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//Levinson- Durbin Recurssion Algorithm
////calling syntax
//a = levinson(r)
//a = levinson(r,n)
//[a,e] = levinson(r,n)
//[a,e,k] = levinson(r,n)
// where
// a is the coefficients of a length(r)-1 order autoregressive linear process
//e is the prediction error when order is n
// k is a column vector containing the reflection coefficients of length n
//Example :
//a = [1 0.1 -0.8]; //Estimate the coefficients of an autoregressive process given by x(n) = 0.1x(n-1) - 0.8x(n-2) + w(n)
//
//v = 0.4;
//w = sqrt(v)*rand(15000,1,"normal");
//x = filter(1,a,w);
//
//[r,lg] = xcorr(x,'biased');
//r(lg<0) = [];
//
//ar = levinson(r,length(a)-1)
// // Output :---
// ar =
//
// 1. 0.0983843 - 0.7929775
//
//**************************************************************************************************
//______________________________________________version1 code (not working)_________________________
//__________________________________________________________________________________________________
//**************************************************************************************************
//function [a, v_f, ref_f] = levinson (acf, p)
//if ( argn(2)<1 )
// error("Too few input arguments");
// elseif( length(acf)<2 )
// error( "levinson: arg 1 (acf) must be vector of length >1\n");
//elseif ( argn(2)>1 & ( ~isscalar(p) | fix(p)~=p ) )
// error( "levinson: arg 2 (p) must be integer >0\n");
//elseif (isempty(acf))
// error("R cannot be empty");
//else
// if ((argn(2) == 1)|(p>=length(acf)))
// p = length(acf) - 1;
// end
// if( size(acf,1)==1 & size(acf,2)>1 ) then
// acf=acf(:);
//
// end force a column vector
// if size(acf,1)>=1 then handles matrix i/p
//
// acf=acf';
// a=acf;
// rows=size(acf,1);
// for i=1:rows
// acf_temp=acf(i,:);
// acf_temp=acf_temp(:);
// p=length(acf_temp)-1;
// disp(acf_temp);
// if argn(1) < 3 & p < 100
//
// // Kay & Marple Eqn (2.39)
//
// R = toeplitz(acf_temp(1:p), conj(acf_temp(1:p)));
// an = R \ -acf_temp(2:p+1);
// an= [ 1, an.' ];
// v_f(i,:)= real( an*conj(acf_temp(1:p+1)) );
// a(i,:)=an;
// an=[];
//
// else
//
// // Kay & Marple Eqns (2.42-2.46)
//
// ref = zeros(p,1);
// g = -acf_temp(2)/acf_temp(1);
//
// an = [ g ];
// v= real( ( 1 - g*conj(g)) * acf_temp(1) );
// ref(1) = g;
// for t = 2 : p
// g = -(acf_temp(t+1) + an * acf_temp(t:-1:2)) / v;
// an = [ an+g*conj(an(t-1:-1:1)), g ];
// v = v * ( 1 - real(g*conj(g)) ) ;
// ref(t) = g;
// end
// v_f(i,:)=v;
// v=[];
// ref_f(:,i)=ref;
// an = [1, an];
// a(i,:)=an;
// an=[];
// end end if
// end end for
// end
//end
//
//endfunction
//**************************************************************************************************
//______________________________________________version2 code ( working)____________________________
//__________________________________________________________________________________________________
//**************************************************************************************************
function [a, v, ref] = levinson(bcf, p)
funcprot(0);
nargin = argn(2);
nargout = argn(1);
[rows columns] = size(bcf)
if ( nargin<1 )
error("Wrong input argument ");
elseif( ~isvector(bcf) | length(bcf)<2 )
error( "levinson: arg 1 (bcf) must be vector of length >1\n");
elseif ( nargin>1 & ( ~isscalar(p) | fix(p)~=p ) )
error( "levinson: arg 2 (p) must be integer >0\n");
else
if ((nargin == 1)|(p>=length(bcf))) p = length(bcf) - 1; end
if( columns >1 ) bcf=bcf(:); end
if nargout < 3 & p < 100
// ## direct solution [O(p^3), but no loops so slightly faster for small p]
// ## Kay & Marple Eqn (2.39)
R = toeplitz(bcf(1:p), conj(bcf(1:p)));
a = R \ -bcf(2:p+1);
a = [ 1, a.' ];
v = real( a*conj(bcf(1:p+1)) );
else
// ## durbin-levinson [O(p^2), so significantly faster for large p]
// ## Kay & Marple Eqns (2.42-2.46)
ref = zeros(p,1);
g = -bcf(2)/bcf(1);
a = [ g ];
v = real( ( 1 - g*conj(g)) * bcf(1) );
ref(1) = g;
for t = 2 : p
g = -(bcf(t+1) + a * bcf(t:-1:2)) / v;
a = [ a+g*conj(a(t-1:-1:1)), g ];
v = v * ( 1 - real(g*conj(g)) ) ;
ref(t) = g;
end
a = [1, a];
end
end
endfunction
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