Chris@1: % -*- mode: latex; TeX-master: "Vorbis_I_spec"; -*-
Chris@1: %!TEX root = Vorbis_I_spec.tex
Chris@1: % $Id$
Chris@1: \section{Floor type 0 setup and decode} \label{vorbis:spec:floor0}
Chris@1: 
Chris@1: \subsection{Overview}
Chris@1: 
Chris@1: Vorbis floor type zero uses Line Spectral Pair (LSP, also alternately
Chris@1: known as Line Spectral Frequency or LSF) representation to encode a
Chris@1: smooth spectral envelope curve as the frequency response of the LSP
Chris@1: filter.  This representation is equivalent to a traditional all-pole
Chris@1: infinite impulse response filter as would be used in linear predictive
Chris@1: coding; LSP representation may be converted to LPC representation and
Chris@1: vice-versa.
Chris@1: 
Chris@1: 
Chris@1: 
Chris@1: \subsection{Floor 0 format}
Chris@1: 
Chris@1: Floor zero configuration consists of six integer fields and a list of
Chris@1: VQ codebooks for use in coding/decoding the LSP filter coefficient
Chris@1: values used by each frame.
Chris@1: 
Chris@1: \subsubsection{header decode}
Chris@1: 
Chris@1: Configuration information for instances of floor zero decodes from the
Chris@1: codec setup header (third packet).  configuration decode proceeds as
Chris@1: follows:
Chris@1: 
Chris@1: \begin{Verbatim}[commandchars=\\\{\}]
Chris@1:   1) [floor0\_order] = read an unsigned integer of 8 bits
Chris@1:   2) [floor0\_rate] = read an unsigned integer of 16 bits
Chris@1:   3) [floor0\_bark\_map\_size] = read an unsigned integer of 16 bits
Chris@1:   4) [floor0\_amplitude\_bits] = read an unsigned integer of six bits
Chris@1:   5) [floor0\_amplitude\_offset] = read an unsigned integer of eight bits
Chris@1:   6) [floor0\_number\_of\_books] = read an unsigned integer of four bits and add 1
Chris@1:   7) array [floor0\_book\_list] = read a list of [floor0\_number\_of\_books] unsigned integers of eight bits each;
Chris@1: \end{Verbatim}
Chris@1: 
Chris@1: An end-of-packet condition during any of these bitstream reads renders
Chris@1: this stream undecodable.  In addition, any element of the array
Chris@1: \varname{[floor0\_book\_list]} that is greater than the maximum codebook
Chris@1: number for this bitstream is an error condition that also renders the
Chris@1: stream undecodable.
Chris@1: 
Chris@1: 
Chris@1: 
Chris@1: \subsubsection{packet decode} \label{vorbis:spec:floor0-decode}
Chris@1: 
Chris@1: Extracting a floor0 curve from an audio packet consists of first
Chris@1: decoding the curve amplitude and \varname{[floor0\_order]} LSP
Chris@1: coefficient values from the bitstream, and then computing the floor
Chris@1: curve, which is defined as the frequency response of the decoded LSP
Chris@1: filter.
Chris@1: 
Chris@1: Packet decode proceeds as follows:
Chris@1: \begin{Verbatim}[commandchars=\\\{\}]
Chris@1:   1) [amplitude] = read an unsigned integer of [floor0\_amplitude\_bits] bits
Chris@1:   2) if ( [amplitude] is greater than zero ) \{
Chris@1:        3) [coefficients] is an empty, zero length vector
Chris@1:        4) [booknumber] = read an unsigned integer of \link{vorbis:spec:ilog}{ilog}( [floor0\_number\_of\_books] ) bits
Chris@1:        5) if ( [booknumber] is greater than the highest number decode codebook ) then packet is undecodable
Chris@1:        6) [last] = zero;
Chris@1:        7) vector [temp\_vector] = read vector from bitstream using codebook number [floor0\_book\_list] element [booknumber] in VQ context.
Chris@1:        8) add the scalar value [last] to each scalar in vector [temp\_vector]
Chris@1:        9) [last] = the value of the last scalar in vector [temp\_vector]
Chris@1:       10) concatenate [temp\_vector] onto the end of the [coefficients] vector
Chris@1:       11) if (length of vector [coefficients] is less than [floor0\_order], continue at step 6
Chris@1: 
Chris@1:      \}
Chris@1: 
Chris@1:  12) done.
Chris@1: 
Chris@1: \end{Verbatim}
Chris@1: 
Chris@1: Take note of the following properties of decode:
Chris@1: \begin{itemize}
Chris@1:  \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.
Chris@1:  \item An end-of-packet condition during decode should be considered a
Chris@1: nominal occruence; if end-of-packet is reached during any read
Chris@1: operation above, floor decode is to return 'unused' status as if the
Chris@1: \varname{[amplitude]} value had read zero at the beginning of decode.
Chris@1: 
Chris@1:  \item The book number used for decode
Chris@1: can, in fact, be stored in the bitstream in \link{vorbis:spec:ilog}{ilog}( \varname{[floor0\_number\_of\_books]} -
Chris@1: 1 ) bits.  Nevertheless, the above specification is correct and values
Chris@1: greater than the maximum possible book value are reserved.
Chris@1: 
Chris@1:  \item The number of scalars read into the vector \varname{[coefficients]}
Chris@1: may be greater than \varname{[floor0\_order]}, the number actually
Chris@1: required for curve computation.  For example, if the VQ codebook used
Chris@1: for the floor currently being decoded has a
Chris@1: \varname{[codebook\_dimensions]} value of three and
Chris@1: \varname{[floor0\_order]} is ten, the only way to fill all the needed
Chris@1: scalars in \varname{[coefficients]} is to to read a total of twelve
Chris@1: scalars as four vectors of three scalars each.  This is not an error
Chris@1: condition, and care must be taken not to allow a buffer overflow in
Chris@1: decode. The extra values are not used and may be ignored or discarded.
