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📄 matlab routines for linear predictive coding (lpc).htm

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<H1>Matlab routines for Linear Predictive Coding (LPC)</H1>
<HR>

<P>Return to <A 
href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html">voicebox home 
page</A></P>
<HR>

<H2>Data Format</H2>
<P>All the LPC routines described in this section can process several frames 
together. Each frame corresponds to a single row of the data matrix; if there is 
only one frame then the matrix must be a row vector rather than a column 
vector.</P>
<H2>LPC Analysis</H2>
<P>Two routines are provided: <A 
href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcauto.txt">lpcauto</A> 
for autocorrelation analysis and <A 
href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpccovar.txt">lpccovar</A> 
for covariance analysis.</P>
<P>The analysis order, <I>p</I>, denotes the number of poles in the resultant 
autoregressive filter. The appropriate value for <I>p</I> is typically 
2+<I>f<SUB>s</SUB></I>/1kHz where <I>f<SUB>s</SUB></I> is the sample frequency. 
This expression assumes that sound takes about = ms to travel the length of the 
vocal tract.</P>
<H3>Fancy versions of LPC</H3>
<P>Although the analysis order must be the same for all frames, the individual 
frame lengths can vary; this allows <I>pitch-synchronous</I> analysis. It is 
also possible to restrict the analysis interval to particular segments of each 
frame to allow <I>closed-phase</I> analysis. For high pitched voices, the closed 
phases may be very short: it is possible in this case to combine the data from 
two or more consecutive cycles to give <I>multi-cycle closed-phase</I> analysis. 
To obtain reliable estimates of the AR coefficients must be based on at least 2 
ms of data.</P>
<H2>LPC Coefficient Representations</H2>
<P>The coefficients generated by LPC analysis can be represented in many 
equivalent forms. Voicebox recognizes the coefficient sets listed below and 
denotes each with a two-letter mnemonic. The number of coefficients varies: for 
an analysis of order <I>p</I> there can be <I>p</I>, <I>p</I>+1, or <I>p</I>+2 
coefficients. This is indicated in the table. The meaning of the coefficient 
sets is explained below with reference to the lossless tube model of speech 
production. Routines are provided to convert each representation to the other 
forms indicated: the routine that converts from representation <EM>xx</EM> to 
representation <EM>yy</EM> is called lpc<EM>xx</EM>2<EM>yy</EM>.</P>
<P>The routine <A 
href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcconv.txt">lpcconv 
</A>can be used to figure out the sequence of calls needed to convert between 
any pair of these representations.</P>
<TABLE border=0 cellPadding=6 cellSpacing=0 width="100%">
  <TBODY>
  <TR>
    <TD vAlign=top width=50><B>Code</B></TD>
    <TD vAlign=top width=70><B>Size</B></TD>
    <TD vAlign=top width=120><B>Convert from</B></TD>
    <TD vAlign=top width=120><B>Convert to</B></TD>
    <TD vAlign=top><B>Description</B></TD></TR>
  <TR>
    <TD vAlign=top width=50>aa</TD>
    <TD vAlign=top width=70><I>p</I>+2</TD>
    <TD vAlign=top width=120><A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcdl2aa.txt">dl</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcrf2aa.txt">rf</A></TD>
    <TD vAlign=top width=120><A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcaa2ao.txt">ao</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcaa2dl.txt">dl</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcaa2rf.txt">rf</A></TD>
    <TD vAlign=top>The <I>area coefficients</I> represent the cross-sectional 
      areas of the vocal tract segments. The areas are normalised so that 
      aa(<I>p</I>+2), the effective area of the free space beyond the lips, is 
      equal to 1. aa(1) is the area at the glottis and is usually near 0.</TD></TR>
  <TR>
    <TD vAlign=top width=50>am</TD>
    <TD vAlign=top width=70>(<EM>p</EM>+1)<SUP>2</SUP></TD>
    <TD vAlign=top width=120><A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcar2am.txt">ar</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcrr2am.txt">rr</A></TD>
    <TD vAlign=top width=120></TD>
    <TD vAlign=top>An upper unit-triangular matrix containing the AR 
      coefficients for all orders 0,...,<EM>p</EM>. This matrix is a diagonal 
      multiple of the hermitian square root of the symmetric toeplitz matrix 
