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    1 <?xml version="1.0" standalone="no"?>
    2 <!DOCTYPE section PUBLIC "-//OASIS//DTD DocBook XML V4.2//EN"
    3                 "http://www.oasis-open.org/docbook/xml/4.2/docbookx.dtd" [
    4 
    5 ]>
    6 
    7 <section id="vorbis-spec-helper">
    8 <sectioninfo>
    9 <releaseinfo>
   10  $Id: 09-helper.xml 7186 2004-07-20 07:19:25Z xiphmont $
   11 </releaseinfo>
   12 </sectioninfo>
   13 <title>Helper equations</title>
   14 
   15 <section>
   16 <title>Overview</title>
   17 
   18 <para>
   19 The equations below are used in multiple places by the Vorbis codec
   20 specification.  Rather than cluttering up the main specification
   21 documents, they are defined here and referenced where appropriate.
   22 </para>
   23 
   24 </section>
   25 
   26 <section>
   27 <title>Functions</title>
   28 
   29 <section id="vorbis-spec-ilog">
   30 <title>ilog</title>
   31 
   32 <para>
   33 The "ilog(x)" function returns the position number (1 through n) of the highest set bit in the two's complement integer value
   34 <varname>[x]</varname>.  Values of <varname>[x]</varname> less than zero are defined to return zero.</para>
   35 
   36 <programlisting>
   37   1) [return_value] = 0;
   38   2) if ( [x] is greater than zero ){
   39       
   40        3) increment [return_value];
   41        4) logical shift [x] one bit to the right, padding the MSb with zero
   42        5) repeat at step 2)
   43 
   44      }
   45 
   46    6) done
   47 </programlisting>
   48 
   49 <para>
   50 Examples:
   51 
   52 <itemizedlist>
   53  <listitem><simpara>ilog(0) = 0;</simpara></listitem>
   54  <listitem><simpara>ilog(1) = 1;</simpara></listitem>
   55  <listitem><simpara>ilog(2) = 2;</simpara></listitem>
   56  <listitem><simpara>ilog(3) = 2;</simpara></listitem>
   57  <listitem><simpara>ilog(4) = 3;</simpara></listitem>
   58  <listitem><simpara>ilog(7) = 3;</simpara></listitem>
   59  <listitem><simpara>ilog(negative number) = 0;</simpara></listitem>
   60 </itemizedlist>
   61 </para>
   62 
   63 </section>
   64 
   65 <section id="vorbis-spec-float32_unpack">
   66 <title>float32_unpack</title>
   67 
   68 <para>
   69 "float32_unpack(x)" is intended to translate the packed binary
   70 representation of a Vorbis codebook float value into the
   71 representation used by the decoder for floating point numbers.  For
   72 purposes of this example, we will unpack a Vorbis float32 into a
   73 host-native floating point number.</para>
   74 
   75 <programlisting>
   76   1) [mantissa] = [x] bitwise AND 0x1fffff (unsigned result)
   77   2) [sign] = [x] bitwise AND 0x80000000 (unsigned result)
   78   3) [exponent] = ( [x] bitwise AND 0x7fe00000) shifted right 21 bits (unsigned result)
   79   4) if ( [sign] is nonzero ) then negate [mantissa]
   80   5) return [mantissa] * ( 2 ^ ( [exponent] - 788 ) )
   81 </programlisting>
   82 
   83 </section>
   84 
   85 <section id="vorbis-spec-lookup1_values">
   86 <title>lookup1_values</title>
   87 
   88 <para>
   89 "lookup1_values(codebook_entries,codebook_dimensions)" is used to
   90 compute the correct length of the value index for a codebook VQ lookup
   91 table of lookup type 1.  The values on this list are permuted to
   92 construct the VQ vector lookup table of size
   93 <varname>[codebook_entries]</varname>.</para>
   94 
   95 <para>
   96 The return value for this function is defined to be 'the greatest
   97 integer value for which <varname>[return_value] to the power of
