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<h1><font color=#000070>
Stereo Channel Coupling in the Vorbis CODEC
</font></h1>

<em>Last update to this document: June 27, 2001</em><br> 

<h2>Abstract</h2> 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.

<h2>Terminology</h2> Terminology as used in this document is based on
common terminology associated with contemporary CODECs such as MPEG I
audio layer 3 (mp3).  However, some differences in terminology are
useful in the context of Vorbis as Vorbis functions somewhat
differently than most current formats.  For clarity, a few terms are
defined beforehand here, and others will be defined where they first
appear in context.<p>

<h3>Subjective and Objective</h3>

<em>Objective</em> fidelity is a measure, based on a computable,
mechanical metric, of how carefully an output matches an input.  For
example, a stereo amplifier may claim to introduce less that .01%
total harmonic distortion when amplifying an input signal; this claim
is easy to verify given proper equipment, and any number of testers are
likely to arrive at the same, exact results.  One need not listen to
the equipment to make this measurement.<p>

However, given two amplifiers with identical, verifiable objective
specifications, listeners may strongly prefer the sound quality of one
over the other.  This is actually the case in the decades old debate
[some would say jihad] among audiophiles involving vacuum tube versus
solid state amplifiers.  There are people who can tell the difference,
and strongly prefer one over the other despite seemingly identical,
measurable quality.  This preference is <em>subjective</em> and
difficult to measure but nonetheless real.

Individual elements of subjective differences often can be qualified,
but overall subjective quality generally is not measurable.  Different
observers are likely to disagree on the exact results of a subjective
test as each observer's perspective differs.  When measuring
subjective qualities, the best one can hope for is average, empirical
results that show statistical significance across a group.<p>

Perceptual codecs are most concerned with subjective, not objective,
quality.  This is why evaluating a perceptual codec via distortion
measures and sonograms alone is useless; these objective measures may
provide insight into the quality or functioning of a codec, but cannot
answer the much squishier subjective question, "Does it sound
good?". The tube amplifier example is perhaps not the best as very few
people can hear, or care to hear, the minute differences between tubes
and transistors, whereas the subjective differences in perceptual
codecs tend to be quite large even when objective differences are
not.<p>

<h3>Fidelity, Artifacts and Differences</h3> Audio <em>artifacts</em>
and loss of fidelity or more simply put, audio <em>differences</em>
are not the same thing.<p>

A loss of fidelity implies differences between the perceived input and
output signal; it does not necessarily imply that the differences in
output are displeasing or that the output sounds poor (although this
is often the case).  Tube amplifiers are <em>not</em> higher fidelity
than modern solid state and digital systems.  They simply produce a
form of distortion and coloring that is either unnoticeable or actually
pleasing to many ears.<p>

As compared to an original signal using hard metrics, all perceptual
codecs [ASPEC, ATRAC, MP3, WMA, AAC, TwinVQ, AC3 and Vorbis included]
lose objective fidelity in order to reduce bitrate.  This is fact. The
idea is to lose fidelity in ways that cannot be perceived.  However,
most current streaming applications demand bitrates lower than what
can be achieved by sacrificing only objective fidelity; this is also
fact, despite whatever various company press releases might claim.
Subjective fidelity eventually must suffer in one way or another.<p>

The goal is to choose the best possible tradeoff such that the
fidelity loss is graceful and not obviously noticeable.  Most listeners
of FM radio do not realize how much lower fidelity that medium is as
compared to compact discs or DAT.  However, when compared directly to
source material, the difference is obvious.  A cassette tape is lower
fidelity still, and yet the degredation, relatively speaking, is
graceful and generally easy not to notice.  Compare this graceful loss
of quality to an average 44.1kHz stereo mp3 encoded at 80 or 96kbps.
The mp3 might actually be higher objective fidelity but subjectively
sounds much worse.<p>

Thus, when a CODEC <em>must</em> sacrifice subjective quality in order
to satisfy a user's requirements, the result should be a
<em>difference</em> that is generally either difficult to notice
without comparison, or easy to ignore.  An <em>artifact</em>, on the
other hand, is an element introduced into the output that is
immediately noticeable, obviously foreign, and undesired.  The famous
'underwater' or 'twinkling' effect synonymous with low bitrate (or
poorly encoded) mp3 is an example of an <em>artifact</em>.  This
working definition differs slightly from common usage, but the coined
distinction between differences and artifacts is useful for our
discussion.<p>

The goal, when it is absolutely necessary to sacrifice subjective
fidelity, is obviously to strive for differences and not artifacts.
The vast majority of CODECs today fail at this task miserably,
predictably, and regularly in one way or another.  Avoiding such
failures when it is necessary to sacrifice subjective quality is a
fundamental design objective of Vorbis and that objective is reflected
in Vorbis's channel coupling design.<p>

<h2>Mechanisms</h2>

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 it's analogous
uncoupled multichannel modes.

