
cannam@86: % -*- mode: latex; TeX-master: "Vorbis_I_spec"; -*-
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cannam@86: \section{Floor type 0 setup and decode} \label{vorbis:spec:floor0}
cannam@86: 
cannam@86: \subsection{Overview}
cannam@86: 
cannam@86: Vorbis floor type zero uses Line Spectral Pair (LSP, also alternately
cannam@86: known as Line Spectral Frequency or LSF) representation to encode a
cannam@86: smooth spectral envelope curve as the frequency response of the LSP
cannam@86: filter.  This representation is equivalent to a traditional all-pole
cannam@86: infinite impulse response filter as would be used in linear predictive
cannam@86: coding; LSP representation may be converted to LPC representation and
cannam@86: vice-versa.
cannam@86: 
cannam@86: 
cannam@86: 
cannam@86: \subsection{Floor 0 format}
cannam@86: 
cannam@86: Floor zero configuration consists of six integer fields and a list of
cannam@86: VQ codebooks for use in coding/decoding the LSP filter coefficient
cannam@86: values used by each frame.
cannam@86: 
cannam@86: \subsubsection{header decode}
cannam@86: 
cannam@86: Configuration information for instances of floor zero decodes from the
cannam@86: codec setup header (third packet).  configuration decode proceeds as
cannam@86: follows:
cannam@86: 
cannam@86: \begin{Verbatim}[commandchars=\\\{\}]
cannam@86:   1) [floor0\_order] = read an unsigned integer of 8 bits
cannam@86:   2) [floor0\_rate] = read an unsigned integer of 16 bits
cannam@86:   3) [floor0\_bark\_map\_size] = read an unsigned integer of 16 bits
cannam@86:   4) [floor0\_amplitude\_bits] = read an unsigned integer of six bits
cannam@86:   5) [floor0\_amplitude\_offset] = read an unsigned integer of eight bits
cannam@86:   6) [floor0\_number\_of\_books] = read an unsigned integer of four bits and add 1
cannam@86:   7) array [floor0\_book\_list] = read a list of [floor0\_number\_of\_books] unsigned integers of eight bits each;
cannam@86: \end{Verbatim}
cannam@86: 
cannam@86: An end-of-packet condition during any of these bitstream reads renders
cannam@86: this stream undecodable.  In addition, any element of the array
cannam@86: \varname{[floor0\_book\_list]} that is greater than the maximum codebook
cannam@86: number for this bitstream is an error condition that also renders the
cannam@86: stream undecodable.
cannam@86: 
cannam@86: 
cannam@86: 
cannam@86: \subsubsection{packet decode} \label{vorbis:spec:floor0-decode}
cannam@86: 
cannam@86: Extracting a floor0 curve from an audio packet consists of first
cannam@86: decoding the curve amplitude and \varname{[floor0\_order]} LSP
cannam@86: coefficient values from the bitstream, and then computing the floor
cannam@86: curve, which is defined as the frequency response of the decoded LSP
cannam@86: filter.
cannam@86: 
cannam@86: Packet decode proceeds as follows:
cannam@86: \begin{Verbatim}[commandchars=\\\{\}]
cannam@86:   1) [amplitude] = read an unsigned integer of [floor0\_amplitude\_bits] bits
cannam@86:   2) if ( [amplitude] is greater than zero ) \{
cannam@86:        3) [coefficients] is an empty, zero length vector
cannam@86:        4) [booknumber] = read an unsigned integer of \link{vorbis:spec:ilog}{ilog}( [floor0\_number\_of\_books] ) bits
cannam@86:        5) if ( [booknumber] is greater than the highest number decode codebook ) then packet is undecodable
cannam@86:        6) [last] = zero;
cannam@86:        7) vector [temp\_vector] = read vector from bitstream using codebook number [floor0\_book\_list] element [booknumber] in VQ context.
cannam@86:        8) add the scalar value [last] to each scalar in vector [temp\_vector]
cannam@86:        9) [last] = the value of the last scalar in vector [temp\_vector]
cannam@86:       10) concatenate [temp\_vector] onto the end of the [coefficients] vector
cannam@86:       11) if (length of vector [coefficients] is less than [floor0\_order], continue at step 6
cannam@86: 
cannam@86:      \}
cannam@86: 
cannam@86:  12) done.
cannam@86: 
cannam@86: \end{Verbatim}
cannam@86: 
cannam@86: Take note of the following properties of decode:
cannam@86: \begin{itemize}
cannam@86:  \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.
cannam@86:  \item An end-of-packet condition during decode should be considered a
cannam@86: nominal occruence; if end-of-packet is reached during any read
cannam@86: operation above, floor decode is to return 'unused' status as if the
cannam@86: \varname{[amplitude]} value had read zero at the beginning of decode.
cannam@86: 
cannam@86:  \item The book number used for decode
cannam@86: can, in fact, be stored in the bitstream in \link{vorbis:spec:ilog}{ilog}( \varname{[floor0\_number\_of\_books]} -
cannam@86: 1 ) bits.  Nevertheless, the above specification is correct and values
cannam@86: greater than the maximum possible book value are reserved.
cannam@86: 
cannam@86:  \item The number of scalars read into the vector \varname{[coefficients]}
cannam@86: may be greater than \varname{[floor0\_order]}, the number actually
cannam@86: required for curve computation.  For example, if the VQ codebook used
cannam@86: for the floor currently being decoded has a
cannam@86: \varname{[codebook\_dimensions]} value of three and
cannam@86: \varname{[floor0\_order]} is ten, the only way to fill all the needed
cannam@86: scalars in \varname{[coefficients]} is to to read a total of twelve
cannam@86: scalars as four vectors of three scalars each.  This is not an error
cannam@86: condition, and care must be taken not to allow a buffer overflow in
cannam@86: decode. The extra values are not used and may be ignored or discarded.
