Generalised FSMC model for radio channels with correlated fading

Y.L.Guan and L.F.Turner

Abstract: Finite-state Markov channel (FSMC) models are useful for analysing radio channels with nonindependent fading, which give rise to bursty channel errors. In the paper the FSMC model is constructed by partitioning the dynamic range of the fade amplitude into finite number of intervals and representing each interval as a ‘state’ in the model. Transitions between these channel states thus characterise the physical fading process. Based on the Nakagami-rn distribution, a new analytical formulation for the state transition probability is presented and validated for channels with arbitrary degrees of time-interleaving. The resultant FSMC model is shown to be applicable to a wide range of practical fading channels, including diversity-combined channels and multicarrier channels.

1 Introduction

Finite-state Markov chains have been widely used in the analysis of radio channels exhibiting correlated fading [l- 51. Channels of this type are characterised by multipath in which the fade amplitudes at successive epochs of interest are not strictly independent of one another. Such channels are also known as ‘channel with dependent fading’. The resulting channel model is often termed a finite-state Markov channel (FSMC) model. Conventional FSMC approaches have centred on modelling the ‘error process’ observed at the receiver output, by statistically characteris- ing the ‘birth’ and ‘death of error-burst and error-free events [2]. A more modern approach [l] focuses on the root rather than the consequence of the problem, by modelhng the physical dynamics of the fading process rather than the error process. Since the signal fading mechanism contrib- utes to the generation of bursty channel errors, the modem model is capable of generating the conventional models [4], thus it is a more generalised model. This paper is concerned with the modem model.

An example of a FSMC model for a noninterleaved fad- ing process is sketched in Fig. 1. Basically, the dynamic range of the fade amplitude, r, is partitioned into several nonoverlapping intervals, for example, LO, rl), [rl, r2), ..., [TA-,, w). The fading channel is said to be in channel state ‘U’ if the instantaneous fade amplitude falls into the interval [r,-l, Y,). During the transmission of a sequence of data symbols, the channel takes on different channel states dur- ing different symbol epochs and makes transitions from one state to another according to the physical fading proc- ess. If the fading process is wide-sense stationary (i.e. the

OEE, 1999 IEE Proceedings online no. 19990130 DOL 10.1049/com:19990130 Paper fmt meived 12th January and in revid form 14th October 1998 Y.L. Guan was with the Imperial College of Snence, Technology and Medi- cine, and is now with the School of EEE (Block Sl), Nanyang Technological University, Nanyang Avenue, Singapre 639798 L.F. Turner is with the Deparlment of Electrical and Electronic Engineering, Imprial College of Science, Technology and Mdcine, Exhibition Road, Lon- don SW7 2BT. UK

fading statistics remain unchanged over the short term [6]), the overall equilibrium probability of the channel being in any state and the transition probabilities between states are well defined and are such that [l]

P T = P (1)

where: p = row vector with elements p(a), the equilibrium proba- bility of channel state ‘U’ T = matrix with elements p(bla), the transition probability from channel state ‘U’ to ‘b’ a, b E {I , 2, ...) A ) A = total number of channel states

time

Fig. 1 Finite-stute Markov-chin model of u noninterleavedfading c h l

It is p(bla) that characterises the degree of fading correla- tion in the channel. In a mobile radio channel, p(&) depends on many physical parameters such as the vehicular speed, carrier wavelength, channel symbol duration, overall fading statistics etc. Although p(bla) can be obtained by simulation ([l, 4]), an analytical formulation is highly desir- able.

IEE Proc-Commun., Vol. 146, No. 2, April 1999 133

In [ 1, 71, approximate analytical expressions for p(blu) are derived based on level-crossing rate considerations, under the assumption that no channel state makes transitions beyond its immediate neighbouring states, that is, p(blu) s 0 if (b - a( > 1. The resultant FSMC model is hence valid for nonin terleaved channels with slow fading (low Doppler fre- quency to symbol-transmission rate ratio) only. Moreover, since Rayleigh fading statistics are assumed, the model is also inadequate for representing many practical channels with non-Rayleigh statistics, such as line-of-sight channels and diversity-combined channels ([8, 91). In the following account, a new analytical p(&) formulation that renders the FSMC model applicable to a wider variety of corre- lated-fading channels wdl be presented and validated. It will be shown that, besides conventional time-interleaved channels, the proposed formulation can also model some diversity-combined channels and multicarrier channels, thus offering more flexibility than the existing formulations

2 Formulation of state transition probabilities

Based on probabihty and conditional probability concepts, the equilibrium channel state probability, p(u), and state transition probability, p(blu), can be expressed as follows:

p ( a ) = Pr(r,-l 5 r < r,)

of[1, 71.

