# 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