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The authors compare different finite-state Markov channel (FSMC) models used to approximate the Rayleigh fading channel. The criterion used to compare the different Markov models is the error performance of corresponding FSMC receivers performing joint maximum a posteriori (MAP) sequence detection and channel estimation, where the sufficient statistics are obtained from the Jakes-Clarke (1993) fading channel. To put the results in perspective, the results of these Markov receivers are compared with those of a Kalman filtering receiver based on an ARMA model of the Jakes-Clarke fading channel. There is a moderate improvement in Markov receiver performance when based on a second-order model compared to a first-order model, and the number of Markov states is normalised by model order. This does not justify a second-order model, however, as the complexity of implementing a Markov receiver increases exponentially with model order. Furthermore, the error performance floor of a first-order Markov receiver increases linearly with the number of Markov states. Based on the performance of Markov receivers, it is concluded that a first-order Markov model is sufficient for representing the memory of the fading channel.