'fontsize', 16); grid on; 运行结果：
部分MUSIC算法原理参考张贤达先生的《矩阵分析与应用》。
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