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73 <h1>Ogg Vorbis stereo-specific channel coupling discussion</h1>
77 <p>The Vorbis audio CODEC provides a channel coupling
78 mechanisms designed to reduce effective bitrate by both eliminating
79 interchannel redundancy and eliminating stereo image information
80 labeled inaudible or undesirable according to spatial psychoacoustic
81 models. This document describes both the mechanical coupling
82 mechanisms available within the Vorbis specification, as well as the
83 specific stereo coupling models used by the reference
84 <tt>libvorbis</tt> codec provided by xiph.org.</p>
88 <p>In encoder release beta 4 and earlier, Vorbis supported multiple
89 channel encoding, but the channels were encoded entirely separately
90 with no cross-analysis or redundancy elimination between channels.
91 This multichannel strategy is very similar to the mp3's <em>dual
92 stereo</em> mode and Vorbis uses the same name for its analogous
93 uncoupled multichannel modes.</p>
95 <p>However, the Vorbis spec provides for, and Vorbis release 1.0 rc1 and
96 later implement a coupled channel strategy. Vorbis has two specific
97 mechanisms that may be used alone or in conjunction to implement
98 channel coupling. The first is <em>channel interleaving</em> via
99 residue backend type 2, and the second is <em>square polar
100 mapping</em>. These two general mechanisms are particularly well
101 suited to coupling due to the structure of Vorbis encoding, as we'll
102 explore below, and using both we can implement both totally
103 <em>lossless stereo image coupling</em> [bit-for-bit decode-identical
104 to uncoupled modes], as well as various lossy models that seek to
105 eliminate inaudible or unimportant aspects of the stereo image in
106 order to enhance bitrate. The exact coupling implementation is
107 generalized to allow the encoder a great deal of flexibility in
108 implementation of a stereo or surround model without requiring any
109 significant complexity increase over the combinatorially simpler
110 mid/side joint stereo of mp3 and other current audio codecs.</p>
112 <p>A particular Vorbis bitstream may apply channel coupling directly to
113 more than a pair of channels; polar mapping is hierarchical such that
114 polar coupling may be extrapolated to an arbitrary number of channels
115 and is not restricted to only stereo, quadraphonics, ambisonics or 5.1
116 surround. However, the scope of this document restricts itself to the
117 stereo coupling case.</p>
120 <h3>Square Polar Mapping</h3>
122 <h4>maximal correlation</h4>
124 <p>Recall that the basic structure of a a Vorbis I stream first generates
125 from input audio a spectral 'floor' function that serves as an
126 MDCT-domain whitening filter. This floor is meant to represent the
127 rough envelope of the frequency spectrum, using whatever metric the
128 encoder cares to define. This floor is subtracted from the log
129 frequency spectrum, effectively normalizing the spectrum by frequency.
130 Each input channel is associated with a unique floor function.</p>
132 <p>The basic idea behind any stereo coupling is that the left and right
133 channels usually correlate. This correlation is even stronger if one
134 first accounts for energy differences in any given frequency band
135 across left and right; think for example of individual instruments
136 mixed into different portions of the stereo image, or a stereo
137 recording with a dominant feature not perfectly in the center. The
138 floor functions, each specific to a channel, provide the perfect means
139 of normalizing left and right energies across the spectrum to maximize
140 correlation before coupling. This feature of the Vorbis format is not
141 a convenient accident.</p>
143 <p>Because we strive to maximally correlate the left and right channels
144 and generally succeed in doing so, left and right residue is typically
145 nearly identical. We could use channel interleaving (discussed below)
146 alone to efficiently remove the redundancy between the left and right
147 channels as a side effect of entropy encoding, but a polar
148 representation gives benefits when left/right correlation is
151 <h4>point and diffuse imaging</h4>
153 <p>The first advantage of a polar representation is that it effectively
154 separates the spatial audio information into a 'point image'
155 (magnitude) at a given frequency and located somewhere in the sound
156 field, and a 'diffuse image' (angle) that fills a large amount of
157 space simultaneously. Even if we preserve only the magnitude (point)
158 data, a detailed and carefully chosen floor function in each channel
159 provides us with a free, fine-grained, frequency relative intensity
160 stereo*. Angle information represents diffuse sound fields, such as
161 reverberation that fills the entire space simultaneously.</p>
163 <p>*<em>Because the Vorbis model supports a number of different possible
164 stereo models and these models may be mixed, we do not use the term
165 'intensity stereo' talking about Vorbis; instead we use the terms
166 'point stereo', 'phase stereo' and subcategories of each.</em></p>
168 <p>The majority of a stereo image is representable by polar magnitude
169 alone, as strong sounds tend to be produced at near-point sources;
170 even non-diffuse, fast, sharp echoes track very accurately using
171 magnitude representation almost alone (for those experimenting with
172 Vorbis tuning, this strategy works much better with the precise,
173 piecewise control of floor 1; the continuous approximation of floor 0
174 results in unstable imaging). Reverberation and diffuse sounds tend
175 to contain less energy and be psychoacoustically dominated by the
176 point sources embedded in them. Thus, we again tend to concentrate
177 more represented energy into a predictably smaller number of numbers.
