Development of Outage Tolerant FSM Model for Fading Channels
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1 Development of Outage Tolerant FSM Model for Fading Channels Ms. Anjana Jain 1 P. D. Vyavahare 1 L. D. Arya 2 1 Department of Electronics and Telecomm. Engg., Shri G. S. Institute of Technology and Science, Indore, M.P., India 2 Department of Electrical Engineering, Shri G. S. Institute of Technology and Science, Indore, M.P., India Abstract - Finite State Markov (FSM) models for fading channels need to be revised for more realistic design of emerging mobile networks and their performance evaluation. In this paper a Outage Tolerant FSM Model (OTFSM) is proposed based on concept of certain tolerable outage times, which are defined as Tolerance time. These are the short duration of outage time which is considered as satisfactory times over the channel. In this paper, a statistical approach is being presented for the development of OTFSM model and evaluation of its fading parameters such as Average Fade Duration (AFD), outage probability and outage frequency. Derived results may be used for higher layer performance evaluation and selection of physical layer parameters of wireless networks. Keywords: FSM model, Tolerable outage time, Outage probability, Fading channel, Fade Duration Distribution 1 Introduction Various approaches for characterization of fading radio channels as Finite State Markov (FSM) model have appeared in the literature over last five decades. Initial channel models assumed fading as a Gaussian process and the resultant envelope with Gaussian probability density function (pdf). Mobile wireless channels suffer from multipath propagation and therefore the received signal envelope is approximated according to certain pdf s like Rayleigh, Rican, and Nakagami [1, 2]. For reliable characterization of the fading channels,various approaches were presented to model the channel. Started with the pioneer work of Gilbert and Elliot [3, 4] for two state model of fading channel, the FSM channel models are classified as finite N state and variable state Markov chain model. [5]. Wang and Mayori [6] proposed FSMC model with more than two states based on Signal to Noise Ratio (SNR) partitioning for Rayleigh channel. Binary Symmetric channel (BSC) is associated with each state and transitions with Markov property are assumed between states. It is pointed out that fading speed of the channel decides the SNR and its partitions to specify the states. Deterministic channel modeling and long range prediction of fast fading mobile radio channels has also appeared in [7]. The relationship between a physical channel and its FSM model for bit and packet error probabilities are demonstrated in [8, 9, 10, 11]. Babich demonstrated a technique to improve FSM model description based on Context Tree Pruning (CTP) algorithm [12]. Tan and Beaulieu proposed fading simulation via filtering methods and quantization of resulting process [13]. Further First order Markov chain analysis for short and long time duration of time is examined for the Rayleigh Channel by Chockolingam and Milstein [14]. Bai [15] presented various error modeling schemes for fading channels. Stochastic channel models were used to compute the fading parameters such as Level Crossing Rate (LCR) and Average Fade Duration (AFD) for fading channels [16, 17]. The Fade Duration Distribution (FDD) function and Minimum Duration Outage ( MDO) in Weibull fading channel is described in [18]. Concept of repair time is elaborated in [19]. Reig and Rubio have shown the modeling of Fade Depth and the Fade Margin in UWB Channels applicable to emerging mobile networks [20]. It is revealed from the literature survey that a major contribution to channel characterization studies goes to FSM channel modeling and evaluation of traditional fading statistical parameters such as AFD, outage probability and level crossing rate. These parameters give the insight of mean behavior of the wireless systems. However in recent years mobile systems are emerging as 3G and 4G networks. For such systems the different frequency components contained in the transmitted bandwidth experience different propagation environments and operate at high data rate, ultra high frequency and in a hostile channel environment. Therefore mean behavior is not sufficient and more precise characterization of fading process is needed. Existing FSM models are therefore needed to be reformed with evaluation of additional channel characterization parameters. Apart from the traditional fading parameters outage time distribution is also identified as one of the important parameter for investigation. This paper presents the methodology to develop Outage Tolerant FSM (OTFSM) channel model based on the
2 concept of tolerable outage time. Second-order fading parameters such as AFD (Average duration of outage time), outage probability (probability of being in outage) and outage frequency (number of times received signal crosses the threshold level and experience outage) are derived for proposed model. The proposed model considers the discrete value of the observation time of the channel. The concept is applied to Rayleigh and Nakagami channel with the variation in parameters like Doppler spread and fade margin. Paper is organized as follows. Section II presents formulation of tolerance time. Development of OTFSM model is discussed in section III. Fading parameters of the OTFSM model are derived in section IV with results and discussions in section V. Paper is concluded in section VI. 2 Formulation of tolerance time 3G and 4G wireless communication systems are supposed to offer variety