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72
73<h1>Ogg Vorbis stereo-specific channel coupling discussion</h1>
74
75<h2>Abstract</h2>
76
77<p>The Vorbis audio CODEC provides a channel coupling
78mechanisms designed to reduce effective bitrate by both eliminating
79interchannel redundancy and eliminating stereo image information
80labeled inaudible or undesirable according to spatial psychoacoustic
81models. This document describes both the mechanical coupling
82mechanisms available within the Vorbis specification, as well as the
83specific stereo coupling models used by the reference
84<tt>libvorbis</tt> codec provided by xiph.org.</p>
85
86<h2>Mechanisms</h2>
87
88<p>In encoder release beta 4 and earlier, Vorbis supported multiple
89channel encoding, but the channels were encoded entirely separately
90with no cross-analysis or redundancy elimination between channels.
91This multichannel strategy is very similar to the mp3's <em>dual
92stereo</em> mode and Vorbis uses the same name for its analogous
93uncoupled multichannel modes.</p>
94
95<p>However, the Vorbis spec provides for, and Vorbis release 1.0 rc1 and
96later implement a coupled channel strategy. Vorbis has two specific
97mechanisms that may be used alone or in conjunction to implement
98channel coupling. The first is <em>channel interleaving</em> via
99residue backend type 2, and the second is <em>square polar
100mapping</em>. These two general mechanisms are particularly well
101suited to coupling due to the structure of Vorbis encoding, as we'll
102explore below, and using both we can implement both totally
103<em>lossless stereo image coupling</em> [bit-for-bit decode-identical
104to uncoupled modes], as well as various lossy models that seek to
105eliminate inaudible or unimportant aspects of the stereo image in
106order to enhance bitrate. The exact coupling implementation is
107generalized to allow the encoder a great deal of flexibility in
108implementation of a stereo or surround model without requiring any
109significant complexity increase over the combinatorially simpler
110mid/side joint stereo of mp3 and other current audio codecs.</p>
111
112<p>A particular Vorbis bitstream may apply channel coupling directly to
113more than a pair of channels; polar mapping is hierarchical such that
114polar coupling may be extrapolated to an arbitrary number of channels
115and is not restricted to only stereo, quadraphonics, ambisonics or 5.1
116surround. However, the scope of this document restricts itself to the
117stereo coupling case.</p>
118
119<h3>Square Polar Mapping</h3>
120
121<h4>maximal correlation</h4>
122 
123<p>Recall that the basic structure of a a Vorbis I stream first generates
124from input audio a spectral 'floor' function that serves as an
125MDCT-domain whitening filter. This floor is meant to represent the
126rough envelope of the frequency spectrum, using whatever metric the
127encoder cares to define. This floor is subtracted from the log
128frequency spectrum, effectively normalizing the spectrum by frequency.
129Each input channel is associated with a unique floor function.</p>
130
131<p>The basic idea behind any stereo coupling is that the left and right
132channels usually correlate. This correlation is even stronger if one
133first accounts for energy differences in any given frequency band
134across left and right; think for example of individual instruments
135mixed into different portions of the stereo image, or a stereo
136recording with a dominant feature not perfectly in the center. The
137floor functions, each specific to a channel, provide the perfect means
138of normalizing left and right energies across the spectrum to maximize
139correlation before coupling. This feature of the Vorbis format is not
140a convenient accident.</p>
141
142<p>Because we strive to maximally correlate the left and right channels
143and generally succeed in doing so, left and right residue is typically
144nearly identical. We could use channel interleaving (discussed below)
145alone to efficiently remove the redundancy between the left and right
146channels as a side effect of entropy encoding, but a polar
147representation gives benefits when left/right correlation is
148strong.</p>
149
150<h4>point and diffuse imaging</h4>
151
152<p>The first advantage of a polar representation is that it effectively
153separates the spatial audio information into a 'point image'
154(magnitude) at a given frequency and located somewhere in the sound
155field, and a 'diffuse image' (angle) that fills a large amount of
156space simultaneously. Even if we preserve only the magnitude (point)
157data, a detailed and carefully chosen floor function in each channel
158provides us with a free, fine-grained, frequency relative intensity
159stereo*. Angle information represents diffuse sound fields, such as
160reverberation that fills the entire space simultaneously.</p>
161
162<p>*<em>Because the Vorbis model supports a number of different possible
163stereo models and these models may be mixed, we do not use the term
164'intensity stereo' talking about Vorbis; instead we use the terms
165'point stereo', 'phase stereo' and subcategories of each.</em></p>
166
167<p>The majority of a stereo image is representable by polar magnitude
168alone, as strong sounds tend to be produced at near-point sources;
169even non-diffuse, fast, sharp echoes track very accurately using
170magnitude representation almost alone (for those experimenting with
171Vorbis tuning, this strategy works much better with the precise,
172piecewise control of floor 1; the continuous approximation of floor 0
173results in unstable imaging). Reverberation and diffuse sounds tend
174to contain less energy and be psychoacoustically dominated by the
175point sources embedded in them. Thus, we again tend to concentrate
176more represented energy into a predictably smaller number of numbers.
