Finite-state Markov modeling of fading channels - a survey of principles and applications <s> NUMBER OF FSMC STATES <s> An adaptive coding scheme for digital communication over time-varying channels is presented. The scheme is based on a finite-state Markov channel model. Emphasis is on the adaptation of the error protection to the actual channel state. The throughput gains that are achieved by the adaptive scheme relative to the conventional nonadaptive coding methods are demonstrated by several examples. Of special interest is the use of punctured convolutional codes with maximum-likelihood Viterbi algorithm to enable adaptive encoding and decoding without modifying the basic structure of the encoder and the decoder. > <s> BIB001 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> NUMBER OF FSMC STATES <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> NUMBER OF FSMC STATES <s> The finite-state Markov channel (FSMC) is a discrete time-varying channel whose variation is determined by a finite-state Markov process. These channels have memory due to the Markov channel variation. We obtain the FSMC capacity as a function of the conditional channel state probability. We also show that for i.i.d. channel inputs, this conditional probability converges weakly, and the channel's mutual information is then a closed-form continuous function of the input distribution. We next consider coding for FSMCs. In general, the complexity of maximum-likelihood decoding grows exponentially with the channel memory length. Therefore, in practice, interleaving and memoryless channel codes are used. This technique results in some performance loss relative to the inherent capacity of channels with memory. We propose a maximum-likelihood decision-feedback decoder with complexity that is independent of the channel memory. We calculate the capacity and cutoff rate of our technique, and show that it preserves the capacity of certain FSMCs. We also compare the performance of the decision-feedback decoder with that of interleaving and memoryless channel coding on a fading channel with 4PSK modulation. <s> BIB003 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> NUMBER OF FSMC STATES <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> BIB004 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> NUMBER OF FSMC STATES <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> BIB005 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> NUMBER OF FSMC STATES <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> NUMBER OF FSMC STATES <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> BIB007 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> NUMBER OF FSMC STATES <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> BIB008 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> NUMBER OF FSMC STATES <s> This paper addresses the design and performance evaluation with respect to capacity of M-PSK turbo-coded systems operating in frequency-flat time-selective Rayleigh fading. The receiver jointly performs channel estimation and turbo decoding, allowing the two processes to benefit from each other. To this end, we introduce a suitable Markov model with a finite number of states, designed to approximate both the values and the statistical properties of the correlated flat fading channel phase, which poses a more severe challenge to PSK transmission than amplitude hiding. Then, the forward-backward algorithm determines both the maximum a posteriori probability (MAP) value for each symbol in the data sequence and the MAP channel phase in each iteration. Simulations show good performance in standard correlated Rayleigh fading channels. A sequence of progressively tighter upper bounds to the capacity of a simplified Markov-phase channel is derived, and performance of a turbo code with joint iterative channel estimation and decoding is demonstrated to approach these capacity bounds. <s> BIB009 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> NUMBER OF FSMC STATES <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> BIB010 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> NUMBER OF FSMC STATES <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> BIB011 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> NUMBER OF FSMC STATES <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> BIB012 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> NUMBER OF FSMC STATES <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> NUMBER OF FSMC STATES <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> NUMBER OF FSMC STATES <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> NUMBER OF FSMC STATES <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> NUMBER OF FSMC STATES <s> In this paper, Rayleigh fading channel with an OFDMA system is modeled as a finite-state Markov channel (FSMC) by partitioning the received signal envelope into several intervals. With the aid of sub-band formation and property of lumpability, the size of feedback information can be reduced. This approach involves reducing an exponentially increased states of Markov channel to multiple lumpable Markov channels. The corresponding state transition probabilities and steady-state probabilities are used to predict channel states information in multiple symbol durations ahead. Some simulation examples are presented to illustrate the capability of the lumpable FSMC in channel prediction. <s> BIB017 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> NUMBER OF FSMC STATES <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> BIB018
Before concluding this section, we comment on the required number of FSMC states for representing the TV-FFC. There is no clear-cut solution on how to select the number of fading states in the FSMC model. This may depend on many factors including application, model complexity, required accuracy and modulation/demodulation format, and the coding scheme. There is often a trade-off between model accuracy and its complexity. In the first-order FSMC models considered in the literature (either in the developed theory or in provided numerical examples), the number of considered FSMC states K ranges from 2-4 BIB001 , BIB004 , BIB005 , BIB003 , BIB010 , BIB013 , BIB017 , BIB011 to 1,000-2,000 BIB008 , BIB014 . However, the middle range of K = 8 to K = 64 states is more widely used BIB002 ,  , BIB015 , BIB014 , BIB006 , BIB012 , BIB018 , BIB007 - BIB009 , BIB016 and is generally accepted to have a better balance between accuracy and model complexity than FSMC models with smaller number of states. The number of input symbols P in the FSMC model is also a factor for the physical layer analysis of fading channels. The number of FSMC states K = 8 or K = 16 is aligned with binary signaling P = 2. For example, it is shown in BIB016 that the first-order FSMC capacity saturates with D = 8 to D = 16 partition regions for the TV-FFC in binary signaling. For higher dimensional modulations ( P > 2), a larger number of partition regions D may be required for the fading channel gain. While the number of partitions D is mainly influenced by application (such as the modulation dimensionality), the FSMC memory order M is mainly affected by the required accuracy. In the intermediate fading region, model accuracy should be improved by increasing the FSMC memory order M. The effect of FSMC memory order on model accuracy will be discussed in more detail in the section devoted to FSMC applications.