1 % -*- mode: latex; TeX-master: "Vorbis_I_spec"; -*- 2 %!TEX root = Vorbis_I_spec.tex 3 % $Id$ 4 \section{Helper equations} \label{vorbis:spec:helper} 5 6 \subsection{Overview} 7 8 The equations below are used in multiple places by the Vorbis codec 9 specification. Rather than cluttering up the main specification 10 documents, they are defined here and referenced where appropriate. 11 12 13 \subsection{Functions} 14 15 \subsubsection{ilog} \label{vorbis:spec:ilog} 16 17 The "ilog(x)" function returns the position number (1 through n) of the highest set bit in the two's complement integer value 18 \varname{[x]}. Values of \varname{[x]} less than zero are defined to return zero. 19 20 \begin{programlisting} 21 1) [return_value] = 0; 22 2) if ( [x] is greater than zero ) { 23 24 3) increment [return_value]; 25 4) logical shift [x] one bit to the right, padding the MSb with zero 26 5) repeat at step 2) 27 28 } 29 30 6) done 31 \end{programlisting} 32 33 Examples: 34 35 \begin{itemize} 36 \item ilog(0) = 0; 37 \item ilog(1) = 1; 38 \item ilog(2) = 2; 39 \item ilog(3) = 2; 40 \item ilog(4) = 3; 41 \item ilog(7) = 3; 42 \item ilog(negative number) = 0; 43 \end{itemize} 44 45 46 47 48 \subsubsection{float32_unpack} \label{vorbis:spec:float32:unpack} 49 50 "float32_unpack(x)" is intended to translate the packed binary 51 representation of a Vorbis codebook float value into the 52 representation used by the decoder for floating point numbers. For 53 purposes of this example, we will unpack a Vorbis float32 into a 54 host-native floating point number. 55 56 \begin{programlisting} 57 1) [mantissa] = [x] bitwise AND 0x1fffff (unsigned result) 58 2) [sign] = [x] bitwise AND 0x80000000 (unsigned result) 59 3) [exponent] = ( [x] bitwise AND 0x7fe00000) shifted right 21 bits (unsigned result) 60 4) if ( [sign] is nonzero ) then negate [mantissa] 61 5) return [mantissa] * ( 2 ^ ( [exponent] - 788 ) ) 62 \end{programlisting} 63 64 65 66 \subsubsection{lookup1_values} \label{vorbis:spec:lookup1:values} 67 68 "lookup1_values(codebook_entries,codebook_dimensions)" is used to 69 compute the correct length of the value index for a codebook VQ lookup 70 table of lookup type 1. The values on this list are permuted to 71 construct the VQ vector lookup table of size 72 \varname{[codebook_entries]}. 73 74 The return value for this function is defined to be 'the greatest 75 integer value for which \varname{[return_value]} to the power of 76 \varname{[codebook_dimensions]} is less than or equal to 77 \varname{[codebook_entries]}'. 78 79 80 81 \subsubsection{low_neighbor} \label{vorbis:spec:low:neighbor} 82 83 "low_neighbor(v,x)" finds the position \varname{n} in vector \varname{[v]} of 84 the greatest value scalar element for which \varname{n} is less than 85 \varname{[x]} and vector \varname{[v]} element \varname{n} is less 86 than vector \varname{[v]} element \varname{[x]}. 87 88 \subsubsection{high_neighbor} \label{vorbis:spec:high:neighbor} 89 90 "high_neighbor(v,x)" finds the position \varname{n} in vector [v] of 91 the lowest value scalar element for which \varname{n} is less than 92 \varname{[x]} and vector \varname{[v]} element \varname{n} is greater 93 than vector \varname{[v]} element \varname{[x]}. 94 95 96 97 \subsubsection{render_point} \label{vorbis:spec:render:point} 98 99 "render_point(x0,y0,x1,y1,X)" is used to find the Y value at point X 100 along the line specified by x0, x1, y0 and y1. This function uses an 101 integer algorithm to solve for the point directly without calculating 102 intervening values along the line. 103 104 \begin{programlisting} 105 1) [dy] = [y1] - [y0] 106 2) [adx] = [x1] - [x0] 107 3) [ady] = absolute value of [dy] 108 4) [err] = [ady] * ([X] - [x0]) 109 5) [off] = [err] / [adx] using integer division 110 6) if ( [dy] is less than zero ) { 111 112 7) [Y] = [y0] - [off] 113 114 } else { 115 116 8) [Y] = [y0] + [off] 117 118 } 119 120 9) done 121 \end{programlisting} 122 123 124 125 \subsubsection{render_line} \label{vorbis:spec:render:line} 126 127 Floor decode type one uses the integer line drawing algorithm of 128 "render_line(x0, y0, x1, y1, v)" to construct an integer floor 129 curve for contiguous piecewise line segments. Note that it has not 130 been relevant elsewhere, but here we must define integer division as 131 rounding division of both positive and negative numbers toward zero. 132 133 134 \begin{programlisting} 135 1) [dy] = [y1] - [y0] 136 2) [adx] = [x1] - [x0] 137 3) [ady] = absolute value of [dy] 138 4) [base] = [dy] / [adx] using integer division 139 5) [x] = [x0] 140 6) [y] = [y0] 141 7) [err] = 0 142 143 8) if ( [dy] is less than 0 ) { 144 145 9) [sy] = [base] - 1 146 147 } else { 148 149 10) [sy] = [base] + 1 150 151 } 152 153 11) [ady] = [ady] - (absolute value of [base]) * [adx] 154 12) vector [v] element [x] = [y] 155 156 13) iterate [x] over the range [x0]+1 ... [x1]-1 { 157 158 14) [err] = [err] + [ady]; 159 15) if ( [err] >= [adx] ) { 160 161 16) [err] = [err] - [adx] 162 17) [y] = [y] + [sy] 163 164 } else { 165 166 18) [y] = [y] + [base] 167 168 } 169 170 19) vector [v] element [x] = [y] 171 172 } 173 \end{programlisting} 174 175 176 177 178 179 180 181 182
