DOA——ESPRIT算法

相位phei = 2*pi*f*d*sind(theta)，因此理论上来讲测向的算法都可以用来测频。

ESPRIT：Estimating signal parameters viarotational invariance techniques，中文为旋转因子不变法。

ESPRIT用于测频：

%Example_esprit

clear all;clc;close all;

N=200;     %signal length

m = N/2;

L=N-m;

s_amp=[1.31*exp(1i*pi/4),2.07*exp(1i*pi/3),1.88*exp(1i*pi/5)];

s_omega=[0.12*pi,0.37*pi,0.72*pi];

eps_sigma=0;

sigma=0.5;

x=zeros(1,N);%initialize

w = sqrt(sigma)*randn(1,N);

n = [1:N];

for slen=1:length(s_omega)

    x = x+s_amp(slen)*exp(1j*s_omega(slen)*n)  ;

end

x=x+w;

for n = 1:L

    X(:,n) = x(n:(n+m-1));

end

for n = 1:L

    Y(:,n) = x((n+1):(n+m));

end

%Rxx\Rxy

Rxx = 0;

for i = 1:L

    Rxx = Rxx+X(:,i)*X(:,i)';

end

Rxx = Rxx/L;

Rxy = 0;

for i = 1:L

    Rxy = Rxy+X(:,i)*Y(:,i)';

end

Rxy = Rxy/L;

[A,B] = eig(Rxx);

var = min(diag(B));

I = eye(m);

Z = diag(ones(1,m-1),-1);

Cxx = Rxx - I*var;

Cxy = Rxy - Z*var;

[~,B] = eig(Cxx,Cxy);

f=angle(diag(B));

[~,fpos]=sort(abs(abs(diag(B))-1));

f=f(fpos);

fval=f(f>0);

omega_est=sort(fval(1:length(s_amp)));

disp('结果对比：')

[omega_est,s_omega']

　　

ESPRIT用于测向：

clear all

clf

set(0,'defaultaxesfontsize',22);

M=12;

Nb=32;

P=3;

%angles=[25 80 135]*(pi/180);

angdeg=[80 90 115];

angles=angdeg*(pi/180);

dlambda=0.5;

%generate random bits of information

poles=zeros(1,P);

X=zeros(M,Nb);

Rideal=zeros(M,M);

for k=1:P,

mu=pi*cos(angles(1,k));

poles(1,k)=exp(j*mu);

a=exp(j*mu*(0:M-1)).';

Rideal=Rideal+a*a';

br=ones(1,Nb);

temp=rand(1,Nb);

br(find(temp<.5))=-1;

bi=ones(1,Nb);

temp=rand(1,Nb);

bi(find(temp<.5))=-1;

b=br+j*bi;

X=X+a*b;

end

%add some noise

X=X+0.6*(randn(M,Nb)+j*randn(M,Nb));

Rxx=X*X'/Nb;

[E,D,V]=svd(Rxx);

%ESPRIT algorithm:

Es=E(:,1:P);

Es1=Es(1:M-1,:); Es2=Es(2:M,:);

Psi=Es1\Es2;

[T,Phi]=eig(Psi);

Phivec=diag(Phi);

%plot eigenvalues from ESPRIT and compare with true frequencies

polar(0,1,'.')

hold on

plot(real(poles),imag(poles),'kx','MarkerSize',12,'Linewidth',2);

plot(real(Phivec),imag(Phivec),'ro','MarkerSize',12,'Linewidth',2);

hold off

legend('True "poles"','ESPRIT eigenvalues')