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OC_Gaussian.m
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OC_Gaussian.m
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% OC_Gaussian - Function for executing the complex-valued MMSE estimate of
% a Gaussian distributed sparse signal using orthoganal clustering approach.
%
% This function implements the algorithm described in "Structure-Based
% Bayesian Sparse Reconstruction", Ahmed A. Quadeer & Tareq Y. Al-Naffouri.
% OC_Gaussian will find a MMSE estimate of sparse solutions "x"
% to the underdetermined system of linear equations
% yp = Psi*x + n,
% where n is complex Gaussian, while x is Gaussian distributed.
%
% SYNTAX: xmmse_final = OC_Gaussian(yp, Psi, sigma_imp, N0, pp, index)
%
% Outputs: xmmse_final = The MMSE estimate of the sparse vector%
%
% Inputs: yp = n x 1 observation vector
% Psi = n x m measurement/sensing matrix
% sigma_imp = Non-zero elements' variance in the sparse vector
% N0 = Noise variance
% pp = Probability of non-zero elements occurence
% index = n-length vector representing indices of continuous
% sensing carriers
%
% Coded by: Ahmed Abdul Quadeer
% E-mail: [email protected]
% Last change: Dec. 12, 2012
% Copyright (c) Ahmed Abdul Quadeer, Tareq Y. Al-Naffouri, 2012
function xmmse_final = OC_Gaussian(yp,Psi,sigma_imp,N0,pp,index)
%% Initialization
sig2s = [0; sigma_imp]; %Signal variance
sig2w = N0; %Noise variance
%% Step 1: Determine dominant positions using correlation
Phi_mtrx = Psi';
[n,m] = size(Phi_mtrx); %Size of Phi
yp_corr = Phi_mtrx'*yp; %Performing correlation
% Pz = 1e-2; %Probability of noise to be greater than thresh
thresh = 3; %Calculated from pdf of chisquare distribution
indx_gt_thresh = find(abs(yp_corr)>thresh); %Indices greater than threshold
%% Step 2: Form semi-orthogonal clusters using chi-square threshold
if isempty(indx_gt_thresh) %If no index greater than thereshold
xmmse_final=zeros(m,1);
else
[J_cluster,cluster_indices] = clustering(indx_gt_thresh,m);
%% Step 3: Find the dominant supports and their likelihoods (Main algorithm)
%Initialization
D = cellfun('length',J_cluster).'; %Finding length of each cluster
P = length(D); %Number of clusters
P_c = 2; %Initial P for cluster
AA = Psi';
A = AA(:,cluster_indices(1:D(1))); %Measurement matrix corresponding to first cluster
T = cell(P,1); %Indices of active taps
nu = cell(P,1); %Metrics
xmmse = cell(P,1); %Initial MMSE estimate of x
Omega = cell(D(1),P_c+1); %
Xi = cell(D(1),P_c+1); %
xmmse_p = zeros(m,P); %MMSE estimate of each cluster p
for p = 1:P %Loop for each cluster p
ppp = pp/m*D(p);
ps = [1 - ppp; ppp];
T{p} = cell(D(p),P_c); %Indices of active taps in cluster p
nu{p} = -inf*ones(D(p),P_c); %Metrics in cluster p
xmmse{p} = cell(D(p),P_c); %Initial MMSE estimate of x in cluster p
if p == 1 %If first cluster
Omega_root_1 = A/sig2w;
Xi_root_1 = abs(sig2s(2)*(1 + sig2s(2)*sum(conj(A).*Omega_root_1)).^(-1));
Omega_root = Omega_root_1;
Xi_root = Xi_root_1;
z = yp;
else %If subsequent clusters
Omega_root = Omega_root_1(:,1:(D(p))); %Selecting according to length of cluster
Xi_root = Xi_root_1(1:(D(p))); %Selecting according to length of cluster
dd = mod(J_cluster{p}(1)-J_cluster{1}(1),m); %Finding the difference delta between cluster p and cluster 1
wd = (exp(-sqrt(-1)*(2*pi)/m*(index(1:n)-1)*dd)).'; %Calculating the modulation vector
z = yp.*conj(wd); %Modulation
end
nu_root = -norm(yp)^2/sig2w + D(p)*log(ps(1)); %Root node
nuxt_root = zeros(1,D(p));
nuxt_root(1:D(p)) = nu_root + log(Xi_root/sig2s(2)) + ...
