Exploring Joint Stereo Encoding, with Non-Negative Integers

A concept can exist, by which a stereo signal consists of a left channel L and a right channel R, and by which it gets translated in the time-domain, into sample streams M and S, such that M = L+R and S = L-R. In this case, L and R can be reconstructed as


L = (M+S) / 2
R = (M-S) / 2



This seems trivial. but a more specific context for this set of equations could be, the variables could be frequency coefficients, and

L >= 0
R >= 0
M >= 0
L, R, M, S are all Integers.


Because the equations for L and R are truly the inverse, of the definition of M and S, it would follow that in order for them to be true, (M+S) and (M-S) must also be even integers.

If we were encoding the integers M and S in a variable-length scheme, then the bit-length of S has already been compromised by 1 bit, because somewhere we need to state its sign. Yet, we might want to be certain, that the encoding of (M,S) is not longer than that of (L,R).

And so an implication of this which we might want to take advantage of, is knowing that


If M is Even, S Must Also Be Even.
If M is Odd,  S Must Also Be Odd.



And so one idea that might be helpful, would be to define a derived value S’ , such that


S' = S / 2, Rounded Down,



meaning, rounded to the More Negative, If S was Odd.

We could then store (M,S’). The length of S’ is the length of S reduced by at least one bit. Then, when the time comes to decode the stream, we could compute


IF M Is Even, S = (S' * 2)

and,

IF M Is Odd,  S = (S' * 2) + 1



Thereby not wasting any bits. And, depending on what type of variable-length encoding was being used, shortening the length of the integer S’ by 1 bit, may in fact shorten its encoding by more than 1 bit.

Dirk

Not Being Sure about the Sign Bit

One format in which MP3 can encode stereo, is in the form of “Joint Stereo”. In this form, the signal is sent as a sum-channel, and a difference-channel. The left and right channels can be reconstructed from them, just as easily as a sum and a difference, could be computed. The reason this is done, is to save on the bit-rate of the difference channel, along the argument that human stereo-directionality is more limited in frequencies, than straightforward hearing is.

But this is also one example, in which the difference channel needs to have a defined sign, as either being positive when left is more so, or being more positive when right is more so.

And so one way in which this could be encoded, if the decision was made to include doing so into the standard, could be to encode an additional sign-bit into each non-zero frequency coefficient of the difference channel. But doing so would also affect the overall bit-rate of the signal enough, that this deters professionals from doing it. Also, the argument is made that lower bit-rates can lead to higher sound quality, because if they want, users can increase the bit-rate anyway, resulting in greater definition of the more audible components of the sound.

And so with Joint Stereo, a feature built-in to MP3 is, “Side-Bits”. These side-bits are included if the mode is enabled, and declare as header information, how the signal should be panned, either to the left or the right, as well as what the sign of the difference-channel might be. ( :1 )

Well there is an implementation detail about the side-bits, which I do not specifically know: Whether they are stored only once per frame, or once per frequency sub-band, since the compression schemes do divide the spectrum into sub-bands already. Thus, if the side-bits were encoded for each sub-band, for each frame, then a good compromise could be reached, in terms of how much data should be spent on that.

The only coefficient which I imagine would have a sign-bit to itself each time, would be the zero coefficient, that corresponds to DC, but that also corresponds to F = Sampling Interval / 2.

This question becomes more relevant for surround-sound encoding. If, rather than using the Pro Logic method, some other scheme was decided on, for defining a ‘Back-Front’ channel, then it would suddenly become critical, that this channel have correct sign information. And then it would also become possible for this channel to correlate negatively with frequency components that also belong to the stereo channels. Hence, a more familiar method of designing the servos of Pro Logic II would be effective again, than what would be needed, if none of the cosine transforms could produce negative coefficients. ( :2 )

Dirk

P.S. When we listen to accurately-reproduced signals on stereo headphones, If the signals are perfectly in-phase, humans tend to hear them as if they came from inside our own heads. If they are 180 degrees out-of-phase, we tend to hear them as coming from a non-specific place outside our heads.

I have had a personal friend complain, that when he listens to Classical Music via MP3 Files, he cannot ascertain the direction which sounds are coming from – i.e. the positions of the instruments in the orchestra, but can hear panning and this in-versus-out condition. Listening to Classical Music via MP3 is a very bad idea.

What this tells me is that my friend has very good hearing, which is tuned to the needs of Classical Music.

The reason he hears some of the sound as being in the out-of-phase position, may well just be due to the fact that Joint Stereo was being selected by the encoder, and that certain frequency components in the difference-channel, did not have substantial counterparts in the sum-channel. Mathematically, this results in a correlation of zero between the encoded channels, but in a correlation of -1 between the reconstructed left and right…

Subjectively, I would say that I have observed better sound quality in this regard, when using OGG Compression, and at a higher bit-rate. I found that “Enya” required 192kbps, while “Strauss” did not sound good until I had reached 256kbps.

But I do not know objectively, what it is in the OGG Files, that gives me this better experience. I do not have the precision hearing, which said friend has. I have used FLAC to encode some “Beethoven” and “Schubert”, but mainly just in order to archive their music without any loss in information at all, and not as a testament, to the listening experience really being that much better.

1: ) In the case of Joint Stereo with MP3, what I would expect, is that the ‘pan-number’ will also direct the decoder to set the polarity of the difference-channel, to be positive with whichever side the sum-channel is being panned-towards more strongly. And I expect this to happen, regardless of what the phase information was when encoding.

If there was explicit sign information here, such information would first also have had to be measured, when encoding, as relative to the phase-position of the sum-channel. Since phase-information is generally relative. And I do not hear speak, of correlation information being collected first when encoding, between the stereo and difference channels.

2: ) This subject peaked my interest into how OGG Compression deals with multi-channel sound. I did an experiment using “Audacity”, in which I prepared a 6-channel project, chose to export it in a custom channel-configuration, and then chose different settings in the channel-meta-data window.

While AC3 was ‘limited’ to allowing a maximum of 7 channels, OGG allows a compressed stream with up to 32 channels. But, I seem to have observed that when compressing more than 2 channels, OGG forgoes even joint stereo optimization, instead only compressing each channel individually. This seems to follow from the observation, that If I mix channels 3 and 4, assigned to a hypothetical front-center and LFE, I should have turned those two into a monaural signal repeated once. But doing so does not improve the OGG File size.

There was a 3m45s stream, which took 5.2MB as a 6-channel AC3. The same stream takes up 18.1MB as a 6-channel OGG. And these bit-rates result from choosing a rate of 192kbps for the AC3 File, while choosing ‘Quality Level 8/10′ for the OGG.

I think that one reason for the big difference in bit-rates, is the fact that my stream consisted of a stereo signal originally, of which there were merely 3 copies. The AC3 File takes advantage of the correlations to compress, while OGG is not as able to do so.

Further, I read somewhere that OGG takes the remarkable approach, to convert the stereo into joint stereo, after quantization (in the frequency domain), while MP3 does so before quantization. This makes the conversion which OGG performs, of a signal into stereo, a lossless process, and also seems to imply, encoding one sign bit with each coefficient of the difference-channel. Any advantage OGG gives to the bit-rate would need to stem, from the majority of low-amplitude coefficients in the difference-channel, as well as from limiting its frequencies.

By contrast, this would seem to suggest, that MP3 will compute an ‘FFT’ of each channel, also in order to determine the side-bits, after which it will compute a sum and a difference channel in the time-domain, and then compute the ‘DCT’ of each…