# v_distitar

## PURPOSE

V_DISTITAR calculates the Itakura distance between AR coefficients D=(AR1,AR2,MODE)

## SYNOPSIS

function d=v_distitar(ar1,ar2,mode)

## DESCRIPTION

    V_DISTITAR calculates the Itakura distance between AR coefficients D=(AR1,AR2,MODE)

     Inputs: AR1,AR2     AR coefficient sets to be compared. Each row contains a set of coefficients.
                         AR1 and AR2 must have the same number of columns.

             MODE        Character string selecting the following options:
                             'x'  Calculate the full distance matrix from every row of AR1 to every row of AR2
                             'd'  Calculate only the distance between corresponding rows of AR1 and AR2
                                  The default is 'd' if AR1 and AR2 have the same number of rows otherwise 'x'.
                              'e'  Calculates exp(d) instead of d (quicker because no log is necessary)

     Output: D           If MODE='d' then D is a column vector with the same number of rows as the shorter of AR1 and AR2.
                         If MODE='x' then D is a matrix with the same number of rows as AR1 and the same number of columns as AR2'.

     If ave() denotes the average over +ve and -ve frequency, the Itakura spectral distance is

                                   log(ave(pf1/pf2)) - ave(log(pf1/pf2))

     The Itakura distance is gain-independent, i.e. v_distitpf(f*pf1,g*pf2) is independent of f and g.

     The Itakura distance may be expressed as log(ar2*toeplitz(lpcar2rr(ar1))*ar2') where the ar1 and ar2 polynomials
     have first been normalised by dividing through by their 0'th order coefficients.

## CROSS-REFERENCE INFORMATION

This function calls:

- v_lpcar2ra V_LPCAR2RA Convert ar filter to inverse filter autocorrelation coefs. RA=(AR)
- v_lpcar2rr V_LPCAR2RR Convert autoregressive coefficients to autocorrelation coefficients RR=(AR,P)

This function is called by:

- v_fxrapt V_FXRAPT RAPT pitch tracker [FX,VUV]=(S,FS,M,Q)

## SOURCE CODE

    0001 function d=v_distitar(ar1,ar2,mode)
    0002 %V_DISTITAR calculates the Itakura distance between AR coefficients D=(AR1,AR2,MODE)
    0003 %
    0004 % Inputs: AR1,AR2     AR coefficient sets to be compared. Each row contains a set of coefficients.
    0005 %                     AR1 and AR2 must have the same number of columns.
    0006 %
    0007 %         MODE        Character string selecting the following options:
    0008 %                         'x'  Calculate the full distance matrix from every row of AR1 to every row of AR2
    0009 %                         'd'  Calculate only the distance between corresponding rows of AR1 and AR2
    0010 %                              The default is 'd' if AR1 and AR2 have the same number of rows otherwise 'x'.
    0011 %                          'e'  Calculates exp(d) instead of d (quicker because no log is necessary)
    0012 %
    0013 % Output: D           If MODE='d' then D is a column vector with the same number of rows as the shorter of AR1 and AR2.
    0014 %                     If MODE='x' then D is a matrix with the same number of rows as AR1 and the same number of columns as AR2'.
    0015 %
    0016 % If ave() denotes the average over +ve and -ve frequency, the Itakura spectral distance is
    0017 %
    0018 %                               log(ave(pf1/pf2)) - ave(log(pf1/pf2))
    0019 %
    0020 % The Itakura distance is gain-independent, i.e. v_distitpf(f*pf1,g*pf2) is independent of f and g.
    0021 %
    0022 % The Itakura distance may be expressed as log(ar2*toeplitz(lpcar2rr(ar1))*ar2') where the ar1 and ar2 polynomials
    0023 % have first been normalised by dividing through by their 0'th order coefficients.
    0024
    0025 % Since the power spectrum is the fourier transform of the autocorrelation, we can calculate
    0026 % the average value of p1/p2 by taking the 0'th order term of the convolution of the autocorrelation
    0027 % functions associated with p1 and 1/p2. Since 1/p2 corresponds to an FIR filter, this convolution is
    0028 % a finite sum even though the autocorrelation function of p1 is infinite in extent.
    0029 % The average value of log(pf1) is equal to log(ar1(1)^-2) where ar1(1) is the 0'th order AR coefficient.
    0030
    0031 % The Itakura distance can also be calculated directly from the power spectra; providing np is large
    0032 % enough, the values of d0 and d1 in the following will be very similar:
    0033 %
    0034 %         np=255; d0=v_distitar(ar1,ar2); d1=v_distitpf(v_lpcar2pf(ar1,np),v_lpcar2pf(ar2,np))
    0035 %
    0036
    0037 % Ref: A.H.Gray Jr and J.D.Markel, "Distance measures for speech processing", IEEE ASSP-24(5): 380-391, Oct 1976
    0038 %      L. Rabiner abd B-H Juang, "Fundamentals of Speech Recognition", Section 4.5, Prentice-Hall 1993, ISBN 0-13-015157-2
    0039 %      F. Itakura, "Minimum prediction residual principle applied to speech recognition", IEEE ASSP-23: 62-72, 1975
    0040
    0041 %      Copyright (C) Mike Brookes 1997
    0042 %      Version: $Id: v_distitar.m 10865 2018-09-21 17:22:45Z dmb $
    0043 %
    0044 %   VOICEBOX is a MATLAB toolbox for speech processing.
    0045 %   Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html
    0046 %
    0047 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    0048 %   This program is free software; you can redistribute it and/or modify
    0049 %   it under the terms of the GNU Lesser General Public License as published by
    0050 %   the Free Software Foundation; either version 3 of the License, or
    0051 %   (at your option) any later version.
    0052 %
    0053 %   This program is distributed in the hope that it will be useful,
    0054 %   but WITHOUT ANY WARRANTY; without even the implied warranty of
    0055 %   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    0056 %   GNU Lesser General Public License for more details.
    0057 %
    0058 %   You can obtain a copy of the GNU Lesser General Public License from
    0059 %   https://www.gnu.org/licenses/ .
    0060 %
    0061 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    0062
    0063 [nf1,p1]=size(ar1);
    0064 nf2=size(ar2,1);
    0065 m2=v_lpcar2ra(ar2);
    0066 m2(:,1)=0.5*m2(:,1);
    0067 if nargin<3 | isempty(mode) mode='0'; end
    0068 if any(mode=='d') | (mode~='x' & nf1==nf2)
    0069    nx=min(nf1,nf2);
    0070    d=2*sum(v_lpcar2rr(ar1(1:nx,:)).*m2(1:nx,:),2).*((ar1(1:nx,1)./ar2(1:nx,1)).^2);
    0071 else
    0072    d=2*v_lpcar2rr(ar1)*m2'.*((ar1(:,1)*ar2(:,1)'.^(-1)).^2);
    0073 end
    0074 if all(mode~='e')
    0075     d=log(d);
    0076 end