% -*- 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}