of services such as voice, data, image transmission and internet browsing at high data rate in a relatively severe fading environment. The paper proposes a new FSM channel model as Outage Tolerant FSM (OTFSM) model while introducing tolerance time as a new statistical fading parameter. In the proposed model the fading channel is described by discrete time and discrete amplitude Markov chain, X (n) with sample space S 1 = {1 k} (1) S 2 = {k + 1 N} (2) Subset S 1 consists of outage states and subset S 2 consists of satisfactory states. X (n) can be represented as sampled PSD of the channel with the sampling period denoted as slot duration t. A threshold value of received signal PSD is selected to decide the state of Markov process. When received signal during slot duration is more than the specified threshold, state of the process belongs to subset S 2, otherwise in subset S 1. The finite number of slot durations which are in outage and are tolerated by the channel with acceptable with acceptable Packet Error Rate (PER) target, is defined as the tolerable error event length and the time duration is denoted as tolerance time. In OTFSM model discrete time approach is proposed for the tolerance time and outage is considered when the received signal strength stays below the threshold value longer than tolerance time. Based on the concept, OTFSM model considers the states which are in tolerable outages as satisfactory states rather than outage states. OTFSM channel model is developed in this section with its state diagram, steady state probabilities of being in satisfactory, outage and tolerant states and fading parameters. It is proposed that channel may tolerate maximum duration of time denoted as t tol which is equal discrete number of time slot τ. Fading parameters for OTFSM channel model are denoted as p out(outage probability ), (Average outage time) and f out (Frequency of outage). The choice of tolerance time depends on the past outage statistics, error correction time and acceptable PER. Tolerance time may be considered as a non-negative continuous random variable with certain distribution functions like Beta or Exponential [18]. The range of tolerance time τ is selected from τ 1 to τ 2 with τ 1 < τ < τ 2 and tolerance time is considered as continuous variable. If τ = 0, The OTFSM model is the same as the original model τ =, The OTFSM model will never be in fading state. 3 Development of outage tolerant FSM Channel model In the proposed that in the OTFSM model short tolerable outages are considered as tolerant state along with satisfactory and outage state. States 1 to N- k are considered as outage states. States N- k+ 1 to N- k + m is considered as satisfactory states and state N k + m +1 to N are considered as tolerant states. Figure one shows the plot for channel state versus time for OTFSM model. The model can be represented by three states. Figure 1- The timing diagram of proposed OTFSM model O - Outage state, represented by subset S1 S - Satisfactory state, represented by subset S2 T- Tolerant state, represented by subset 3 S T O Figure 2- State diagram of OTFSM model with satisfactory outage and tolerate state State diagram shows the probability of transition from satisfactory state to tolerant state p st and from tolerant state
3 to satisfactory state is p ts is assumed to be nonzero. Probability of transition from satisfactory state to outage state p so is nonzero however probability of transition from an outage to tolerant state p ot is assumed to be zero. Channel is modeled as OTFSM where during tolerance time channel is supposed to be nonfading state as described above and made available to user. Model is based on following assumptions - i. This is a Markov model of fading channel subject to multiple outages. ii. The duration between the satisfactory instant and occurrence of an outage event is the satisfactory time and the time for the channel being in outage is random variable. iii. Figure two is the state diagram of this model. iv. Tolerant and outage state cannot occur at the same time. The state space of the fading process of OTFSM model is given as S 1 = {1, N- k} (3) S 2 = {N - k + 1, {N- k + m } (4) S 3 = {N k + m +1, -----N} (5) The one step transition probability matrix of the model is defined as For the proposed model and are assumed to be zero. The one step probability matrix for the OTFSM model can be represented as follows (6) (17) (18) Substituting above equations in equation (9) and equation (10), steady state probabilities would be same as two state model, shown as below (19) 4 Estimation of fading parameters (20) For OTFSM model outage is considered when channel is in satisfactory state at time zero followed by n t slots of outage state, equal to tolerance time, t tol. AFD can be rewritten as below [17] P B (I - P B ) -1 (21) P B (22) ( I - P B ) (23) (I - P B ) -1 (24) (7) Error probability matrix, P B, in which all entries of being in satisfactory state is set to zero, is given as Steady state probability can be shown as (9) (10) (11) 1 (12) As and are assumed to be zero, solving the equations for steady state probabilities (13) (8) (14) (15) For existing model tolerant states are considered as outage states, therefore (16) P B (I - P B ) -1 Substituting in equation (21), AFD of OTFSM would be Frequency of outage f out is given as [16] (25) (26) (27) (P B k-1 P) e B (28) For OTFSM model ) and e B = are considered and hence equation (28) can be rewritten as p out can be given as ) (29) = Frequency of outage. AFD (30) Result are in agreement with the earler results for two stage FSM model [16]. Probabilities of being in any of the states
