Finite-state Markov modeling of fading channels - a survey of principles and applications <s> THE EFFECT OF MARKOV MEMORY ORDER <s> The use of received signal-to-noise ratio (SNR) as the side information in communication systems has been widely accepted especially when the channel quality is time-varying. In many occasions, this side information is treated as the received SNR of the current channel symbol or that of previous symbols. In particular, the finite-state Markov channel (FSMC) provides a mathematically tractable model for time-varying channels and uses only the received SNR of the symbol immediately preceding the current one. The FSMC model is constructed by partitioning the range of the received SNR into a finite number of intervals. Each state of the channel corresponds to one of these intervals. For a first-order Markovian model, given the information of the state immediately preceding the current one, any other previous state should be independent of the current state. Although the experimental measurement confirms the usefulness of the first-order Markovian assumption, one may argue that second or higher-order Markov processes should provide a more accurate model. We answer this question by showing that given the information corresponding to the previous symbol, the amount of uncertainty remaining in the current symbol should be negligible. > <s> BIB001 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> THE EFFECT OF MARKOV MEMORY ORDER <s> In this paper, we study the correlation properties of the fading mobile radio channel. Based on these studies, we model the channel as a one-step Markov process whose transition probabilities are a function of the channel characteristics. Then we present the throughput performance of the Go-Back-N and selective-repeat automatic repeat request (ARQ) protocols with timer control, using the Markov model for both forward and feedback channels. This approximation is found to be very good, as confirmed by simulation results. <s> BIB002 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> THE EFFECT OF MARKOV MEMORY ORDER <s> Fading is modeled by applying context tree pruning (CTP) to the sequence of fading states that results from quantizing a sequence of real-valued fading measurements. There results a family of finite state Markov chains of increasing order that represent the channel with increasing fidelity. A fading state may be the union of chain states. A particular model may be chosen from that family according to an application-specific criterion. The resulting models facilitate analysis and simulation of wireless systems that exploit channel memory. An example criterion compares each model's fading state sojourn time statistics with those of the original sequence. <s> BIB003 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> THE EFFECT OF MARKOV MEMORY ORDER <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> BIB004 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> THE EFFECT OF MARKOV MEMORY ORDER <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> BIB005 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> THE EFFECT OF MARKOV MEMORY ORDER <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> BIB006 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> THE EFFECT OF MARKOV MEMORY ORDER <s> This paper introduces a novel and efficient method based on information theoretic metrics to evaluate the accuracy of the first-order Markov approximation for block errors in transmissions over fading channels. Numerical results show that our approach provides significant improvements in both accuracy and simplicity over previously proposed methods. <s> BIB007 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> THE EFFECT OF MARKOV MEMORY ORDER <s> The performance studies of wireless systems often require a wireless channel model. A common one, for Rayleigh fading environments, is founded upon variations of the finite state Markov channel model (FSMC). This model partitions the received signal to noise ratio (SNR) into a finite number of intervals and represents each interval as a state of a Markov process. Previous studies have elaborated on the accuracy and appropriateness of the model under several assumptions. A key aspect that remains to be studied is the effect of the partitioning scheme. We study the effects of several proposed schemes and investigate the performance of a new scheme that takes into account the average fade duration at various SNR levels. <s> BIB008 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> THE EFFECT OF MARKOV MEMORY ORDER <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> BIB009
The suitable choice of Markov memory order in the FSMC-based system performance analysis and evaluation has been the subject of research since 1996 BIB001 . The first-order Markovian assumption for the TV-FFC amplitude means that given a −1 , a is independent of any other previous amplitude realizations a −2 , . . . . In BIB001 , the authors proposed an information-theoretic measure for validating the first-order Markovian assumption in the FSMC model. The authors stated that when the information in a −2 about a given a −1 is small, then the first-order FSMC model is sufficient for describing the TV-FFC amplitude. In particular, they investigated the ratio between mutual information between a and a −2 given a −1 , I(A ; A −2 |A −1 ), to the mutual information between a and the pair {a −1 , a −2 }, I(A ; A −2 , A −1 ). Based on this criterion, BIB001 concluded that the first-order FSMC model is accurate for TV-FFCs for the considered normalized fading rate of f D T s < 0.002. This criterion was subsequently used in BIB004 , BIB002 or validating the first-order block success/failure for wireless data protocols. This information-theoretic measure was later criticized in BIB005 where it was shown that the mutual information ratio can only have some validity in justifying whether a second-order FSMC model is marginally better than a first-order FSMC model. By means of comparing the ACF, BER, and fade duration distribution, the authors in BIB005 then concluded that first-order FSMC models are only accurate for very slow fading channels and short-duration applications (such as analysis of BER with short block lengths) and inaccurate for long-duration applications (such as analysis of fade duration distribution). An improved version of the information-theoretic criterion that takes into account more memory of the block success/failure was later investigated in BIB007 . By comparing the PED in the first-order and second-order FSMC models with simulation results in the reference Clarke's model, the authors in  verified that it is possible to obtain more accurate evaluation of PED using the secondorder VQ-based FSMC model. By using the CT-based Markov modeling for two-to four-level quantization of the TV-FFC amplitude, the authors in BIB006 and BIB009 The above observations are intuitively justifiable to some extent. When the fading channel is very slow, the most recent realization of the fading channel gain may be enough to statistically represent its history. For example, fading channel amplitudes a , a −1 , a −2 , . . . are expected to be similar. Therefore, we can only keep track of a −1 at time index and drop the rest. Based on this argument, we can only conclude that among FSMC models of various orders, more accurate accounts of the slow fading regime may not be found by increasing the memory order. However, we cannot rule out the existence of completely different types of (non-Markovian) channels that could provide more accurate representations of slow fading channels. The fact that first-order FSMC models are still approximations of slow TV-FFCs can be confirmed by observing the differences between the ACF in the slow TV-FFC in Clarke's model, given in (4), and the ACF in the first-order FSMC model BIB005 , BIB006 , BIB009 , BIB008 . On the other hand, when the fading channel is very fast, fading channel gain changes in an almost uncorrelated manner. For example, a has little dependence on a −1 , a −2 , . . . . Hence, there is almost no memory in the channel. However, in the intermediate regime, the channel changes neither very slowly to be represented by its most recent realization nor very fast to be memoryless. In this case, a longer sequence of previous fading channel gains are required to represent channel's evolution in time. The authors in BIB006 , BIB003 also mention that high-order FSMC models may be more useful in applications related to analyzing packet error behavior or channel statistics (such as the ACF), whereas for wireless protocol evaluation low-order FSMC model may suffice BIB006 . The requirement of having high-order FSMC models for accurate representation of fading channel statistics and packet error behavior was later confirmed by BIB009 for f D T s < 0.1.