Xi_root.*abs(z'*Omega_root).^2 + log(ps(2)/ps(1));
p_c = 1;
for d = 1:D(p) %Within a cluster
nstar = d;
T{p}{d,p_c} = J_cluster{p}(nstar);
nu{p}(d,p_c) = nuxt_root(nstar);
if p == 1 %Only need to update Omega and Xi for first cluster. Remaining clusters use the same values
Omega{d,p_c} = Omega_root; %Omega{1,:} will all be equal to Omega_root
Xi{d,p_c} = Xi_root;
Omega{d,p_c+1} = Omega{d,p_c} - ...
Omega{d,p_c}(:,nstar)*Xi{d,p_c}(nstar)*( Omega{d,p_c}(:,nstar)'*A );
Xi{d,p_c+1} = abs(sig2s(2)*(1 + sig2s(2)*sum(conj(A).*Omega{d,p_c+1})).^(-1));
end
xmmse{p}{d,p_c} = zeros(m,1);
xmmse{p}{d,p_c}(T{p}{d,p_c}) = sig2s(2)*Omega{d,p_c+1}(:,nstar)'*z;
nuxt = zeros(1,D(p));
nuxt(1:D(p)) = nu{p}(d,p_c) + log(Xi{d,p_c+1}(1:D(p))/sig2s(2)) ...
+ Xi{d,p_c+1}(1:D(p)).*abs(z'*Omega{d,p_c+1}(:,1:D(p))).^2 ...
+ log(ps(2)/ps(1));
% can't activate an already activated coefficient!
nuxt(nstar) = -inf*ones(size(T{p}{d,p_c}));
[nustar,nqstar] = max(nuxt); % find best extension
while sum(abs(nustar-nu{p}(1:d-1,p_c+1)) < 1e-8) % if same as explored node...
nuxt(nqstar) = -inf; % ... mark extension as redundant
[nustar, nqstar] = max(nuxt); % ... and find next best extension
end
nstar2 = mod(nqstar - 1, m) + 1; % coef index of best extension
nu{p}(d,p_c+1) = nustar; % replace worst explored node...
T{p}{d,p_c+1} = [T{p}{d,p_c}, J_cluster{p}(nstar2)];
Omega{d,p_c+2} = Omega{d,p_c+1}(:,1:D(p)) - ...
Omega{d,p_c+1}(:,nstar2)*Xi{d,p_c+1}(nstar2)*...
( Omega{d,p_c+1}(:,nstar2)'*A(:,1:D(p)));
Xi{d,p_c+2} = abs(sig2s(2)*(1 + sig2s(2)*sum(conj(A(:,1:D(p)))...
.*Omega{d,p_c+2})).^(-1));
xmmse{p}{d,p_c+1} = zeros(m,1);
xmmse{p}{d,p_c+1}(T{p}{d,p_c+1}) = sig2s(2)*Omega{d,p_c+2}(:,[nstar,nstar2])'*z;
end
%Finding the best supp. of hamming weight 3 from the best supp. of hamming weight 2
[a b]=max(nu{p}(:,2)); %a is the max likelihood of supp. of size and b is the index
nstar = find(J_cluster{p}==T{p}{b,2}(1));
nstar2 = find(J_cluster{p}==T{p}{b,2}(2));
nuxt3 = a + log(Xi{b,p_c+2}(1:D(p))/sig2s(2)) ...
+ (Xi{b,p_c+2}(1:D(p))).*abs(z'*Omega{b,p_c+2}(:,1:D(p))).^2 ...
+ log(ps(2)/ps(1));
nuxt3([nstar nstar2]) = -inf*ones(size(T{p}{b,2}));
[nustar3,nqstar3] = max(nuxt3);
while ismember(nqstar3,[nstar nstar2])==1 % if same as explored node...
nuxt3(nqstar3) = -inf; % ... mark extension as redundant
[nustar3, nqstar3] = max(nuxt3); % ... and find next best extension
end
nstar3 = mod(nqstar3 - 1, m) + 1;
T_3 = [T{p}{b,2}, J_cluster{p}(nstar3)];
Omega3 = Omega{b,p_c+2}(:,1:D(p)) - Omega{b,p_c+2}(:,nstar3)*Xi{b,p_c+2}...
