ESPRIT

假设D个中心频率相同的远场窄带信号入射到一个M个阵元的均匀圆阵列中，则按逆时针来看，第l个阵元输出为

x_{l} (t) = \sum_{k = 1}^{D} s_{k} (t) e^{c o s φ_{k} c o s (\frac{2 π (l - 1)}{M} - θ_{k})} + n_{l} (t)

$x_{l}(t)=\sum_{k=1}^{D}s_{k}(t)e^{cos\varphi_{k}cos(\frac{2\pi(l-1)}{M}-\theta_{k})}+n_{l}(t)$
其中

s_{k} (t)

$s_{k}(t)$ 表示第

k

$k$ 个源信号，

θ_{k}, φ_{k}

$\theta_{k},\varphi_{k}$ 分别表示第

k

$k$ 个信号的俯仰角和方位角，

n_{l} (t)

$n_{l}(t)$ 表示第

l

$l$ 个阵元上的噪声。将上式写成矩阵形式

X (t) = A S (t) + N (t)

$X(t)=AS(t)+N(t)$
其中

X (t) = [x_{1} (t), \dots, x_{M} (t)]^{T}

$X(t)=[x_{1}(t),\cdots,x_{M}(t)]^{T}$ 为阵元的输出矢量，

N (t) = [n_{1} (t), \dots, n_{M} (t)]^{T}

$N(t)=[n_{1}(t),\cdots,n_{M}(t)]^{T}$ 为加性白噪声矢量，

S (t) = [s_{1} (t), \dots, s_{D} (t)]^{T}

$S(t)=[s_{1}(t),\cdots,s_{D}(t)]^{T}$ 表示源信号矢量，

A = [a (θ_{1}, φ_{1}), \dots, a (θ_{D}, φ_{D})]

$A=[\mathbf{a}(\theta_{1},\varphi_{1}),\cdots,\mathbf{a}(\theta_{D},\varphi_{D})]$ 表示阵列方向矩阵。并且其中

a (θ_{k}, φ_{k}) = [a_{1} (θ_{k}, φ_{k}, \dots, a_{M} (θ_{k}, φ_{k}))]^{T}, a_{l} (θ_{k}, φ_{k}) = e^{i 2 π c o s (φ_{k}) c o s (\frac{2 π (l - 1)}{M} - θ_{k]})}

$\mathbf{a}(\theta_{k},\varphi_{k})=[a_{1}(\theta_{k},\varphi_{k},\cdots,a_{M}(\theta_{k},\varphi_{k}))]^{T},a_{l}(\theta_{k},\varphi_{k})=e^{i2\pi cos(\varphi_{k})cos(\frac{2\pi(l-1)}{M}-\theta_{k]})}$

因为环阵结构并不能直接利用ESPRIT算法去进行测向，因而需要进行波束空间转换，从而利用贝塞尔函数的递归关系构造等式，从而进行测向。假设经过转换矩阵 $F_{o}^{H}$ ,将均匀圆阵导向矢量变换成为 $\mathbf{b}(\theta,\varphi)$ ，那么转换矩阵的具体定义可以表示为

F_{o}^{H} = C_{o} C_{v} V^{H}

$\mathbf{F_{o}^{H}=C_{o}C_{v}V^{H}}$
其中相应的矩阵具体定义为

C_{0} = d i a g {(- 1)^{N}, \dots, (- 1)^{1}, \dots, 1^{N}} C_{v} = d i a g {j^{- N}, \dots, j^{- 1}, \dots, j^{- N}} V = \sqrt{M} [w_{- N}, \dots, w_{N}] w_{m}^{H} = \frac{1}{M} [1, e^{j 2 π m / M}, \dots, e^{j 2 π m (M - 1) / V}]

