We can consider finite-state Markov modeling of fading channels from five distinct viewpoints: history, parameter design, applications/applicability, and information theory. A very brief summary of these topics is provided below. See this feature article for a detailed tutorial/survey.

FSMC History: In late 1950s and early 1960s, Gilbert and Elliott at Bell Labs were modeling burst-noise telephone circuits with a very simple two-state channel model with memory. This simple model allowed them to evaluate channel capacity and error rate performance through bursty wireline telephone circuits. However, it took another 30 years for the so-called Gilbert-Elliott channel (GEC) and its generalized finite-state Markov channel (FSMC) to be applied in the design of second-generation (2G) wireless communication systems.

It was the work by Wang and Moayeri in 1995 that is widely viewed to have revived FSMC modeling of fading channels in mobile radio communications. One of the novelties of their work was to explicitly establish the link between the statistical Clarke’s model for fading channels and the FSMC states. In particular, each FSMC state in their model represented a range of received signal-to-noise ratio (SNR), which in turn determined the error probability in that state. Based on this assumption, Wang and Moayeri provided analytical expressions for states, state transition probabilities, and error probabilities in each state. The original FSMC modeling of fading channels as proposed in or its variations are still widely used in the literature.

FSMC Parameter Design: The main question is: how can we model a fading channel with an FSMC? To answer this question, we must first divide this general question into three more specific questions.

  • STATES: What is the relationship between states in the FSMC model and the TV-FFC gain in the fading model? One of the most common practices since 1995 has been to divide or partition the magnitude of the fading gain into a number of non-overlapping regions. However, FSMC modelling of fading channel phase is also possible.
  • STATE TRANSITION PROBABILITY: What is the relationship between state transition probability in the FSMC model and the TV-FFC channel gain dynamics and statistics? The state transition probability is the probability that the fading partition moves from region k at time index n to region m at time index n+1.
  • CHANNEL OBSERVATION LAW: What is the relationship between channel observation law in each FSMC state and the observation equation in the fading channel? The channel observation law in the kth FSMC state is the probability that we observe channel output y given the channel input x provided that the fading amplitude is in the kth region.

    • FSMC memory order selection is also an important issue in the design of FSMC models for fading channels. In the literature, first-order FSMC modeling of the fading is most commonly used. However, higher order FSMC models are needed for some applications. The main difference between the first-order FSMC model and higher-order FSMC model for the fading channel is that a higher order FSMC model keeps track of the fading channel gain deeper in the past. There are three distinct ways to obtain Mth-order FSMC models for the fading channel:

        1) Cartesian product (CP) method
        2) vector quantization (VQ) method
        3) context-tree pruning (CTP) method.

    Applications: While a wide range of FSMC applications for channels with memory has been proposed and studied in the literature, it is often hard to find a unified and explicit classification of these applications, especially in terms of their practical significance. We can generally identify four main applications for FSMC modeling of channels with memory, as follows:

        1) Modeling channel error bursts for analytical or simulation-based performance evaluation.
        2) Decoding in channels with memory using FSMC models
        3) Mismatched decoding setup
        4) Adaptive transmission

    There is no clear-cut solution on when (under what fading conditions) FSMC models are accurate representations of fading channels. Whether an FSMC model is suitable to represent a fading channel greatly depends on the application.

    Information Theory: Evaluating and analysing the FSMC information capacity still remains a challenging issue and active area of research, despite many recent advances in the field in the last 20 years. One relatively recent and practical advance in the field is the numerical computation of the FSMC capacity with linear computational complexity, which was formulated in 2001.
    One main issue is whether an FSMC model is applicable to analyze the information rates through more complicated channels such as the fading channels that may neither be Markov nor finite state. The applicability of FSMC models for the TV-FFC has been analyzed in the literature from an information-theoretic viewpoint and the effect of channel parameters, such as the normalized fading rate, SNR, Markov memory order M, and the number of states on the resulting information rates is investigated. Both fading amplitude and phase are mapped into FSMC models and the resulting information rates or information rate bounds are analyzed. It is concluded that for binary signalling [such as BPSK or binary frequency shift keying (BFSK)] the capacity is rapidly saturated beyond D = 16 fading amplitude or phase partition regions in the FSMC model. This gives an indication on the number of required FSMC states for the FSMC-based decoding at the receiver, which was discussed as application types two and three in the previous section. In other words, the optimum number of FSMC states for joint channel estimation and decoding is where the information rate stops to increase noticeably by increasing the number of FSMC states.

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