= 1 pdf,(r)dr ( 2 ) T a - 1

- Pr ( n - 1 5 r < rb,r,-~ 5 f < r,) - P(a)

T b T a

j d f T , f ( r , f ) d r d f

(3) T b - 1 ? - - - I - -

7 pdfr (r )dr T a - l

where r, ? = fade amplitudes of concern ra-17 ra = lower and upper boundaries of fade-inter-

PrP) = probability that the event occurs PrP, A) = probability that both events and A occur P d W ) = probability density function of r jdfr,r(., A) =joint density function of r and r“

The overall fading statistics wdl be modelled by the Nak- agami-m distribution [lo], which is a generalised fading model that includes the Rayleigh distribution as a special case and approximates the Rician and lognormal distribu- tions [I 11. It also provides greater flexibllity in fitting empir- ical measurements [ 1 11 and gives useful representations for the signal statistics of many diversity-combining systems [lo]. The probability and joint density functions of Nakag- ami-m distribution can be obtained from [lo]. @!fir, r ” ) is obtained by setting Ql, Q2 + Plm; RI - r; R2 - r” and p2 + pin eqn. 126 of [lo]).:

Val corresponding to channel state ‘U’

(5) where

m = P2/E[(? - P)2] re) = Euler gamma function h{*} = mth-order moditied Bessel function of the first

p = correlation coeficient between r and r” For narrowband land-mobile channels with isotropic

scattering and Rayleigh fading (i.e. Nakagami fading with m = l), the correlation coefficient p between two fade amplitudes separated in time by t seconds has been estab- lished to be [6, 91:

where .I0[.] = zero-order Bessel function of the first kind fd = Doppler frequency (= vehicular speedkarrier wave-

In noninterleaved channels, z is equal to the channel sym- bol duration. In time-interleaved channels, z is equal to the channel symbol duration x interleaving depth [8].

Substituting eqn. 4 into eqn. 2, a closed-form expression for p(u) can be obtained using ‘Mathematica’ (a symbolic mathematical software package):

P = E[?] = E[P]

kind

P = J,[27&7] (6)

length)

r m , ~ , m x 2 r m , ~ , m x .>- ( c1 1 r (m)

( P(a) =

(7 ) where re, A, +) is the generalised incomplete gamma function. Unfortunately, the double integral in the numera- tor of eqn. 3, after being substituted with eqn. 5, does not have a simpler form, so p(blu) has to be evaluated numeri- cally.

3 Validation

3. I Noninterleaved channels Simulation data (from Table I1 of [l]) can be used to vali- date the proposed p(blu) expression on noninterleaved channels with slow fading. Pertinent simulation parameters include: m = 1 (i.e. Rayleigh fading) z = channel symbol duration = lC5s fd = 10 and 100. Since eight equi-probable channel states were used in the simulation, the lower (ra-l) and upper (r,) boundaries of the channel states can be obtained by setting all p(a)s to 1/8 and solving for the state boundaries, starting from ro = 0 and ending with r8 = W. Numerical computation of the proposed generalised p(b(u) values can then follow using eqn. 3 4 inclusive. Without loss of generality, the average fading signal power is set to unity (i.e. P = 1).

