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# lpcrf2ar

## PURPOSE

LPCRF2AR Convert reflection coefs to autoregressive coefs [AR,ARP,ARU,G]=(RF)

## SYNOPSIS

function [ar,arp,aru,g]=lpcrf2ar(rf)

## DESCRIPTION

```LPCRF2AR Convert reflection coefs to autoregressive coefs [AR,ARP,ARU,G]=(RF)

Input:  RF(:,p+1) gives reflection coefficients of one or more p-section lossless tubes
Ouputs: G is the gain of the all-pole AR filter
AR/G is the transfer function from U_in to the glottal input wave, U_g.
AR(:,1)=1 always.
ARP*K is the transfer function from U_in to the pressure just after the glottis
where K = rho*c/Alips: rho = air density 1.23 kg/m^3, c=sound speed 340 m/s,
Alips = effective area of free space beyond the lips.
ARU is the transfer function from U_in to the total volume velocity through the glottis

where U_in=z^(p/2)*U_lips is the time-advanced volume velocity at the lips

Energy into the vcal tract is equal to K*filter(ARP,1,Ulips).*filter(ARU,1,Ulips)
reverse glottal flows divided by 1-r0 where r0 is the glottal reflection coefficient.
The scale factor is included to avoid a zero answer when the glottis is closed giving r0=1.

The transfer functions have ar(:,1)=art(:,1)=1
They should both be multiplied by z^(p/2)/prod(1+rf) to correct the absolute
gain and to compensate for the delay of p/2 samples along the length of the tube.

The energy into the vocal tract is given by ars(speech) * are(speech)

Ref: D. M. Brookes and H. P. Loke. "Modelling energy flow in the vocal tract with
applications to glottal closure and opening detection." In Proc ICASSP'99, pages 213-216, Mar 1999.```

## CROSS-REFERENCE INFORMATION

This function calls:
This function is called by:
• lpcrand generate n random stable polynomials of order p with a minimum pole

## SOURCE CODE

```0001 function [ar,arp,aru,g]=lpcrf2ar(rf)
0002 %LPCRF2AR Convert reflection coefs to autoregressive coefs [AR,ARP,ARU,G]=(RF)
0003 %
0004 % Input:  RF(:,p+1) gives reflection coefficients of one or more p-section lossless tubes
0005 % Ouputs: G is the gain of the all-pole AR filter
0006 %         AR/G is the transfer function from U_in to the glottal input wave, U_g.
0007 %               AR(:,1)=1 always.
0008 %         ARP*K is the transfer function from U_in to the pressure just after the glottis
0009 %               where K = rho*c/Alips: rho = air density 1.23 kg/m^3, c=sound speed 340 m/s,
0010 %               Alips = effective area of free space beyond the lips.
0011 %         ARU is the transfer function from U_in to the total volume velocity through the glottis
0012 %
0013 %              where U_in=z^(p/2)*U_lips is the time-advanced volume velocity at the lips
0014 %
0015 %         Energy into the vcal tract is equal to K*filter(ARP,1,Ulips).*filter(ARU,1,Ulips)
0016 %              reverse glottal flows divided by 1-r0 where r0 is the glottal reflection coefficient.
0017 %              The scale factor is included to avoid a zero answer when the glottis is closed giving r0=1.
0018 %
0019 % The transfer functions have ar(:,1)=art(:,1)=1
0020 % They should both be multiplied by z^(p/2)/prod(1+rf) to correct the absolute
0021 % gain and to compensate for the delay of p/2 samples along the length of the tube.
0022 %
0023 % The energy into the vocal tract is given by ars(speech) * are(speech)
0024 %
0025 % Ref: D. M. Brookes and H. P. Loke. "Modelling energy flow in the vocal tract with
0026 %           applications to glottal closure and opening detection." In Proc ICASSP'99, pages 213-216, Mar 1999.
0027
0028
0029 %      Copyright (C) Mike Brookes 1997
0030 %      Version: \$Id: lpcrf2ar.m 713 2011-10-16 14:45:43Z dmb \$
0031 %
0032 %   VOICEBOX is a MATLAB toolbox for speech processing.
0034 %
0035 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0036 %   This program is free software; you can redistribute it and/or modify
0038 %   the Free Software Foundation; either version 2 of the License, or
0039 %   (at your option) any later version.
0040 %
0041 %   This program is distributed in the hope that it will be useful,
0042 %   but WITHOUT ANY WARRANTY; without even the implied warranty of
0043 %   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
0044 %   GNU General Public License for more details.
0045 %
0046 %   You can obtain a copy of the GNU General Public License from
0047 %   http://www.gnu.org/copyleft/gpl.html or by writing to
0048 %   Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA.
0049 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0050
0051 [nf,p1]=size(rf);
0052 p2=p1+1;
0053 p=p1-1;
0054 pm=p-1;
0055 arf=[ones(nf,1) zeros(nf,p)];
0056 arr=[zeros(nf,p) rf(:,p1)];
0057 cr=zeros(nf,p);
0058 for k=1:p-1
0059   rk=rf(:,(p1-k)*ones(1,k));
0060   cr(:,1:k)=arr(:,p2-k:p1);
0061   arr(:,p1-k:p)=arr(:,p1-k:p)+rk.*arf(:,1:k);
0062   arf(:,2:k+1)=arf(:,2:k+1)+rk.*cr(:,1:k);
0063 end
0064 r1=rf(:,1);
0065 ar=arf+r1(:,ones(1,p1)).*arr;
0066 if nargout>1
0067    kp=prod(1-rf(:,2:p1),2);
0068    arp=(arf-arr)./kp(:,ones(1,p1));
0069    if nargout>2
0070       g=prod(1+rf(:,2:p1),2);
0071       aru=(arf+arr)./g(:,ones(1,p1));
0072       if nargout>3
0073          g=g.*(1+rf(:,1));
0074       end
0075    end
0076 end
0077
0078```

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