# v_lpcra2ar

## PURPOSE

V_LPCRA2AR Convert inverse filter autocorrelation coefs to AR filter. AR=(RA)

## SYNOPSIS

function ar=v_lpcra2ar(ra,tol)

## DESCRIPTION

    V_LPCRA2AR Convert inverse filter autocorrelation coefs to AR filter. AR=(RA)

     Usage: (1) ar0=poly([0.5 0.2]);  % ar0=[1 -0.7 0.1]
                ra0=v_lpcar2ra(ar0);    % ra0=[1.5 -0.77 0.1]
                ar1=v_lpcra2ar(ra0);    % ar1 = ar0

            (2) ar0=poly([0.5 0.2]);                   % ar0=[1 -0.7 0.1]
                arx=xcorr(ar0);                        % arx=[0.1 -0.77 1.5 -0.77 0.1]
                ar1=v_lpcra2ar(arx(length(ar0):end));    % ar1 = ar0

      Inputs: ra(n,p+1) each row is the second half of the autocorrelation of
                        the coefficients of a stable AR filter of order p
              tol       tolerance relative to ra(1) [1e-8]

     Outputs: ar(n,p+1) each row gives coefficients of an AR filter of order p

     This routine uses a Newton-Raphson iteration described in [1] to invert
     the cross-correlation function (as in the second usage example above).

     Refs:
     [1]  G. Wilson. Factorization of the covariance generating function of a pure moving average process.
          SIAM Journal on Numerical Analysis, 6 (1): 1�7, 1969.

## CROSS-REFERENCE INFORMATION

This function calls:

This function is called by:

## SOURCE CODE

    0001 function ar=v_lpcra2ar(ra,tol)
    0002 %V_LPCRA2AR Convert inverse filter autocorrelation coefs to AR filter. AR=(RA)
    0003 %
    0004 % Usage: (1) ar0=poly([0.5 0.2]);  % ar0=[1 -0.7 0.1]
    0005 %            ra0=v_lpcar2ra(ar0);    % ra0=[1.5 -0.77 0.1]
    0006 %            ar1=v_lpcra2ar(ra0);    % ar1 = ar0
    0007 %
    0008 %        (2) ar0=poly([0.5 0.2]);                   % ar0=[1 -0.7 0.1]
    0009 %            arx=xcorr(ar0);                        % arx=[0.1 -0.77 1.5 -0.77 0.1]
    0010 %            ar1=v_lpcra2ar(arx(length(ar0):end));    % ar1 = ar0
    0011 %
    0012 %  Inputs: ra(n,p+1) each row is the second half of the autocorrelation of
    0013 %                    the coefficients of a stable AR filter of order p
    0014 %          tol       tolerance relative to ra(1) [1e-8]
    0015 %
    0016 % Outputs: ar(n,p+1) each row gives coefficients of an AR filter of order p
    0017 %
    0018 % This routine uses a Newton-Raphson iteration described in [1] to invert
    0019 % the cross-correlation function (as in the second usage example above).
    0020 %
    0021 % Refs:
    0022 % [1]  G. Wilson. Factorization of the covariance generating function of a pure moving average process.
    0023 %      SIAM Journal on Numerical Analysis, 6 (1): 1�7, 1969.
    0024
    0025 %      Copyright (C) Mike Brookes 2015
    0026 %      Version: $Id: v_lpcra2ar.m 10865 2018-09-21 17:22:45Z dmb $
    0027 %
    0028 %   VOICEBOX is a MATLAB toolbox for speech processing.
    0029 %   Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html
    0030 %
    0031 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    0032 %   This program is free software; you can redistribute it and/or modify
    0033 %   it under the terms of the GNU Lesser General Public License as published by
    0034 %   the Free Software Foundation; either version 3 of the License, or
    0035 %   (at your option) any later version.
    0036 %
    0037 %   This program is distributed in the hope that it will be useful,
    0038 %   but WITHOUT ANY WARRANTY; without even the implied warranty of
    0039 %   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    0040 %   GNU Lesser General Public License for more details.
    0041 %
    0042 %   You can obtain a copy of the GNU Lesser General Public License from
    0043 %   https://www.gnu.org/licenses/ .
    0044 %
    0045 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    0046 persistent p0 i1 i2 j1 j2
    0047 if nargin<2
    0048     tol=1e-8;  % tolerance in ra/ra(1)
    0049 end
    0050 imax=20;
    0051 [nf,pp]=size(ra);
    0052 if ~numel(p0) || p0~=pp % create index lists for hankel and toeplitz matrices
    0053     p0=pp;
    0054     ix=zeros(1,(pp)*(pp+1)/2);
    0055     nn=1:pp;
    0056     ix(1+(nn-1).*nn/2)=1;
    0057     j1=cumsum(ix);
    0058     i1=cumsum(pp-1+(j1*(1-pp)+pp).*ix)-pp+1;
    0059     j2=pp+1-j1;
    0060     i2=cumsum(((pp+1)*j1-1).*ix-pp-1)+pp+1;
    0061 end
    0062 ar=zeros(nf,pp);    % space for output filter coefficients
    0063 t1=zeros(pp,pp);    % space for hankel coefficient matrix
    0064 t2=t1;              % space for toeplitz lower triangular coefficient matrix
    0065 ax0=zeros(1,pp);    % temporary filter coefficient row vector
    0066 for n=1:nf          % process the input matrix one row at a time
    0067     xa=ra(n,:);     % pick out row n
    0068     ax=ax0;         % initialize ax to have all roots at zero
    0069     ax(1)=sqrt(xa(1)+2*sum(xa(2:end)));
    0070     i=imax;         % maximum number of iterations
    0071     while(i>0)
    0072         t1(i1)=ax(j1); % t1=hankel(ax)
    0073         t2(i2)=ax(j2); % t2=toeplitz(ax,[ax(1) zeros(1,p)])
    0074         ct=ax*t1;
    0075         ax=(xa+ct)/(t1+t2);
    0076         i=min(i-1,i*(max(abs(ct-xa))>tol*xa(1))+1); % do one final iteration after tolerance reached
    0077     end
    0078     ar(n,:)=ax;
    0079 end