In the simulation trials, it was found that the p(alu) val- ues (i.e. the self-transitioning probabilities) tend to suffer from numerical instability due to exploding values of the exponential and Bessel terms in eqn. 5. This problem can however be bypassed by first evaluating all the other p(blu) values before subtracting their sum from 1 to obtain p(alu), that is,

p(ala) = 1 - C P ( b l 4 (8) a l l b b f a

IEE Proc-Commun., Vol. I46, No. 2, April 1999 134

The values of p(blu) thus obtained for f d = lOOHz are listed in Table 1 alongside the simulated values and approximate analytical values from [l]. It is found that the proposed analytical p(b)u) values match the simulated data better than the approximate analytical values of El]. For the case of fd = lOHz, both sets of analytical p(blu) values agree exactly.

Table 1: Channel-state transition probabilities of a noninter- leaved Rayleigh fading channel with channel symbol dura- tion of IO-%, fd = 100Hz

p (blab Proposed Approximate [ I ] Simulated [I1

p ( l l 1 ) 0.993546 0.993588 0.993198

p(112)

P(211)

p (212)

P (213)

p (312)

p (313)

P (314)

P (413)

P (414)

P (415)

P (514)

P (515)

P (516)

P (615)

P (616)

P (617)

P (716)

P (717)

P (718)

P (817)

0.006454

0.006454

0.985426

0.008 120

0.008 120

0.983231

0.008649

0.008649

0.982948 0.008403

0.008403

0.984101

0.007497

0.007497

0.986562

0.005942

0.005942 0.990420

0.003638

0.003638

0.006412

0.006412

0.985521 0.008067

0.008067

0.983341 0.008592

0.008592

0.983060

0.008348

0.008348

0.984205

0.007447

0.007447

0.986650

0.005903

0.005903 0.990483

0.00361 5

0.003615

0.006774

0.006802

0.984704 0.008570

0.008523

0.982362 0.008973

0.009068

0.982441

0.008705

0.008586

0.983559

0.007672

0.007736

0.986202

0.006295

0.006 126

0.989790

0.003878

0.003914

p (818) 0.996362 0.996385 0.996 122 Zero-valued p(bla)s are not shown

3.2 Time-interleaved channels To validate the proposed p(bla) expression on time-inter- leaved Rayleigh fading channels, separate Monte Carlo simulations are carried out using the fading simulator described in [9]. Channel symbol duration was maintained at lk5s and a Doppler frequency of lOOHz was assumed. The resulting p(blu) values for an interleaving depth of 30 channel symbols are shown in Table 2 together with corre- sponding predictions of the proposed p(blu) formulation. It is found that the analytical and simulated data agree rea- sonably well. It is further noted that the channel states are now beginning to make transitions beyond their immediate neighbouring states, a situation also expected of a noninter- leaved channcl with fast fading, but which cannot be ade- quately modelled by the approximate formulation of [I].

The accuracy of the proposed p(bla) expression at other interleaving depths can be investigated in a more compact manner by comparing the state-transition entropies of the proposed and simulated FSMC models. Denoted by H(blu), the state-transition entropy of a FSMC model is herein defined as follows:

a b

The comparison results are given in Fig. 2, which shows that, with eight equiprobable channel states, discrepancies

IEE Proc.-Commun., Vol. 146, No. 2, April 1999

2nfdT

Fig.2 channel with Ruyleigh fading and IR5s channel symbol hut ion -x- proposed -0- simulated Boxed numbers indicate the interleaving depth

Simulated and unalyticul state-transition entropies of a mobile radio

Table 2: Channel-state transition probabilities of an inter- leaved Rayleigh fading channel with channel symbol dura- tion of 10%. fd = 100Hz

~ ~~

p (bla) Proposed Simulated (after [91)

0.8091

0.1835

0.0074 0.1835

0.5828

0.2188 0.0149

0.0074

0.2188

0.5318

0.2261 0.0160

0.0149

0.2261

0.5253

0.2224

0.01 14

0.0160

0.2224

0.5501

0.2072

0.0044 0.01 14

0.2072

0.6091 0.1721

0.0003

0.0044

0.1721

0.7 156

0.1079

0.0003

0.1079 0.8918

0.8179

0.1700

0.0049

0.1771

0.6095

0.2046

0.0102

0.0050

0.2101 0.5624

0.2138 0.0104

0.0104

0.2173

0.5540

0.2091

0.0067

0.0108

0.2149

0.5815

0.1900 0.0017

0.0072

0.1972

0.6457

0.1529 -

0.0019 0.1576

0.7512

0.1034 -

0.0942 0.8966

Zero-valued p(bla)s are not shown

135

between the analytical and simulated p(blu) data remain small and bounded at all interleaving depths. These dis- crepancies can be reduced by increasing the number of sim- ulation runs and channel states (i.e. quantisation levels).

