1% -*- mode: latex; TeX-master: "Vorbis_I_spec"; -*-
2%!TEX root = Vorbis_I_spec.tex
3% $Id$
4\section{Floor type 1 setup and decode} \label{vorbis:spec:floor1}
5
6\subsection{Overview}
7
8Vorbis floor type one uses a piecewise straight-line representation to
9encode a spectral envelope curve. The representation plots this curve
10mechanically on a linear frequency axis and a logarithmic (dB)
11amplitude axis. The integer plotting algorithm used is similar to
12Bresenham's algorithm.
13
14
15
16\subsection{Floor 1 format}
17
18\subsubsection{model}
19
20Floor type one represents a spectral curve as a series of
21line segments.  Synthesis constructs a floor curve using iterative
22prediction in a process roughly equivalent to the following simplified
23description:
24
25\begin{itemize}
26 \item  the first line segment (base case) is a logical line spanning
27from x_0,y_0 to x_1,y_1 where in the base case x_0=0 and x_1=[n], the
28full range of the spectral floor to be computed.
29
30\item the induction step chooses a point x_new within an existing
31logical line segment and produces a y_new value at that point computed
32from the existing line's y value at x_new (as plotted by the line) and
33a difference value decoded from the bitstream packet.
34
35\item floor computation produces two new line segments, one running from
36x_0,y_0 to x_new,y_new and from x_new,y_new to x_1,y_1. This step is
37performed logically even if y_new represents no change to the
38amplitude value at x_new so that later refinement is additionally
39bounded at x_new.
40
41\item the induction step repeats, using a list of x values specified in
42the codec setup header at floor 1 initialization time.  Computation
43is completed at the end of the x value list.
44
45\end{itemize}
46
47
48Consider the following example, with values chosen for ease of
49understanding rather than representing typical configuration:
50
51For the below example, we assume a floor setup with an [n] of 128.
52The list of selected X values in increasing order is
530,16,32,48,64,80,96,112 and 128.  In list order, the values interleave
54as 0, 128, 64, 32, 96, 16, 48, 80 and 112.  The corresponding
55list-order Y values as decoded from an example packet are 110, 20, -5,
56-45, 0, -25, -10, 30 and -10.  We compute the floor in the following
57way, beginning with the first line:
58
59\begin{center}
60\includegraphics[width=8cm]{floor1-1}
61\captionof{figure}{graph of example floor}
62\end{center}
63
64We now draw new logical lines to reflect the correction to new_Y, and
65iterate for X positions 32 and 96:
66
67\begin{center}
68\includegraphics[width=8cm]{floor1-2}
69\captionof{figure}{graph of example floor}
70\end{center}
71
72Although the new Y value at X position 96 is unchanged, it is still
73used later as an endpoint for further refinement.  From here on, the
74pattern should be clear; we complete the floor computation as follows:
75
76\begin{center}
77\includegraphics[width=8cm]{floor1-3}
78\captionof{figure}{graph of example floor}
79\end{center}
80
81\begin{center}
82\includegraphics[width=8cm]{floor1-4}
83\captionof{figure}{graph of example floor}
84\end{center}
85
86A more efficient algorithm with carefully defined integer rounding
87behavior is used for actual decode, as described later.  The actual
88algorithm splits Y value computation and line plotting into two steps
89with modifications to the above algorithm to eliminate noise
90accumulation through integer roundoff/truncation.
91
92
93
94\subsubsection{header decode}
95
96A list of floor X values is stored in the packet header in interleaved
97format (used in list order during packet decode and synthesis).  This
98list is split into partitions, and each partition is assigned to a
99partition class.  X positions 0 and [n] are implicit and do not belong
100to an explicit partition or partition class.
101
102A partition class consists of a representation vector width (the
103number of Y values which the partition class encodes at once), a
104'subclass' value representing the number of alternate entropy books
105the partition class may use in representing Y values, the list of
106[subclass] books and a master book used to encode which alternate
107books were chosen for representation in a given packet.  The
108master/subclass mechanism is meant to be used as a flexible
109representation cascade while still using codebooks only in a scalar
110context.
