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<h1>Ogg Vorbis stereo-specific channel coupling discussion</h1>

<h2>Abstract</h2>

<p>The Vorbis audio CODEC provides a channel coupling
mechanisms designed to reduce effective bitrate by both eliminating
interchannel redundancy and eliminating stereo image information
labeled inaudible or undesirable according to spatial psychoacoustic
models. This document describes both the mechanical coupling
mechanisms available within the Vorbis specification, as well as the
specific stereo coupling models used by the reference
<tt>libvorbis</tt> codec provided by xiph.org.</p>

<h2>Mechanisms</h2>

<p>In encoder release beta 4 and earlier, Vorbis supported multiple
channel encoding, but the channels were encoded entirely separately
with no cross-analysis or redundancy elimination between channels.
This multichannel strategy is very similar to the mp3's <em>dual
stereo</em> mode and Vorbis uses the same name for its analogous
uncoupled multichannel modes.</p>

<p>However, the Vorbis spec provides for, and Vorbis release 1.0 rc1 and
later implement a coupled channel strategy. Vorbis has two specific
mechanisms that may be used alone or in conjunction to implement
channel coupling. The first is <em>channel interleaving</em> via
residue backend type 2, and the second is <em>square polar
mapping</em>. These two general mechanisms are particularly well
suited to coupling due to the structure of Vorbis encoding, as we'll
explore below, and using both we can implement both totally
<em>lossless stereo image coupling</em> [bit-for-bit decode-identical
to uncoupled modes], as well as various lossy models that seek to
eliminate inaudible or unimportant aspects of the stereo image in
order to enhance bitrate. The exact coupling implementation is
generalized to allow the encoder a great deal of flexibility in
implementation of a stereo or surround model without requiring any
significant complexity increase over the combinatorially simpler
mid/side joint stereo of mp3 and other current audio codecs.</p>

<p>A particular Vorbis bitstream may apply channel coupling directly to
more than a pair of channels; polar mapping is hierarchical such that
polar coupling may be extrapolated to an arbitrary number of channels
and is not restricted to only stereo, quadraphonics, ambisonics or 5.1
surround. However, the scope of this document restricts itself to the
stereo coupling case.</p>

<a name="sqpm"></a>
<h3>Square Polar Mapping</h3>

<h4>maximal correlation</h4>
 
<p>Recall that the basic structure of a a Vorbis I stream first generates
from input audio a spectral 'floor' function that serves as an
MDCT-domain whitening filter. This floor is meant to represent the
rough envelope of the frequency spectrum, using whatever metric the
encoder cares to define. This floor is subtracted from the log
frequency spectrum, effectively normalizing the spectrum by frequency.
Each input channel is associated with a unique floor function.</p>

<p>The basic idea behind any stereo coupling is that the left and right
channels usually correlate. This correlation is even stronger if one
first accounts for energy differences in any given frequency band
across left and right; think for example of individual instruments
mixed into different portions of the stereo image, or a stereo
recording with a dominant feature not perfectly in the center. The
floor functions, each specific to a channel, provide the perfect means
of normalizing left and right energies across the spectrum to maximize
correlation before coupling. This feature of the Vorbis format is not
a convenient accident.</p>

<p>Because we strive to maximally correlate the left and right channels
and generally succeed in doing so, left and right residue is typically
nearly identical. We could use channel interleaving (discussed below)
alone to efficiently remove the redundancy between the left and right
channels as a side effect of entropy encoding, but a polar
representation gives benefits when left/right correlation is
strong.</p>

<h4>point and diffuse imaging</h4>

<p>The first advantage of a polar representation is that it effectively
separates the spatial audio information into a 'point image'
(magnitude) at a given frequency and located somewhere in the sound
field, and a 'diffuse image' (angle) that fills a large amount of
space simultaneously. Even if we preserve only the magnitude (point)
data, a detailed and carefully chosen floor function in each channel
provides us with a free, fine-grained, frequency relative intensity
stereo*. Angle information represents diffuse sound fields, such as
reverberation that fills the entire space simultaneously.</p>

<p>*<em>Because the Vorbis model supports a number of different possible
stereo models and these models may be mixed, we do not use the term
'intensity stereo' talking about Vorbis; instead we use the terms
'point stereo', 'phase stereo' and subcategories of each.</em></p>

