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diff --git a/doc/06-floor0.tex b/doc/06-floor0.tex new file mode 100644 index 0000000..f3042a6 --- /dev/null +++ b/doc/06-floor0.tex @@ -0,0 +1,201 @@ +% -*- mode: latex; TeX-master: "Vorbis_I_spec"; -*- +%!TEX root = Vorbis_I_spec.tex +\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} + |