# v_lpcar2am

## PURPOSE

V_LPCAR2AM Convert ar coefs to ar coef matrix [AM,EM]=(AR,P)

## SYNOPSIS

function [am,em]=v_lpcar2am(ar,p);

## DESCRIPTION

    V_LPCAR2AM Convert ar coefs to ar coef matrix [AM,EM]=(AR,P)
    AM is a 3-dimensional matrix of size (p+1,p+1,nf) where p is the lpc order
    and nf the number of frames.
    The matrix AM(:,:,*) is upper triangular with 1's on the main diagonal
    and contains the lpc coefficients for all orders from p down to 0.

    For lpc order p+1-r, AM(r,r:p+1,*), AM(p+1:-1:r,p+1,*) and EM(*,r) contain
    the lpc coefficients, reflection coefficients and the residual energy respectively.
    EM(:,1) is always 1.

    If A=am(:,:,*), R=toeplitz(rr(*,:)) and E=diag(em(*,:)), then
     A*R*A'=E; inv(R)=A'*(1/E)*A; A*R is lower triangular with the same diagonal as E

     For each j in 1:P we have AM(j:end:-1:j+1,*) = AM(j:end-1,end,*)'*am(j+1:end,j+1:end,*)

     Also em(*,1:end)' = em(*,2:end)'.*(1-am(1:end-1,end,*).^2)

## CROSS-REFERENCE INFORMATION

This function calls:

This function is called by:

## SOURCE CODE

    0001 function [am,em]=v_lpcar2am(ar,p);
    0002 %V_LPCAR2AM Convert ar coefs to ar coef matrix [AM,EM]=(AR,P)
    0003 %AM is a 3-dimensional matrix of size (p+1,p+1,nf) where p is the lpc order
    0004 %and nf the number of frames.
    0005 %The matrix AM(:,:,*) is upper triangular with 1's on the main diagonal
    0006 %and contains the lpc coefficients for all orders from p down to 0.
    0007 %
    0008 %For lpc order p+1-r, AM(r,r:p+1,*), AM(p+1:-1:r,p+1,*) and EM(*,r) contain
    0009 %the lpc coefficients, reflection coefficients and the residual energy respectively.
    0010 %EM(:,1) is always 1.
    0011 %
    0012 %If A=am(:,:,*), R=toeplitz(rr(*,:)) and E=diag(em(*,:)), then
    0013 % A*R*A'=E; inv(R)=A'*(1/E)*A; A*R is lower triangular with the same diagonal as E
    0014 %
    0015 % For each j in 1:P we have AM(j:end:-1:j+1,*) = AM(j:end-1,end,*)'*am(j+1:end,j+1:end,*)
    0016 %
    0017 % Also em(*,1:end)' = em(*,2:end)'.*(1-am(1:end-1,end,*).^2)
    0018
    0019 % we should be able to do this more directly using the step down algorithm
    0020
    0021 %      Copyright (C) Mike Brookes 1997
    0022 %      Version: $Id: v_lpcar2am.m 10865 2018-09-21 17:22:45Z dmb $
    0023 %
    0024 %   VOICEBOX is a MATLAB toolbox for speech processing.
    0025 %   Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html
    0026 %
    0027 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    0028 %   This program is free software; you can redistribute it and/or modify
    0029 %   it under the terms of the GNU Lesser General Public License as published by
    0030 %   the Free Software Foundation; either version 3 of the License, or
    0031 %   (at your option) any later version.
    0032 %
    0033 %   This program is distributed in the hope that it will be useful,
    0034 %   but WITHOUT ANY WARRANTY; without even the implied warranty of
    0035 %   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    0036 %   GNU Lesser General Public License for more details.
    0037 %
    0038 %   You can obtain a copy of the GNU Lesser General Public License from
    0039 %   https://www.gnu.org/licenses/ .
    0040 %
    0041 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    0042
    0043 [nf,p1] = size(ar);
    0044 if any(ar(:,1)~=1)
    0045   ar=ar./ar(:,ones(1,p1));
    0046 end
    0047 p0=p1-1;
    0048 if nargin < 2
    0049    p=p0;
    0050 end
    0051 am=zeros(nf,p+1,p+1);
    0052 em=ones(nf,p+1);
    0053 e=ones(nf,1);
    0054 rf=ar;
    0055 if p>=p0
    0056    for jj=1:p+1-p0
    0057       am(:,jj:jj+p0,jj)=ar;
    0058    end
    0059 else
    0060    for j=p0:-1:p+2
    0061       k = rf(:,j+1);
    0062       d = (1-k.^2).^(-1);
    0063       e = e.*d;
    0064       wj=ones(1,j-1);
    0065       rf(:,2:j) = (rf(:,2:j)-k(:,wj).*rf(:,j:-1:2)).*d(:,wj);
    0066    end
    0067    jj=0;
    0068 end
    0069 for jj=jj+1:p
    0070    j = p+2-jj;
    0071    k = rf(:,j+1);
    0072    d = (1-k.^2).^(-1);
    0073    e = e.*d;
    0074    wj=ones(1,j-1);
    0075    rf(:,2:j) = (rf(:,2:j)-k(:,wj).*rf(:,j:-1:2)).*d(:,wj);
    0076    am(:,jj:end,jj)=rf(:,1:j);
    0077    em(:,jj)=e;
    0078 end
    0079 em(:,end)=e./(1-rf(:,2).^2);
    0080 am(:,end,end)=1;
    0081 am=permute(am,[3 2 1]);
    0082