111
112\begin{Verbatim}[commandchars=\\\{\}]
113
114  1) [floor1_partitions] = read 5 bits as unsigned integer
115  2) [maximum_class] = -1
116  3) iterate [i] over the range 0 ... [floor1_partitions]-1 \{
117
118        4) vector [floor1_partition_class_list] element [i] = read 4 bits as unsigned integer
119
120     \}
121
122  5) [maximum_class] = largest integer scalar value in vector [floor1_partition_class_list]
123  6) iterate [i] over the range 0 ... [maximum_class] \{
124
125        7) vector [floor1_class_dimensions] element [i] = read 3 bits as unsigned integer and add 1
126	8) vector [floor1_class_subclasses] element [i] = read 2 bits as unsigned integer
127        9) if ( vector [floor1_class_subclasses] element [i] is nonzero ) \{
128
129             10) vector [floor1_class_masterbooks] element [i] = read 8 bits as unsigned integer
130
131           \}
132
133       11) iterate [j] over the range 0 ... (2 exponent [floor1_class_subclasses] element [i]) - 1 \{
134
135             12) array [floor1_subclass_books] element [i],[j] =
136                 read 8 bits as unsigned integer and subtract one
137           \}
138      \}
139
140 13) [floor1_multiplier] = read 2 bits as unsigned integer and add one
141 14) [rangebits] = read 4 bits as unsigned integer
142 15) vector [floor1_X_list] element [0] = 0
143 16) vector [floor1_X_list] element [1] = 2 exponent [rangebits];
144 17) [floor1_values] = 2
145 18) iterate [i] over the range 0 ... [floor1_partitions]-1 \{
146
147       19) [current_class_number] = vector [floor1_partition_class_list] element [i]
148       20) iterate [j] over the range 0 ... ([floor1_class_dimensions] element [current_class_number])-1 \{
149             21) vector [floor1_X_list] element ([floor1_values]) =
150                 read [rangebits] bits as unsigned integer
151             22) increment [floor1_values] by one
152           \}
153     \}
154
155 23) done
156\end{Verbatim}
157
158An end-of-packet condition while reading any aspect of a floor 1
159configuration during setup renders a stream undecodable.  In addition,
160a \varname{[floor1_class_masterbooks]} or
161\varname{[floor1_subclass_books]} scalar element greater than the
162highest numbered codebook configured in this stream is an error
163condition that renders the stream undecodable.  All vector
164[floor1_x_list] element values must be unique within the vector; a
165non-unique value renders the stream undecodable.
166
167\paragraph{packet decode} \label{vorbis:spec:floor1-decode}
168
169Packet decode begins by checking the \varname{[nonzero]} flag:
170
171\begin{Verbatim}[commandchars=\\\{\}]
172  1) [nonzero] = read 1 bit as boolean
173\end{Verbatim}
174
175If \varname{[nonzero]} is unset, that indicates this channel contained
176no audio energy in this frame.  Decode immediately returns a status
177indicating this floor curve (and thus this channel) is unused this
178frame.  (A return status of 'unused' is different from decoding a
179floor that has all points set to minimum representation amplitude,
180which happens to be approximately -140dB).