(nstar3)*( Omega{b,p_c+2}(:,nstar3)'*A(:,1:D(p)));
Xi3 = abs(sig2s(2)*(1 + sig2s(2)*sum(conj(A(:,1:D(p))).*Omega3)).^(-1));
xmmse_3 = zeros(m,1);
xmmse_3(T_3) = sig2s(2)*Omega3(:,[nstar,nstar2,nstar3])'*z;
% Continuing the greedy approach for hamming weight greater than 3
% depending on the length of cluster
Pc = min(n, 1 + ceil(D(p)*pp + erfcinv(1e-1)*sqrt(2*D(p)*pp*(1 - pp))));
p_cc = 4;
nnustar = cell(1,Pc);
TT = cell(1,Pc);
Omega_gt_3 = cell(1,Pc);
Xi_gt_3 = cell(1,Pc);
NSSTAR = cell(1,Pc);
xmmsee = cell(1,Pc);
nnustar{p_cc-1} = nustar3;
TT{p_cc-1}=T_3;
Omega_gt_3{p_cc-1} = Omega3;
Xi_gt_3{p_cc-1}=Xi3;
NSSTAR{p_cc-1}=[nstar nstar2 nstar3];
xmmsee{p_cc-1} = xmmse_3;
clear nxt
while p_cc <= Pc
nxt{p_cc} = zeros(1,D(p));
nxt{p_cc} = nnustar{p_cc-1} + log(Xi_gt_3{p_cc-1}(1:D(p))/sig2s(2)) ...
+ (Xi_gt_3{p_cc-1}(1:D(p))).*abs(z'*Omega_gt_3{p_cc-1}(:,1:D(p))).^2 + log(ps(2)/ps(1));
nxt{p_cc}(NSSTAR{p_cc-1}) = -inf*ones(1,p_cc-1);
[nnustar{p_cc},nnqstar{p_cc}] = max(nxt{p_cc});
nnstar{p_cc} = mod(nnqstar{p_cc} - 1, m) + 1;
TT{p_cc} = [TT{p_cc-1} J_cluster{p}(nnstar{p_cc})];
Omega_gt_3{p_cc} = Omega_gt_3{p_cc-1}(:,1:D(p)) - Omega_gt_3{p_cc-1}(:,nnstar{p_cc})* Xi_gt_3{p_cc-1}(nnstar{p_cc})*...
( Omega_gt_3{p_cc-1}(:,nnstar{p_cc})'*A(:,1:D(p)));
Xi_gt_3{p_cc} = abs(sig2s(2)*(1 + sig2s(2)*sum(conj(A(:,1:D(p))).*Omega_gt_3{p_cc})).^(-1));
NSSTAR{p_cc} = [NSSTAR{p_cc-1} nnstar{p_cc}];
xmmsee{p_cc} = zeros(m,1);
xmmsee{p_cc}(TT{p_cc}) = sig2s(2)*Omega_gt_3{p_cc}(:,NSSTAR{p_cc})'*z;
p_cc = p_cc+1;
end
%% Step 4: Evaluate the estimate of x
if Pc > 3
nup = [nu_root ; nu{p}(:); [nnustar{3:end}].'];
%including nu_root to include the case when no impulse is present in cluster
p_nup = exp(nup-max(nup))/sum(exp(nup-max(nup)));
xmmse_p(:,p) = [zeros(m,1) xmmse{p}{:} [xmmsee{3:end}]]*p_nup;
else
nup = [nu_root ; nu{p}(:); nustar3];
%including nu_root to include the case when no impulse is present in cluster
p_nup = exp(nup-max(nup))/sum(exp(nup-max(nup)));
xmmse_p(:,p) = [zeros(m,1) xmmse{p}{:} xmmse_3]*p_nup;
end
end
xmmse_final = sum(xmmse_p,2); %Final MMSE estimate of sparse signal
end