Commit 65a4ae98 authored by Ralph Giles's avatar Ralph Giles
Browse files

Add latex versions of the lsp equations.

svn path=/trunk/vorbis/; revision=3999
parent 1c23ffc1
<section id="vorbis-spec-floor0">
<sectioninfo>
<releaseinfo>
$Id: 06-floor0.xml,v 1.1 2002/10/12 20:37:11 giles Exp $
$Id: 06-floor0.xml,v 1.2 2002/10/13 14:32:16 giles Exp $
<emphasis>Last update to this document: July 19, 2002</emphasis>
</releaseinfo>
</sectioninfo>
......@@ -137,7 +137,35 @@ assuming the following definitions for the given vector to be
synthesized:</para>
<informalequation>
<alt>[latex for lspmap goes here]</alt>
<alt>
[lsp map equation]
\begin{math}
\mathrm{map}_i = \left\{
\begin{array}{ll}
\min (
\mathtt{floor0\_bark\_map\_size} - 1,
foobar
) & \textrm{for } i \in [0,n-1] \\
-1 & \textrm{for } i = n
\end{array}
\right.
\end {math}
where
\begin{math}
foobar =
\left\lfloor
\mathrm{bark}\left(\frac{\mathtt{floor0\_rate} \cdot i}{2n}\right) \cdot \frac{\mathtt{floor0\_bark\_map\_size}} {\mathrm{bark}(.5 \cdot \mathtt{floor0\_rate})}
\right\rfloor
\end{math}
and
\begin{math}
\mathrm{bark}(x) = 13.1 \arctan (.00074x) + 2.24 \arctan (.0000000158x^2)+.0001x
\end{math}
</alt>
<graphic fileref="lspmap.png"/>
</informalequation>
......@@ -153,7 +181,13 @@ Similarly, the below calculation synthesizes the output LSP curve <varname>[outp
<orderedlist>
<listitem><para>calculate <varname>[p]</varname> and <varname>[q]</varname> according to:
<informalequation>
<alt>[latex for oddlsp here]</alt>
<alt>
[equation for odd lsp]
\begin{eqnarray*}
p & = & (1 - \cos^2\omega)\prod_{j=0}^{(\mathtt{order}-3)/2} 4 (\cos c_{2j+1} - \cos \omega)^2 \\
q & = & \frac{1}{4} \prod_{j=0}^{(\mathtt{order}-1)/2} 4 (\cos c_{2j+1} - \cos \omega)^2
\end{eqnarray*}
</alt>
<graphic fileref="oddlsp.png"/>
</informalequation>
</para></listitem>
......@@ -162,7 +196,13 @@ Similarly, the below calculation synthesizes the output LSP curve <varname>[outp
<orderedlist>
<listitem><para>calculate <varname>[p]</varname> and <varname>[q]</varname> according to:
<informalequation>
<alt>[latex for evenlsp here]</alt>
<alt>
[equation for even lsp]
\begin{eqnarray*}
p & = & \frac{(1 - \cos^2\omega)}{2} \prod_{j=0}^{(\mathtt{order}-2)/2} 4 (\cos c_{2j} - \cos \omega)^2 \\
q & = & \frac{(1 + \cos^2\omega)}{2} \prod_{j=0}^{(\mathtt{order}-2)/2} 4 (\cos c_{2j} - \cos \omega)^2
\end{eqnarray*}
</alt>
<graphic fileref="evenlsp.png"/>
</informalequation>
</para></listitem>
......@@ -171,7 +211,13 @@ Similarly, the below calculation synthesizes the output LSP curve <varname>[outp
</para></listitem>
<listitem><para>calculate <varname>[linear_floor_value]</varname> according to:
<informalequation>
<alt>[latex for floorval here]</alt>
<alt>
[expression for floorval]
\begin{math}
\exp \left( .11512925 \left(\frac{\mathtt{amplitude} \cdot \mathtt{floor0\_amplitute\_offset}}{(2^{\mathtt{floor0\_amplitude\_bits}}-1)\sqrt{p+q}}
- \mathtt{floor0\_amplitude\_offset} \right) \right)
\end{math}
</alt>
<graphic fileref="floorval.png"/>
</informalequation>
</para></listitem>
......
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