Ogg Vorbis stereo-specific channel coupling discussion

Abstract

Mechanisms

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 channel interleaving via residue backend type 2, and the second is square polar mapping. 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 lossless stereo image coupling [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.

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.

Square Polar Mapping

maximal correlation

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.

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.

point and diffuse imaging

*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.

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.

controlling bit leakage and symbol crosstalk

eliminating trigonometry and rounding

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.

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):

      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;
        }
    }

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:

 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

Channel interleaving

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.

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.

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:

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

Stereo Models

Dual Stereo

Lossless Stereo

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.

Phase Stereo

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.

'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)

example: eight phase stereo

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.

example: four phase stereo

example: point stereo

Mixed Stereo

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

Vorbis Stereo Modes