$\mathbf{C_{0}}=diag\{(-1)^{N},\cdots,(-1)^{1},\cdots,1^{N}\}\\\mathbf{C_{v}}=diag\{j^{-N},\cdots,j^{-1},\cdots,j^{-N}\}\\\mathbf{V}=\sqrt{M}[\mathbf{w}_{-N},\cdots,\mathbf{w}_{N}]\\\mathbf{w}_{m}^{H}=\frac{1}{M}[1,e^{j2\pi m/M},\cdots,e^{j2\pi m(M-1)/V}]$
其中

N

$N$ 表示所能激励的最大阶数。那么我们可以得到

b (θ, φ)

$\mathbf{b}(\theta,\varphi)$ 的具体表达式为

b (θ, φ) = F_{o}^{H} a (θ, φ) = \sqrt{M} J_{η} v (φ)

$\mathbf{b}(\theta,\varphi)=\mathbf{F}_{o}^{H}\mathbf{a}(\theta,\varphi)=\sqrt{M}\mathbf{J}_{\eta}\mathbf{v}(\varphi)$
这里

v (φ) = [e^{- j N φ}, \dots, 1, \dots, e^{j N φ}], J_{η} = d i a g {J_{- N} (η), \dots, J_{N} (η)}

$\mathbf{v}(\varphi)=[e^{-jN\varphi},\cdots,1,\cdots,e^{jN\varphi}],\mathbf{J}_{\eta}=diag\{J_{-N}(\eta),\cdots,J_{N}(\eta)\}$ 其中

η

$\eta$ 的具体表达式为

η = k r s i n θ

$\eta=krsin\theta$ ,然后我们从

b (θ, φ)

$\mathbf{b}(\theta,\varphi)$ 抽取三个大小为

2 N - 1

$2N-1$ ，让

b^{0}, b^{1}, b^{- 1}

$\mathbf{b^{0},\mathbf{b^{1}},\mathbf{b}^{-1}}$ 分别表示

b (θ, φ)

$\mathbf{b}(\theta,\varphi)$ 的最后

2 N - 1

$2N-1$ 的元素、中间

2 N - 1

$2N-1$ 以及前面

2 N - 1

$2N-1$ 元素，根据贝塞尔函数的递归关系

J_{m - 1} (ξ) + J_{m + 1} (ξ) = \frac{2 m}{ξ} (ξ)

$J_{m-1}(\xi)+J_{m+1}(\xi)=\frac{2m}{\xi}(\xi)$
我们可以得到关系

Ξ b^{0} = ζ b^{- 1} + ζ^{*} b^{1}

$\mathbf{\Xi}\mathbf{b}^{0}=\zeta\mathbf{b}^{-1}+\zeta^{*}\mathbf{b}^{1}$
其中

ξ = s i n θ e^{j φ}

$\xi=sin\theta e^{j\varphi}$ ,

Ξ = (λ / r) d i a g {- (N - 1), \dots, N - 1}

$\mathbf{\Xi}=(\lambda/r)diag\{-(N-1),\cdots,N-1\}$ ,然后对我们阵元所接收到信号求相关矩阵，得到信号子空间

S_{u}

$\mathbf{S}_{u}$ 和噪声子空间.根据参考文献[1]

这里写图片描述

最后得到 $\theta_{i},\varphi{i}$ 就是我们需要的测向角。相应的代码如下

clear all;
close all;
j=sqrt(-1);
% ======= elevation and azimuth in degree ======== %
azimuth   = [-35 -40 -15  13  19  ];          % from -180 - 180
elevation = [ 60  41  28  17  2];         % from 0 - 90
% ================================================ %
snapshot = 512;
n =  size( azimuth, 2 ); % number of  signals
SNR = 20;
P = 10.^(SNR/10);
Ps = diag( ones( 1, n )*P ) ;        % power of singal
Pn = 1;         % power of noise    
N = 13;         %  number of sensors N > 2n , test for N = 5, 6 ,8 , 10 ,...
Border = 6;%ceil(N/2);%5;%6;     % order of first kind Bessel function 
if Border > N/2
    Border = Border - 1;
end
M = -Border : Border;
M1 = 2*Border + 1;              
Ntry = 200; 
for mm = 1:Ntry   
    %  X is an observation data
    X = ucadata( azimuth ,elevation ,Ps,snapshot,Pn,N);
    Rx = X*X'/snapshot;     %   Covariance of observation datas

