# 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.
0030 %
0031 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0032 %   This program is free software; you can redistribute it and/or modify
0034 %   the Free Software Foundation; either version 2 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 General Public License for more details.
0041 %
0042 %   You can obtain a copy of the GNU General Public License from
0043 %   http://www.gnu.org/copyleft/gpl.html or by writing to
0044 %   Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA.
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```