181
182
183Assuming \varname{[nonzero]} is set, decode proceeds as follows:
184
185\begin{Verbatim}[commandchars=\\\{\}]
186  1) [range] = vector \{ 256, 128, 86, 64 \} element ([floor1_multiplier]-1)
187  2) vector [floor1_Y] element [0] = read \link{vorbis:spec:ilog}{ilog}([range]-1) bits as unsigned integer
188  3) vector [floor1_Y] element [1] = read \link{vorbis:spec:ilog}{ilog}([range]-1) bits as unsigned integer
189  4) [offset] = 2;
190  5) iterate [i] over the range 0 ... [floor1_partitions]-1 \{
191
192       6) [class] = vector [floor1_partition_class]  element [i]
193       7) [cdim]  = vector [floor1_class_dimensions] element [class]
194       8) [cbits] = vector [floor1_class_subclasses] element [class]
195       9) [csub]  = (2 exponent [cbits])-1
196      10) [cval]  = 0
197      11) if ( [cbits] is greater than zero ) \{
198
199             12) [cval] = read from packet using codebook number
200                 (vector [floor1_class_masterbooks] element [class]) in scalar context
201          \}
202
203      13) iterate [j] over the range 0 ... [cdim]-1 \{
204
205             14) [book] = array [floor1_subclass_books] element [class],([cval] bitwise AND [csub])
206             15) [cval] = [cval] right shifted [cbits] bits
207	     16) if ( [book] is not less than zero ) \{
208
209	           17) vector [floor1_Y] element ([j]+[offset]) = read from packet using codebook
210                       [book] in scalar context
211
212                 \} else [book] is less than zero \{
213
214	           18) vector [floor1_Y] element ([j]+[offset]) = 0
215
216                 \}
217          \}
218
219      19) [offset] = [offset] + [cdim]
220
221     \}
222
223 20) done
224\end{Verbatim}
225
226An end-of-packet condition during curve decode should be considered a
227nominal occurrence; if end-of-packet is reached during any read
228operation above, floor decode is to return 'unused' status as if the
229\varname{[nonzero]} flag had been unset at the beginning of decode.
230
231
232Vector \varname{[floor1_Y]} contains the values from packet decode
233needed for floor 1 synthesis.
234
235
236
237\paragraph{curve computation} \label{vorbis:spec:floor1-synth}
238
239Curve computation is split into two logical steps; the first step
240derives final Y amplitude values from the encoded, wrapped difference
241values taken from the bitstream.  The second step plots the curve
242lines.  Also, although zero-difference values are used in the
243iterative prediction to find final Y values, these points are
244conditionally skipped during final line computation in step two.
245Skipping zero-difference values allows a smoother line fit.
246
247Although some aspects of the below algorithm look like inconsequential
248optimizations, implementors are warned to follow the details closely.
249Deviation from implementing a strictly equivalent algorithm can result
250in serious decoding errors.
251
252\begin{description}
253\item[step 1: amplitude value synthesis]
254
255Unwrap the always-positive-or-zero values read from the packet into
256+/- difference values, then apply to line prediction.
257
258\begin{Verbatim}[commandchars=\\\{\}]
259  1) [range] = vector \{ 256, 128, 86, 64 \} element ([floor1_multiplier]-1)
260  2) vector [floor1_step2_flag] element [0] = set
261  3) vector [floor1_step2_flag] element [1] = set
262  4) vector [floor1_final_Y] element [0] = vector [floor1_Y] element [0]
263  5) vector [floor1_final_Y] element [1] = vector [floor1_Y] element [1]
264  6) iterate [i] over the range 2 ... [floor1_values]-1 \{
265
266       7) [low_neighbor_offset] = \link{vorbis:spec:low:neighbor}{low_neighbor}([floor1_X_list],[i])
267       8) [high_neighbor_offset] = \link{vorbis:spec:high:neighbor}{high_neighbor}([floor1_X_list],[i])
268
269       9) [predicted] = \link{vorbis:spec:render:point}{render_point}( vector [floor1_X_list] element [low_neighbor_offset],
270				      vector [floor1_final_Y] element [low_neighbor_offset],
271                                      vector [floor1_X_list] element [high_neighbor_offset],
272				      vector [floor1_final_Y] element [high_neighbor_offset],
