Finite-state Markov modeling of fading channels - a survey of principles and applications <s> APPROXIMATION OF STATE TRANSITION PROBABILITIES <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> BIB001 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> APPROXIMATION OF STATE TRANSITION PROBABILITIES <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> BIB002 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> APPROXIMATION OF STATE TRANSITION PROBABILITIES <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> BIB003 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> APPROXIMATION OF STATE TRANSITION PROBABILITIES <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> BIB004 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> APPROXIMATION OF STATE TRANSITION PROBABILITIES <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> BIB005 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> APPROXIMATION OF STATE TRANSITION PROBABILITIES <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> BIB006
In BIB001 , the authors proposed a method to approximate state transition probabilities P k→m in the first-order FSMC model for the fading amplitude or the received SNR in Rayleigh fading channels. The advantage of the approximate method is that it provides single-letter solutions for FSMC state transition probabilities and can replace numerical evaluation of the double integration in (15) in real-time applications. Referring to Figure 7 , BIB001 assumed that from time − 1 to , the fading channel amplitude either stays in the same region r k or it transits to its immediate neighboring regions r k−1 or r k+1 . Therefore, FSMC states can only loop back to themselves or move to immediate previous/next states. In other words, The above assumptions are reasonably accurate when the rate of temporal variations in the TV-FFC (determined by the normalized fading rate f D T s ) is relatively slow and the number of states is not so high. As the number of regions for the fading amplitude increases, the possibility of state transition among nonneighboring states increases and should be taken into account. Let us denote the level crossing rate (LCR) at amplitude boundary threshold v k by N k , which is the number of times per second that the fading amplitude a crosses v k in a downward direction and is given as BIB001  where f D is the maximum Doppler frequency shift and σ 2 is the variance of the fading channel gain per complex dimension. The transition probability from state k to state k − 1 was approximated in BIB001 to be the ratio of the LCR at point v k to the average number of symbols per second that is transmitted in state k, which is equal to R k = π k / T s . In summary, Similarly, the transition probability from state k to state k + 1 was approximated in BIB001 to be the ratio of the LCR at point v k+1 to the average number of symbols per second that is transmitted in state k. That is, Other state transition probabilities are given as , and The authors in BIB001 verified the validity of approximate state transition probabilities through computer simulations of the TV-FFC for the normalized fading rates of f D T s = 0.0001 and f D T s = 0.001 and K = 8 equiprobable amplitude states. Other researchers including BIB003 , BIB005 , BIB002 , BIB004 , and BIB006 have used this approximation method.