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Theo, M/S is simply another way for two signals to "represent" an X/Y
(coincident) stereo recording--so as Scott says, it is another form of
intensity-based stereo.
If you take any X/Y recording, you can derive a signal which represents
the difference between its left and right channels, and quite easily
you can also obtain the sum of those two channels. Because there are no
significant arrival time differences between left and right channels in
a coincident stereo recording, the sum should give you a very
listenable mono signal with no significant phase cancellation in the
direct sound. The difference channel on the other hand won't be
directly listenable in the usual sense; it will have an inferior
balance of direct to reflected sound, and the phase conditions within
the signal will be quite odd.
You could then (I'll explain the motive in a moment; for now let's just
focus on the technical feasibility) transmit or record the signals in
that form, and recreate the original stereo left and right signals in
playback, again by deriving the difference between the two signals that
you have, and also (in a separate channel) deriving their sum. One
would be left; the other would be right. If there are no losses in the
system and the gains remain unity all the way through, the original
left and right stereo signals should come out sounding like the
originals.
(That is, in fact, how stereo FM works, so there's nothing really
radical or new about any of this.)
Now as to the benefit of all these contortions: In playback you can
control how much of the difference signal you use when reconstructing
the left and right stereo signals; this gives you control over the
apparent width of the stereo image, so that you can set it to be
narrow, realistic, wide, bizarre, or fully mono but the overall sound
of the pickup (the timbre and balance of the instruments or voices)
won't change much. That again is a boon for radio broadcasts (and LP
records, during the first years of stereo) where some of the audience
is/was listening in mono and some in stereo.
You can also apply EQ to just the difference channel, e.g. boosting the
bass to increase the sense of spaciousness in the hall. If the
difference channel is noisy (as it often is in fringe FM radio
reception) you can apply a noise gate to its high frequency region,
etc.; there are many interesting possibilities.
Now to get back to fundamentals, you can greatly simplify the above
system as follows (and this is what is usually done in practice):
Instead of recording with an X/Y pair and taking the sum and difference
to produce an M/S "representation" of the recording, you can _start_
with a special coincident microphone pair, in which one microphone (if
it is directional) faces directly "front and center" (the "mid"
microphone) while the other one, which must be a bidirectional
(figure-8) microphone, faces 90 degrees away from the front/center of
whatever you are trying to record (the "side" microphone). The signals
from those two microphones will then give you your sum and difference
channels directly; you can record them, then use the same playback
setup exactly as you did before, in which you obtain the left and right
stereo channels by deriving the sum and difference of "M" (mid) and "S"
(side) signals.
Or you can matrix (obtain the sum and difference) before recording if
you like, but that commits you to a certain decision as to the amount
of "S" signal in the matrix, and you might want to change your mind
later. (Of course that's supported, too, if you matrix back to M/S,
readjust the M and S gains and matrix yet again back to stereo.)
--best regards
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