273                                      vector [floor1_X_list] element [i] )
274
275      10) [val] = vector [floor1_Y] element [i]
276      11) [highroom] = [range] - [predicted]
277      12) [lowroom]  = [predicted]
278      13) if ( [highroom] is less than [lowroom] ) \{
279
280            14) [room] = [highroom] * 2
281
282          \} else [highroom] is not less than [lowroom] \{
283
284            15) [room] = [lowroom] * 2
285
286          \}
287
288      16) if ( [val] is nonzero ) \{
289
290            17) vector [floor1_step2_flag] element [low_neighbor_offset] = set
291            18) vector [floor1_step2_flag] element [high_neighbor_offset] = set
292            19) vector [floor1_step2_flag] element [i] = set
293            20) if ( [val] is greater than or equal to [room] ) \{
294
295                  21) if ( [highroom] is greater than [lowroom] ) \{
296
297                        22) vector [floor1_final_Y] element [i] = [val] - [lowroom] + [predicted]
298
299		      \} else [highroom] is not greater than [lowroom] \{
300
301                        23) vector [floor1_final_Y] element [i] = [predicted] - [val] + [highroom] - 1
302
303                      \}
304
305                \} else [val] is less than [room] \{
306
307		  24) if ([val] is odd) \{
308
309                        25) vector [floor1_final_Y] element [i] =
310                            [predicted] - (([val] + 1) divided by  2 using integer division)
311
312                      \} else [val] is even \{
313
314                        26) vector [floor1_final_Y] element [i] =
315                            [predicted] + ([val] / 2 using integer division)
316
317                      \}
318
319                \}
320
321          \} else [val] is zero \{
322
323            27) vector [floor1_step2_flag] element [i] = unset
324            28) vector [floor1_final_Y] element [i] = [predicted]
325
326          \}
327
328     \}
329
330 29) done
331
332\end{Verbatim}
333
334
335
336\item[step 2: curve synthesis]
337
338Curve synthesis generates a return vector \varname{[floor]} of length
339\varname{[n]} (where \varname{[n]} is provided by the decode process
340calling to floor decode).  Floor 1 curve synthesis makes use of the
341\varname{[floor1_X_list]}, \varname{[floor1_final_Y]} and
342\varname{[floor1_step2_flag]} vectors, as well as [floor1_multiplier]
343and [floor1_values] values.
344
345Decode begins by sorting the scalars from vectors
346\varname{[floor1_X_list]}, \varname{[floor1_final_Y]} and
347\varname{[floor1_step2_flag]} together into new vectors
348\varname{[floor1_X_list]'}, \varname{[floor1_final_Y]'} and
349\varname{[floor1_step2_flag]'} according to ascending sort order of the
350values in \varname{[floor1_X_list]}.  That is, sort the values of
351\varname{[floor1_X_list]} and then apply the same permutation to
352elements of the other two vectors so that the X, Y and step2_flag
353values still match.
354
355Then compute the final curve in one pass:
356
357\begin{Verbatim}[commandchars=\\\{\}]
358  1) [hx] = 0
359  2) [lx] = 0
360  3) [ly] = vector [floor1_final_Y]' element [0] * [floor1_multiplier]
361  4) iterate [i] over the range 1 ... [floor1_values]-1 \{
362
363       5) if ( [floor1_step2_flag]' element [i] is set ) \{
364
365             6) [hy] = [floor1_final_Y]' element [i] * [floor1_multiplier]
366 	     7) [hx] = [floor1_X_list]' element [i]
367             8) \link{vorbis:spec:render:line}{render_line}( [lx], [ly], [hx], [hy], [floor] )
368             9) [lx] = [hx]
369	    10) [ly] = [hy]
370          \}
371     \}
372
373 11) if ( [hx] is less than [n] ) \{
374
375        12) \link{vorbis:spec:render:line}{render_line}( [hx], [hy], [n], [hy], [floor] )
376
377     \}
378
379 13) if ( [hx] is greater than [n] ) \{
380
381            14) truncate vector [floor] to [n] elements
382
383     \}
384
385 15) for each scalar in vector [floor], perform a lookup substitution using
386     the scalar value from [floor] as an offset into the vector \link{vorbis:spec:floor1:inverse:dB:table}{[floor1_inverse_dB_static_table]}
387
388 16) done
389
390\end{Verbatim}
391
392\end{description}
393