182 lines
4.8 KiB
TeX
Executable File
182 lines
4.8 KiB
TeX
Executable File
% -*- mode: latex; TeX-master: "Vorbis_I_spec"; -*-
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%!TEX root = Vorbis_I_spec.tex
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% $Id$
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\section{Helper equations} \label{vorbis:spec:helper}
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\subsection{Overview}
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The equations below are used in multiple places by the Vorbis codec
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specification. Rather than cluttering up the main specification
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documents, they are defined here and referenced where appropriate.
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\subsection{Functions}
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\subsubsection{ilog} \label{vorbis:spec:ilog}
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The "ilog(x)" function returns the position number (1 through n) of the highest set bit in the two's complement integer value
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\varname{[x]}. Values of \varname{[x]} less than zero are defined to return zero.
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\begin{programlisting}
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1) [return_value] = 0;
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2) if ( [x] is greater than zero ) {
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3) increment [return_value];
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4) logical shift [x] one bit to the right, padding the MSb with zero
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5) repeat at step 2)
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}
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6) done
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\end{programlisting}
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Examples:
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\begin{itemize}
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\item ilog(0) = 0;
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\item ilog(1) = 1;
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\item ilog(2) = 2;
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\item ilog(3) = 2;
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\item ilog(4) = 3;
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\item ilog(7) = 3;
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\item ilog(negative number) = 0;
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\end{itemize}
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\subsubsection{float32_unpack} \label{vorbis:spec:float32:unpack}
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"float32_unpack(x)" is intended to translate the packed binary
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representation of a Vorbis codebook float value into the
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representation used by the decoder for floating point numbers. For
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purposes of this example, we will unpack a Vorbis float32 into a
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host-native floating point number.
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\begin{programlisting}
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1) [mantissa] = [x] bitwise AND 0x1fffff (unsigned result)
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2) [sign] = [x] bitwise AND 0x80000000 (unsigned result)
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3) [exponent] = ( [x] bitwise AND 0x7fe00000) shifted right 21 bits (unsigned result)
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4) if ( [sign] is nonzero ) then negate [mantissa]
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5) return [mantissa] * ( 2 ^ ( [exponent] - 788 ) )
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\end{programlisting}
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\subsubsection{lookup1_values} \label{vorbis:spec:lookup1:values}
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"lookup1_values(codebook_entries,codebook_dimensions)" is used to
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compute the correct length of the value index for a codebook VQ lookup
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table of lookup type 1. The values on this list are permuted to
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construct the VQ vector lookup table of size
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\varname{[codebook_entries]}.
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The return value for this function is defined to be 'the greatest
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integer value for which \varname{[return_value]} to the power of
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\varname{[codebook_dimensions]} is less than or equal to
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\varname{[codebook_entries]}'.
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\subsubsection{low_neighbor} \label{vorbis:spec:low:neighbor}
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"low_neighbor(v,x)" finds the position \varname{n} in vector \varname{[v]} of
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the greatest value scalar element for which \varname{n} is less than
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\varname{[x]} and vector \varname{[v]} element \varname{n} is less
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than vector \varname{[v]} element \varname{[x]}.
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\subsubsection{high_neighbor} \label{vorbis:spec:high:neighbor}
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"high_neighbor(v,x)" finds the position \varname{n} in vector [v] of
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the lowest value scalar element for which \varname{n} is less than
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\varname{[x]} and vector \varname{[v]} element \varname{n} is greater
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than vector \varname{[v]} element \varname{[x]}.
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\subsubsection{render_point} \label{vorbis:spec:render:point}
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"render_point(x0,y0,x1,y1,X)" is used to find the Y value at point X
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along the line specified by x0, x1, y0 and y1. This function uses an
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integer algorithm to solve for the point directly without calculating
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intervening values along the line.
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\begin{programlisting}
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1) [dy] = [y1] - [y0]
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2) [adx] = [x1] - [x0]
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3) [ady] = absolute value of [dy]
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4) [err] = [ady] * ([X] - [x0])
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5) [off] = [err] / [adx] using integer division
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6) if ( [dy] is less than zero ) {
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7) [Y] = [y0] - [off]
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} else {
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8) [Y] = [y0] + [off]
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}
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9) done
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\end{programlisting}
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\subsubsection{render_line} \label{vorbis:spec:render:line}
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Floor decode type one uses the integer line drawing algorithm of
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"render_line(x0, y0, x1, y1, v)" to construct an integer floor
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curve for contiguous piecewise line segments. Note that it has not
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been relevant elsewhere, but here we must define integer division as
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rounding division of both positive and negative numbers toward zero.
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\begin{programlisting}
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1) [dy] = [y1] - [y0]
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2) [adx] = [x1] - [x0]
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3) [ady] = absolute value of [dy]
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4) [base] = [dy] / [adx] using integer division
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5) [x] = [x0]
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6) [y] = [y0]
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7) [err] = 0
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8) if ( [dy] is less than 0 ) {
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9) [sy] = [base] - 1
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} else {
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10) [sy] = [base] + 1
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}
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11) [ady] = [ady] - (absolute value of [base]) * [adx]
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12) vector [v] element [x] = [y]
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13) iterate [x] over the range [x0]+1 ... [x1]-1 {
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14) [err] = [err] + [ady];
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15) if ( [err] >= [adx] ) {
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16) [err] = [err] - [adx]
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17) [y] = [y] + [sy]
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} else {
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18) [y] = [y] + [base]
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}
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19) vector [v] element [x] = [y]
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}
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\end{programlisting}
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