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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html>
+<head>
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+<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-15"/>
+<title>Ogg Vorbis Documentation</title>
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+<body>
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+<div id="xiphlogo">
+ <a href="http://www.xiph.org/"><img src="fish_xiph_org.png" alt="Fish Logo and Xiph.Org"/></a>
+</div>
+
+<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>
+
+<div id="copyright">
+ The Xiph Fish Logo is a
+ trademark (&trade;) of Xiph.Org.<br/>
+
+ These pages &copy; 1994 - 2005 Xiph.Org. All rights reserved.
+</div>
+
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+
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