177Separating representation of point and diffuse imaging also allows us
178to model and manipulate point and diffuse qualities separately.</p>
179
180<h4>controlling bit leakage and symbol crosstalk</h4>
181
182<p>Because polar
183representation concentrates represented energy into fewer large
184values, we reduce bit 'leakage' during cascading (multistage VQ
185encoding) as a secondary benefit. A single large, monolithic VQ
186codebook is more efficient than a cascaded book due to entropy
187'crosstalk' among symbols between different stages of a multistage cascade.
188Polar representation is a way of further concentrating entropy into
189predictable locations so that codebook design can take steps to
190improve multistage codebook efficiency. It also allows us to cascade
191various elements of the stereo image independently.</p>
192
193<h4>eliminating trigonometry and rounding</h4>
194
195<p>Rounding and computational complexity are potential problems with a
196polar representation. As our encoding process involves quantization,
197mixing a polar representation and quantization makes it potentially
198impossible, depending on implementation, to construct a coupled stereo
199mechanism that results in bit-identical decompressed output compared
200to an uncoupled encoding should the encoder desire it.</p>
201
202<p>Vorbis uses a mapping that preserves the most useful qualities of
203polar representation, relies only on addition/subtraction (during
204decode; high quality encoding still requires some trig), and makes it
205trivial before or after quantization to represent an angle/magnitude
206through a one-to-one mapping from possible left/right value
207permutations. We do this by basing our polar representation on the
208unit square rather than the unit-circle.</p>
209
210<p>Given a magnitude and angle, we recover left and right using the
211following function (note that A/B may be left/right or right/left
212depending on the coupling definition used by the encoder):</p>
213
214<pre>
215      if(magnitude>0)
216        if(angle>0){
217          A=magnitude;
218          B=magnitude-angle;
219        }else{
220          B=magnitude;
221          A=magnitude+angle;
222        }
223      else
224        if(angle>0){
225          A=magnitude;
226          B=magnitude+angle;
227        }else{
228          B=magnitude;
229          A=magnitude-angle;
230        }
231    }
232</pre>
233
234<p>The function is antisymmetric for positive and negative magnitudes in
235order to eliminate a redundant value when quantizing. For example, if
236we're quantizing to integer values, we can visualize a magnitude of 5
237and an angle of -2 as follows:</p>
238
239<p><img src="squarepolar.png" alt="square polar"/></p>
240
241<p>This representation loses or replicates no values; if the range of A
242and B are integral -5 through 5, the number of possible Cartesian
243permutations is 121. Represented in square polar notation, the
244possible values are:</p>
245
246<pre>
247 0, 0
248
249-1,-2  -1,-1  -1, 0  -1, 1
250
251 1,-2   1,-1   1, 0   1, 1
252
253-2,-4  -2,-3  -2,-2  -2,-1  -2, 0  -2, 1  -2, 2  -2, 3 
254
255 2,-4   2,-3   ... following the pattern ...
256
257 ...   5, 1   5, 2   5, 3   5, 4   5, 5   5, 6   5, 7   5, 8   5, 9
258
259</pre>
260
261<p>...for a grand total of 121 possible values, the same number as in
262Cartesian representation (note that, for example, <tt>5,-10</tt> is
263the same as <tt>-5,10</tt>, so there's no reason to represent
264both. 2,10 cannot happen, and there's no reason to account for it.)
265It's also obvious that this mapping is exactly reversible.</p>
266
267<h3>Channel interleaving</h3>
268
269<p>We can remap and A/B vector using polar mapping into a magnitude/angle
270vector, and it's clear that, in general, this concentrates energy in
271the magnitude vector and reduces the amount of information to encode
272in the angle vector. Encoding these vectors independently with
273residue backend #0 or residue backend #1 will result in bitrate
274savings. However, there are still implicit correlations between the
275magnitude and angle vectors. The most obvious is that the amplitude
276of the angle is bounded by its corresponding magnitude value.</p>
277
278<p>Entropy coding the results, then, further benefits from the entropy
279model being able to compress magnitude and angle simultaneously. For
280this reason, Vorbis implements residue backend #2 which pre-interleaves
281a number of input vectors (in the stereo case, two, A and B) into a
282single output vector (with the elements in the order of
283A_0, B_0, A_1, B_1, A_2 ... A_n-1, B_n-1) before entropy encoding. Thus
284each vector to be coded by the vector quantization backend consists of
285matching magnitude and angle values.</p>
286
287<p>The astute reader, at this point, will notice that in the theoretical
288case in which we can use monolithic codebooks of arbitrarily large
289size, we can directly interleave and encode left and right without
290polar mapping; in fact, the polar mapping does not appear to lend any
291benefit whatsoever to the efficiency of the entropy coding. In fact,
292it is perfectly possible and reasonable to build a Vorbis encoder that
293dispenses with polar mapping entirely and merely interleaves the
294channel. Libvorbis based encoders may configure such an encoding and
295it will work as intended.</p>
296
297<p>However, when we leave the ideal/theoretical domain, we notice that
298polar mapping does give additional practical benefits, as discussed in
299the above section on polar mapping and summarized again here:</p>
300
301<ul>
302<li>Polar mapping aids in controlling entropy 'leakage' between stages
303of a cascaded codebook.</li>
304<li>Polar mapping separates the stereo image
305into point and diffuse components which may be analyzed and handled
306differently.</li>
307</ul>
308
309<h2>Stereo Models</h2>
310
311<h3>Dual Stereo</h3>
312
313<p>Dual stereo refers to stereo encoding where the channels are entirely
314separate; they are analyzed and encoded as entirely distinct entities.
