The Structure of Efficient Demodulators for Multidimensional Phase Modulated Signals

The structure of optimum receivers for multilink diversity reception of analog, nonlinear modulated signals is outlined. The receivers are optimum in the sense that they are maximum a posteriori probability estimates (an unconditional maximum-likelihood estimate) of the signal over a finite time interval. For the single link case, Youla has developed a pair of integral equations which specify the optimum estimator. For the multiple-signal, multiple-link case, the corresponding two matrix integral equations have been developed by Thomas and Wong. In both cases, the authors treat the AM case in detail and indicate that the equations for the PM case suggest mechanization schemes. This paper is, in essence, a study of the multidimensional implementation schemes and some of their ramifications. Several useful modulation schemes are discussed. For the multilink PM case with additive white Gaussian noise, we show that the optimum demodulator reduces to a generalized maximal-ratio combiner cascaded with an optimum one-dimensional demodulator. The optimum one-dimensional demodulator contains a nonrealizable filter in the feedback loop. By replacing the loop with an iterative structure the nonrealizability is eliminated. Next, a common multiple-carrier modulation scheme
PM_{n}/PM
is discussed. The optimum demodulators are shown. Once again, these loops contain nonrealizable filters. Here it is also possible to replace the feedback structure with an iterative structure to eliminate the nonrealizability. Alternately, one can use realizable filters and have a suboptimal demodulation scheme. In currently used demodulation schemes, the threshold of the main carrier is a primary system limitation. A heuristic explanation of the apparent threshold improvement of these suboptimum demodulators is given. An extension of the analytical development from no-memory modulation schemes such as PM to integral modulation schemes such as FM requires some justification. However, in most modulation schemes, the modulation waveforms are passed through pre-emphasis networks before being modulated onto the carrier. For this case, we show that our estimators are essentially optimum for the multilink FM, and FM,/FM case.