EM and MAP Methods for Joint Path Delay and Complex Gain Estimation of a Slowly Varying Fading Channel for CPM Signals

This paper addresses the joint path delay and time-varying complex gain estimation for continuous phase modulation (CPM), over a time-selective slowly varying Rayleigh flat fading channel. We propose two estimation methods: an expectation-maximization (EM) algorithm for path delay estimation in a Kalman framework, and a Maximum a Posteriori (MAP) method for joint path delay and complex gain estimation. The time-varying complex gains are modeled by a first order autoregressive (AR) process. Such a modeling yields to the representation of the problem by a dynamic Bayesian system in a state-space model, which allows the use of the EM algorithm in the context of unobserved data for obtaining an estimate of the path delay, coupled with a Kalman smoother for the complex gain estimation. Also, the direct joint estimation using a MAP estimator over the observation block is given. We derive analytically a closed-form expression of the modified hybrid Cramér-Rao bound (MHCRB) for the path delay and complex gain estimation problem. Some numerical examples are presented to illustrate the performance of the proposed methods compared to both the conventional generalized correlation method and the MHCRB. Keywords-Continuous phase modulation (CPM), EM algorithm, hybrid Cram\'er-Rao bound, path delay estimation, maximum-likelihood (ML) estimation, MAP estimator, Kalman smoother filter, fading channels.