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OC_nonGaussian_seq.m
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OC_nonGaussian_seq.m
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% OC_nonGaussian_seq - Function for executing the complex-valued MMSE estimate of
% a non-Gaussian distributed sparse signal using orthoganal clustering approach
% where "clusters are formed sequentially".
%
% This function implements the algorithm described in "Structure-Based
% Bayesian Sparse Reconstruction", Ahmed A. Quadeer & Tareq Y. Al-Naffouri.
% OC_nonGaussian_seq will find a MMSE estimate of sparse solutions "x"
% for the underdetermined system of linear equations
% yp = Psi*x + n,
% where n is complex Gaussian additive noise,
% x is non-Gaussian distributed, and
% Psi is the partial DFT measurement matrix
%
% SYNTAX: xmmse_final = OC_nonGaussian_seq(yp, Psi, sigma_imp, N0, pp, index)
%
% Outputs: xls_mmse_final = The MMSE estimate of the sparse vector
% xls_map_final = The MAP estimate of the sparse vector
%
% Inputs: yp = n x 1 observation vector
% Psi = n x m measurement/sensing matrix (partial DFT matrix)
% N0 = Noise variance
% p = Probability of non-zero elements occurence
% index = "n-length" vector representing indices of continuous
% sensing columns
%
% Coded by: Ahmed Abdul Quadeer
% E-mail: [email protected]
% Last change: Dec. 12, 2012
% OC_Gaussian_seq version 1.0
% Copyright (c) Ahmed Abdul Quadeer, Tareq Y. Al-Naffouri, 2012
function [xls_mmse_final,xls_map_final] = OC_nonGaussian_seq(y,Psi,N0,p,index)
%% Determine dominant positions using correlation
[n,m] = size(Psi); %Size of Psi
yp_corr = Psi'*y; %Performing correlation
%% Step 2: Form semi-orthogonal clusters using sequential method
[J_cluster,cluster_indices] = clustering_seq(yp_corr,p,m,n);
%% 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
A = Psi(:,cluster_indices(1:D(1))); %Measurement matrix corresponding to first cluster
p_c = 1;
T = cell(P,1); %Indices of active taps
nu = cell(P,1); %Likelihood
xls = cell(P,1); %Initial LS estimate
xls_k_mmse = zeros(m,P);%MMSE estimate of each cluster k
xls_k_map = zeros(m,P); %MAP estimate of each cluster k
for k = 1:P %Loop for each cluster k
pp = p/m*D(k); %Probability of non-zero signal in each cluster
ps = [1 - pp; pp];
T{k} = cell(D(k),P_c); %Indices of active taps in cluster k
nu{k} = -inf*ones(D(k),P_c); %Likelihood in cluster k
xls{k} = cell(D(k),P_c); %Initial LS estimate of x in cluster k
if k == 1; %If first cluster
z = y;
else %If subsequent clusters
dd = mod(J_cluster{k}(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 = y.*conj(wd); %Modulation
end
nu_root = -norm(z)^2/N0 ; %Root node
nuxt_root = (-norm(z)^2 + (abs(z'*A).^2))/N0 ; %Likelihood of support size 1
for d = 1:D(k) %Within a cluster
nstar = d;
T{k}{d,p_c} = J_cluster{k}(nstar);
nu{k}(d,p_c) = nuxt_root(nstar);
Lambda=1; %As columns of psi are unit-norm
psi = Psi(:,T{1}{d,p_c});
if k == 1 %Only need to update eta, omega and xi for first cluster. Remaining clusters use the same values
eta{d,p_c} = psi'*A;
omega{d,p_c} = Lambda*eta{d,p_c};
xi{d,p_c} = (1 - sum((conj(omega{d,p_c}).*eta{d,p_c}),1)).^(-1); %defining xi as inverse
end
xls{k}{d,p_c} = zeros(m,1);
xls{k}{d,p_c}(T{k}{d,p_c}) = Lambda*psi'*z;
alpha = z'*psi*omega{d,p_c}(1:D(k));
beta = z'*A(:,1:D(k));
nuxt = zeros(1,D(k));
nuxt(1:D(k)) = nu{k}(d,p_c) + (xi{d,p_c}(1:D(k))/N0).*(abs(alpha).^2 ... %Likelihood of support size 2
- 2*real(beta.*conj(alpha)) + abs(beta).^2) ;
% can't activate an already activated coefficient
nuxt(nstar) = -inf*ones(size(T{k}{d,p_c}));
[nustar,nqstar] = max(nuxt); % find best extension
while sum(abs(nustar-nu{k}(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{k}(d,p_c+1) = nustar; % replace worst explored node...
