在Matlab中使用MVNRND创建2个高斯相关矩阵

Question

这是我在这里询问的第一个问题，请耐心等待。我对Matlab并不陌生，但之前从未使用过MVNRND函数，而且我的统计知识并不强大。我试图做的总结如下：我试图创建一个函数，生成2个相关的相位屏幕（NxN矩阵），将用于电磁高斯谢尔模型光束传播模拟。光束需要用于X和Y偏振状态的单独的随机相位屏幕。我到目前为止的代码如下。

function [phz_x,phz_y]=GSM_phase_screen_2(l_phi_x,l_phi_y,sigma_phi_x, ... 
sigma_phi_y,gamma,N,delta) 
%GSM_PHASE_SCREEN_2 
% This code generates two correlated 2-D NxN Gaussian Schell-Model (GSM) 
% phase screens (matrices) of unit variance circular complex Gaussian 
% random numbers for the X and Y polarization states provided l_phi_x, 
% l_phi_y, sigma_phi_x, sigma_phi_y, and gamma. It utilizes the MVNRND 
% Matlab function. 
% 
% l_phi_x:  [m] correlation length in X phase screen 
% l_phi_y:  [m] correlation length in Y phase screen 
% sigma_phi_x: phase variance in X phase screen 
% sigma_phi_y: phase variance in Y phase screen 
% gamma:   correlation coefficient 
% N:    number of samples per side of grid 
% delta:   [m] sample grid spacing 
% 
% phz_x:   [rad] 2-D phase screen for X polarization state 
% phz_y:   [rad] 2-D phase screen for Y polarization state 

% ORIGINAL AUTHOR: Santasri Basu 
% MODIFIED BY: Matthew Gridley 
%  Added input arguments needed to generate 2 correlated phase 
%  screens, updated PSD equations, and replaced RANDN with MVNRND 
% 

% Setup the Power Spectral Density (PSD) 
del_f = 1/(N*delta); % frequency grid spacing [1/m] 
fx = (-N/2 : N/2-1) * del_f; 

% Frequency grid 
[fx,fy] = meshgrid(fx); 

% GSM phase PSD 
PSD_phi_x = (sigma_phi_x^2) * pi * (l_phi_x^2) * gamma * ... 
    exp(-((pi * l_phi_x)^2) * (fx.^2 + fy.^2)); 
PSD_phi_y = (sigma_phi_y^2) * pi * (l_phi_y^2) * gamma * ... 
    exp(-((pi * l_phi_y)^2) * (fx.^2 + fy.^2)); 

% Random draws of Fourier series coefficients 
% (zero mean Gaussian random numbers) 
% 
% the 2 lines of code below need changed to generate the correlated random 
% draws using MVNRND and GAMMA 
cn_x = (randn(N) + 1i*randn(N)) .* sqrt(PSD_phi_x) * del_f; 
cn_y = (randn(N) + 1i*randn(N)) .* sqrt(PSD_phi_y) * del_f; 

% Synthesize the phase screens 
phz_x = real(ift2(cn_x,1)); 
phz_y = real(ift2(cn_y,1)); 

end 



function [g, x] = ift2(G, df) 
% [g, x] = ift2(G, df) 
% 2-D inverse Fourier transform that keeps the origin at the center of 
% the grid. 
% 
% G: Complex field in frequency space 
% df: Spacing in frequency space [m^-1] 
% g: Complex field in coordinate space 

% Core function written by Jason Schmidt 
% Modified: 17 Apr 2010 
% By: Daniel J. Wheeler 
% 
% x output added functionality by Michael Steinbock 6/8/2014 
% 

g = ifftshift(ifft2(ifftshift(G))) * (length(G) * df)^2; 

%% Calc x: 
if nargout == 2 
    N = size(G, 1); 

    x = (0 : N-1)/(N*delta_f); 
end 

在上面的代码，代码的两行与开始的评论下面的“％随机抽取傅立叶级数系数​​”是，我需要帮助。我以前用你看到的代码做了两个矩阵，但意识到它们不是高斯相关的。从我的学术顾问的建议，我应该使用MVNRND生成这些阶段屏幕。在查看了MVNRND的帮助文件之后，我迷失在如何使用它的目的。我在这里搜索，试图找到类似的问题和答案，没有运气，我也搜索了谷歌。任何人都可以协助改变这两行代码来利用MVNRND。谢谢！

Answer 1

在这里输入代码经过大量的研究，我想出了如何使用MVNRND来满足我的目的。我的意图是创建4个随机NxN矩阵来替换下面的代码片段中randn（N）的4个用法。我需要用MVNRND替换这些随机矩阵的原因是它们是相关的。在MVNRND中，你必须提供一个协方差矩阵。这就是困扰我的东西。

cn_x = (randn(N) + 1i*randn(N)) .* sqrt(PSD_phi_x) * del_f; 
cn_y = (randn(N) + 1i*randn(N)) .* sqrt(PSD_phi_y) * del_f; 

要创建它，我有4个不同的随机值，其中我必须计算组合对的方差（协方差）：rx_real，rx_imag，ry_real，和ry_imag。一旦我明白了这一点，我就可以创建协方差矩阵。

下一个问题是确定需要设置MVNRND中的'个案'值。我需要4个相关的NxN矩阵，所以我确定案例需要是4xN^2的矩阵。然后，我可以使用'reshape'命令将MVNRND输出转换为需要的4个NxN矩阵。

请参阅下面的代码。希望这可以帮助别人！

% Multivariate normal parameters 
mu = zeros([1,4]); % Zero mean Gaussian 
% Covariance matrix for 4 circular complex Gaussian random numbers: 
% rx_real, rx_imag, ry_real, ry_imag 
% 
% [<rx_real rx_real> <rx_real rx_imag> <rx_real ry_real> <rx_real ry_imag>; 
% <rx_imag rx_real> <rx_imag rx_imag> <rx_imag ry_real> <rx_imag ry_imag>; 
% <ry_real rx_real> <ry_real rx_imag> <ry_real ry_real> <ry_real ry_imag>; 
% <ry_imag rx_real> <ry_imag rx_imag> <ry_imag ry_real> <ry_imag ry_imag>] 
sigma = [1 0 gamma 0; 
     0 1 0 gamma; 
     gamma 0 1 0; 
     0 gamma 0 1]; 
cases = N^2; % matrix of random vectors 

r = mvnrnd(mu, sigma, cases); % gives a 512^2x4 double matrix 

rx_real = reshape(r(:,1),[N N]); 
rx_imag = reshape(r(:,2),[N N]); 
ry_real = reshape(r(:,3),[N N]); 
ry_imag = reshape(r(:,4),[N N]); 

% Correlated random draws of Fourier series coefficients 
cn_x = (rx_real + 1i*rx_imag) .* sqrt(PSD_phi_x) * del_f; 
cn_y = (ry_real + 1i*ry_imag) .* sqrt(PSD_phi_y) * del_f;