相位phei = 2*pi*f*d*sind(theta)，因此理論上來講測向的算法都可以用來測頻。

ESPRIT：Estimating signal parameters viarotational invariance techniques，中文為旋轉因子不變法。

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ESPRIT用于測頻：

%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:LX(:,n) = x(n:(n+m-1)); end for n = 1:LY(:,n) = x((n+1):(n+m)); end %Rxx\Rxy Rxx = 0; for i = 1:LRxx = Rxx+X(:,i)*X(:,i)'; end Rxx = Rxx/L; Rxy = 0; for i = 1:LRxy = 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('結果對比：') [omega_est,s_omega']

　　

ESPRIT用于測向：

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')

　　

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