Finite-State Markov Modeling of Flat Fading Channels

Transcription

1 International Telecounications Syposiu ITS, Natal, Brazil Finite-State Markov Modeling of Flat Fading Channels Cecilio Pientel, Tiago Falk and Luciano Lisbôa Counications Research Group - CODEC Departent of Electronics and Systes Federal University of Pernabuco P.O. Box Recife - Brazil E.ail: cecilio@npd.ufpe.br Abstract The stochastic properties of the binary channel that describe the successes and the failures of the transission of a odulated signal over a tie correlated flat fading channel is considered for investigation. This analysis is eployed to develop K-th order Markov odels for such a burst channel. The order of the Markov odel that generates accurate analytical odels is estiated for a broad range of fading environents. The paraeterization and the accuracy of an iportant class of hidden Markov odels known as the Gilbert-Elliott channel are also investigated. Fading rates are identified in which the K-th order and the Gilbert-Elliott channel odel approxiate the fading channel with siilar accuracy. The latter odel is useful for approxiating slowly fading processes, since it provides a far ore copact paraeterization. I. INTRODUCTION In a typical obile counication channel the transitted signal undergoes attenuation and distortion caused by ultipath propagation and shadowing. The non-frequency selective (flat) fading channel iposes ultiplicative narrow-band coplex Gaussian noise (referred to as fading process) on the transitted signal. As a consequence, abrupt changes in the ean received signal level ay occur, and the autocorrelation function of the fading process ay lead to the occurrence of a burst of bit errors. The analytical analysis of a counication syste operating over such a correlated fading process is difficult and there are no analytical expressions for several statistics relevant to syste perforance evaluation. Finite state channel (FSC) odels have been widely accepted as an effective approach to characterize the correlation structure of the fading process [1]-[11]. An FSC is described by a deterinistic or probabilistic function of a first-order Markov chain, where each state ay be associated with a particular channel quality. The strategy adopted by any researchers to design FSC odels for fading channels consists in representing each state of a first-order Markov chain by a non-overlapping interval of the received instantaneous signal to noise ratio []-[7]. Criteria to partitioning the signal to noise ratio are discussed in [], [5], [7]. The odulation and deodulation schees are incorporated into the odel through the crossover probability of the binary syetric channel associated to each fading state. An inforation theoretic etric was proposed in [3] to validate the first-order odel. The liitations of this criterion and the applicability of the first-order assuption have been discussed in [7]. Other odel structures have also been proposed to represent the quantized signal to noise ratio, including, higher-order Markov This work received partial support fro the Brazilian National Council for Scientific and Technological Developent (CNPq) under Grant 3987/96-. odels [8], general Hidden Markov odels [11], and Gilbert- Elliott channels [1]. This paper concerns the developent of FSC odels for a discrete counication syste coposed by a odulator, a tie correlated flat fading channel, and a hard quantized deodulator. The FSC odel describes the successes and the failures of the sybol transitted over a fading channel, which is represented atheatically as a binary error sequence. We consider two classes of FSC odels coonly used to characterize fading channels: The Kth-order Markov odel and the Gilbert- Elliott channel () odel. We first describe a ethodology to estiate the paraeters of these odels directly fro the binary error sequence. A ethod recently proposed in [7] to judge odel accuracy is applied to identify the order of the Markov odel that satisfactorily approxiates the channel, for several fading regies. A siilar study evaluates the accuracy of the odel. The results presented here allow us to study coding perforance on correlated fading channels using the analytical techniques