Finite-state Markov modeling of fading channels - a survey of principles and applications <s> THE TIME-VARYING NATURE OF THE FADING CHANNEL <s> The statistical characteristics of the fields and signals in the reception of radio frequencies by a moving vehicle are deduced from a scattering propagation model. The model assumes that the field incident on the receiver antenna is composed of randomly phased azimuthal plane waves of arbitrary azimuth angles. Amplitude and phase distributions and spatial correlations of fields and signals are deduced, and a simple direct relationship is established between the signal amplitude spectrum and the product of the incident plane waves' angular distribution and the azimuthal antenna gain. The coherence of two mobile-radio signals of different frequencies is shown to depend on the statistical distribution of the relative time delays in the arrival of the component waves, and the coherent bandwidth is shown to be the inverse of the spread in time delays. Wherever possible theoretical predictions are compared with the experimental results. There is sufficient agreement to indicate the validity of the approach. Agreement improves if allowance is made for the nonstationary character of mobile-radio signals. <s> BIB001 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> THE TIME-VARYING NATURE OF THE FADING CHANNEL <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
The main characteristic of the fading channel under consideration is that it is a correlated and time-varying random process. In other words, the communication channel is a dynamic system with the fading channel gain h being a random process that changes over time in a correlated manner. Tractable mathematical models are required to accurately describe time variations of the fading channel gain and its dynamics. In this article, we consider two such models: Clarke's model and the autoregressive (AR) model. Clarke's model BIB001 is one of the most widely used statistical models for describing the fading random process. In Clarke's model, the in-phase and imaginary components of the fading channel gain are intra-correlated and the autocorrelation function (ACF) for each component is given as BIB001  where f D is the maximum Doppler frequency shift, T s is symbol period, and J 0 (·) is the zero-order Bessel function of the first kind. Because the fading channel in (1) is a frequency-flat channel and is varying with time, we refer to these class of fading channels as time-varying, flat-fading channels (TV-FFC). For the observation equation in (1) to be valid, we have assumed that the channel fading gain h stays more or less constant during one symbol transmission. This excludes extremely fast TV-FFCs, where the channel coherence time T c ≈ 1/f D is less than the symbol period T s  . Extremely fast fading channels, however, are not common in the majority of practical wireless communication systems, where the channel coherence time is at least a few times larger than the symbol period. In this article, we will often refer to the fading channel memory order. The notion of fading channel memory order can be best explained in AR modeling of the TV-FFC BIB002 . In the AR model of order M, the fading channel gain h is statistically related to its M past realizations through where w is a complex-valued, white coefficients (the superscript T denotes matrix transpose). Vector d can be derived from: Rd = −v. Matrix R is the M × M covariance matrix of the AR process, whose element at row m and column p is R |m−p| . Also, Clarke's TV-FFC model is inherently non-Markovian and does not fit into AR modeling. In fact, the power spectral density of the fading process in Clarke's model is a nonrational function of frequency, whereas the power spectral density of the fading process in an AR model is a rational function of frequency BIB002 . However, it is possible to think of the fading process in Clarke's model as an infinite-order AR process BIB002 . In practice, a large, but finite memory order M may suffice for accurate AR modeling of the Clarke's fading process. In this case, one may use the first M coefficients of the ACF in Clarke's model, defined in (4), to solve the Yule-Walker equation and to obtain an AR model of order M for the fading channel. More technical details can be found in BIB002 . Figure 3 shows 500 samples of the Rayleigh fading channel amplitude a = |h |, where the fading channel gain sequence {h } is generated according to an AR model with memory order M = 500 and the normalized Doppler frequency shift f D T s = 0.01 using the technique described in BIB002 . It is noted that a normalized Doppler frequency shift f D T s = 0.01 means that the channel coherence time T c ≈ 1/f D is 100 times the symbol period T s . (4)- (6), the received signal y in the observation (1) only depends on the latest realization of the TV-FFC gain h .