   98 [codebook_dimensions] is less than or equal to
   99 [codebook_entries]</varname>'.</para>
  100 
  101 </section>
  102 
  103 <section id="vorbis-spec-low_neighbor">
  104 <title>low_neighbor</title>
  105 
  106 <para>
  107 "low_neighbor(v,x)" finds the position <varname>n</varname> in vector <varname>[v]</varname> of
  108 the greatest value scalar element for which <varname>n</varname> is less than
  109 <varname>[x]</varname> and vector <varname>[v]</varname> element <varname>n</varname> is less
  110 than vector <varname>[v]</varname> element <varname>[x]</varname>.</para>
  111 
  112 <section id="vorbis-spec-high_neighbor">
  113 <title>high_neighbor</title>
  114 
  115 <para>
  116 "high_neighbor(v,x)" finds the position <varname>n</varname> in vector [v] of
  117 the lowest value scalar element for which <varname>n</varname> is less than
  118 <varname>[x]</varname> and vector <varname>[v]</varname> element <varname>n</varname> is greater
  119 than vector <varname>[v]</varname> element <varname>[x]</varname>.</para>
  120 
  121 </section>
  122 
  123 <section id="vorbis-spec-render_point">
  124 <title>render_point</title>
  125 
  126 <para>
  127 "render_point(x0,y0,x1,y1,X)" is used to find the Y value at point X
  128 along the line specified by x0, x1, y0 and y1.  This function uses an
  129 integer algorithm to solve for the point directly without calculating
  130 intervening values along the line.</para>
  131 
  132 <programlisting>
  133   1)  [dy] = [y1] - [y0]
  134   2) [adx] = [x1] - [x0]
  135   3) [ady] = absolute value of [dy]
  136   4) [err] = [ady] * ([X] - [x0])
  137   5) [off] = [err] / [adx] using integer division
  138   6) if ( [dy] is less than zero ) {
  139 
  140        7) [Y] = [y0] - [off]
  141 
  142      } else {
  143 
  144        8) [Y] = [y0] + [off]
  145   
  146      }
  147 
  148   9) done
  149 </programlisting>
  150 
  151 </section>
  152 
  153 <section id="vorbis-spec-render_line">
  154 <title>render_line</title>
  155 
  156 <para>
  157 Floor decode type one uses the integer line drawing algorithm of
  158 "render_line(x0, y0, x1, y1, v)" to construct an integer floor
  159 curve for contiguous piecewise line segments. Note that it has not
  160 been relevant elsewhere, but here we must define integer division as
  161 rounding division of both positive and negative numbers toward zero.
  162 </para>
  163 
  164 <programlisting>
  165   1)   [dy] = [y1] - [y0]
  166   2)  [adx] = [x1] - [x0]
  167   3)  [ady] = absolute value of [dy]
  168   4) [base] = [dy] / [adx] using integer division
  169   5)    [x] = [x0]
  170   6)    [y] = [y0]
  171   7)  [err] = 0
  172 
  173   8) if ( [dy] is less than 0 ) {
  174 
  175         9) [sy] = [base] - 1
  176 
  177      } else {
  178 
  179        10) [sy] = [base] + 1
  180 
  181      }
  182 
  183  11) [ady] = [ady] - (absolute value of [base]) * [adx]
  184  12) vector [v] element [x] = [y]
  185 
  186  13) iterate [x] over the range [x0]+1 ... [x1]-1 {
  187 
  188        14) [err] = [err] + [ady];
  189        15) if ( [err] >= [adx] ) {
  190 
  191              16) [err] = [err] - [adx]
  192              17)   [y] = [y] + [sy]
  193 
  194            } else {
  195 
  196              18) [y] = [y] + [base]
  197    
  198            }
  199 
  200        19) vector [v] element [x] = [y]
  201 
  202      }
  203 </programlisting>
  204 
  205 </section>
  206 
  207 </section>
  208 
  209 </section>
  210 
  211 </section>