The easiest way to understand the MS matrix is by studying these two simple calculations:
Left channel=M + S
Right channel=M - S
The MS system will give directional information since the bi-directional microphone captures two directions and only puts out one signal with mutual opposite polarity. When sound approaches the MS arrray from the right, it enters the bi-directional Side mic in its phase inversed side. The matrix calculation says that Right output is M minus S. If you subtract a phase inversed signal from something it is like adding it. Mathematically speaking minus minus is plus. So, voila, sound approaching the MS array from the right side creates a signal in the right output of the matrix due to a simple subtraction.
For the left side, the sound meets the Side mic in its in-phase lobe and is simply added to the Mid signal.
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