Wireless Channel Model with Markov Chains Using MATLABDiana Alejandra Sanchez-Salas1, Jose Luis Cuevas-Ruiz1 and Miguel Gonzalez-Mendoza1[1] Tecnologico de Monterrey, Campus Estado de Mexico, Mexico1. Engineering is one branch of science that is hugely popular among the students in the Indian subcontinent. Indian Institute of Technology is an independent public engineering college of India. IIT in India is spread over 16 states namely Kanpur, Indore, Kharagpur, Mandi, Mumbai, Patna, Ropar, Roorkee, Jodhpur, Chennai, Delhi, Gandhinagar, Guwahati, Hyderabad and Bhubaneswar. Indian Institute of Technology(IIT), Kanpur was established in 1959 and is located in Grand Trunk Road near the village of Kalyanpur Kanpur, Uttar Pradesh. IIT Kanpur is benefited from the Kanpur Indo-American Programme (KIAP) which is a combined association of nine US universities viz.

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The institution also provides continuous educational program for working professionals and distance education programs to the students so that they can learn from any part of India. Another option to describe a channel is by using statistical models which are based on probability density functions (pdf). Although they do not describe the behavior of the sent signal in a complete way, they give a good approximation of the conditions of the channel in a certain moment. States of Markov chains could be defined in agreement to channel conditions of an scenario, like it was done by E. Lutz who defined a two state channel model, one classified as good and the other one classified as bad depending on the conditions and therefore the attenuation level. Vucetic which characterizes the wireless channel of a certain area in Australia with a four state Markov chain by combining four different types of statistical channels.Next sections explain in detail how to create a wireless channel model using Markov chains and its application in a communication system with a coder and modulator.

An analysis with a semimarkovian channel is explained in the last section and a comparative analysis is done between this model and that from Markov chains. Markov chainsOne type of stochastic process is Markov chains, named after Andrei Markov who studied the transitions between consonants and vowels in a poem at the beginning of the 20th century. X(t) = i represents that the process is in state i at an instant of time t and take discrete values. It is said that there is a fixed probability pij that chain goes from a state i to a state j in the next time t. This statement is illustrated in Figure 1 which shows that the system can change from one condition to another or continues in the same state. Probability transition matrixThe switch between states is established in the probability transition matrix P. Transition diagramA probability transition matrix can be also represented as a diagram called transition diagram. Each node 0 represents a state of the Markov chain indicated with a number inside; an arrow ? connects state i with state j if a transition exists and the transition probability pij is written on that connecting arrow, even if the transition is to the same state. Steady-state vectorAnother important element in Markov chains is the steady-state vector ?, which represents the total appearing percentage of a state in a Markov chain.

Creation of a Markov chains in MatlabIn this section you will find the steps and the code to create a Markov chain using Matlab for two and three states. The first step for the creation of a chain is to establish a transition matrix that indicates the switch between states. Since this is a stochastic event, the first sample and the switch between stated will be obtained in a random way but always following the characteristics of P.

The precise steps to create the Matlab code are listed below.Generate a random integer number between 1 and 2 (because of the two state markovian chain).

The input is the 3 x 3 size probability transition matrix, like the one shown in (13) and the output vector will be also the vector containing the Markov chain with the characteristics set by P. The steps are listed next.Generate a random integer number between 1 and 3 (because now it is a three-state model).

To verify that the chain follows also the steady-state vector it is necessary to count the number of times each state appeared in the chain and divide each count by the total number of samples (size of the output vector).Two and three-state Markov chains Matlab programs described above represent the wireless transmission channel that will be implemented in a communication system.

Implementation of a Markov chains in MatlabAnother alternative to characterize a wireless channel is using Markov chains as it was said before. This model is a better option than statistic channels (AWGN, Rice, Rayleigh) because the former represent a statistical channel combination of the latter.

Markov chains will be used as a base to create a wireless channel model and will represent the multiplicative fading, which is right after the transmitter, and will be switching between the possible channels according to the probability transition matrix, as it is shown in Figure 4.

From now, this model will be called as a “markovian channel”.To have a better understanding of this markovian model, several examples with different probability transition with its steady-state vector, calculated using (9), are proposed. The codes for the creation of a two and three state Markov chains are used to produce results. Interpretations are based on BER curves obtained by implementing the communication system and the markovian channel of figures below. Two-state Markov channel modelAs it was said before, this is the simplest model in Markov chains.

Conversely, channel P2chan2 establish that channel 2 (Rayleigh) will appear all the time, this channels can be seen as an urban environment where there are a lot of obstacles between transmitter and receiver that will cause the multipath or also it can represents an user who is moving from one point to another one.

It can be appreciated in Figure 5 that the resulting BER curve for P2chan1 is exactly the same as the one resulting from the simulation of the Rice K = 14 dB statistical channel. Likewise, BER curve generated from P2chan2 is exactly the same to the simulation where only Rayleigh statistical channel is implemented.To proof the Markov channel follows the characteristics of (16) and (17), a counting of the transitions and the number of times both channels appeared is realized from the created Markov chain. The size of the simulated sent message was 100,000 bits, so the results of the count are given as percentages, which is equivalent to the probabilities of the transition matrixes. This means that 100% of chances there was a transition from channel 1 to 1 and from channel 2 to 1.

For channel P2chan4 probabilities are opposite than the other channel, in other words, there is a greater probability of having LOS than multipath, like in a rural area. Resulting BER curves for both channels are shown in Figure 8, where it shows that BER curves from markovian channels of (20) and (22) are very close to that curves resulting from simulating only the correspondent statistical channel but they are not the same, proving the combination of the channels according to the probability transition matrixes.

A probability transition matrix shown in (24) with its steady-state vector in (25) are presented as a new wireless channel model.

In this channel model the user has LOS the most part of the time but in some instants experiments multipath. This seems a semi urban environment because it can exists some building that cause multipath but they are not that tall to block the line of sight.

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