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<HTML><HEAD><TITLE>xiph.org: Ogg Vorbis documentation</TITLE>
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<BODY bgcolor="#ffffff" text="#202020" link="#006666" vlink="#000000">
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<nobr><img src="white-ogg.png"><img src="vorbisword2.png"></nobr><p>
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<h1><font color=#000070>
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Ogg Vorbis I format specification: helper equations
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</font></h1>
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<em>Last update to this document: July 18, 2002</em><p>
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<h1>Overview</h1>
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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 linked int he main documents
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where appropriate.<p>
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<a name=log><h2>ilog</h2></a>
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The "ilog(x)" function returns the highest set bit in the two's complement integer value
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<tt>[x]</tt>. Values of <tt>[x]</tt> less than zero are defined to return zero.
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<pre>
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1) [return_value] = 0;
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2) if ( [x] is greater than nonzero ){
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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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</pre>
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Examples:
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<ul><li> ilog(0) = 0;
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<li> ilog(1) = 1;
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<li> ilog(2) = 2;
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<li> ilog(3) = 2;
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<li> ilog(4) = 3;
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<li> ilog(7) = 3;
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<li> ilog(negative number) = 0;
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</uL>
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<a name=float32_unpack><h2>float32_unpack</h2></a>
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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.<p>
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<pre>
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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 left 21 bits (unsigned result)
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4) if ( [mantissa] is nonzero ) then negate [mantissa]
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5) return [mantissa] * ( 2 ^ ( [exponent] - 788 ) )
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</pre>
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<a name=lookup1_values><h2>lookup1_values</h2></a>
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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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<tt>[codebook_entries]</tt>.<p>
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The return value for this function is defined to be 'the greatest
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integer value for which <tt>[return_value] to the power of
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[codebook_dimensions] is less than or equal to
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[codebook_entries]</tt>'.
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<a name=low_neighbor><h2>low_neighbor</h2></a>
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"low_neighbor(v,x)" finds the position <i>n</i> in vector [v] of
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the greatest value scalar element for which <i>n</i> is less than
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<tt>[x]</tt> and <tt>vector [v] element <i>n</i> is less
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than vector [v] element [x]</tt>.
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<a name=high_neighbor><h2>high_neighbor</h2></a>
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"high_neighbor(v,x)" finds the position <i>n</i> in vector [v] of
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the lowest value scalar element for which <i>n</i> is less than
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<tt>[x]</tt> and <tt>vector [v] element <i>n</i> is greater
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than vector [v] element [x]</tt>.
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<a name=render_point><h2>render_point</h2></a>
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"render_point(x0,x1,y0,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.<p>
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<pre>
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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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</pre>
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<a name=render_line><h2>render_line</h2></a>
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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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<pre>
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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] - [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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15) [err] = [err] - [adx]
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16) [y] = [y] + [sy]
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} else {
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17) [y] = [y] + [base]
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}
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18) vector [v] element [x] = [y]
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}
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</pre>
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<hr>
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<a href="http://www.xiph.org/">
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<img src="white-xifish.png" align=left border=0>
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</a>
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<font size=-2 color=#505050>
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Ogg is a <a href="http://www.xiph.org">Xiph.org Foundation</a> effort
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to protect essential tenets of Internet multimedia from corporate
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hostage-taking; Open Source is the net's greatest tool to keep
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everyone honest. See <a href="http://www.xiph.org/about.html">About
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the Xiph.org Foundation</a> for details.
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<p>
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Ogg Vorbis is the first Ogg audio CODEC. Anyone may freely use and
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distribute the Ogg and Vorbis specification, whether in a private,
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public or corporate capacity. However, the Xiph.org Foundation and
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the Ogg project (xiph.org) reserve the right to set the Ogg Vorbis
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specification and certify specification compliance.<p>
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Xiph.org's Vorbis software CODEC implementation is distributed under a
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BSD-like license. This does not restrict third parties from
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distributing independent implementations of Vorbis software under
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other licenses.<p>
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Ogg, Vorbis, Xiph.org Foundation and their logos are trademarks (tm)
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of the <a href="http://www.xiph.org/">Xiph.org Foundation</a>. These
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pages are copyright (C) 1994-2002 Xiph.org Foundation. All rights
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reserved.<p>
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</body>
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