3.3 First-order versus higher-order Markov process Strictly speaking, the output sequence of a channel receiver should follow a higher-order Markov process, where some channel states depend not only on those immediately pre- ceding them, but also on earlier ones [12]. This is a conse- quence of the chain expansion rule in multivariate probability distribution theory [8]:

P ( S n , sn-I, ’ . . , s2, s1) = P ( S n ( S n - 1 , . . f , s 2 , s1)

x P(%-1 Isn-2,. . . 1 a ) x . ‘ . x P(S2lSl) x P(S1)

(9) where sk denotes the channel state of the kth receiver out- put symbol.

For noninterleaved channels with slow fading, it has been shown in [12] that all higher-order conditional proba- bility terms can be adequately approximated by first-order terms, that is,

P(SklSk-1, SIC-2, . . .) 2 P(SklSk-1) (10) Therefore, the first-order Markov formulation proposed herein is suficient for characterising these channels. How- ever, for time-interleaved channels with large interleaving depths, or equivalently, noninterleaved channels with fast fading, eqn. 10 is no longer valid. Nonetheless, to render the proposed first-order Markov formulation usable on these channels, the p ( ~ ~ l s ~ - ~ , sk-2, ...) term can be expanded by introducing intermediate channel states in between adja- cent symbol epochs of the actual de-interleaver output, as illustrated below:

P(SklSk-l,Sk-2,. . .)

(11) In the above example, .sk-k-f denotes the intermediate chan- nel state which prevails at a time instant that is midway between the kth and (k - I)th de-interleaved symbol epochs. By virtue of the smaller time separation involved, first-order Markov approximation becomes more justified for the channel-state pairs (sk, sk-f) and (sk-f, In prin- ciple, more and more of such ‘channel-state interpolation’ can be carried out to minimise the errors introduced by invokmg first-order Markov approximation on the p ( s k l ~ ~ - - ~ , sk-2, ...) terms. However, computation round-off errors may start accumulating as more and more interme- diate conditional probability terms with near-zero or near- one magnitudes are inserted, multiplied and summed together. Computational load is also expected to increase with each additional level of interpolation introduced. Obviously, a balance needs to be struck depending on the requirement of the problem at large, but this is not within the scope of this paper.

136

4 Other applications

It is not dificult to see that the proposed FSMC model can be used on any type of time-varying channels so long as the pertinent correlation coef€icient, p, is known and the overall statistics of the channel output can be modelled after the generalised Nakagami distribution. Two examples will be elaborated in this Section.

4. I Channels with equal-gain diversity combining Linear diversity combining is an established technique for mitigating the undesirable effects of Rayleigh fading [9]. In systems with equal-gain diversity combining, L uncorre- lated copies of the signal of interest (obtained from, say, L suitably spaced receive-antennas) are directly summed together to produce a single output. Since it is unlikely for all signal copies to fade simultaneously, the combined out- put signal will have the advantages of not only an increased average power but also a non-Rayleigh fading distribution. More generally, if each of the L diversity branches follows a Nakagami-m distribution with m-parameter mo and aver- age power Po, their equal-gain-combined output will approximately follow a new Nakagami-m distribution with the following parameters (see Section 4.2 of [lo]. Note also that the cumulative distributions of other types of diversity- combined outputs can he found in Section 4.3 of [lo]).

m S Lmo (12)

This approximation has been shown in [I31 to be ade- quately tight.