4 can be computed either using simulation using MATLAB or mathematically for different values of Doppler shift and fade margin. The work can be extended for continuous and arbitrary distributed tolerance time [19] tol1 tol2 tol3 5 Results and discussions Observations were made for estimating AFD of OTFSM model. Further AFD of the proposed model is evaluated for the different values of one step transition probability. It has been demonstrated that AFD is the continuous increasing function of tolerance time. With tolerance time AFD decreases almost by ten percent as the value of ρ changes from one to two, as demonstrated in figure 3. Similarly, figure 4 demonstrates that the fading rate has an large impact on AFD. When fading rate is low i.e. low value of Doppler spread results in large value of AFD. The channel is likely to stay for larger time in the state once it crosses the threshold. Twenty percent fall in AFD results with the decrease in the Doppler spread from 120 Hz to 60 Hz. AFD of OTFSM model Fade depth = 0 db Fade depth=20 db tol time Figure 3: Variation in AFD of OTFSM with Tolerance time with various fade depth AFD of OTFSM M odel Doppler spread = 20 Hz Doppler spread = 60 Hz fdoppler spread =120 Hz tol time Figure 4: Variation in AFD of OTFSM with tolerance time for various Doppler spread in Hz. Frequency of outage Doppler spread Figure 5: Variation in frequency of outage with Doppler frequency with tol1 < tol2 < tol3 Figure 5 shows the variations in frequency of outage with Doppler frequency with tol1 < tol2 < tol3. There is a slow change in frequency of outage with large value of tolerance time. 6 Conclusion The paper presents the development of Outage Tolerant Finite State Markov model, which is based on the concept of tolerable outage time. Various fading parameters of the channel such as AFD, outage probability and outage frequency are evaluate and their behavior is studied using proposed OTFSM model. Simulation of the proposed model is carried out and the results are validated with the earlier literature. Channel can be made available to the user under the tolerable outage times and hence will result in higher spectral efficiency. Presented work may be extended for relaying fading channels and descrete sampled fading channels of emerging mobile networks [21]. Results may be used for higher layer Protocol performance evaluation and physical layer parameters selection of wireless networks. 7 References [1] Rappport T. S., Wireless Communication, Pearson Education, Second Edition, [2] Goldsmith A., Wireless Communication, Cambridge Press, [3] Gilbert E., Capacity of a burst-noise channel, Bell System Technical Journal, vol. 39, pp , Sept [4] Elliott E. O., Estimates of error rates for codes on burst-error channels, Bell System Technical Journal, vol. 42, pp , September [5] Aulin T., A modified Markov Model for the fading signal at a mobile radio channel, IEEE Transactions on
5 Vehicular Technology, vol.28, no. 3, pp , August [6] Wang H.S. and Mouyeri N, Finite State Markov Channel- A useful model for radio communication channels, IEEE Transactions of Vehicular Technology, vol. 44, no. 1, pp , Feb [7] Eyceoz T., Hallen A., and Hallen H., Deterministic channel modeling and long range prediction of fast fading mobile radio channels, IEEE Communication Letter, vol. 2, no. 9, pp , Sept [8] Wang H.S. and Chang P.C., On Verifying the First Order Markovian Assumption for a Rayleigh Fading Channel Model, IEEE Transactions on Vehicular Technology, vol. 45, no. 2, pp , May [9] Ross S. M., Introduction to Probability Models, Elsevier Publication, IX edition, [10] Michele Zorzi, On the Statistics of Block Errors in Bursty Channel, IEEE Transactions on Communication, vol. 45, no.6, pp , June1997. [11] Zhang Q. and Kassam S. A., Finite State Markov Model for Rayleigh Fading Channels, IEEE Transactions on Communication, vol. 47, no. 11, pp , Nov [12] Babich F., Kelly O. E, Lombardi G, Generalized Markov Modeling for Flat Fading, IEEE Transactions on Communication, vol. 48, no.4, pp , April [13] Tan C.C and Beaulieu N.C, On First Order Markov Modeling for Rayleigh Fading Channel, IEEE Transactions on Communication, vol. 48, no. 12, pp , Dec [14] Ramesh A. Chockolingam, and Laurence B. Milstein, SNR Estimation in Macadam-m Fading With Diversity Combining and Its Application to Turbo Decoding, IEEE Transactions on Communication, vol. 50, no. 11, pp , Nov [15] Bai H and Atiquzzaman M., Error modeling schemes for fading channel in Wireless Communication A Survey, IEEE Electronics Magazine on Communication, vol. 5, no.2, pp. 2-8, Fourth Quarter [16] Zorzi M., Minimum duration outage in Markov channel, IEEE Transactions on Communication, vol. 54, no. 6, pp , [17] Beaulieu N. C. and Dong X., Level Crossing Rate & Average Fade Duration of MRC (maximum ratio combiner) and EGC (Equal gain combiner) Diversity in Rician Fading, IEEE Transactions on Communication, vol. 51, no. 5, pp , May [18] Wang Miao Liu et al., Fade Duration Distribution and Minimum Duration Outage in Weibull Fading Channels, Vehicular Technology Conference, [19] Zheng Z. Cui L., and Hawakes A. G., A Study on a Single-Unit Markov Repairable System with Repair Time Omission, IEEE Transactions on Reliability, vol. 55, no.2, pp , June [20] Lano G. Reig and J. Rubio L., Analytical Approach to Model the Fade Depth and the Fade Margin in UWB Channels, IEEE Transactions on Vehicular Technology, vol. 59, no. 9, pp , [21] Javier L Martinez F. et al., Higher order statistics of sampled fading channels with applications, IEEE Transactions of Vehicular Technology, vol. 61, no. 7, pp , Sept
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