Finite-state Markov modeling of fading channels - a survey of principles and applications <s> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE AND PHASE <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> FIRST-ORDER FSMC MODELS FOR THE TV-FFC AMPLITUDE AND PHASE <s> Autoregressive stochastic models for the computer simulation of correlated Rayleigh fading processes are investigated. The unavoidable numerical difficulties faced by this method are elucidated and a simple heuristic approach is adopted to enable the synthesis of accurately correlated, bandlimited Rayleigh variates. Performance comparisons are made with a popular generation technique to demonstrate the merits of the method. <s> BIB002
First-order FSMC modeling of the TV-FFC amplitude and phase can be suitable for tracking time variations of channel phase for certain fading rates. The instantaneous TV-FFC amplitude a is partitioned into K 1 nonoverlapping regions, where the k 1 th region is r k 1 = [v k 1 , v k 1 +1 ) as before (see Figure 7) . The range of fading phase [0, 2π) is also partitioned into K 2 non-overlapping regions, where the k 2 th phase region is ). This is shown in Figure 7 for K 2 = 4 phase regions. The FSMC is said to be in state s = k if the instantaneous TV- Phase-only FSMC modeling of the TV-FFC is a special case where there is a single region for fading amplitude: a ∈ [0, ∞) (K 1 = 1) 2 . Steady-state probability in joint phase and amplitude FSMC modeling of the TV-FFC is the probability that the fading amplitude resides in the region ) and the fading phase resides in the region ). This probability is given by integrating the pdf of the TV-FFC amplitude and phase over the desired regions where we have assumed that fading amplitude and phase are independent of each other (as in the Rayleigh fading). Strictly speaking, the state-transition probabilities P k→m should be derived from integrating the joint pdf of the TV-FFC amplitude and phase over two consecutive time indices and over the desired regions. Although the joint pdf f (a −1 , a , θ −1 , θ ) is available in closed form for the Rayleigh fading channel, numerical computation of the quadruple integral involving f (a −1 , a , θ −1 , θ ) over the regions is time consuming. Therefore, we can use the following Monte Carlo method to obtain an estimate of P k→m . ■ Generate a long sequence of simulated complex-valued TV-FFC gains h N 1 , according to the simulation method in BIB002 or the references therein. ■ For each state index m, k ∈ {0, . . . , K − 1}, find the number of joint occurrences that two consecutive fading amplitude and phase a , a −1 , θ , and θ −1 belong to regions ). For obtaining the channel observation law in each FSMC state, we need the probability of observing channel output y given channel input x, the instantaneous channel amplitude a and the instantaneous channel phase θ , which is denoted as Pr (y|x, a, θ) or f(y|x, a, θ) . This depends on the type of modulation at the transmitter and detection at the receiver. For example, according to (1), f(y|x, a, θ) is the pdf of the complex-valued Gaussian noise z with variance N 0 /2 per dimension and mean m = a cos(θ)x + ja sin(θ)x. The channel observation law in each FSMC state is the normalized average of instantaneous observation probability over the desired TV-FFC amplitude and phase regions. That is, where the denominator is the steady-state probability of being in state k as defined in BIB001 .