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BAd_GVAMP_c_known.m
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BAd_GVAMP_c_known.m
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function [theta_b, dMSE_b] = BAd_GVAMP_c_known(Ai, A0, y, T_LMMSE, T_VN_denoising, x_true,b_true, NumBits,delta)
% Code for paper:"Bilinear Adaptive Generalized Adaptive Vector Approximate
% Message Passing", IEEE Access, 2018.
% main function for BAd-GVAMP algorithm (L = 1 case) with c known
% Code written by Xiangming Meng and Jiang Zhu
% Email: [email protected], [email protected]
% 2018, Sep. 27
global T
[m, n] = size(A0);
% Perform estimation
computeMse = @(noise) 20*log10(norm(noise(:))/norm(x_true));
computeMseb = @(noise) 20*log10(norm(noise(:))/norm(b_true));
dMSE = zeros();
dMSE_b = zeros();
noise_var = zeros();
% initialization
x_hat_1k= zeros(n,1);
Q = length(b_true);
theta_b = zeros(Q,1);
% initialization of EM parameters
AQ_est = gen_matrix(Ai,b_true,Q);
A = A0+AQ_est;
wvar_hat = 1e1;
mu0 = x_true;
vx = 1e-12;
% Initialization for nonlinear case
lar_num = 1e6;
sma_num = 1e-6;
z_A_ext = zeros(m,1);
v_A_ext = lar_num;
gamma2k = sma_num;
r2k = zeros(size(x_true));
damp = 0.8; % damping factor
for t = 1:T
if(NumBits==1)
c0 = x_hat_1k'*x_true/(x_hat_1k'*x_hat_1k+eps);
dMSE(t) = computeMse(c0*x_hat_1k-x_true);
cb = theta_b'*b_true/(theta_b'*theta_b+eps);
% cb = 1;
dMSE_b(t) = computeMseb(cb*theta_b-b_true);
else
dMSE(t) = computeMse(x_hat_1k-x_true);
dMSE_b(t) = computeMseb(theta_b-b_true);
end
if NumBits < inf % nonlinear observations
% obtain the equivalent linear observations
[z_B_post, v_B_post] = outputUpdate(y, z_A_ext, v_A_ext*ones(m,1), sqrt(wvar_hat), NumBits,delta);
v_B_post = mean(v_B_post);
sigma2_tilde = v_B_post.*v_A_ext./(v_A_ext-v_B_post); %
sigma2_tilde = lar_num*(sigma2_tilde<0)+sigma2_tilde.*(sigma2_tilde>0);
sigma2_tilde = min(sigma2_tilde,lar_num);
sigma2_tilde = max(sigma2_tilde,sma_num);
y_tilde = sigma2_tilde.*(z_B_post./v_B_post-z_A_ext./v_A_ext); %
sigma2_tilde = mean(sigma2_tilde);
else
y_tilde = y;
if t==1
sigma2_tilde = wvar_hat; % for unquantized measurements, the noise variance is initialized
end
end
noise_var(t) = sigma2_tilde;
% LMMSE estimation
for k0 = 1:T_LMMSE
gammaw = 1/sigma2_tilde;
AQ_est = gen_matrix(Ai,theta_b,Q);
A_eq = A0+AQ_est;
Cov = gammaw*(A_eq'*A_eq)+gamma2k*eye(n);
Cov_inv = inv(Cov);
xhat2k = Cov\(gammaw*A_eq'*y_tilde+gamma2k*r2k);
eta2k = n/trace(inv(Cov));
H = zeros(Q,Q);
beta = zeros(Q,1);
for qi = 1:Q
for qj = 1:Q
H(qi,qj) = trace(Ai(:,:,qj)'*Ai(:,:,qi)*(Cov_inv+xhat2k*xhat2k'));
end
beta(qi) = trace(y_tilde'*Ai(:,:,qi)*xhat2k)-trace(A0'*Ai(:,:,qi)*(Cov_inv+xhat2k*xhat2k'));
end
theta_b = H\beta;
% EM learning of noise variance
sigma2_tilde = 1/m*((y_tilde-A_eq*xhat2k)'*(y_tilde-A_eq*xhat2k)+trace(A_eq*Cov_inv*A_eq'));
end
if(t>1)
gamma1k_new = (eta2k-gamma2k);
r1k_new = (eta2k*xhat2k-gamma2k*r2k)/gamma1k_new;
gamma1k = damp*(gamma1k_new)+(1-damp)*gamma1k_old;
gamma1k = max(gamma1k,sma_num);
gamma1k = min(gamma1k,lar_num);
r1k = damp*(r1k_new) + (1-damp)*r1k_old;
else
gamma1k = eta2k-gamma2k;
gamma1k = max(gamma1k,sma_num);
gamma1k = min(gamma1k,lar_num);
r1k = (eta2k*xhat2k-gamma2k*r2k)/gamma1k;
end
gamma1k_old = gamma1k;