<p>The majority of a stereo image is representable by polar magnitude
alone, as strong sounds tend to be produced at near-point sources;
even non-diffuse, fast, sharp echoes track very accurately using
magnitude representation almost alone (for those experimenting with
Vorbis tuning, this strategy works much better with the precise,
piecewise control of floor 1; the continuous approximation of floor 0
results in unstable imaging). Reverberation and diffuse sounds tend
to contain less energy and be psychoacoustically dominated by the
point sources embedded in them. Thus, we again tend to concentrate
more represented energy into a predictably smaller number of numbers.
Separating representation of point and diffuse imaging also allows us
to model and manipulate point and diffuse qualities separately.</p>

<h4>controlling bit leakage and symbol crosstalk</h4>

<p>Because polar
representation concentrates represented energy into fewer large
values, we reduce bit 'leakage' during cascading (multistage VQ
encoding) as a secondary benefit. A single large, monolithic VQ
codebook is more efficient than a cascaded book due to entropy
'crosstalk' among symbols between different stages of a multistage cascade.
Polar representation is a way of further concentrating entropy into
predictable locations so that codebook design can take steps to
improve multistage codebook efficiency. It also allows us to cascade
various elements of the stereo image independently.</p>

<h4>eliminating trigonometry and rounding</h4>

<p>Rounding and computational complexity are potential problems with a
polar representation. As our encoding process involves quantization,
mixing a polar representation and quantization makes it potentially
impossible, depending on implementation, to construct a coupled stereo
mechanism that results in bit-identical decompressed output compared
to an uncoupled encoding should the encoder desire it.</p>

<p>Vorbis uses a mapping that preserves the most useful qualities of
polar representation, relies only on addition/subtraction (during
decode; high quality encoding still requires some trig), and makes it
trivial before or after quantization to represent an angle/magnitude
through a one-to-one mapping from possible left/right value
permutations. We do this by basing our polar representation on the
unit square rather than the unit-circle.</p>

<p>Given a magnitude and angle, we recover left and right using the
following function (note that A/B may be left/right or right/left
depending on the coupling definition used by the encoder):</p>

<pre>
      if(magnitude>0)
        if(angle>0){
          A=magnitude;
          B=magnitude-angle;
        }else{
          B=magnitude;
          A=magnitude+angle;
        }
      else
        if(angle>0){
          A=magnitude;
          B=magnitude+angle;
        }else{
          B=magnitude;
          A=magnitude-angle;
        }
    }
</pre>

<p>The function is antisymmetric for positive and negative magnitudes in
order to eliminate a redundant value when quantizing. For example, if
we're quantizing to integer values, we can visualize a magnitude of 5
and an angle of -2 as follows:</p>

<p><img src="squarepolar.png" alt="square polar"/></p>

<p>This representation loses or replicates no values; if the range of A
and B are integral -5 through 5, the number of possible Cartesian
permutations is 121. Represented in square polar notation, the
possible values are:</p>

<pre>
 0, 0

-1,-2  -1,-1  -1, 0  -1, 1

 1,-2   1,-1   1, 0   1, 1

-2,-4  -2,-3  -2,-2  -2,-1  -2, 0  -2, 1  -2, 2  -2, 3  

 2,-4   2,-3   ... following the pattern ...

 ...   5, 1   5, 2   5, 3   5, 4   5, 5   5, 6   5, 7   5, 8   5, 9

</pre>

<p>...for a grand total of 121 possible values, the same number as in
Cartesian representation (note that, for example, <tt>5,-10</tt> is
the same as <tt>-5,10</tt>, so there's no reason to represent
both. 2,10 cannot happen, and there's no reason to account for it.)
It's also obvious that this mapping is exactly reversible.</p>

<h3>Channel interleaving</h3>

<p>We can remap and A/B vector using polar mapping into a magnitude/angle
vector, and it's clear that, in general, this concentrates energy in
the magnitude vector and reduces the amount of information to encode
in the angle vector. Encoding these vectors independently with
residue backend #0 or residue backend #1 will result in bitrate
savings. However, there are still implicit correlations between the
magnitude and angle vectors. The most obvious is that the amplitude
of the angle is bounded by its corresponding magnitude value.</p>

<p>Entropy coding the results, then, further benefits from the entropy
model being able to compress magnitude and angle simultaneously. For
this reason, Vorbis implements residue backend #2 which pre-interleaves
a number of input vectors (in the stereo case, two, A and B) into a
single output vector (with the elements in the order of
A_0, B_0, A_1, B_1, A_2 ... A_n-1, B_n-1) before entropy encoding. Thus
each vector to be coded by the vector quantization backend consists of
matching magnitude and angle values.</p>

