function [posterioriErrorVector,...
prioriErrorVector,...
coefficientVector] = Fast_RLS(desired,input,S)
% Fast_RLS.m
% Implements the Fast Transversal RLS algorithm for REAL valued data.
% (Algorithm 8.1 - book: Adaptive Filtering: Algorithms and Practical
% Implementation, 3rd Ed., Diniz)
%
% Syntax:
% [posterioriErrorVector,prioriErrorVector,coefficientVector] = Fast_RLS(desired,input,S)
%
% Input Arguments:
% . desired : Desired Signal. (ROW Vector)
% . input : Signal fed into the adaptive filter. (ROW Vector)
% . S : Structure with the following fields
% - lambda : Forgetting factor. (0 << lambda < 1)
% - predictorOrder : refered as N in the textbook.
% - epsilon : Initilization of xiMin_backward and xiMin_forward. (usually 0 < epsilon <= 1)
%
% Output Arguments:
% . posterioriErrorVector : Store the a posteriori error for each iteration. (COLUMN vector)
% . prioriErrorVector : Store the a priori error for each iteration. (COLUMN vector)
% . coefficientVector : Store the estimated coefficients for each iteration.
% (Coefficients at one iteration are COLUMN vector)
%
% Authors:
% . Guilherme de Oliveira Pinto - guilhermepinto7@gmail.com & guilherme@lps.ufrj.br
% . Markus VinÃcius Santos Lima - mvsl20@gmailcom & markus@lps.ufrj.br
% . Wallace Alves Martins - wallace.wam@gmail.com & wallace@lps.ufrj.br
% . Luiz Wagner Pereira Biscainho - cpneqs@gmail.com & wagner@lps.ufrj.br
% . Paulo Sergio Ramirez Diniz - diniz@lps.ufrj.br
%
%################################################
% Data Initialization
%################################################
% Basic Parameters
nCoefficients = S.predictorOrder +1;
nIterations = length(desired);
% Pre Allocations
xiMin_f_Curr = 0;
xiMin_f_Prev = 0;
xiMin_b = 0;
gamma_Np1 = 0;
gamma_N = 0;
w_f = zeros(nCoefficients, 1);
w_b = zeros(nCoefficients, 1);
coefficientVector = zeros(nCoefficients, nIterations+1);
error_f = 0;
error_f_line = 0;
error_b = 0;
error_b_line = 0;
posterioriErrorVector = zeros(nIterations,1);
prioriErrorVector = zeros(nIterations,1);
phiHatN = zeros(nCoefficients,1);
phiHatNPlus1 = zeros(nCoefficients+1,1);
regressor = zeros(nCoefficients+1,1);
%################################################
% Computations
%################################################
%%Initialize Parameters
w_f = zeros(nCoefficients, 1);
w_b = zeros(nCoefficients, 1);
coefficientVector(:,1) = zeros(nCoefficients, 1);
%
phiHatN = zeros(nCoefficients, 1);
%
gamma_N = 1;
%
xiMin_f_Prev = S.epsilon;
xiMin_b = S.epsilon;
%
prefixedInput = [ zeros(1,nCoefficients) input];
%-------------------------------------------------------------
%||||||||||||||||||||||| - Main Loop - |||||||||||||||||||||||
%------------------------------------------------------------
for it = 1:nIterations
regressor = prefixedInput(it+nCoefficients:-1:it).';
error_f_line = regressor.'*[ 1 ; -w_f];
error_f = error_f_line*gamma_N;
xiMin_f_Curr = S.lambda*xiMin_f_Prev + error_f_line*error_f;
phiHatNPlus1 = [0 ; phiHatN ] + ...
1/(S.lambda*xiMin_f_Prev)*[1 ; -w_f]*error_f_line;
w_f = w_f + phiHatN*error_f;
gamma_Np1 = (S.lambda*xiMin_f_Prev*gamma_N)/xiMin_f_Curr;
error_b_line = S.lambda*xiMin_b*phiHatNPlus1(end);
gamma_N = 1/(1/gamma_Np1 -phiHatNPlus1(end)*error_b_line);
error_b = error_b_line*gamma_N;
xiMin_b = S.lambda*xiMin_b + error_b*error_b_line;
phiHatN = phiHatNPlus1(1:end-1) +phiHatNPlus1(end)*w_b;
w_b = w_b + phiHatN*error_b;
% Joint Process Estimation
prioriErrorVector(it) = desired(it) -...
coefficientVector(:,it).'*regressor(1:end-1);
posterioriErrorVector(it) = prioriErrorVector(it)*gamma_N;
coefficientVector(:,it+1) = coefficientVector(:,it) + ...
phiHatN*posterioriErrorVector(it);
% K- 1 Info
xiMin_f_Prev = xiMin_f_Curr;
end % END OF LOOP
%EOF
