% -*- mode: latex; TeX-master: "Vorbis_I_spec"; -*-
%!TEX root = Vorbis_I_spec.tex
% $Id$
\section{Floor type 0 setup and decode} \label{vorbis:spec:floor0}

\subsection{Overview}

Vorbis floor type zero uses Line Spectral Pair (LSP, also alternately
known as Line Spectral Frequency or LSF) representation to encode a
smooth spectral envelope curve as the frequency response of the LSP
filter.  This representation is equivalent to a traditional all-pole
infinite impulse response filter as would be used in linear predictive
coding; LSP representation may be converted to LPC representation and
vice-versa.



\subsection{Floor 0 format}

Floor zero configuration consists of six integer fields and a list of
VQ codebooks for use in coding/decoding the LSP filter coefficient
values used by each frame.

\subsubsection{header decode}

Configuration information for instances of floor zero decodes from the
codec setup header (third packet).  configuration decode proceeds as
follows:

\begin{Verbatim}[commandchars=\\\{\}]
  1) [floor0\_order] = read an unsigned integer of 8 bits
  2) [floor0\_rate] = read an unsigned integer of 16 bits
  3) [floor0\_bark\_map\_size] = read an unsigned integer of 16 bits
  4) [floor0\_amplitude\_bits] = read an unsigned integer of six bits
  5) [floor0\_amplitude\_offset] = read an unsigned integer of eight bits
  6) [floor0\_number\_of\_books] = read an unsigned integer of four bits and add 1
  7) array [floor0\_book\_list] = read a list of [floor0\_number\_of\_books] unsigned integers of eight bits each;
\end{Verbatim}

An end-of-packet condition during any of these bitstream reads renders
this stream undecodable.  In addition, any element of the array
\varname{[floor0\_book\_list]} that is greater than the maximum codebook
number for this bitstream is an error condition that also renders the
stream undecodable.



\subsubsection{packet decode} \label{vorbis:spec:floor0-decode}

Extracting a floor0 curve from an audio packet consists of first
decoding the curve amplitude and \varname{[floor0\_order]} LSP
coefficient values from the bitstream, and then computing the floor
curve, which is defined as the frequency response of the decoded LSP
filter.

Packet decode proceeds as follows:
\begin{Verbatim}[commandchars=\\\{\}]
  1) [amplitude] = read an unsigned integer of [floor0\_amplitude\_bits] bits
  2) if ( [amplitude] is greater than zero ) \{
       3) [coefficients] is an empty, zero length vector
       4) [booknumber] = read an unsigned integer of \link{vorbis:spec:ilog}{ilog}( [floor0\_number\_of\_books] ) bits
       5) if ( [booknumber] is greater than the highest number decode codebook ) then packet is undecodable
       6) [last] = zero;
       7) vector [temp\_vector] = read vector from bitstream using codebook number [floor0\_book\_list] element [booknumber] in VQ context.
       8) add the scalar value [last] to each scalar in vector [temp\_vector]
       9) [last] = the value of the last scalar in vector [temp\_vector]
      10) concatenate [temp\_vector] onto the end of the [coefficients] vector
      11) if (length of vector [coefficients] is less than [floor0\_order], continue at step 6

     \}

 12) done.

\end{Verbatim}

Take note of the following properties of decode:
\begin{itemize}
 \item An \varname{[amplitude]} value of zero must result in a return code that indicates this channel is unused in this frame (the output of the channel will be all-zeroes in synthesis).  Several later stages of decode don't occur for an unused channel.
 \item An end-of-packet condition during decode should be considered a
nominal occruence; if end-of-packet is reached during any read
operation above, floor decode is to return 'unused' status as if the
\varname{[amplitude]} value had read zero at the beginning of decode.

 \item The book number used for decode
can, in fact, be stored in the bitstream in \link{vorbis:spec:ilog}{ilog}( \varname{[floor0\_number\_of\_books]} -
1 ) bits.  Nevertheless, the above specification is correct and values
greater than the maximum possible book value are reserved.

 \item The number of scalars read into the vector \varname{[coefficients]}
may be greater than \varname{[floor0\_order]}, the number actually
required for curve computation.  For example, if the VQ codebook used
for the floor currently being decoded has a
\varname{[codebook\_dimensions]} value of three and
\varname{[floor0\_order]} is ten, the only way to fill all the needed
scalars in \varname{[coefficients]} is to to read a total of twelve
scalars as four vectors of three scalars each.  This is not an error
condition, and care must be taken not to allow a buffer overflow in
decode. The extra values are not used and may be ignored or discarded.
\end{itemize}




\subsubsection{curve computation} \label{vorbis:spec:floor0-synth}

Given an \varname{[amplitude]} integer and \varname{[coefficients]}
vector from packet decode as well as the [floor0\_order],
[floor0\_rate], [floor0\_bark\_map\_size], [floor0\_amplitude\_bits] and
[floor0\_amplitude\_offset] values from floor setup, and an output
vector size \varname{[n]} specified by the decode process, we compute a
floor output vector.