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 #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>, 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 model without requiring any significant complexity increase
over the combinatorically simpler mid/side joint stereo of mp3 and
other current audio codecs.<p>

Channel interleaving may be applied directly to more than a single
channel and polar mapping is hierarchical such that polar coupling may be
extrapolated to an arbitrary number of channels and is not restricted
to only stereo, quadriphonics, ambisonics or 5.1 surround.  However,
the scope of this document restricts itself to the stereo coupling
case.<p>

<h3>Square Polar Mapping</h3>

<h4>maximal correlation</h4>
 
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>

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>

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>

The first advantage of a polar representation is that it effectively
seperates 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>

*<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>

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> 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>

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>

Vorbis uses a mapping that preserves the most useful qualities of
polar representation, relies only on addition/subtraction, 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>

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>

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>

<img src="squarepolar.png">

<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:

<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>

...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>

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 substantial
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>

Entropy coding the results, then, further benefits from the entropy
model being able to compress magnitude and angle simultaneously.  For
this reason, Vorbis implements residuebackend #2 which preinterleaves
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>

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>

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>Polar mapping separates the stereo image
into point and diffuse components which may be analyzed and handled
differently.
</ul>

<h2>Stereo Models</h2>

<h3>Dual Stereo</h3>

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>

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 offers this
mode.<p>

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

<h3>Phase Stereo</h3>

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

It's often quoted that the human ear is nearly entirely 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
demostration before finding the counterexample.<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.  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>eight phase stereo</h4>

Vorbis implements 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>

<img src="eightphase.png"><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>four phase stereo</h4>

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

<img src="fourphase.png"><p>

<h3>Point Stereo</h3>

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 degrdation to the sound quality and is thus used in frequency
ranges where it is least noticeable.<p>

<h3>Mixed Stereo</h3>

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>

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>

Vorbis, for the most part, uses lossless stereo and a number of mixed
modes constructed out of the above models.  As of the current pre-1.0
testing version of the encoder, oggenc supports the following modes.
Oggenc's default choice varies by bitrate and each mode is selectable
by the user:<p>

<dl>
<dt>dual stereo
<dd>uncoupled stereo encoding<p>

<dt>lossless stereo
<dd>lossless stereo coupling; produces exactly equivalent output to dual stereo<p>

<dt>eight phase stereo
<dd>a mixed mode combining lossless stereo for frequencies to approximately 4 kHz (and all strong pure tones) and eight phase stereo above<p>

<dt>aggressive eight phase stereo
<dd>a mixed mode combining lossless stereo for frequencies to approximately 2 kHz (and for all strong pure tones) and eight phase stereo above<p>

<dt>eight/four phase stereo <dd>A mixed mode combining lossless stereo
for bass, eight phase stereo for noisy content and lossless stereo for
tones to approximately 4kHz and four phase stereo above 4kHz.<p>

<dt>eight phase/point stereo <dd>A mixed mode combining lossless stereo
for bass, eight phase stereo for noisy content and lossless stereo for
tones to approximately 4kHz and point stereo above 4kHz.<p>

<dt>aggressive eight phase/point stereo
<dd>A mixed mode combining lossless stereo
for bass, eight phase stereo to approximately 2kHz and point stereo above 2kHz.<p>

<dt>point stereo
<dd>A mixed mode combining lossless stereo to approximately 4kHz and point stereo above 4kHz.<p>

<dt>aggressive point stereo
<dd>A mixed mode combining lossless stereo to approximately 1-2kHz and point stereo above.<p>

</dl>

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<p>

Ogg Vorbis is the first Ogg audio CODEC.  Anyone may
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Xiphophorus and the Ogg project (xiph.org) reserve the right to set
the Ogg/Vorbis specification and certify specification compliance.<p>

Xiphophorus's Vorbis software CODEC implementation is distributed
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distributing independent implementations of Vorbis software under
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