developed to analyze burst channels represented as specific FSC odels [1]-[15]. II. THE CHANNEL MODEL We consider a counication syste that eploys M-ary FSK odulation, a tie correlated flat Rician fading channel and non-coherent deodulation. The coplex envelope of the received signal at the input to the deodulator is corrupted by a ultiplicative Rician fading and by an additive white Gaussian noise with zero ean and autocorrelation function 1Ef N(t ~ + fi ) N ~? (t)g = N ffi(fi ). The coplex envelope of the fading process, G(t) ~ = G ~ I (t) + G ~ Q (t), is a coplex, wide-sense stationary, p Gaussian process with real constant ean, where = 1, and the quadrature coponents G ~ I (t) and G ~ Q (t) are utually independent Gaussian process with the sae covariance function, naed C(fi ). Although the analysis carried out here can be applied to a fading process with arbitrary covariance function, we adopted the Clarke s odel [16], [17] for C(fi ): C(fi )= 1 Ef[ ~ G(t + fi ) ][ ~ G? (t) ]g = ff g J (ßf D fi ); where J (x) is the zero-order Bessel function of the first kind, f D is the axiu Doppler frequency, and ff g is the variance of G(t). ~ For a fixed tie instant, the fading envelope A k =

2 International Telecounications Syposiu ITS, Natal, Brazil q ~G I (kt )+ G ~ Q (kt ) (where T is the sybol interval) has the Rician probability density function given by: p a A (a) = e ( +a )=ff ffg g I a ( ); (1) ffg where I (x) is the zero-order odified Bessel function of the first kind. When the in-phase process is zero-ean (K R =), the fading envelope follows the Rayleigh probability density function: p A (a) = a ff g exp a ff g : () At each signaling interval of length T, the deodulator fors the M decision variables and decides which signal was ore likely to have been transitted. We define a binary error process fe k g 1 k=1, where E k = indicates no sybol error at the kth interval, and E k = 1 indicates a sybol error. It can be shown that the probability of an error sequence of length n, e n = e 1 e :::e n, ay be expressed as [9]: P (e n ) = M1 ::: l 1=e 1 expf Es M1 l n=en ψ ny k=1 M1 l k (1) l k+e k l k +1 N K R 1 T F((K R +1)I + C? F) 1 1g det(i + N Es (1 + K R ) 1 C? ; F) (3) where C? is the coplex conjugate of the noralized n n covariance atrix of G(t) ~ (the (i; j)th entry of C? is J (ßf D (i j)t ), F is a diagonal atrix defined as F = diag( l1 l ;::: ; l n 1+1 l n+1 ), E s is the energy of the transitted sybol, K R = =ffg is the Rician factor, 1 is a colun vector of ones, and the superscript [ ] T indicates the transpose of a atrix. We will call this discrete fading odel fro the odulator input to the deodulator output the discrete channel with Clarke s autocorrelation () odel. Hereafter, we consider binary odulation (M =), so the odel has three paraeters K R, f D T, and E s =N. Equation (3) can be used to calculate the probability of any error event relevant to the analysis of the fading odel. For exaple, the probability the error bit is a 1 is:! Equation (3) will be eployed to paraeterize an FSC odel that accurately reflects the statistical description of the real error process. A brief description of FSC odels is given next. Consider fs k g 1 k= an N-state, first-order Markov chain with a finite state space N N = f; 1;::: ;N 1g. Let P be an N N transition probability atrix, whose (i; j) th entry is the transition probability p i;j = P (S k = j j S k1 = i), i; j N N. The FSC odel generates an error sybol according to the following probabilistic echanis. At the k th tie interval, the chain akes a transition fro state S k1 = i to S k = j with probability p i;j and generates an output (error) sybol e k N (independent of i), with probability b j;ek = P (E k = e k j S k = j). Conditioned to the state process, Q the error process is n eoryless, that is, P (e n j s n )= P (e k=1 k j s k ). We are assuing that the distribution of the initial state is the stationary distribution Π =[ß ;ß 1 ;:::;ß N1 ] T. The probability of an error sequence generated by the FSC odel, conditioned to the initial state, is: Hence P (e n j s ) = P (e n ) = = N1 s = s nn n N s nnn n k=1 ß s P (e n j s n ;s ) P (s n j s ) ny b sk ;e k p sk1 ;s k : ny s nnn n k=1 b sk ;e k p sk1 ;s k : (7) Equation (7) can be rewritten in a atrix for. Define two N N atrices, P() and P(1), where P = P() + P(1). The (i; j)th entry of the atrix P(e k ), e k f; 1g, isp (E k = e k ;S k = j j S k1 = i) = b j;ek p i;j, which is the probability that the output sybol is e k when the chain akes a transition fro state i to j. Equation (7) has a atrix for given by: P (e n ) = Π T ψ ny k=1 P(e k )! 