      toeplitz(rr).</TD></TR>
  <TR>
    <TD vAlign=top width=50>ao</TD>
    <TD vAlign=top width=70><I>p</I>+1</TD>
    <TD vAlign=top width=120><A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcaa2ao.txt">aa</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcrf2ao.txt">rf</A></TD>
    <TD vAlign=top width=120><A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcao2rf.txt">rf</A></TD>
    <TD vAlign=top>The <I>area ratios</I> give the ratio of one tube segment 
      to that of the following segment.</TD></TR>
  <TR>
    <TD vAlign=top width=50>ar</TD>
    <TD vAlign=top width=70><I>p</I>+1</TD>
    <TD vAlign=top width=120><A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpccc2ar.txt">cc</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcim2ar.txt">im</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcls2ar.txt">ls</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcrf2ar.txt">rf</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcrr2ar.txt">rr, 
      </A><A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpczz2ar.txt">zz</A><A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcrr2ar.txt"> 
    </A></TD>
    <TD vAlign=top width=120><A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcar2am.txt">am</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcar2cc.txt">cc</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcar2db.txt">db</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcar2ff.txt">ff</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcar2im.txt">im</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcar2ls.txt">ls</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcar2pf.txt">pf</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcar2pp.txt">pp</A>,<A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcar2ra.txt"> 
      ra</A>, <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcar2rf.txt">rf</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcar2rr.txt">rr</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcar2zz.txt">zz</A></TD>
    <TD vAlign=top>The <I>autoregressive coefficients</I> or <I>AR 
      coefficients</I> represent the transfer function from the output flow of 
      the vocal tract to the input flow. The coefficients are usually normalised 
      so that ar(1)=1.</TD></TR>
  <TR>
    <TD vAlign=top width=50>cc</TD>
    <TD vAlign=top width=70><I>p</I></TD>
    <TD vAlign=top width=120><A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcar2cc.txt">ar</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcpf2cc.txt">pf</A>, 
      <A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcpf2cc.txt"></A><A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpczz2cc.txt">zz</A></TD>
    <TD height=145 vAlign=top width=120><A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpccc2ar.txt">ar</A></TD>
    <TD vAlign=top>The <I>complex cepstrum coefficients</I> are actually real 
      despite their name. They equal the inverse fourier transform of the log 
      frequency response of the autoregressive filter. These coefficients do not 
      include cc(0) which is the DC component of the log frequency 
  response.</TD></TR>
  <TR>
    <TD vAlign=top width=50>cw</TD>
    <TD vAlign=top width=70><I>p</I></TD>
    <TD vAlign=top width=120><A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcpp2cw.txt">pp</A></TD>
    <TD vAlign=top width=120><A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpccw2zz.txt">zz</A></TD>
    <TD vAlign=top>The roots of the power spectrum polynomial <I>pp</I>. These 
      are the, normally complex, values of cos(w) that make the power spectrum 
      of the inverse filter equal to zero. </TD></TR>
  <TR>
    <TD vAlign=top width=50>db</TD>
    <TD vAlign=top width=70><I>p</I>+2</TD>
    <TD vAlign=top width=120><A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcar2db.txt">ar</A></TD>
    <TD vAlign=top width=120><A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcdb2pf.txt">pf</A></TD>
    <TD vAlign=top>The <I>power spectrum</I> of the AR filter expressed in 
      decibels. The first and last elements of ff() are respectively the DC and 
      nyquist terms.</TD></TR>
  <TR>
    <TD vAlign=top width=50>dl</TD>
    <TD vAlign=top width=70><I>p</I></TD>
    <TD vAlign=top width=120><A 
      href="http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/txt/lpcaa2dl.txt">aa</A></TD>
    <TD vAlign=top width=120><A
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