178 Separating representation of point and diffuse imaging also allows us
179 to model and manipulate point and diffuse qualities separately.</p>
181 <h4>controlling bit leakage and symbol crosstalk</h4>
184 representation concentrates represented energy into fewer large
185 values, we reduce bit 'leakage' during cascading (multistage VQ
186 encoding) as a secondary benefit. A single large, monolithic VQ
187 codebook is more efficient than a cascaded book due to entropy
188 'crosstalk' among symbols between different stages of a multistage cascade.
189 Polar representation is a way of further concentrating entropy into
190 predictable locations so that codebook design can take steps to
191 improve multistage codebook efficiency. It also allows us to cascade
192 various elements of the stereo image independently.</p>
194 <h4>eliminating trigonometry and rounding</h4>
196 <p>Rounding and computational complexity are potential problems with a
197 polar representation. As our encoding process involves quantization,
198 mixing a polar representation and quantization makes it potentially
199 impossible, depending on implementation, to construct a coupled stereo
200 mechanism that results in bit-identical decompressed output compared
201 to an uncoupled encoding should the encoder desire it.</p>
203 <p>Vorbis uses a mapping that preserves the most useful qualities of
204 polar representation, relies only on addition/subtraction (during
205 decode; high quality encoding still requires some trig), and makes it
206 trivial before or after quantization to represent an angle/magnitude
207 through a one-to-one mapping from possible left/right value
208 permutations. We do this by basing our polar representation on the
209 unit square rather than the unit-circle.</p>
211 <p>Given a magnitude and angle, we recover left and right using the
212 following function (note that A/B may be left/right or right/left
213 depending on the coupling definition used by the encoder):</p>
235 <p>The function is antisymmetric for positive and negative magnitudes in
236 order to eliminate a redundant value when quantizing. For example, if
237 we're quantizing to integer values, we can visualize a magnitude of 5
238 and an angle of -2 as follows:</p>
240 <p><img src="squarepolar.png" alt="square polar"/></p>
242 <p>This representation loses or replicates no values; if the range of A
243 and B are integral -5 through 5, the number of possible Cartesian
244 permutations is 121. Represented in square polar notation, the
245 possible values are:</p>
250 -1,-2 -1,-1 -1, 0 -1, 1
254 -2,-4 -2,-3 -2,-2 -2,-1 -2, 0 -2, 1 -2, 2 -2, 3
256 2,-4 2,-3 ... following the pattern ...
258 ... 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6 5, 7 5, 8 5, 9
262 <p>...for a grand total of 121 possible values, the same number as in
263 Cartesian representation (note that, for example, <tt>5,-10</tt> is
264 the same as <tt>-5,10</tt>, so there's no reason to represent
265 both. 2,10 cannot happen, and there's no reason to account for it.)
266 It's also obvious that this mapping is exactly reversible.</p>
268 <h3>Channel interleaving</h3>
270 <p>We can remap and A/B vector using polar mapping into a magnitude/angle
271 vector, and it's clear that, in general, this concentrates energy in
272 the magnitude vector and reduces the amount of information to encode
273 in the angle vector. Encoding these vectors independently with
274 residue backend #0 or residue backend #1 will result in bitrate
275 savings. However, there are still implicit correlations between the
276 magnitude and angle vectors. The most obvious is that the amplitude
277 of the angle is bounded by its corresponding magnitude value.</p>
279 <p>Entropy coding the results, then, further benefits from the entropy
280 model being able to compress magnitude and angle simultaneously. For
281 this reason, Vorbis implements residue backend #2 which pre-interleaves
282 a number of input vectors (in the stereo case, two, A and B) into a
283 single output vector (with the elements in the order of
284 A_0, B_0, A_1, B_1, A_2 ... A_n-1, B_n-1) before entropy encoding. Thus
285 each vector to be coded by the vector quantization backend consists of
286 matching magnitude and angle values.</p>
288 <p>The astute reader, at this point, will notice that in the theoretical
289 case in which we can use monolithic codebooks of arbitrarily large
290 size, we can directly interleave and encode left and right without
291 polar mapping; in fact, the polar mapping does not appear to lend any
292 benefit whatsoever to the efficiency of the entropy coding. In fact,
293 it is perfectly possible and reasonable to build a Vorbis encoder that
294 dispenses with polar mapping entirely and merely interleaves the
295 channel. Libvorbis based encoders may configure such an encoding and
296 it will work as intended.</p>
298 <p>However, when we leave the ideal/theoretical domain, we notice that
299 polar mapping does give additional practical benefits, as discussed in
300 the above section on polar mapping and summarized again here:</p>
303 <li>Polar mapping aids in controlling entropy 'leakage' between stages
304 of a cascaded codebook.</li>
305 <li>Polar mapping separates the stereo image
306 into point and diffuse components which may be analyzed and handled
310 <h2>Stereo Models</h2>
314 <p>Dual stereo refers to stereo encoding where the channels are entirely
315 separate; they are analyzed and encoded as entirely distinct entities.
