Optimal HMM

In this paper conditional hidden Markov model (HMM) lters and conditional Kalman lters (KF) are coupled together to improve demodulation of di erential encoded signals in noisy fading channels. We present an indicator matrix representation for di erential encoded signals and the optimal HMM lter for demodulation. The lter requires O(N) calculations per time iteration, where N is the number of message symbols. Decision feedback equalisation is investigated via coupling the optimal HMM lter for estimating the message, conditioned on estimates of the channel parameters, and a KF for estimating the channel states, conditioned on soft infomation message estimates. Here the soft message estimates are conditional mean estimates. The conditional KF is an adaptive channel estimation scheme based on modelling the phase and amplitude variations as a stochastic linear system. The key to our coupled HMM-KF lter approach is that the HMM lter provides immediate soft information message estimates and our Kalman Filter exploits the idempotent nature of Markov chains. The particular di erential encoding scheme examined in this paper is di erential phase shift keying (DPSK). However, the techniques developed can be extended to other forms of di erential modulation. The channel model we use allows for multiplicative channel distortions and additive white Gaussian noise. Simulation studies are also presented.