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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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