Finite-state Markov modeling of fading channels - a survey of principles and applications <s> ANALYTICAL PERFORMANCE EVALUATION <s> The error structure on communication channels used for data transmission may be so complex as to preclude the feasibility of accurately predicting the performance of given codes when employed on these channels. Use of an approximate error rate as an estimate of performance allows the complex statistics of errors to be reduced to a manageable table of parameters and used in an economical evaluation of large collections of error detecting codes. Exemplary evaluations of error detecting codes on the switched telephone network are included in this paper. On channels which may be represented by Gilbert's model of a burst-noise channel, the probabilities of error or of retransmission may be calculated without approximations for both error correcting and error detecting codes <s> BIB001 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> ANALYTICAL PERFORMANCE EVALUATION <s> The throughput performance of a wireless media access protocol taking into account the effect of correlated channel fading, capture, and propagation delay is analyzed. For efficient access on the uplink (mobile-to-base-station link), the protocol makes use of the uplink channel status information which is conveyed to the mobiles through a busy/idle flag broadcast on the downlink (base-station-to-mobile link). A first-order Markov model is used to describe the correlation in the packet success/failure process on a Rayleigh-fading channel. The analytical results obtained through the first-order Markov approximation of the channel are compared to those obtained from an independent and identically distributed (i.i.d.) channel model. The Markovian-fading channel model is shown to provide better performance results than the i.i.d. channel model. Simulations show that a first-order Markov approximation of the Rayleigh-fading process is quite accurate. An enhanced version of the access protocol to take advantage of the memory in the fading channel behavior is proposed and analyzed. The effect of retransmission of erroneous data packets and propagation delay on the throughput is also analyzed. It is shown that the access protocol with an error detect (ED) feature is efficient in slow fading (e.g., pedestrian user speeds), whereas a retransmission protocol is more efficient in fast fading (e.g., vehicular user speeds). <s> BIB002 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> ANALYTICAL PERFORMANCE EVALUATION <s> This paper investigates the properties of a method for obtaining Markov models of unspecified order to be applied to narrow-band fading channels with additive white Gaussian noise. The models are obtained by applying the context tree pruning algorithm to experimental or simulated sequences. Fading environments are identified in which the extension from first-order to higher order models is justified. The paper presents, as examples, the evaluation of the covariance function and the packet error distribution. <s> BIB003 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> ANALYTICAL PERFORMANCE EVALUATION <s> Previous models for the received signal amplitude of the flat-fading channel that use first-order finite-state Markov chains are examined. The stochastic properties of a proposed first-order model based on these models are examined. The limitations of using an information theoretic metric, which is sometimes used to justify a first-order Markov chain as a sufficient model for very slowly fading channels, are discussed. A simple method of qualitatively comparing autocorrelation functions is instead proposed. The usefulness of the first-order Markov chain in representing the flat-fading channel is examined by looking at two specific problems in wireless system applications that represent two disparate cases. The first case involves analysis over a short duration of time, relative to the inverse of the normalized Doppler frequency, while the second involves analysis over a long duration of time. Contrary to previous reports, the results indicate that first-order Markov chains are not generally suitable for very slowly fading channels. Rather, first-order Markov chains can be suitable for very slowly fading applications, which require analysis over only a short duration of time. <s> BIB004 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> ANALYTICAL PERFORMANCE EVALUATION <s> A finite-state Markov model is fitted to the mobile propagation channel by the use of contingency tables. Narrow-band Rayleigh and Ricean fading are considered in detail, but the techniques extend to other types of fading. Several criteria may indicate that a first-order finite-state Markov model sufficiently characterizes the channel behavior. The parameters of the model are obtained both from theoretical analysis and simulation. Functional dependence between the transition probabilities and the steady-state probabilities is found in the slow fading case. <s> BIB005 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> ANALYTICAL PERFORMANCE EVALUATION <s> Stochastic