1: (8) An FSC odel is copletely specified by the atrices P() and P(1). We define in the next section soe properties of FSC odels and discuss the evaluation of its paraeters. P (1) = 1+K R +K R + Es N e K R N Es +K R + Es and the probability of two consecutive ones (errors) is: P (11) = (1 + K R ) +K R + Es N ρ Es N e N ; () K Es R N +K R +(ρ+1) N Es ; where ρ is the correlation coefficient of two consecutive saples of the fading process ~ G(t): (5) ρ = J (ßf D T ): (6) III. PARAMETERIZATION OF SPECIFIC FSC MODELS We consider two classes of FSC odels: Kth-order Markov odels and the odel. Following the ideas introduced in [9], the paraeters of each FSC odel will be expressed as functions of the probabilities of binary sequences generated by the odel. Then, we apply (3) to estiate these probabilities and to paraeterize FSC odels that approxiates the correlated fading odel. A. Kth-order Markov Models A discrete stochastic process fe n g 1 n=1 is a Markov process of Kth-order if it obeys the relation: P (e n j e 1 e e n1 ) = P (e n j e nk e n1 ): (9)

3 International Telecounications Syposiu ITS, Natal, Brazil A first-order binary Markov odel is an FSC odel with space state f; 1g. An error sybol is produced when the chain transitions to state 1. Otherwise, if the chain transition is to state, a correct sybol is produced. Therefore, b ; = 1 and b 1;1 = 1. Using (7) for error sequences of length 1 and, we get, p i;j = P (ij)=p (i), i; j f; 1g. The stationary vector is Π = [P () P (1)] T, and the atrices P() and P(1) for the first-order Markov odel are: P() = P () P () P (1) P (1) 3 5 ; P(1) = P (1) P () P (11) P (1) 3 5 : In general, the Kth-order Markov odel can be represented as a function of a first-order Markov chain [18]. Each state of the Kth-order odel is represented by a binary string of length K. Given two states u = u 1 u u K and v = v 1 v v K, we say that u and v overlap progressively if u u 3 u K = v 1 v v K1.Ifuand v overlap progressively, then, there is a transition fro u to v with probability P (u 1 v 1 v v k )=P (u). Otherwise, the state transition probability is zero. Given a state v = v 1 v v K, b v;ek = v K. For exaple, the atrices P(), P(1) and Π for the second-order Markov odel are: P() = P(1) = 6 6 P () P () P (1) P (1) P (1) P () P (11) P (1) P (1) P (1) P (11) P (11) P (11) P (1) P (111) P (11) ; Π =[P () P (1) P (1) P (11)] T : Clearly, the nuber of states grows exponentially with the order K. B. Gilbert-Elliott Channel The is a two-state FSC odel coposed of state, which produces errors with sall probability, b ;1 = g, and state 1, where errors occur with higher probability, b 1;1 = b, where g<<b. The transition probabilities of the Markov chain are p ;1 = Q and p1; = q. The atrices P(), P(1), where P() + P(1) = P, for the odel are given by: P() =» (1 Q)(1 g) Q (1 b) q (1 g) (1 q)(1 b) P(1) =» (1 Q) g Qb qg (1 q) b ; ; (1) : (11) We define next the notation required in this subsection. Consider ff any binary sequence of finite length, Q( ff ;i) = P (E n = ff ;S n+1 = i). Let ffi be an epty sequence, i.e, a sequence of length zero that possesses the properties ffiff = ffffiand P (ffi) =1. Let ffl and ffi be binary sybols. If i is a particular state, i? indicates the other state of the Markov chain, that is, if i =, then i? =1. The paraeterization of the Gilbert-Elliott channel is based on the following lea. Lea 1: The probability of any sequence ff generated by the Gilbert-Elliott channel satisfies the following recurrence equation: where P (fffflffi) =c(ffl; ffi)p (ffffl) +d(ffl; ffi)p (ff); (1) c(ffl; ffi) = d(ffl; ffi) = b i;ffl (p i;i Λb iλ ;ffi + p i;i b i;ffi ) b i;ffl b b b i;ffl b Λb i Λ ;ffi + p iλ ;ib i;ffi ); (13) b i;ffl b b i;ffl b Λb i Λ ;ffi + p iλ ;ib i;ffi ) b b i;ffl (p i;i Λb iλ ;ffi + p i;i b i;ffi ): (1) b i;ffl b Proof: The probability