Linear time-invariant 4-2-4 matrix recording systems are defined mathematically. Mono and stereo compatibility objectives are states; the class of compatible encoders is then synthesized that meets these objectives. To each of these encoders there corresponds a decoder that yields the smallest possible mean-square error in its quadraphonic outputs. An explicity formula for this minimum-error decoder is derived, which shows that the minimum-error decoder is time varying (uses -logic- to track the correlations of the quadraphonic signals). The optimal time-invariant (no logic) decoder, which minimizes the mean-square error for worst case inputs, is then synthesized, and several of its most important properties are discussed. The conditions for optimum time-invariant decoding are then combined with the conditions for compatible encoding; this yields the class of optimized compatible recording systems, which is termed the SQ family. Two well-known members of this family are basic SQ and forward-oriented SQ.
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