properties of the binary channel that describe the successes and failures of the transmission of a modulated signal over a time-correlated flat-fading channel are considered for investigation. This analysis is employed to develop Kth-order Markov models for such a burst channel. The order of the Markov model that generates accurate analytical models is estimated for a broad range of fading environments. The parameterization and accuracy of an important class of hidden Markov models, known as the Gilbert-Elliott channel (GEC), are also investigated. Fading rates are identified in which the Kth-order Markov model and the GEC model approximate the fading channel with similar accuracy. The latter model is useful for approximating slowly fading processes, since it provides a more compact parameterization. <s> BIB006 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> ANALYTICAL PERFORMANCE EVALUATION <s> This paper shows the existence of the optimal training, in terms of achievable mutual information rate, for an output feedback implicit estimator for finite-state Markov communication channels. A proper quantification of source redundancy information, implicitly used for channel estimation, is performed. This enables an optimal training rate to be determined as a tradeoff between input signal entropy rate reduction (source redundancy) and channel process entropy rate reduction (channel estimation). The maximal mutual information rate, assuming the optimal implicit training and the presence of channel noise, is shown to be strictly below the ergodic channel information capacity. It is also shown that this capacity penalty, caused by noisy time-varying channel process estimation, vanishes only if the channel process is known or memoryless (channel estimation cannot improve system performance). <s> BIB007
Traditionally, FSMC models have been used to evaluate the error rate performance of coded communication systems through channels with memory. The research activity in this area can be traced back to the early work by Elliott in 1963 BIB001 , in which he studied and compared error correcting capability of various channel codes used in burst-noise telephone circuits. One novel aspect of Elliott's work was that using the mathematical tractability of the GEC model, he provided closed form and recursive expressions for PED, which is defined as the probability of m errors in a coded block of length n and is denoted by P(m, n) . In BIB001 , P (m, n) is expressed in terms of two quantities: 1) the probability of m errors in a coded block of length n, given that the channel is in the good state at the start of block, which is denoted by G (m, n) and 2) the probability of m errors in a coded block of length n, given that the channel is in the bad state at the start of block, which is denoted by B (m, n) . Quantities G (m, n) and B (m, n) are, in turn, recursively related to PED in good and bad states in a smaller coded block of size n − 1: , and B (m − 1, n − 1). The recursive expressions for PED in BIB001 can be readily extended to FSMC models with more number of states and the expressions are given in BIB003 . The analytical solution for the PED enables system designers to evaluate PER in a block-coded system that is capable of correcting t errors in a block of size n, which is given as PER and PED for the FSMC modeling of fading channels have been investigated in BIB004 - BIB006 . FSMC models also enable analytical study of packet throughput in wireless data protocols operating through correlated time-varying fading channels. In BIB002 , a first-order Markov model is proposed for the block sucess/failure, which is based on two-level quantization of the square of the continuous-valued channel amplitude a 2 . In particular, the block or packet success/failure is a binary random variable β, where β = 0 and β = 1 represent packet success and failure, respectively, and are related to a 2 as where 1/F is coined as the fading margin. It is assumed that block success/failure β is a first-order Markov process that only depends on β −1 . Using the above model for block success/failure, the authors in BIB002 provided closed-form expressions for the throughput of proposed data protocols. The authors also indicate that by taking channel memory into account in the design of data protocols, better throughput performance can be obtained compared to using an independent (memoryless) channel model. In BIB005 , the authors provide some theoretical insight about the applicability of the twolevel quantized model in BIB007 to represent block success/failure. FSMC models also enable closed-form analysis of the fade duration distribution in the TV-FFC BIB004 ,  . Fade duration is the period of time that the power of a fading process spends below the fading margin 1/F. Fade duration distribution is of interest in the coding/decoding design stages, but does not have a closed-form solution in Clarke's model and requires simulation-based analysis of fade duration distribution. Therefore, an FSMC model can be used to provide a more efficient way for system analysis and design.