T{k}{d,p_c+1} = [T{k}{d,p_c}, J_cluster{k}(nstar2)];
xls{k}{d,p_c+1} = zeros(m,1);
xls{k}{d,p_c+1}(T{k}{d,p_c+1}) = [xls{k}{d,p_c}(T{k}{d,p_c}) + ...
xi{d,p_c}(nstar2)*omega{d,p_c}(nstar2)*eta{d,p_c}(nstar2)'*xls{k}{d,p_c}(T{k}{d,p_c}) ...
- xi{d,p_c}(nstar2)*omega{d,p_c}(nstar2)*A(:,nstar2)'*z ;...
-xi{d,p_c}(nstar2)*eta{d,p_c}(nstar2)'*xls{k}{d,p_c}(T{k}{d,p_c}) + xi{d,p_c}(nstar2)*A(:,nstar2)'*z];
end
Pc = min(n, 1 + ceil(D(k)*p + erfcinv(1e-2)*sqrt(2*D(k)*p*(1 - p)))); %Dependent on actual p, not pp
p_cc = 3;
nnustar = zeros(1,Pc); %Best Likelihood value of previous support size
TT = cell(1,Pc); %Actual supports index
nxt = cell(1,Pc); %New likelihood values
nnstar = cell(1,Pc); %Best next support index in a cluster
NNSTAR = cell(1,Pc); %Best supports index in a cluster
nnqstar = cell(1,Pc); %Temporary storage of nnstar
xxls = cell(1,Pc);
[nnustar(p_cc-1),b] = max(nu{k}(:,2));
TT{p_cc-1} = T{k}{b,2};
nstar = find(J_cluster{k}==T{k}{b,2}(1));
nstar2 = find(J_cluster{k}==T{k}{b,2}(2));
NNSTAR{p_cc-1} = [nstar nstar2];
xxls{p_cc-1} = xls{k}{b,2};
while p_cc <= Pc
psi2 = Psi(:,[TT{p_cc-1}]);
Lambda2=(psi2'*psi2)^(-1);
B = Psi(:,[J_cluster{k}]);
eta2 = psi2'*B;
omega2 = Lambda2*eta2;
xi2 = (1 - sum((conj(omega2).*eta2),1)).^(-1); %defining xi as inverse
alpha2 = y'*psi2*omega2;
beta2 = y'*B;
nxt{p_cc} = nnustar(p_cc-1) + (xi2/N0).*(abs(alpha2).^2 ...
- 2*real(beta2.*conj(alpha2)) + abs(beta2).^2);
nxt{p_cc}(NNSTAR{p_cc-1}) = -inf*ones(1,p_cc-1);
[nnustar(p_cc),nnqstar{p_cc}] = max(real(nxt{p_cc}));
nnstar{p_cc} = mod(nnqstar{p_cc} - 1, m) + 1;
TT{p_cc} = [TT{p_cc-1}, J_cluster{k}([nnstar{p_cc}])];
NNSTAR{p_cc} = [NNSTAR{p_cc-1} nnstar{p_cc}];
xxls{p_cc} = zeros(m,1);
xxls{p_cc}(TT{p_cc}) = [xxls{p_cc-1}(TT{p_cc-1}) + ...
xi2(nnstar{p_cc})*omega2(:,nnstar{p_cc})*eta2(:,nnstar{p_cc})'*xxls{p_cc-1}(TT{p_cc-1}) ...
- xi2(nnstar{p_cc})*omega2(:,nnstar{p_cc})*B(:,nnstar{p_cc})'*y ;...
-xi2(nnstar{p_cc})*eta2(:,nnstar{p_cc})'*xxls{p_cc-1}(TT{p_cc-1}) + xi2(:,nnstar{p_cc})*B(:,nnstar{p_cc})'*y];
p_cc = p_cc+1;
end
%% Step 4: Evaluate the estimate of x
%MMSE
p_of_S = [1:size([nu{k}],2) ;D(k)-(1:size([nu{k}],2))].'*[log(ps(2)) log(ps(1))].'; %p(S) of size (no. of supports size x 1)
nu_p = [nu{k}]+repmat(p_of_S.',size([nu{k}]) - [0 1]);
p_of_S_gt_3 = [1:length(nnustar) ;D(k)-(1:length(nnustar))].'*[log(ps(2)) log(ps(1))].';
nup_gt_3 = nnustar.' + p_of_S_gt_3;
nup = [nu_root+D(k)*log(ps(1)) ; nu_p(:) ; nup_gt_3(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)));
xls_k_mmse(:,k) = [zeros(m,1) xls{k}{:} xxls{3:end}]*p_nup;
%MAP
[~,indx_max_nup] = max(nup);
p_nup_map = zeros(size(nup));
p_nup_map(indx_max_nup) = 1;
xls_k_map(:,k) = [zeros(m,1) xls{k}{:} xxls{3:end}]*p_nup_map;
end
xls_mmse_final = sum(xls_k_mmse,2);
xls_map_final = sum(xls_k_map,2);