Finite-state Markov modeling of fading channels - a survey of principles and applications <s> CP METHOD <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> BIB001 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> CP METHOD <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 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> CP METHOD <s> The problem researched is the comparison of different finite-state Markov channel (FSMC) models used to approximate the Rayleigh fading channel. The criterion used to compare the Markov models is the error performance of receivers based on the FSMC, where the sufficient statistics are obtained from the Jakes-Clarke (1974) fading channel. The FSMC receivers perform joint maximum a posteriori (MAP) sequence detection and channel estimation using the Viterbi algorithm. Receiver outputs of bit decision and channel estimation error are used as the measure of comparison between various models and a well-developed reference receiver. For this research 1/sup st/ and 2/sup nd/ order models were implemented. Simulation results suggest that there can be a significant improvement in receiver performance when the receiver is based on a 2/sup nd/ order model over a 1/sup st/ order model. Likewise, an improvement in performance is observed when the number of Markov states is increased. <s> BIB003 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> CP METHOD <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> BIB004
A schematic of the CP method is shown in Figure 8 . Each state s in the Mth-order FSMC is a composite state that keeps track of M first-order FSMC states, which are now referred to as substates and denoted by u to u −M . In other words, the CP method performs sample-by-sample partitioning of the individual fading channel gain or its components to D regions according to the methods described in the previous section and then inflates the state space to K = D M . Therefore, in the CP method, the number of FSMC states grows exponentially with memory order M. The authors in  , BIB004 , and BIB003 have considered constructing Mth-order FSMC models using the CP method. The Linde-Buzo-Gray (LBG) algorithm (see [64, p. 117 ] for the original LBG reference) is a widely used iterative technique to obtain the codevectors and their associated regions in the VQ method. For the LBG algorithm to work properly, a relatively large number of training vectors are required. The training vectors used to obtain quantization regions are shown in Figure 9 . Training vectors for fading channels can be obtained using Monte-Carlo techniques to generate a fading process. For example, refer to BIB002 and the references therein. Given the training vectors, the LBG design criterion is to find partition regions and codevectors that minimize the average distortion across all training vectors. The distortion is defined as the square distance between training vectors and the associated centroids. Figure 9 also compares the CP and VQ methods. As mentioned before, the CP method is a sample-by-sample quantization of the fading channel. To keep the number of states in both cases equal, D = 4 equiprobable amplitude partition regions are used for each a and a −1 component [see BIB001 ], resulting in K = 4 2 states. The shape of each region in the CP method is forced to be rectangular. Therefore, the CP method is a very special case of the VQ method. The same training vectors used for the VQ method are overlayed on the rectangular CP regions on the right. Since the  CP method fails to capture correlations in the fading channel gain, it results in a less efficient representation of the fading channel in the FSMC states. In  and BIB003 , the authors have compared constructing Mth FSMC models using the CP and VQ methods for two different applications of channel error burst modeling and decoding at the receiver, respectively, and have concluded that with the same number of states K in both FSMC models, the VQ method results in more accurate FSMC models for the TV-FFC or in better receiver BER performance. In a VQ-based FSMC model for the TV-FFC, channel observation law in the kth state is approximated in terms of the centroid of the associated region. According to (1), the channel observation, given the current TV-FFC gain realization h and current channel input x , is independent of previous channel realizations. Therefore, the channel observation law in the VQ-based FSMC model only depends on the last component of the region centroid. For example, in the second-order VQ-based FSMC model for the TV-FFC amplitude in Figure 9 , we can write where (a ,k , a −1,k ) is the centroid for region r k . This approximation becomes more accurate as the number of states K becomes larger.