    temp1 = ones( N,1 )*M;
    R = 2*pi*j*[ 0: N-1 ]'/N ; 
    temp2 = R*ones( 1, M1 );
    Wm = exp( temp1.*temp2 )/N; %   Weight matrix wm

    Cv = diag(j.^( -abs( M ) ) );
    V = sqrt( N )*Wm;           %   Weight matrix V
    FeH = Cv*V';                %   Eq.( 9 ) in reference

    alpha = 2*pi*M/M1;
    alpha = ones( M1, 1 )*alpha;
    temp3 = M'*ones( 1,M1 );
    W = exp( j*temp3.*alpha )/sqrt( M1 );   % Weight matrix
    FrH = W'*FeH;               % Real Beam Former 

    Ry = FrH*Rx*FrH';
    Rr = real( Ry );            % Real Beam space covariance matrix

%    [ U, D, Vec ] = svd( Rr );
    [Vec, Deig] = eig(Rr);
    S = Vec( :, 1: n );         % Real Beam space signal subspace
    % G = Vec( :, n + 1 : end );  % Real Beam space noise subspace

    Co = diag( [( -1 ).^( [ Border:-1:1 ]) ones( 1, M1 - Border ) ] );
    % FoH = Co*W*FeH;
    So_hat = Co*W*S;
    % ---------------------------------- %
    Me = M1 - 2;
    S_1 = So_hat( 1: Me,: );
    S0 = So_hat( 2: Me + 1,: );
    temp = abs( 2 - Border : Border );
    D = diag( (-1).^temp );
    In = zeros( Me,Me );

    for i = 1:Me
        In(i, Me +1-i) = 1;
    end
    E2 = D*In*( conj( S_1 ) ); % 
    E = [ S_1,E2 ];  
    T = diag( [ -( Border - 1 ):( Border - 1 ) ] )/pi;
    Ls = E\( T*S0 );
    y = Ls( 1:end/2 , : )';
    [ Ui, Vi ] = eigs( y );
    EV = diag( Vi );
    AZIMUTH(mm,:) = ( angle( EV ) * 180 /pi )';
    ELEVATION(mm,:) = ( asin( abs( EV ) ) * 180/pi )';
end
AzimuthMean = mean(AZIMUTH);
ElevationMean = mean(ELEVATION);
clc
fprintf( ' @----------------------------- UCA ESPRIT -----------------------------@\n ' );
Info = sprintf('Number of elements = %d \nNumber of snapshots = %d \nNumber of signals = %d \nSNR = %d dB \nBessel = %d ',N ,snapshot,n ,SNR,Border)
fprintf( 'Original Angles are as follows: ( in degree )\n' ),
azimuth,elevation

fprintf( ' @---------------------------- Simulation Result ---------------------------@ \n' );
fprintf( 'Estimated Angles are as follows: ( in degree )\n' ),
AzimuthMean, ElevationMean
fprintf( 'Maximum Estimation Error are as follows: ( in degree )\n' ),
AzimuthMaxError = max(abs(sort(AzimuthMean) - sort(azimuth))),
ElevationMaxError = max(abs(sort(ElevationMean) - sort(elevation)))
fprintf( ' @---------------------------------- END  -----------------------------------@ ' );

相应的结果如下

这里写图片描述

reference

[1] C. P. Mathews and M. D. Zoltowski, “Eigenstructure techniques for 2-D angle estimation with uniform circular arrays,” in IEEE Transactions on Signal Processing, vol. 42, no. 9, pp. 2395-2407, Sep 1994.

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