315This terminology is familiar from mp3.</p>
316
317<h3>Lossless Stereo</h3>
318
319<p>Using polar mapping and/or channel interleaving, it's possible to
320couple Vorbis channels losslessly, that is, construct a stereo
321coupling encoding that both saves space but also decodes
322bit-identically to dual stereo. OggEnc 1.0 and later uses this
323mode in all high-bitrate encoding.</p>
324
325<p>Overall, this stereo mode is overkill; however, it offers a safe
326alternative to users concerned about the slightest possible
327degradation to the stereo image or archival quality audio.</p>
328
329<h3>Phase Stereo</h3>
330
331<p>Phase stereo is the least aggressive means of gracefully dropping
332resolution from the stereo image; it affects only diffuse imaging.</p>
333
334<p>It's often quoted that the human ear is deaf to signal phase above
335about 4kHz; this is nearly true and a passable rule of thumb, but it
336can be demonstrated that even an average user can tell the difference
337between high frequency in-phase and out-of-phase noise. Obviously
338then, the statement is not entirely true. However, it's also the case
339that one must resort to nearly such an extreme demonstration before
340finding the counterexample.</p>
341
342<p>'Phase stereo' is simply a more aggressive quantization of the polar
343angle vector; above 4kHz it's generally quite safe to quantize noise
344and noisy elements to only a handful of allowed phases, or to thin the
345phase with respect to the magnitude. The phases of high amplitude
346pure tones may or may not be preserved more carefully (they are
347relatively rare and L/R tend to be in phase, so there is generally
348little reason not to spend a few more bits on them)</p>
349
350<h4>example: eight phase stereo</h4>
351
352<p>Vorbis may implement phase stereo coupling by preserving the entirety
353of the magnitude vector (essential to fine amplitude and energy
354resolution overall) and quantizing the angle vector to one of only
355four possible values. Given that the magnitude vector may be positive
356or negative, this results in left and right phase having eight
357possible permutation, thus 'eight phase stereo':</p>
358
359<p><img src="eightphase.png" alt="eight phase"/></p>
360
361<p>Left and right may be in phase (positive or negative), the most common
362case by far, or out of phase by 90 or 180 degrees.</p>
363
364<h4>example: four phase stereo</h4>
365
366<p>Similarly, four phase stereo takes the quantization one step further;
367it allows only in-phase and 180 degree out-out-phase signals:</p>
368
369<p><img src="fourphase.png" alt="four phase"/></p>
370
371<h3>example: point stereo</h3>
372
373<p>Point stereo eliminates the possibility of out-of-phase signal
374entirely. Any diffuse quality to a sound source tends to collapse
375inward to a point somewhere within the stereo image. A practical
376example would be balanced reverberations within a large, live space;
377normally the sound is diffuse and soft, giving a sonic impression of
378volume. In point-stereo, the reverberations would still exist, but
379sound fairly firmly centered within the image (assuming the
380reverberation was centered overall; if the reverberation is stronger
381to the left, then the point of localization in point stereo would be
382to the left). This effect is most noticeable at low and mid
383frequencies and using headphones (which grant perfect stereo
384separation). Point stereo is is a graceful but generally easy to
385detect degradation to the sound quality and is thus used in frequency
386ranges where it is least noticeable.</p>
387
388<h3>Mixed Stereo</h3>
389
390<p>Mixed stereo is the simultaneous use of more than one of the above
391stereo encoding models, generally using more aggressive modes in
392higher frequencies, lower amplitudes or 'nearly' in-phase sound.</p>
393
394<p>It is also the case that near-DC frequencies should be encoded using
395lossless coupling to avoid frame blocking artifacts.</p>
396
397<h3>Vorbis Stereo Modes</h3>
398
399<p>Vorbis, as of 1.0, uses lossless stereo and a number of mixed modes
400constructed out of lossless and point stereo. Phase stereo was used
401in the rc2 encoder, but is not currently used for simplicity's sake. It
402will likely be re-added to the stereo model in the future.</p>
403
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