of any sequence generated by a odel satisfies the relations: P (ff) =Q(ff;i Λ )+Q(ff;i): (15) P (ffffl) =Q(ff;i Λ )b + Q(ff;i)b i;ffl : (16) Hence, Q(ff;i) and Q(ff;i Λ ) are expressed as: Q(ff;i)= b i Λ ;ffl b i;ffl b b i;ffl P (ff) + 1 P (ffffl); (17) b i;ffl b Q(ff;i Λ )= P (ff) P (ffffl): (18) b i;ffl b b i;ffl b The following equation also holds for the odel: P (fffflffi) = Q(ff;i Λ )b [p iλ ;i Λb i Λ ;ffi + p iλ ;ib i;ffi ] + Q(ff;i)b i;ffl [p i;i Λb iλ ;ffi + p i;i b i;ffi ]: (19) Substituting (17) into (18) and rearranging the ters, yields: P (fffflffi) = f b i;fflb b i;ffl b Λb i Λ ;ffi + p iλ ;ib i;ffi ) + f b b i;ffl (p i;i Λb iλ ;ffi + p i;i b i;ffi )gp (ff) b i;ffl b b i;ffl (p i;i Λb iλ ;ffi + p i;i b i;ffi ) b i;ffl b b b i;ffl b Λb i Λ ;ffi + p iλ ;ib i;ffi )gp (ffffl) = c(ffl; ffi)p (ffffl) + d(ffl; ffi)p (ff): Equation (1) allows us to express c(ffl; ffi) and d(ffl; ffi), and consequently, b, g, q e Q as functions of the probabilities of error 1

4 International Telecounications Syposiu ITS, Natal, Brazil sequences. Substituting ff = ffi and ff = ffl in (1) yields, respectively: P (fflffi) = c(ffl; ffi)p (ffl) +d(ffl; ffi); () P (fflfflffi) = c(ffl; ffi)p (fflffl) +d(ffl; ffi)p (ffl): (1) Solving this linear syste, we obtain: and c(ffl; ffi) = d(ffl; ffi) = P (fflfflffi) P (fflffi)p (ffl) ; () P (fflffl) P (ffl) P (fflffi)p (fflffl) P (fflfflffi)p (ffl) : (3) P (fflffl) P (ffl) easure to judge if a particular odel approxiates better the fading channel when copared to other candidates. The criteria coonly used to ake this decision include the iniization of a distance easure between the probability of error sequences generated by the odel and by the fading channel (e.g. variational, noralized divergence), the inforation theoretic etric[3], and the coparison of certain statistics of the odels, such as, autocorrelation function, and packet error rate [7]. Motivated by the results presented in [7], we copare next the autocorrelation function (ACF) of the fading odel with the ACF of FSC odels. The ACF of a binary stationary process fe k g 1 k=1 is given by: R[] =EfE i E i+ g = P (E i =1;E i+ =1); (33) The following proposition expresses the paraeters of the odel in ters of c(ffl; ffi) and d(ffl; ffi), or consequently, in ters of the probability of error sequences of length, at ost, 3. Proposition 1: If P (1) 6= P () P (1), the paraeters of the Gilbert-Elliott channel are uniquely deterined by the four probabilities P ();P();P();P(111). The paraeters b and g are the roots of the quadratic equation: [1 +c(1; 1) + c(; )]x +[1 c(1; 1) c(; ) + d(1; 1) d(; )]x d(1; 1) = ; () and the paraeters Q and q are given by: c(; )b c(1; 1)(1 b) +(g b) Q = ; g b c(; )g c(1; 1)(1 g) +(b g) q = : b g Proof: Fro (13) and (1), we have: (5) c(; ) = (1 g)(1 Q) +(1 b)(1 q); (6) c(1; 1) = g(1 Q) +b(1 q); (7) d(; ) = (1 q Q)(1 g)(1 b); (8) d(1; 1) = (1 q Q)gb; (9) Fro (6) and (7), and fro (8) and (9), we get, respectively: 1 c(; ) c(1; 1) = μ; (3) d(; ) gb( 1)=1 b g; (31) d(1; 1) where μ =1 q Q. Cobining (9) and (31) results in the quadratic equation: μb +(μ + d(; ) d(1; 1))b + d(1; 1) = ; (3) where the sae equation holds for g. So, substituting (3) into (3) we conclude that b and g are the roots of (). Once we have deterined b and g, we use (6) and (7) to obtain (5). IV. MODEL EVALUATION This section evaluates the accuracy in which the FSC odels described in the previous section approxiate the correlated fading channel. In general, it is difficult to define a unique where Efg denotes the expected value of a rando variable. A closed-for expression for the ACF of the odel is given by (5), where the correlation coefficient ρ given by (6) is replaced by ρ() =J (ßf D T ). Then, it follows fro (5) that for the special case of Rayleigh fading (K R =): R[] = 1 ( + Es N ) ρ() Es N : (3) The ACF of an FSC odel described by the atrices P() and P(1) is expressed as [13]: R[] = Π T P(1)P 1 P(1) 1; (35) for 1. The ACF over twenty values of of the and the FSC odels are copared in Fig. 1. The paraeters of the are K R =, E s =N = 15 db, and f D T =.1 (a), f D T =. (b), f D T =.1 (c). Markov odels of order up to 6 have been considered. It is observed in Fig. 1(a) that there is a