cannam@86: \end{itemize}
cannam@86: 
cannam@86: 
cannam@86: 
cannam@86: 
cannam@86: \subsubsection{curve computation} \label{vorbis:spec:floor0-synth}
cannam@86: 
cannam@86: Given an \varname{[amplitude]} integer and \varname{[coefficients]}
cannam@86: vector from packet decode as well as the [floor0\_order],
cannam@86: [floor0\_rate], [floor0\_bark\_map\_size], [floor0\_amplitude\_bits] and
cannam@86: [floor0\_amplitude\_offset] values from floor setup, and an output
cannam@86: vector size \varname{[n]} specified by the decode process, we compute a
cannam@86: floor output vector.
cannam@86: 
cannam@86: If the value \varname{[amplitude]} is zero, the return value is a
cannam@86: length \varname{[n]} vector with all-zero scalars.  Otherwise, begin by
cannam@86: assuming the following definitions for the given vector to be
cannam@86: synthesized:
cannam@86: 
cannam@86:    \begin{displaymath}
cannam@86:      \mathrm{map}_i = \left\{
cannam@86:        \begin{array}{ll}
cannam@86:           \min (
cannam@86:             \mathtt{floor0\texttt{\_}bark\texttt{\_}map\texttt{\_}size} - 1,
cannam@86:             foobar
cannam@86:           ) & \textrm{for } i \in [0,n-1] \\
cannam@86:           -1 & \textrm{for } i = n
cannam@86:         \end{array}
cannam@86:       \right.
cannam@86:     \end{displaymath}
cannam@86: 
cannam@86:     where
cannam@86: 
cannam@86:     \begin{displaymath}
cannam@86:     foobar =
cannam@86:       \left\lfloor
cannam@86:         \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})}
cannam@86:       \right\rfloor
cannam@86:     \end{displaymath}
cannam@86: 
cannam@86:     and
cannam@86: 
cannam@86:     \begin{displaymath}
cannam@86:       \mathrm{bark}(x) = 13.1 \arctan (.00074x) + 2.24 \arctan (.0000000185x^2 + .0001x)
cannam@86:     \end{displaymath}
cannam@86: 
cannam@86: The above is used to synthesize the LSP curve on a Bark-scale frequency
cannam@86: axis, then map the result to a linear-scale frequency axis.
cannam@86: Similarly, the below calculation synthesizes the output LSP curve \varname{[output]} on a log
cannam@86: (dB) amplitude scale, mapping it to linear amplitude in the last step:
cannam@86: 
cannam@86: \begin{enumerate}
cannam@86:  \item  \varname{[i]} = 0
cannam@86:  \item  \varname{[$\omega$]} = $\pi$ * map element \varname{[i]} / \varname{[floor0\_bark\_map\_size]}
cannam@86:  \item if ( \varname{[floor0\_order]} is odd ) {
cannam@86:   \begin{enumerate}
cannam@86:    \item calculate \varname{[p]} and \varname{[q]} according to:
cannam@86:            \begin{eqnarray*}
cannam@86:              p & = & (1 - \cos^2\omega)\prod_{j=0}^{\frac{\mathtt{floor0\texttt{\_}order}-3}{2}} 4 (\cos([\mathtt{coefficients}]_{2j+1}) - \cos \omega)^2 \\
cannam@86:              q & = & \frac{1}{4} \prod_{j=0}^{\frac{\mathtt{floor0\texttt{\_}order}-1}{2}} 4 (\cos([\mathtt{coefficients}]_{2j}) - \cos \omega)^2
cannam@86:            \end{eqnarray*}
cannam@86: 
cannam@86:   \end{enumerate}
cannam@86:   } else \varname{[floor0\_order]} is even {
cannam@86:   \begin{enumerate}[resume]
cannam@86:    \item calculate \varname{[p]} and \varname{[q]} according to:
cannam@86:            \begin{eqnarray*}
cannam@86:              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 \\
cannam@86:              q & = & \frac{(1 + \cos\omega)}{2} \prod_{j=0}^{\frac{\mathtt{floor0\texttt{\_}order}-2}{2}} 4 (\cos([\mathtt{coefficients}]_{2j}) - \cos \omega)^2
cannam@86:            \end{eqnarray*}
cannam@86: 
cannam@86:   \end{enumerate}
cannam@86:   }
cannam@86: 
cannam@86:  \item calculate \varname{[linear\_floor\_value]} according to:
cannam@86:          \begin{displaymath}
cannam@86:            \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}}
cannam@86:                   - \mathtt{floor0\texttt{\_}amplitude\texttt{\_}offset} \right) \right)
cannam@86:          \end{displaymath}
cannam@86: 
cannam@86:  \item \varname{[iteration\_condition]} = map element \varname{[i]}
cannam@86:  \item \varname{[output]} element \varname{[i]} = \varname{[linear\_floor\_value]}
cannam@86:  \item increment \varname{[i]}
cannam@86:  \item if ( map element \varname{[i]} is equal to \varname{[iteration\_condition]} ) continue at step 5
cannam@86:  \item if ( \varname{[i]} is less than \varname{[n]} ) continue at step 2
cannam@86:  \item done
cannam@86: \end{enumerate}
cannam@86: 
cannam@86: 
cannam@86: 
cannam@86: 
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