作者：桂。
时间：2018-04-24 22:04:52
链接:
前言
本文为一文的补充，给出另一种矩阵运算的思路。
一、算法流程
对于复数相关矩阵R，通过矩阵变换，在维度不变的情况下，转化为实数矩阵：
对于MUSIC算法，该思路可以降低Jacobi运算复杂度。额外的操作仅仅是少量的乘法操作，即耗费少量硬件资源换取更快速的处理时间。
直接复数转实数，需要将nxn的矩阵扩展为2n x 2n的矩阵，而直接转化的相关矩阵仍然为 n x n，降低了Jacobi的复杂度。
容易证明U*Un为新的特征向量，而U可与导向矢量a提前乘法处理，存储到Ram里。
这里可以看出：（R + J*conj(R)*J）/2等价于中心对称线阵的前、后项空间平滑算法，而斜Hermitian矩阵的特征向量与转化的特征向量等价，因此可以得出特性：对于具备中心对称特性的线阵，复数->实数，既可以降低Jacobi复杂度，又具备了解相干信号的能力。
二、仿真验证
未做实数化处理，code:
clc;clear all;close all%Ref:Narrowband direction of arrival estimation for antenna arraysdoas=[-30 -5 40]*pi/180; %DOA's of signals in rad.P=[1 1 1]; %Power of incoming signalsN=10; %Number of array elementsK=1024; %Number of data snapshotsd=0.5; %Distance between elements in wavelengthsnoise_var=1; %Variance of noiser=length(doas); %Total number of signals% Steering vector matrix. Columns will contain the steering vectors% of the r signalsA=exp(-i*2*pi*d*(0:N-1)'*sin([doas(:).']));% Signal and noise generationsig=round(rand(r,K))*2-1; % Generate random BPSK symbols for each of the% r signalsnoise=sqrt(noise_var/2)*(randn(N,K)+i*randn(N,K)); %Uncorrelated noiseX=A*diag(sqrt(P))*sig+noise; %Generate data matrixR=X*X'/K; %Spatial covariance matrix[Q ,D]= svd(R); %Compute eigendecomposition of covariance matrix[D,I]=sort(diag(D),1,'descend'); %Find r largest eigenvaluesQ=Q(:,I);%Sort?the?eigenvectors?to?put?signal?eigenvectors?firstQs=Q (:,1:r); %Get the signal eigenvectorsQn=Q(:,r+1:N); %Get the noise eigenvectors% MUSIC algorithm%?Define?angles?at?which?MUSIC???spectrum????will?be?computedangles=(-90:0.1:90);%Compute steering vectors corresponding values in anglesa1=exp(-i*2*pi*d*(0:N-1)'*sin([angles(:).']*pi/180));for k=1:length(angles)%Compute?MUSIC???spectrum??music_spectrum(k)= 1/(a1(:,k)'*Qn*Qn'*a1(:,k));endfigure(1)plot(angles,abs(music_spectrum))title('MUSIC Spectrum')xlabel('Angle in degrees') 
实数化处理，code：
clc;clear all;close all%Ref:Narrowband direction of arrival estimation for antenna arraysdoas=[-30 -5 40]*pi/180; %DOA's of signals in rad.P=[1 1 1]; %Power of incoming signalsN=10; %Number of array elementsK=1024; %Number of data snapshotsd=0.5; %Distance between elements in wavelengthsnoise_var=1; %Variance of noiser=length(doas); %Total number of signals% Steering vector matrix. Columns will contain the steering vectors% of the r signalsA=exp(-i*2*pi*d*(0:N-1)'*sin([doas(:).']));% Signal and noise generationsig=round(rand(r,K))*2-1; % Generate random BPSK symbols for each of the% r signalsnoise=sqrt(noise_var/2)*(randn(N,K)+i*randn(N,K)); %Uncorrelated noiseX=A*diag(sqrt(P))*sig+noise; %Generate data matrixR=X*X'/K; %Spatial covariance matrix%% Reconstruct%实数n = size(R);I = eye(n/2);J = fliplr(eye(n));U = 1/sqrt(2)*[I fliplr(I);1j*fliplr(I) -1j*I];R = 0.5*U*(R+J*conj(R)*J)*U';% Reconstruct_end[Q ,D]= svd(R); %Compute eigendecomposition of covariance matrix[D,I]=sort(diag(D),1,'descend'); %Find r largest eigenvaluesQ=Q(:,I);%Sort?the?eigenvectors?to?put?signal?eigenvectors?firstQs=Q (:,1:r); %Get the signal eigenvectorsQn=Q(:,r+1:N); %Get the noise eigenvectors% MUSIC algorithm%?Define?angles?at?which?MUSIC???spectrum????will?be?computedangles=(-90:0.1:90);%Compute steering vectors corresponding values in anglesa1=exp(-i*2*pi*d*(0:N-1)'*sin([angles(:).']*pi/180));for k=1:length(angles)%Compute?MUSIC???spectrum??music_spectrum(k)= 1/(a1(:,k)'*U'*Qn*Qn'*U*a1(:,k));endfigure(1)plot(angles,abs(music_spectrum))title('MUSIC Spectrum')xlabel('Angle in degrees') 