Question

这是我在这里提出的第一个问题，请耐心等待。我不是Matlab的新手，但之前从未使用过MVNRND函数，而且我的统计知识不强。我试图做的总结如下：我正在尝试创建一个生成2个相关相位屏幕（NxN矩阵）的函数，该屏幕将用于电磁高斯谢尔模型光束传播模拟。光束需要单独的随机相位屏幕用于X和Y偏振态。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

在上面的代码中，注释下面的两行代码以＆＃34;％随机抽取傅里叶级数系数＆＃34;是我需要帮助的地方。我以前使用你看到的代码制作了两个矩阵，但发现它们不是高斯相关的。根据我的学术顾问的建议，我应该使用MVNRND来生成这些阶段屏幕。在查看了MVNRND的帮助文件后，我迷失了如何将其用于此目的。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。任何人都可以帮助改变这两行代码以利用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。我的目的是创建4个随机NxN matricies来代替下面代码片段中randn（N）的4种用法。我需要用MVNRND替换这些生成随机矩阵的原因是它们是相关的。在MVNRND中，您必须提供协方差矩阵。这就是困扰我的。

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，我必须计算组合对的方差（协方差）：rx_real，rx_imag，ry_real和ry_imag。一旦我弄明白了，我就能够创建协方差矩阵。

下一个问题是要确定需要设置MVNRND中的“案例”值。我需要4个相关的NxN矩阵，所以我确定需要4xN ^ 2矩阵的情况。然后我就可以使用'reshape'命令将MVNRND输出转换为所需的4个NxN matricies。

请参阅下面的代码。希望这有助于其他人！

% 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;

在Matlab中使用MVNRND创建2个高斯相关矩阵

1 个答案: