function [ladderVector,...
kappaVector,...
posterioriErrorMatrix] = NLRLS_pos(desired,input,S)
% NLRLS_pos.m
% Implements the Normalized Lattice RLS algorithm based on a posteriori error.
% (Algorithm 7.2 - book: Adaptive Filtering: Algorithms and Practical
% Implementation, Diniz)
%
% Syntax:
% [ladderVector,kappaVector,posterioriErrorMatrix] = NLRLS_pos(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)
% - nSectionsLattice : Number of lattice sections, refered as N in the textbook.
% - epsilon : Initilization of xiMin_backward and xiMin_forward.
%
% Output Arguments:
% . ladderVector : Store the ladder coefficients for each iteration, refered
% as v in the textbook.
% (Coefficients at one iteration are column vector)
% . kappaVector : Store the reflection coefficients for each iteration, considering
% only the forward part.
% (Coefficients at one iteration are column vector)
% . posterioriErrorMatrix : Store the a posteriori errors at each section of the lattice for
% each iteration.
% (The errors at one iteration are column vectors)
%
% 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 Procedure
nCoefficients = S.nSectionsLattice +1;
nIterations = length(desired);
% Pre Allocations
deltaVector = zeros(1, S.nSectionsLattice +1);
deltaVector_D = zeros(1, S.nSectionsLattice +1); %acho que deveria ser +2
error_b_Curr = zeros(1, S.nSectionsLattice +2);
error_b_Prev = zeros(1, S.nSectionsLattice +2);
error_f = zeros(1, S.nSectionsLattice +2);
ladderVector = zeros(nCoefficients , nIterations);
kappaVector = zeros(nCoefficients , nIterations);
posterioriErrorMatrix = zeros(S.nSectionsLattice +2, nIterations);
%################################################
% Computations
%################################################
%%Initialize Parameters
%
deltaVector = zeros(1, S.nSectionsLattice+1);
deltaVector_D = zeros(1, S.nSectionsLattice+1);
%
error_b_Prev = zeros(1,S.nSectionsLattice +2);
%
sigma_d_2 = S.epsilon;
sigma_x_2 = S.epsilon;
%-------------------------------------------------------------
%||||||||||||||||||||||| - Main Loop - |||||||||||||||||||||||
%-------------------------------------------------------------
for it = 1:nIterations
sigma_x_2 = S.lambda*sigma_x_2 + input(it)^2;
sigma_d_2 = S.lambda*sigma_d_2 + desired(it)^2;
% Set Values for Section 0(Zero)
error_b_Curr(1) = input(it)/(sigma_x_2^0.5);
error_f(1) = error_b_Curr(1);
posterioriErrorMatrix(1,it) = desired(it)/(sigma_d_2^0.5);
% Propagate the Order Update Equations
for ot = 1:S.nSectionsLattice+1
%Delta Time Update
deltaVector(ot) = deltaVector(ot)*...
sqrt((1 -error_b_Prev(ot)^2)*(1- error_f(ot)^2)) + ...
error_b_Prev(ot)*error_f(ot);
%Order Update Equations
error_b_Curr(ot+1) = (error_b_Prev(ot) -deltaVector(ot)*error_f(ot))/...
sqrt((1 -deltaVector(ot)^2)*(1 -error_f(ot)^2));
error_f(ot+1) = (error_f(ot) - deltaVector(ot)*error_b_Prev(ot))/...
sqrt((1 -deltaVector(ot)^2)*(1 -error_b_Prev(ot)^2));
% Feedforward Filtering
deltaVector_D(ot) = deltaVector_D(ot)*...
sqrt((1 -error_b_Curr(ot)^2)*(1 -posterioriErrorMatrix(ot,it)^2)) + ...
posterioriErrorMatrix(ot,it)*error_b_Curr(ot);
posterioriErrorMatrix(ot+1,it)= 1/sqrt((1 -error_b_Curr(ot)^2)*(1 -deltaVector_D(ot)^2))*...
(posterioriErrorMatrix(ot,it) -deltaVector_D(ot)*error_b_Curr(ot));
% Outputs:
ladderVector(ot,it) = deltaVector_D(ot);
kappaVector(ot,it) = deltaVector(ot);
end
%Upadate K-1 Info
error_b_Prev = error_b_Curr;
end % END OF LOOP
%EOF
