An exact recursive filter for quadrature amplitude modulation dynamics

In certain models for communications signals, such as Quadrature Amplitude Modulation (QAM), circular stochastic processes arise quite naturally. However, much of the literature concerning estimation for communications processes, such as QAM signals, is based upon Cartesian coordinate representations and approximated dynamics, subsequently amenable to the Extended Kalman Filter (EKF). This common approach, using EKFs, is well known to be unstable, for example, in demodulating a QAM signal, one must first estimate timing information. If this information is uncertain, then EKFs can fail profoundly. In this article we compute a general recursive filter for the QAM family of communications signals. This filter is exact and can be configured for any of the standard classes of circular distributions, such as, for example, the von Mises distribution of the wrapped normal distribution. Our filter is computed by using the techniques of reference probability resulting in a recursion in terms of un-normalised probability densities.