significant gain in accuracy when the order of the Markov odel is increased fro K = (eoryless) to K = 1, a little gain is obtained for K =and no further gain is observed for K = 3. Also, the ACF s of the second-order Markov and the odels are very alike. The curves indicate that the first-order Markov odel approxiates satisfactorily the fading channel for f D T =.1. It is worth entioning that we could have chosen either the second-order or the odel to approxiate the odel, since the ACF s of these three odels can barely be distinguished in Fig. 1(a). However, we want to obtain as siple an analytical odel as possible at an acceptable coplexity level. This tradeoff between accuracy and coplexity akes this decision soewhat arbitrary. When the fading rate gets slower the order of the Markov odel increases, as expected. For exaple, we observe (curves not shown) that the second-order Markov is satisfactory for f D T =:5. However, we notice that the ACF of the third-order Markov odel is a bit closer to that of the odel, but this strictness ay not copensate the doubling of the nuber of states. Again, the ACF s of the third-order and the are very siilar. When f D T < :3, the ACF of the odel diverges fro the ACF of the odel. This fact is illustrated in Fig. 1(b), where the curves of the forth and the fifth-order Markov odels approxiate better the ACF of the odel than that of

5 International Telecounications Syposiu ITS, Natal, Brazil R[] R[] R[] TABLE I ORDER OF THE MARKOV MODEL THAT APPROIMATES THE RAYLEIGH FADING CHANNEL FOR SEVERAL VALUES OF f D T. f D T K (E s =N =15dB) K E s =N =5dB) > 6 > K= K= the odel. These values of K can be considered as satisfactory approxiations of the odel for f D T = :. These conclusions hold for greater saple separations. In fact, the ACF s of the Kth-order Markov and the odels atches perfectly over an interval of length K. The curves deonstrate that the ACF criterion is reasonably accurate at indicating the order of the Markov odel that approxiates the fading odel. Markov odels ay not be practical for very slowly fading channels since the nuber of states grows exponentially with K and large data sizes are necessary to paraeterize the odel. Fig. 1(c) illustrates that the odel with non-observable states appears to be useful for approxiating very slowly varying fading, since it provides a far ore copact paraeterization. Fig. displays a siilar coparison for the case E s =N =5dB, f D T =:. It is observed that the ACF of the odel decreases ore rapidly with when copared to Fig. 1 (b), indicating a potential to reduce the order of the Markov approxiation. In order to verify the order K indicated by the ACF ethod using a different perspective, we calculate the variational distance between the n-diensional target easure P (e n ) given by (3) and the easure obtained by the Kth-order Markov odel, naely, P (K) (e n ), which is calculated using (8). The atrices P() and P(1) are described in Section III. The variational distance is defined as: d v (P (e n ) ;P (K) (e n )) = e n jp (e n ) P (K) (en )j: Figure 3 reports the variational distance versus the order K for several values of f D T, for E s =N =15dB (a), E s =N =5dB (b). Note that a lower distance value indicates a ore accurate odel. We say that the order of the Markov chain is K, when the distance converge to approxiately a constant value (roughly zero) for K K. The orders indicated by the convergence of the variational distance, for the range of fading environents investigated, are consistent with those obtained by the ACF ethod. The choice of value K, as entioned before, is soewhat arbitrary and Table I suarizes the order indicated by Fig. 3. We notice that, for slow and ediu rate fading, the increase of the signal to noise ratio fro 15 to 5 db reduces the order of the Markov odel K by 1. V. CONCLUSIONS We have developed FSC odels that characterize the error sequence of a counication syste operating over a fading (a) K= K= K=3 K= K=5 K= (b) K= K= K=3 K= K=5 K= (c) Fig. 1. Coparison of the autocorrelation functions of the fading odel, the Kth-order Markov odel (K = ; 1; ; 6), and the odel. The odel is Rayleigh fading (K R =), with E s=n =15dB, and f D T =.1 (a), f D T =. (b), f D T =.1 (c).

