Optimum Receiver structures for Phase-Multiplexed Modulations

The problem of deriving "optimum" receiver structures for synchronization of BPSK, QPSK, and unbalanced QPSK (UQPSK) modulations has been addressed by many authors starting with the wellknown maximum a posteriori (MAP) estimation technique as motivation. In general, the closed loop structures which result from application of this theory are impractical because of the difficulty of implementing the hyperbolic tangent nonlinearity which is suggested by using the gradient of the likelihood function as an error control signal in the loop. To arrive at practical realizations, one ordinarily approximates this nonlinearity with simpler, more easily implementable functions. This paper reexamines some of these approximations and the optimum structures which result. Using a slightly more sophisticated approximation of the nonlinearity, new and interesting structures are presented which overcome some of the deficiencies posed by MAP receiver structures for QPSK and UQPSK suggested by authors in the past. The equivalence of the QPSK structure with the well-known quadriphase Costas loop is discussed. Furthermore, it is argued that the new UQPSK structure allows carrier reconstruction at all ratios of data rates and powers in the two channels even in the limit of balanced QPSK.