Chris@1: \end{itemize}
Chris@1: 
Chris@1: 
Chris@1: 
Chris@1: 
Chris@1: \subsubsection{curve computation} \label{vorbis:spec:floor0-synth}
Chris@1: 
Chris@1: Given an \varname{[amplitude]} integer and \varname{[coefficients]}
Chris@1: vector from packet decode as well as the [floor0\_order],
Chris@1: [floor0\_rate], [floor0\_bark\_map\_size], [floor0\_amplitude\_bits] and
Chris@1: [floor0\_amplitude\_offset] values from floor setup, and an output
Chris@1: vector size \varname{[n]} specified by the decode process, we compute a
Chris@1: floor output vector.
Chris@1: 
Chris@1: If the value \varname{[amplitude]} is zero, the return value is a
Chris@1: length \varname{[n]} vector with all-zero scalars.  Otherwise, begin by
Chris@1: assuming the following definitions for the given vector to be
Chris@1: synthesized:
Chris@1: 
Chris@1:    \begin{displaymath}
Chris@1:      \mathrm{map}_i = \left\{
Chris@1:        \begin{array}{ll}
Chris@1:           \min (
Chris@1:             \mathtt{floor0\texttt{\_}bark\texttt{\_}map\texttt{\_}size} - 1,
Chris@1:             foobar
Chris@1:           ) & \textrm{for } i \in [0,n-1] \\
Chris@1:           -1 & \textrm{for } i = n
Chris@1:         \end{array}
Chris@1:       \right.
Chris@1:     \end{displaymath}
Chris@1: 
Chris@1:     where
Chris@1: 
Chris@1:     \begin{displaymath}
Chris@1:     foobar =
Chris@1:       \left\lfloor
Chris@1:         \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})}
Chris@1:       \right\rfloor
Chris@1:     \end{displaymath}
Chris@1: 
Chris@1:     and
Chris@1: 
Chris@1:     \begin{displaymath}
Chris@1:       \mathrm{bark}(x) = 13.1 \arctan (.00074x) + 2.24 \arctan (.0000000185x^2 + .0001x)
Chris@1:     \end{displaymath}
Chris@1: 
Chris@1: The above is used to synthesize the LSP curve on a Bark-scale frequency
Chris@1: axis, then map the result to a linear-scale frequency axis.
Chris@1: Similarly, the below calculation synthesizes the output LSP curve \varname{[output]} on a log
Chris@1: (dB) amplitude scale, mapping it to linear amplitude in the last step:
Chris@1: 
Chris@1: \begin{enumerate}
Chris@1:  \item  \varname{[i]} = 0
Chris@1:  \item  \varname{[$\omega$]} = $\pi$ * map element \varname{[i]} / \varname{[floor0\_bark\_map\_size]}
Chris@1:  \item if ( \varname{[floor0\_order]} is odd ) {
Chris@1:   \begin{enumerate}
Chris@1:    \item calculate \varname{[p]} and \varname{[q]} according to:
Chris@1:            \begin{eqnarray*}
Chris@1:              p & = & (1 - \cos^2\omega)\prod_{j=0}^{\frac{\mathtt{floor0\texttt{\_}order}-3}{2}} 4 (\cos([\mathtt{coefficients}]_{2j+1}) - \cos \omega)^2 \\
Chris@1:              q & = & \frac{1}{4} \prod_{j=0}^{\frac{\mathtt{floor0\texttt{\_}order}-1}{2}} 4 (\cos([\mathtt{coefficients}]_{2j}) - \cos \omega)^2
Chris@1:            \end{eqnarray*}
Chris@1: 
Chris@1:   \end{enumerate}
Chris@1:   } else \varname{[floor0\_order]} is even {
Chris@1:   \begin{enumerate}[resume]
Chris@1:    \item calculate \varname{[p]} and \varname{[q]} according to:
Chris@1:            \begin{eqnarray*}
Chris@1:              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 \\
Chris@1:              q & = & \frac{(1 + \cos\omega)}{2} \prod_{j=0}^{\frac{\mathtt{floor0\texttt{\_}order}-2}{2}} 4 (\cos([\mathtt{coefficients}]_{2j}) - \cos \omega)^2
Chris@1:            \end{eqnarray*}
Chris@1: 
Chris@1:   \end{enumerate}
Chris@1:   }
Chris@1: 
Chris@1:  \item calculate \varname{[linear\_floor\_value]} according to:
Chris@1:          \begin{displaymath}
Chris@1:            \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}}
Chris@1:                   - \mathtt{floor0\texttt{\_}amplitude\texttt{\_}offset} \right) \right)
Chris@1:          \end{displaymath}
Chris@1: 
Chris@1:  \item \varname{[iteration\_condition]} = map element \varname{[i]}
Chris@1:  \item \varname{[output]} element \varname{[i]} = \varname{[linear\_floor\_value]}
Chris@1:  \item increment \varname{[i]}
Chris@1:  \item if ( map element \varname{[i]} is equal to \varname{[iteration\_condition]} ) continue at step 5
Chris@1:  \item if ( \varname{[i]} is less than \varname{[n]} ) continue at step 2
Chris@1:  \item done
Chris@1: \end{enumerate}
Chris@1: 
Chris@1: 
Chris@1: 
Chris@1: 
Chris@1: 
Chris@1: 
Chris@1: