Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <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> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <s> The authors first study the behavior of a finite-state channel where a binary symmetric channel is associated with each state and Markov transitions between states are assumed. Such a channel is referred to as a finite-state Markov channel (FSMC). By partitioning the range of the received signal-to-noise ratio into a finite number of intervals, FSMC models can be constructed for Rayleigh fading channels. A theoretical approach is conducted to show the usefulness of FSMCs compared to that of two-state Gilbert-Elliott channels. The crossover probabilities of the binary symmetric channels associated with its states are calculated. The authors use the second-order statistics of the received SNR to approximate the Markov transition probabilities. The validity and accuracy of the model are confirmed by the state equilibrium equations and computer simulation. > <s> BIB002 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <s> We form a finite-state Markov channel model to represent Rayleigh fading channels. We develop and analyze a methodology to partition the received signal-to-noise ratio (SNR) into a finite number of states according to the time duration of each state. Each state corresponds to a different channel quality indicated by the bit-error rate (BER). The number of states and SNR partitions are determined by the fading speed of the channel. Computer simulations are performed to verify the accuracy of the model. <s> BIB003 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <s> A useful model for general time-varying channels is a finite state Markov chain. In this paper, maximum likelihood sequence estimation (MLSE) for signals over finite state Markov channels (FSMCs) is studied. Also studied is the maximum a posteriori (MAP) channel state estimation. When coded signals with interleaving are transmitted, the channel estimates can be used to make soft-decision decoding. The error performance of the proposed sequence and channel state estimation schemes are evaluated through computer simulations. The effect of channel modeling error is also discussed. <s> BIB004 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <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> BIB005 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <s> Finite-state Markov channel (FSMC) models are useful for analysing radio channels with nonindependent fading, which give rise to bursty channel errors. In the paper the FSMC model is constructed by partitioning the dynamic range of the fade amplitude into finite number of intervals and representing each interval as a ‘state’ in the model. Transitions between these channel states thus characterise the physical fading process. Based on the Nakagami-m distribution, a new analytical formulation for the state transition probability is presented and validated for channels with arbitrary degrees of time-interleaving. The resultant FSMC model is shown to be applicable to a wide range of practical fading channels, including diversity-combined channels and multicarrier channels. <s> BIB006 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <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> BIB007 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <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> BIB008 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <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> BIB009 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <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> BIB010 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <s> We propose a decision-feedback decoder for coded signals transmitted over finite-state Markov channels. The decoder achieves maximum-likelihood sequence detection (in the absence of feedback errors) with very low complexity by exploiting previous bit decisions and the Markov structure of the channel. We also propose a similar decoder, the output-feedback decoder, that does not use previous bit decisions and therefore does not suffer from error propagation. The decoder performance is determined using a new sliding window analysis technique as well as by simulation. Both decoders exhibit excellent bit error rate performance with a relatively low complexity that is independent of the channel decorrelation time. <s> BIB011 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <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> BIB012 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <s> We study the effect of training method on the receiver bit error rate (BER) performance in constant envelope phase modulation schemes that are transmitted through flat fading channels. Here we propose a novel method for receiver training by sending information bits with unbalanced probabilities. We call this method implicit training and compare it with traditional (explicit) training scheme in which known periodic training sequences are sent to the receiver. The simulation results show superior performance of implicit training where the gain is 2 dB at information rate of 0.15 bits/channel use. <s> BIB013 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <s> We designed a multi-channel Nakagami fading simulator by modeling the received combined signal-to-noise ratio as a finite-state Markov chain, following a previously proposed approach. Our model generates directly the error process at the output of a diversity receiver and can emulate selection, maximal-ratio, and equal-gain combining. As the order of diversity increases, the savings in computational complexity improve linearly with respect to a traditional waveform simulator. The level crossing rates of the simulated envelope are shown to be very close to their theoretical values. The simulator's performance is also evaluated in terms of the accuracy of the obtained bit error rates, for both uncoded and coded systems. The simulator speeds up the performance evaluation of high-rate communication links where a high number of samples is needed. <s> BIB014 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <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> BIB015 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <s> In this paper, time-varying flat-fading channels are modeled as first-order finite-state Markov channels (FSMC). The effect of this modeling on the channel information capacity is addressed. The approximation accuracy of the first-order memory assumption in the Markov model is validated by comparing the FSMC capacity with the channel capacity assuming perfect state information at the receiver side. The results indicate that the first-order Markovian assumption is accurate for normalized Doppler frequencies f/sub d/T /spl lsim/ 0.01, in amplitude-only quantization of the channel gain for noncoherent binary signaling. In phase-only and joint phase and amplitude quantization of the channel gain for coherent binary signaling, the first-order Markovian assumption is accurate for f/sub d/T /spl lsim/ 0.001. Furthermore, the effect of channel quantization thresholds on the FSMC capacity is studied. In high signal-to-noise ratio (SNR) conditions, nonuniform two-level amplitude quantization scheme outperforms equiprobable quantization method by 0.8-1.5 dB. <s> BIB016 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <s> This paper proposes a novel successive decoding paradigm for fading channels. K independent codewords of different rates and lengths are interleaved at the transmitter according to a pattern, and successively decoded at the receiver. This enables progressively better channel estimates for the receiver, at a minimal cost of pilot symbols. At each decoding stage, channel estimation and decoding are performed separately, or iteratively if the residual channel correlation within a codeword can be exploited. The flexible interleaver pattern, after some level of optimization, achieves near-capacity system performance with only a small number of codewords K, as demonstrated numerically by a finite state Markov modelled flat fading channel. <s> BIB017 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE OR RECEIVED SNR <s> This paper addresses the problem of dynamic control for power and rate allocation in V-BLAST wireless systems over Markovian fading channels. The problem is posed as a controlled Markov decision process problem with the goal of minimizing the average transmission power with the constraint on the average delay that can be interpreted as the Quality of Service (QoS) requirement of a given application. Several structural results on the nature of the optimal randomized policies and costs are derived. In particular, it is shown that number of actions to be considered can be reduced by dividing the rate allocation problem into bit-loading problem across individual antennas and the total rate allocation based on the current buffer and channel state. Further, optimal rate allocation policies are shown to be a mixture of two pure policies that are non-decreasing in the buffer state. These results can be utilized to devise an efficient online learning algorithm for optimal rate allocation policies1. <s> BIB018
States correspond to partitioning the TV-FFC amplitude a or the received SNR γ into K nonoverlapping regions such as [0, , where the v k denote fading amplitude partition thresholds. An example of partitioning the TV-FFC amplitude a into five regions is shown in Figure 7 . Since the instantaneous received SNR γ is proportional to the square of the TV-FFC amplitude a 2 (γ = a 2 E s /N 0 ), partitioning the . Therefore, it suffices to consider FSMC modeling of the TV-FFC amplitude a. In Figure 7 (a), the steady-state probability of each FSMC state is derived by integrating the pdf of the TV-FFC amplitude a over the corresponding region as where f A (a) is the analytical pdf of the fading amplitude. Rayleigh fading pdf is a widely used statistical model for fading amplitude in the FSMC modeling BIB002 , BIB007 ,  , BIB003 , BIB010 , BIB011 , BIB017 , BIB004 - BIB013 , BIB018 - BIB001 , BIB012 , BIB016 . For Rayleigh fading, the pdf f A (a) was given in  . Other statistical models used for the FSMC modeling of the fading amplitude include Rician BIB008 , BIB009 , BIB015 , BIB005 and Nakagami m BIB014 , BIB006 . Having an analytical expression for f A (a) is not necessary for obtaining FSMC steady-state probabilities. Instead, fading channel field measurements or simulated data can be partitioned into the desired K nonoverlapping regions. In this case, the ratio of occurrences of the kth region over the total number of measurements can be used as an estimate of π k . The state transition probability P k→m = Pr(S = m|S −1 = k) is the probability that fading channel amplitude moves from region r k at time index − 1 to region r m at time index . This is loosely shown in Figure 7 (a). Strictly speaking, however, the state transition probabilities should be derived from integrating the joint pdf of the TV-FFC amplitude over two consecutive time indices and over the desired regions as  Magnitude of the Fading Channel Gain, a=|h| where the denominator is the steady-state probability of being in state k as defined in BIB002 . For the Rayleigh fading channel, the joint pdf of the fading amplitude f a −1 , a has a closed form and numerical integration of (15) is possible, which has been used to obtain FSMC state transition probabilities P k→m in BIB007 ,  , BIB004 , and BIB016 . For obtaining channel observation law in each FSMC state, an expression for the probability of observing output y given input x and the instantaneous channel amplitude a is needed, which is denoted by Pr(y|x, a) or f(y|x, a) (the latter is used if y is continuous). This depends on the type of modulation at the transmitter and detection at the receiver. For example, in binary noncoherent detection of frequency shift keying (BFSK), Pr(Y=0|X=1, a) The channel observation law in each FSMC state is the normalized average of instantaneous observation probability over the desired TV-FFC amplitude region. That is, where the denominator is the steady-state probability of being in state k as defined in BIB002 .