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function [Wo, ee, gamma] = qrd_lsl(u, d, M, verbose, lambda, delta)
% function [Wo, ee, gamma] = qrd_lsl(u, d, M, verbose, lambda, delta)
%
% qrd_lsl.m - apply the standard QR-decomposition based LSL algorithm
% using angle-normalized error to predict/estimate complex-valued processes
% written for MATLAB 4.0
%
% For efficiency, the index shifting functions should be replaced with
% their corresponding macro definitions in ms.m and ns.m.
% their corresponding macro definitions in ms.m and ns.m.
%
% Reference: Ch.15 of Haykin, _Adaptive Filter Theory_, 3rd ed., 1991
%
% Input parameters:
%       u       : vector of inputs
%       d       : vector of desired outputs
%       M       : final order of predictor
%       verbose : set to 1 for interactive processing
%       lambda  : the initial value of the forgetting factor
%       delta   : the initial value for the prediction matrices
%
% Output parameters:
%       Wo      : row-wise matrix of Hermitian transposed weights
%                 at each iteration
%       ee      : row vector of a posteriori prediction errors Y - xp
%                 for final prediction order M+1
%       gamma   : row vector of conversion factors for final
%                 prediction order M+1
%       gamma   : row vector of conversion factors for final
%                 prediction order M+1
%
% Copyright (c) 1994-2001, Paul Yee.


% Nout = size(Xi, 2);
Nout = 1;

% length of maximum number of timesteps that can be predicted
N = min(length(u),length(d));
Wo=[];

% prediction initialization
F = zeros(ms(M),ns(N));
F(ms(0):ms(M),ns(0)) = delta * ones(M+1,1);
B = zeros(ms(M),ns(N));
B(ms(0):ms(M),ns(-1)) = delta * ones(M+1,1);
B(ms(M),ns(0)) = delta;
cb = zeros(ms(M),ns(N));
sb = cb;
cf = cb;
sf = cb;
e = zeros(ms(M+1),ns(N));
ee = e;
pfc = zeros(ms(M-1),ns(N));
pbc = pfc;
pc = pfc;
pfc(ms(0):ms(M-1),ns(0)) = zeros(M, 1);
pbc(ms(0):ms(M-1),ns(0)) = zeros(M, 1);
pc(ms(1):ms(M),ns(0)) = zeros(M, 1);
gamma_root = zeros(ms(M+1),ns(N));
gamma_root(ms(0),ns(1):ns(N)) = ones(1, N);
lambda_root = sqrt(lambda);

% set size of reflection coefficients
kappa_f = zeros(ms(M),ns(N));
kappa_b = zeros(ms(M),ns(N));
kappa = zeros(ms(M),ns(N));

for n = 1:N,

  % data initialization
  ef(ms(0),ns(n)) = u(n);
  eb(ms(0),ns(n)) = u(n);
  e(ms(0),ns(n)) = d(n);
  gamma_root(ms(0),ns(n)) = 1;
  
  for m = 1:M,

    % predictions
    B(ms(m-1),ns(n-1)) = lambda * B(ms(m-1),ns(n-2)) +...
abs(eb(ms(m-1),ns(n-1)))^2;
    cb(ms(m-1),ns(n-1)) = lambda_root * sqrt(B(ms(m-1),ns(n-2)) /...
B(ms(m-1),ns(n-1)));
    sb(ms(m-1),ns(n-1)) = eb(ms(m-1),ns(n-1)) / sqrt(B(ms(m-1),ns(n-1)));
    pfc(ms(m-1),ns(n)) = cb(ms(m-1),ns(n-1)) * lambda_root *...
pfc(ms(m-1),ns(n-1)) + conj(sb(ms(m-1),ns(n-1))) * ef(ms(m-1),ns(n));
    ef(ms(m),ns(n)) = cb(ms(m-1),ns(n-1)) * ef(ms(m-1),ns(n)) -...
sb(ms(m-1),ns(n-1)) * lambda_root * pfc(ms(m-1),ns(n-1));
    gamma_root(ms(m),ns(n-1)) = cb(ms(m-1),ns(n-1)) *...
gamma_root(ms(m-1),ns(n-1));
    kappa_f(ms(m),ns(n)) = -conj(pfc(ms(m-1),ns(n))) /...
sqrt(B(ms(m-1),ns(n-1)));