6 R[] International Telecounications Syposiu ITS, Natal, Brazil K= K= K=3 K= Fig.. Coparison of the autocorrelation functions of the fading odel, the Kth-order Markov odel (K = ; 1; ; 6), and the odel. The odel is Rayleigh fading (K R = )with E s=n = 5dB, f D T =.. variational distance variational distance f D T = K (a) f D T = K (b) Fig. 3. Variational distance versus the order K having f D T as a paraeter. Rayleigh fading (K R =), E s=n =15dB (a), E s=n =5dB (b). channel. Markov odels of order up to 6 have been proposed as an approxiation to the odel for a broad range of fading environents. We have used two criteria to estiate the order of the Markov process: The autocorrelation function, and the variational distance. Both criteria lead to a siilar conclusion that the Kth-order Markov odel is a good approxiation to the odel. This analysis reinforces the results in [7] regarding the effectiveness of ACF criterion to estiate the order of Markov odels. It is observed that the first-order approxiation is satisfactory for values of f D T around.1. For fast and ediu fading rates (f D T > :3), the Kth-order (for judiciously selected K) is as accurate as the odel. For slower fading rates (f D T < :) Markov odels of order greater than 6 are required. REFERENCES [1] F. Swarts and H. C. Ferreira, Markov characterization of digital fading obile VHF channels, IEEE Transactions on Vehicular Technology, Vol. 3, pp , Nov [] H. Wang and N. Moayeri, Finite-State Markov channel - A useful odel for radio counication channels, IEEE Transactions on Vehicular Technology, vol., pp , Feb [3] H. Wang and P. Chang, On verifyng the first-order Markovian assuption for a Rayleigh fading channel odel, IEEE Transactions on Vehicular Technology, vol. 5, pp , May [] M. Zorzi, R. Rao, and L. B. Milstein, ARQ error control for fading obile radio channels, IEEE Transactions on Vehicular Technology, vol. 6, pp. 5-55, May [5] Q. Zhang and S. Kassa, Finite-State Markov odel for Rayleigh fading channels, IEEE Transactions on Counications, vol. 7, pp , Nov [6] F. Babich and G. Lobardi, A Markov odel for the obile propagation channel, IEEE Transactions on Vehicular Technology, vol. 9, pp , Jan.. [7] C. Tan and N. C. Beaulieu, On first-order Markov Modeling for the Rayleigh Fading Channel, IEEE Transactions on Counications, vol. 8, pp. 3-, Dec.. [8] F. Babich, O. Kelly, and G. Lobardi Generalized Markov Modeling for Flat Fading, IEEE Transactions on Counications, vol. 8, pp , Apr.. [9] C. Pientel and I. F. Blake, Modeling burst channels using partitioned Fritchan s Markov odels, IEEE Transactions on Vehicular Technology, Vol. 7, No. 3, pp , Aug [1] L. Wilhelsson e L. B. Milstein, On the effect of iperfect interleaving for the Gilbert-Elliott channel, IEEE Transactions on Counications, Vol. 7, pp , May [11] H. Turin and R. van Nobelen, Hidden Markov Modeling of Flat Fading Channels, IEEE Journal on Selected Areas in Counications, vol. 16, pp , Dec [1] M. Zorzi, Soe results on error control for burst-error channels under delay constraints, IEEE Transactions on Vehicular Technology, Vol. 5, pp.1-, Jan. 1. [13] C. Pientel and I. F. Blake, Enueration of Markov chains and burst error statistics for finite state odels, IEEE Transactions on Vehicular Technology, Vol. 8, pp. 15-8, Mar [1] C. Pientel and I. F. Blake, Concatenated coding perforance for FSK odulation on tie-correlated Rician fading channels, IEEE Transactions on Counications, vol. 6, pp , Dec [15] G. Shara, A. Hassan, and A. Dholakia, Perforance evaluation of burst-error correcting codes on a Gilbert-Elliott channel, IEEE Transactions on Counications, vol. 6, pp , Jul [16] R. H. Clarke, A statistical theory of obile-radio reception, Bell Syste Tech. Journal, vol. 7, pp , [17] M. Gans, A power-spectral theory of propagation in the obile radio environent, IEEE Transactions on Vehicular Technology, vol. 1, pp. 7-38, Feb [18] T. W. Anderson and L. A. Goodan, Statistical Inference about Markov Chains, Annals of Matheatical Statistics, vol. 8, pp , Mar