<p>The astute reader, at this point, will notice that in the theoretical
case in which we can use monolithic codebooks of arbitrarily large
size, we can directly interleave and encode left and right without
polar mapping; in fact, the polar mapping does not appear to lend any
benefit whatsoever to the efficiency of the entropy coding. In fact,
it is perfectly possible and reasonable to build a Vorbis encoder that
dispenses with polar mapping entirely and merely interleaves the
channel. Libvorbis based encoders may configure such an encoding and
it will work as intended.</p>

<p>However, when we leave the ideal/theoretical domain, we notice that
polar mapping does give additional practical benefits, as discussed in
the above section on polar mapping and summarized again here:</p>

<ul>
<li>Polar mapping aids in controlling entropy 'leakage' between stages
of a cascaded codebook.</li>
<li>Polar mapping separates the stereo image
into point and diffuse components which may be analyzed and handled
differently.</li>
</ul>

<h2>Stereo Models</h2>

<h3>Dual Stereo</h3>

<p>Dual stereo refers to stereo encoding where the channels are entirely
separate; they are analyzed and encoded as entirely distinct entities.
This terminology is familiar from mp3.</p>

<h3>Lossless Stereo</h3>

<p>Using polar mapping and/or channel interleaving, it's possible to
couple Vorbis channels losslessly, that is, construct a stereo
coupling encoding that both saves space but also decodes
bit-identically to dual stereo. OggEnc 1.0 and later uses this
mode in all high-bitrate encoding.</p>

<p>Overall, this stereo mode is overkill; however, it offers a safe
alternative to users concerned about the slightest possible
degradation to the stereo image or archival quality audio.</p>

<h3>Phase Stereo</h3>

<p>Phase stereo is the least aggressive means of gracefully dropping
resolution from the stereo image; it affects only diffuse imaging.</p>

<p>It's often quoted that the human ear is deaf to signal phase above
about 4kHz; this is nearly true and a passable rule of thumb, but it
can be demonstrated that even an average user can tell the difference
between high frequency in-phase and out-of-phase noise. Obviously
then, the statement is not entirely true. However, it's also the case
that one must resort to nearly such an extreme demonstration before
finding the counterexample.</p>

<p>'Phase stereo' is simply a more aggressive quantization of the polar
angle vector; above 4kHz it's generally quite safe to quantize noise
and noisy elements to only a handful of allowed phases, or to thin the
phase with respect to the magnitude. The phases of high amplitude
pure tones may or may not be preserved more carefully (they are
relatively rare and L/R tend to be in phase, so there is generally
little reason not to spend a few more bits on them)</p>

<h4>example: eight phase stereo</h4>

<p>Vorbis may implement phase stereo coupling by preserving the entirety
of the magnitude vector (essential to fine amplitude and energy
resolution overall) and quantizing the angle vector to one of only
four possible values. Given that the magnitude vector may be positive
or negative, this results in left and right phase having eight
possible permutation, thus 'eight phase stereo':</p>

<p><img src="eightphase.png" alt="eight phase"/></p>

<p>Left and right may be in phase (positive or negative), the most common
case by far, or out of phase by 90 or 180 degrees.</p>

<h4>example: four phase stereo</h4>

<p>Similarly, four phase stereo takes the quantization one step further;
it allows only in-phase and 180 degree out-out-phase signals:</p>

<p><img src="fourphase.png" alt="four phase"/></p>

<h3>example: point stereo</h3>

<p>Point stereo eliminates the possibility of out-of-phase signal
entirely. Any diffuse quality to a sound source tends to collapse
inward to a point somewhere within the stereo image. A practical
example would be balanced reverberations within a large, live space;
normally the sound is diffuse and soft, giving a sonic impression of
volume. In point-stereo, the reverberations would still exist, but
sound fairly firmly centered within the image (assuming the
reverberation was centered overall; if the reverberation is stronger
to the left, then the point of localization in point stereo would be
to the left). This effect is most noticeable at low and mid
frequencies and using headphones (which grant perfect stereo
separation). Point stereo is is a graceful but generally easy to
detect degradation to the sound quality and is thus used in frequency
ranges where it is least noticeable.</p>

<h3>Mixed Stereo</h3>

<p>Mixed stereo is the simultaneous use of more than one of the above
stereo encoding models, generally using more aggressive modes in
higher frequencies, lower amplitudes or 'nearly' in-phase sound.</p>

<p>It is also the case that near-DC frequencies should be encoded using
lossless coupling to avoid frame blocking artifacts.</p>

<h3>Vorbis Stereo Modes</h3>

<p>Vorbis, as of 1.0, uses lossless stereo and a number of mixed modes
constructed out of lossless and point stereo. Phase stereo was used
in the rc2 encoder, but is not currently used for simplicity's sake. It
will likely be re-added to the stereo model in the future.</p>

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