    F(ms(m-1),ns(n)) = lambda * F(ms(m-1),ns(n-1)) + abs(ef(ms(m-1),ns(n)))^2;
    cf(ms(m-1),ns(n)) = lambda_root * sqrt(F(ms(m-1),ns(n-1)) /...
F(ms(m-1),ns(n)));
    sf(ms(m-1),ns(n)) = ef(ms(m-1),ns(n)) / sqrt(F(ms(m-1),ns(n)));
    pbc(ms(m-1),ns(n)) = cf(ms(m-1),ns(n)) * lambda_root * pbc(ms(m-1),ns(n-1))+...
 sf(ms(m-1),ns(n)) * eb(ms(m-1),ns(n-1));
    eb(ms(m),ns(n)) = cf(ms(m-1),ns(n)) * eb(ms(m-1),ns(n-1)) -...
sf(ms(m-1),ns(n)) * lambda_root * pbc(ms(m-1),ns(n-1));
    kappa_b(ms(m),ns(n)) = -conj(pbc(ms(m-1),ns(n))) / sqrt(F(ms(m-1),ns(n)));

  end; % for m
  
end; % for n


for n = 1:N,
  
  for m = 1:M,
      
    % filtering
    pc(ms(m-1),ns(n)) = cb(ms(m-1),ns(n)) * lambda_root * pc(ms(m-1),ns(n-1)) +...
conj(sb(ms(m-1),ns(n))) * e(ms(m-1),ns(n));
   e(ms(m),ns(n)) = cb(ms(m-1),ns(n)) * e(ms(m-1),ns(n)) - sb(ms(m-1),ns(n)) *...
lambda_root * pc(ms(m-1),ns(n-1));
    ee(ms(m),ns(n)) = gamma_root(ms(m),ns(n)) * e(ms(m),ns(n));
    
  end; % for m
    
end; % for n


for n = 1:N,
  
  % handle terminal case m=M separately as in Sayed and Kailath Table 5
  B(ms(M),ns(n)) = lambda * B(ms(M),ns(n-1)) + abs(eb(ms(M),ns(n)))^2;
  cb(ms(M),ns(n)) = lambda_root * sqrt(B(ms(M),ns(n-1)) / B(ms(M),ns(n)));
  sb(ms(M),ns(n)) = eb(ms(M),ns(n)) / sqrt(B(ms(M),ns(n)));
  e(ms(M+1),ns(n)) = cb(ms(M),ns(n)) * e(ms(M),ns(n)) - sb(ms(M),ns(n)) *...
lambda_root * pc(ms(M),ns(n-1));
  pc(ms(M),ns(n)) = cb(ms(M),ns(n)) * lambda_root * pc(ms(M),ns(n-1)) +...
conj(sb(ms(M),ns(n))) * e(ms(M),ns(n));
  gamma_root(ms(M+1),ns(n)) = cb(ms(M),ns(n)) * gamma_root(ms(M),ns(N));  
  ee(ms(M+1),ns(n)) = gamma_root(ms(M+1),ns(n)) * e(ms(M+1),ns(n));  
  


  for m = 1:M,
    
    B(ms(m-1),ns(N)) = lambda * B(ms(m-1),ns(N-2)) + abs(eb(ms(m-1),ns(N)))^2;
    
    % joint-process regression coefficient
    % required to compute weight vector
    kappa(ms(m),ns(n)) = conj(pc(ms(m-1),ns(n))) / sqrt(B(ms(m),ns(n)));

 end; % for m
 
end; % for n


an = zeros(M, M+1);
cn = an;
cn1 = cn;
L_M = eye(M+1, M+1);
for n = 1:N,
  
  an(1, 1:2) = [1 kappa_f(ms(1),ns(n))];
  cn(1, 1:2) = [kappa_b(ms(1),ns(n)) 1];

  for m = 2:M,
  % compute coefficients of forward and backward prediction-error
    % filters for later use in computation of weight vector. Note
    % that we propagate an and cn in row vector form.
    an(m, 1:(m+1)) = [an(m-1, 1:m) 0] + kappa_f(ms(m),ns(n)) * [0 cn1(m-1,1:m)];
    cn(m, 1:(m+1)) = [0 cn1(m-1, 1:m)] + kappa_b(ms(m),ns(n)) * [an(m-1, 1:m) 0];
  end; % for m
  
  for m = 2:(M+1),
    L_M(m, 1:(m-1)) = conj(cn(m-1, 1:(m-1)));
  end; % for m
  
  % note that we store Hermitian-transposed weights
  % row-wise in Wo
  Wo = [Wo ; kappa(ms(0):ms(M),ns(n))' * L_M];
 cn1 = cn;
    
end; % for n


% resize variables before return
Wo = Wo(1:(N-1),:)';
ee = ee(ms(M),ns(1):ns(N-1));
gamma = gamma_root(ms(M),ns(1):ns(N-1)).^2;