316 This terminology is familiar from mp3.</p>
318 <h3>Lossless Stereo</h3>
320 <p>Using polar mapping and/or channel interleaving, it's possible to
321 couple Vorbis channels losslessly, that is, construct a stereo
322 coupling encoding that both saves space but also decodes
323 bit-identically to dual stereo. OggEnc 1.0 and later uses this
324 mode in all high-bitrate encoding.</p>
326 <p>Overall, this stereo mode is overkill; however, it offers a safe
327 alternative to users concerned about the slightest possible
328 degradation to the stereo image or archival quality audio.</p>
330 <h3>Phase Stereo</h3>
332 <p>Phase stereo is the least aggressive means of gracefully dropping
333 resolution from the stereo image; it affects only diffuse imaging.</p>
335 <p>It's often quoted that the human ear is deaf to signal phase above
336 about 4kHz; this is nearly true and a passable rule of thumb, but it
337 can be demonstrated that even an average user can tell the difference
338 between high frequency in-phase and out-of-phase noise. Obviously
339 then, the statement is not entirely true. However, it's also the case
340 that one must resort to nearly such an extreme demonstration before
341 finding the counterexample.</p>
343 <p>'Phase stereo' is simply a more aggressive quantization of the polar
344 angle vector; above 4kHz it's generally quite safe to quantize noise
345 and noisy elements to only a handful of allowed phases, or to thin the
346 phase with respect to the magnitude. The phases of high amplitude
347 pure tones may or may not be preserved more carefully (they are
348 relatively rare and L/R tend to be in phase, so there is generally
349 little reason not to spend a few more bits on them)</p>
351 <h4>example: eight phase stereo</h4>
353 <p>Vorbis may implement phase stereo coupling by preserving the entirety
354 of the magnitude vector (essential to fine amplitude and energy
355 resolution overall) and quantizing the angle vector to one of only
356 four possible values. Given that the magnitude vector may be positive
357 or negative, this results in left and right phase having eight
358 possible permutation, thus 'eight phase stereo':</p>
360 <p><img src="eightphase.png" alt="eight phase"/></p>
362 <p>Left and right may be in phase (positive or negative), the most common
363 case by far, or out of phase by 90 or 180 degrees.</p>
365 <h4>example: four phase stereo</h4>
367 <p>Similarly, four phase stereo takes the quantization one step further;
368 it allows only in-phase and 180 degree out-out-phase signals:</p>
370 <p><img src="fourphase.png" alt="four phase"/></p>
372 <h3>example: point stereo</h3>
374 <p>Point stereo eliminates the possibility of out-of-phase signal
375 entirely. Any diffuse quality to a sound source tends to collapse
376 inward to a point somewhere within the stereo image. A practical
377 example would be balanced reverberations within a large, live space;
378 normally the sound is diffuse and soft, giving a sonic impression of
379 volume. In point-stereo, the reverberations would still exist, but
380 sound fairly firmly centered within the image (assuming the
381 reverberation was centered overall; if the reverberation is stronger
382 to the left, then the point of localization in point stereo would be
383 to the left). This effect is most noticeable at low and mid
384 frequencies and using headphones (which grant perfect stereo
385 separation). Point stereo is is a graceful but generally easy to
386 detect degradation to the sound quality and is thus used in frequency
387 ranges where it is least noticeable.</p>
389 <h3>Mixed Stereo</h3>
391 <p>Mixed stereo is the simultaneous use of more than one of the above
392 stereo encoding models, generally using more aggressive modes in
393 higher frequencies, lower amplitudes or 'nearly' in-phase sound.</p>
395 <p>It is also the case that near-DC frequencies should be encoded using
396 lossless coupling to avoid frame blocking artifacts.</p>
398 <h3>Vorbis Stereo Modes</h3>
400 <p>Vorbis, as of 1.0, uses lossless stereo and a number of mixed modes
401 constructed out of lossless and point stereo. Phase stereo was used
402 in the rc2 encoder, but is not currently used for simplicity's sake. It
403 will likely be re-added to the stereo model in the future.</p>
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