Finite-state Markov modeling of fading channels - a survey of principles and applications <s> SELECTION OF AMPLITUDE REGION BOUNDARIES <s> A finite-state Markov channel (FSMC) model for a Rayleigh fading channel (RFC) is constructed by partitioning the range of its received signal-to-noise ratio (SNR) into K intervals. The crossover probabilities of the K binary symmetric channels associated with its states are calculated. The second-order statistics of the received SNR are used to approximate the Markov transition probabilities. The capacity of the modeled channel can then be calculated with recursive algorithms. A joint source and channel coding scheme based on quantization for this channel model is studied. The decoder is assumed to have access to the channel state information, which it uses to adjust itself to varying channel conditions. It is shown that the use of the finite-state Markov channel model and knowledge of the channel state information significantly improves the performance of the system. <s> BIB001 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> SELECTION OF AMPLITUDE REGION BOUNDARIES <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> SELECTION OF AMPLITUDE REGION BOUNDARIES <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> BIB003 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> SELECTION OF AMPLITUDE REGION BOUNDARIES <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> BIB004 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> SELECTION OF AMPLITUDE REGION BOUNDARIES <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> SELECTION OF AMPLITUDE REGION BOUNDARIES <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> BIB006 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> SELECTION OF AMPLITUDE REGION BOUNDARIES <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> BIB007 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> SELECTION OF AMPLITUDE REGION BOUNDARIES <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> BIB008
Selection of the amplitude region boundaries v k is a key factor that determines the accuracy and applicability of the obtained FSMC model for the fading channel. Finding a clear-cut solution for the best way of selecting the region boundaries is not straightforward. The criterion for optimality or accuracy strongly depends on the application. We now discuss some of the main partitioning methods used in the literature, which will be revisited in light of FSMC applications for fading channels later in the article. Equiprobable partitioning of fading amplitude is a reasonable, but the ad-hoc method that has been used in the literature BIB002 , BIB005 , BIB007 , BIB003 . In this method, steady-state probabilities for all obtained FSMC states are equal to π k = 1/K. In the equiprobable partitioning of the TV-FFC amplitude into K states, region boundaries are given as BIB003  where σ 2 is the variance of the fading channel gain per complex dimension. Another criterion is to minimize the quantization MSE using the Lloyd-Max technique  , BIB001 . In this method, each region r k is represented by a single quantized point in that region denoted by c k = a k , which results in the minimum quantization MSE. If the Lloyd-Max technique is used for the FSMC modeling of Rayleigh fading amplitude, states will not be equiprobable. Due to the nonlinear relationship between SNR and BER performance, nonuniform amplitude partitioning may be useful to obtain more accurate estimates of system performance measures, such as BER, PER, or packet throughput. However, the advantage of this method over equiprobable method may diminish as the number of fading amplitude states K increases BIB007 . Other partitioning schemes have been proposed in the literature. In BIB004 , the authors partitioned the received SNR according to the (average) time duration of each state. The average state duration is chosen so that transitions only happen to current and adjacent states and also each data packet in each state approximately experiences the nominal state BER. In BIB006 , the authors proposed a new partitioning scheme that takes into account the relationship between SNR and fade duration at a given SNR. In BIB008 , the effect of Rayleigh fading amplitude partitioning thresholds on the obtained FSMC capacity is discussed. Selecting a partitioning scheme that results in the highest FSMC capacity can serve as a guideline for designing receivers that use FSMC modeling of fading channels for channel estimation and decoding.