If the value \varname{[amplitude]} is zero, the return value is a
length \varname{[n]} vector with all-zero scalars.  Otherwise, begin by
assuming the following definitions for the given vector to be
synthesized:

   \begin{displaymath}
     \mathrm{map}_i = \left\{
       \begin{array}{ll}
          \min (
            \mathtt{floor0\texttt{\_}bark\texttt{\_}map\texttt{\_}size} - 1,
            foobar
          ) & \textrm{for } i \in [0,n-1] \\
          -1 & \textrm{for } i = n
        \end{array}
      \right.
    \end{displaymath}

    where

    \begin{displaymath}
    foobar =
      \left\lfloor
        \mathrm{bark}\left(\frac{\mathtt{floor0\texttt{\_}rate} \cdot i}{2n}\right) \cdot \frac{\mathtt{floor0\texttt{\_}bark\texttt{\_}map\texttt{\_}size}} {\mathrm{bark}(.5 \cdot \mathtt{floor0\texttt{\_}rate})}
      \right\rfloor
    \end{displaymath}

    and

    \begin{displaymath}
      \mathrm{bark}(x) = 13.1 \arctan (.00074x) + 2.24 \arctan (.0000000185x^2) + .0001x
    \end{displaymath}

The above is used to synthesize the LSP curve on a Bark-scale frequency
axis, then map the result to a linear-scale frequency axis.
Similarly, the below calculation synthesizes the output LSP curve \varname{[output]} on a log
(dB) amplitude scale, mapping it to linear amplitude in the last step:

\begin{enumerate}
 \item  \varname{[i]} = 0
 \item  \varname{[$\omega$]} = $\pi$ * map element \varname{[i]} / \varname{[floor0\_bark\_map\_size]}
 \item if ( \varname{[floor0\_order]} is odd ) {
  \begin{enumerate}
   \item calculate \varname{[p]} and \varname{[q]} according to:
           \begin{eqnarray*}
             p & = & (1 - \cos^2\omega)\prod_{j=0}^{\frac{\mathtt{floor0\texttt{\_}order}-3}{2}} 4 (\cos([\mathtt{coefficients}]_{2j+1}) - \cos \omega)^2 \\
             q & = & \frac{1}{4} \prod_{j=0}^{\frac{\mathtt{floor0\texttt{\_}order}-1}{2}} 4 (\cos([\mathtt{coefficients}]_{2j}) - \cos \omega)^2
           \end{eqnarray*}

  \end{enumerate}
  } else \varname{[floor0\_order]} is even {
  \begin{enumerate}[resume]
   \item calculate \varname{[p]} and \varname{[q]} according to:
           \begin{eqnarray*}
             p & = & \frac{(1 - \cos\omega)}{2} \prod_{j=0}^{\frac{\mathtt{floor0\texttt{\_}order}-2}{2}} 4 (\cos([\mathtt{coefficients}]_{2j+1}) - \cos \omega)^2 \\
             q & = & \frac{(1 + \cos\omega)}{2} \prod_{j=0}^{\frac{\mathtt{floor0\texttt{\_}order}-2}{2}} 4 (\cos([\mathtt{coefficients}]_{2j}) - \cos \omega)^2
           \end{eqnarray*}

  \end{enumerate}
  }

 \item calculate \varname{[linear\_floor\_value]} according to:
         \begin{displaymath}
           \exp \left( .11512925 \left(\frac{\mathtt{amplitude} \cdot \mathtt{floor0\texttt{\_}amplitute\texttt{\_}offset}}{(2^{\mathtt{floor0\texttt{\_}amplitude\texttt{\_}bits}}-1)\sqrt{p+q}}
                  - \mathtt{floor0\texttt{\_}amplitude\texttt{\_}offset} \right) \right)
         \end{displaymath}

 \item \varname{[iteration\_condition]} = map element \varname{[i]}
 \item \varname{[output]} element \varname{[i]} = \varname{[linear\_floor\_value]}
 \item increment \varname{[i]}
 \item if ( map element \varname{[i]} is equal to \varname{[iteration\_condition]} ) continue at step 5
 \item if ( \varname{[i]} is less than \varname{[n]} ) continue at step 2
 \item done
\end{enumerate}

\paragraph{Errata 20150227: Bark scale computation}

Due to a typo when typesetting this version of the specification from the original HTML document, the Bark scale computation previously erroneously read:

    \begin{displaymath}
      \hbox{\sout{$
      \mathrm{bark}(x) = 13.1 \arctan (.00074x) + 2.24 \arctan (.0000000185x^2 + .0001x)
      $}}
    \end{displaymath}

Note that the last parenthesis is misplaced.  This document now uses the correct equation as it appeared in the original HTML spec document:

    \begin{displaymath}
      \mathrm{bark}(x) = 13.1 \arctan (.00074x) + 2.24 \arctan (.0000000185x^2) + .0001x
    \end{displaymath}