Next, the fading-amplitude correlation coefficient of the diversity-combined output must be determined. Let two i.i.d. (independent and identically distributed) diversity branch signals be yl(t) and y2(t), and let their equal-gain combined output be y(t) = yl(t) + y2(t). In a wide-sense sta- tionary environment,

E[YI (t)] = E [ y i ( t + 7-11 1 E [ ~ a ( t ) ] = E[ya(t + 7-)]

(14)

E[Y(t)l = E[Yl(t) + Y2(t)l = 2E[Yl(t)l = E [ d t + 41 (15)

where 4.1 = statistical mean of 0. Hence,

Assuming that all diversity branches have the same time- autocorrelation characteristics, then

E [ Y ( t ) x Y ( t + 7-11

= E { [Y l ( t ) + Y2 (t)l x IYl (t + 7-1 + 92(t + 791) = 2E[Y1 ( t ) x y1 (t + 7-)I + 2E2[y1 (t)]

(16) Based on eqn. 1416, the correlation coefficient of y(t) works out to be

E[Y(t)Y (t + 7-11 - E2 [Y (t>l E[Y2(t)l - E2[Y(t)l - - ~ E [ Y I (~)YI (t + 7-11 + 2E2 [YI (t)l - 4E2 191

WYWI + 2 ~ ~ [ ~ ~ ( t ) l - 4E2[yl @)I

E[Y?(t)l - E2[Yl(t)l - - E [ Y l ( t ) Y l (t + 7-11 - E2[Y1 (t)l

(17) whch is exactly the correlation coefficient that prevails before equal-gain combining. It can now be readily

IEE Proc-Commun., Vol. 146, No. 2, April 1999

deduced that, regardless of the number of diversity branches used, the correlation coefficient of a mobile radio signal with equal-gain combining remains unchanged from eqn. 6, so long as the branch signals are i.i.d. and have the same time-varying statistics.

Having ascertained the fading statistics and correlation coefficient of the combined output, its FSMC model can be constructed using eqn. 2 4 .

4.2 Multicarrier transmission In some communication systems, the information data stream is transmitted over multiple frequency bands in var- ious manner. In FH-SS (frequency-hopping spread spec- trum) systems, for instance, the entire information data stream is transmitted on a carrier which changes its centre frequency from time to time [8]. In OFDM (orthogonal fre- quency division multiplexing) systems, on the other hand, the information data is first split into several parallel sub- streams of lower rates before being transmitted concur- rently by an equivalent number of orthogonal frequency sub-carriers [14].

Since a change in instantaneous carrier frequency is unlikely to aff‘ect the overall fading statistics, the receiver output symbols of these multicarrier systems can be charac- terised by a fixed Nakagami distribution with an appropri- ate m-parameter (say, m = 1 if Rayleigh fading is observed). However, the fading-amplitude correlation coef- ficient p between any pair of adjacent output symbols will have a more elaborate form than eqn. 6, as shown below 191:

where Af = difference in centre frequencies CT = root mean square multipath delay spread In any case, eqn. 18 reduces to eqn. 6 when Af = 0, that is, in the case of a conventional single-carrier system.

5 Conclusions

In this paper, finite-state Markov channel (FSMC) models, with ‘states’ representing different intervals of fading ampli- tude, are used to characterise the dynamic amplitude varia- tions of time-varying multipath fading channels. A new analytical formulation for the state transition probability of the FSMC model is derived based on the Nakagami-m dis- tribution. Its validity for channels with arbitrary degrees of time-interleaving is verified by means of computer simula- tions as well as by comparison with some published data. The applicability of the resultant FSMC model to other forms of fading channels, such as equal-gain-combing

diversity channel and multicarrier channel, is also illus- trated.

FSMC models are useful for analysing the performance of communication systems operating over radio channels with nonindependent fading, which gives rise to bursty channel errors. By virtue of its analytical formulation, the proposed FSMC model can be constructed quickly without the need for intensive computer simulations [4] or laborious empirical measurements, and the effects of pertinent system parameters on the model can be readily deduced. W e other analytical models generally restrict the channel states from making transitions beyond the immediate neighbour- ing states [l, 71, the model proposed herein does not suffer the same constraint. These desirable attributes, coupled with the flexibility to model a wide range of practical com- munication channels such as diversity-combined and multi- carrier channels, render the proposed FSMC model considerably more versatile than those currently found in the open literature.

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