r1k_old = r1k;
% denoising in the variable node
for k1 = 1:T_VN_denoising
% denoising step
vr = 1./gamma1k;
m_t = (r1k.*vx+vr.*mu0)./(vr+vx);
V_t = vr.*vx./(vr+vx);
x_hat_1k = m_t;
x_hat_var_1k = V_t;
eta1k = 1./mean(x_hat_var_1k);
gamma1k = 1/(1/eta1k+mean((x_hat_1k-r1k).^2));
end
if(t>1)
gamma2k_new = eta1k-gamma1k;
r2k_new = (eta1k.*x_hat_1k-gamma1k.*r1k)./gamma2k_new;
gamma2k = damp*(gamma2k_new)+(1-damp)*gamma2k_old;
gamma2k = max(gamma2k,sma_num);
gamma2k = min(gamma2k,lar_num);
r2k = damp*(r2k_new) + (1-damp)*r2k_old;
else
gamma2k = eta1k-gamma1k;
gamma2k = max(gamma2k,sma_num);
gamma2k = min(gamma2k,lar_num);
r2k = (eta1k.*x_hat_1k-gamma1k.*r1k)./gamma2k;
end
gamma2k_old = gamma2k;
r2k_old = r2k;
if NumBits < inf % nonlinear observations
%--- LMMSE step for calculate the extrinsic mean and variance
gammaw = 1/sigma2_tilde;
AQ_est = gen_matrix(Ai,theta_b,Q);
A_eq = A0+AQ_est;
Cov = gammaw*(A_eq'*A_eq)+gamma2k*eye(n);
xhat2k = Cov\(gammaw*A_eq'*y_tilde+gamma2k*r2k);
z_A_post = A_eq*xhat2k;
v_A_post = 1/m*trace(A_eq/(gammaw*(A_eq'*A_eq)+gamma2k*eye(n))*A_eq');%
v_A_ext = v_A_post.*sigma2_tilde./(sigma2_tilde-v_A_post);
v_A_ext = lar_num*(v_A_ext<0)+v_A_ext*(v_A_ext>0);
v_A_ext = min(v_A_ext,lar_num);
v_A_ext = max(v_A_ext,sma_num);
z_A_ext = v_A_ext.*(z_A_post./v_A_post-y_tilde./sigma2_tilde);
if t>1
z_A_ext = (1-damp)*z_A_ext_old+damp*z_A_ext;
v_A_ext = (1-damp)*v_A_ext_old+damp*v_A_ext;
end
z_A_ext_old = z_A_ext;
v_A_ext_old = v_A_ext;
wvar_hat = sigma2_tilde;
end
end
end
function [z_post, vz_post] = outputUpdate(y, z, mz, sigma, NumBits,delta)
% Performs output node update.
%
% NOTE: This function can potentially run into numerical erros. This is due
% to the sub-function evaluateTotalMoment, which performs integration
% of a gaussian in some integral given by quantizer boundaries. In case
% when this inteval is far from the mean of the normal and the normal has a
% small variance moments might result in 0, although in reality they
% represent some small values, ratio of which is definetely non-zero.
% length of the signal to estimate
m = size(y, 1);
% Total effective noise (AWGN + estiamtion)
mtv = mz + (sigma^2);
% Initialize outputs
% comupte the lower and up bounds
r_low = y - delta/2;
r_low(r_low < -(2^NumBits-1/2)*delta) = -1e50;
r_up = y + delta/2;
r_up(r_up > (2^NumBits-1/2)*delta) = 1e50;
% complex-valued case
% ita1 = (sign(y).*z - min(abs(r_low),abs(r_up)))./sqrt(2*mtv);
% ita2 = (sign(y).*z - max(abs(r_low),abs(r_up)))./sqrt(2*mtv);
%
% z_post = z + sign(y).*mz./sqrt(mtv).*((normpdf(ita1) - normpdf(ita2))./(normcdf(ita1) - normcdf(ita2)));
% vz_post = mz/2 - mz.^2./(2*mtv).*((ita1.*normpdf(ita1) - ita2.*normpdf(ita2))./(normcdf(ita1) - normcdf(ita2)) + ((normpdf(ita1) - normpdf(ita2))./(normcdf(ita1) - normcdf(ita2))).^2);
% real-valued case
ita1 = (sign(y).*z - min(abs(r_low),abs(r_up)))./sqrt(mtv);
ita2 = (sign(y).*z - max(abs(r_low),abs(r_up)))./sqrt(mtv);
A = normpdf(ita1) - normpdf(ita2);
B = normcdf(ita1) - normcdf(ita2);
C = ita1.*normpdf(ita1) - ita2.*normpdf(ita2);
D = A./B;
E = C./B + (A./B).^2;
Small_toc = 1e-50;
D(abs(B)<Small_toc) = - ita1(abs(B)<Small_toc);
E(abs(B)<Small_toc) = 1;
z_post = z + sign(y).*mz./sqrt(mtv).*D;
vz_post = mz - mz.^2./(mtv).*(E);
end