A New Simple Model for Land Mobile Satellite Channels

A New Siple Model for Land Mobile Satellite Channels A. Abdi, W. C. Lau, M.-S. Alouini, and M. Kaveh Dept. of Elec. and Cop. Eng., University of Minnesota, Minneapolis, MN 55455 Eails: {abdi, wlau, alouini, kaveh}@ece.un.edu Abstract - In this paper we propose a new shadowed Rice odel for land obile satellite channels. In this odel the aplitude of the line-of-sight is characterized by the Nakagai distribution. The odel provides closed-for atheatically-tractable expressions for the fundaental channel statistics such as envelope probability density function and envelope-squared oent generating function. The odel is very convenient for analytical and nuerical perforance prediction of coplex narrowband and wideband land obile satellite systes, with different types of uncoded/coded odulations, with or without diversity. Coparison of the new odel with published channel data deonstrates the flexibility of the odel in characterizing a variety of channel conditions and propagation echaniss. I. INTRODUCTION Land obile satellite (LMS) systes are an iportant part of the third and fourth generation of wireless systes. The significance of such systes is rapidly growing for a variety of applications such as navigation, counications, broadcast, etc. LMS systes provide services which are not feasible via land obile terrestrial (LMT) systes. As a copleent to LMT systes, LMS systes are able to serve any users over a wide area with low cost. For extensive surveys on LMS systes and services, we refer the readers to [1] [] [3]. The quality of service provided by LMS systes strongly depends on the propagation channel between satellite and the obile user. An accurate odel for the LMS channel is required for calculating fade argins, assessing the perforance of odulation and coding schees, analyzing the efficiency of counication protocols, and so on. The unwanted rando fluctuations of the signal envelope in a LMS channel can be attributed to two types of fading: ultipath fading and shadow fading [4]. In an ideal LMS channel without any types of fading, where there is a clear and clean line-of-sight (LOS) between satellite and user without any obstacle in between, the envelope is a nonrando constant (at a given instant of tie). Due to ultipath fading, caused by weak scattered coponents propagated via different non-los paths, together with the nonblocked LOS coponent, the envelope becoes a Rice rando variable. Shadow fading coes fro the coplete or partial blockage of the LOS by buildings, trees, hills, ountains, etc., which in turn akes the aplitude of the LOS coponent a rando variable. This gives rise to the shadowed Rice distribution for the envelope in LMS channels. Aong the proposed odels for LMS channels, the shadowed Rice odel proposed originally by Loo [5] has found wide applications in different frequency bands such as the UHF-band [6], L-band [6] [7], S-band [8], and Ka-band [6]. In Loo s odel, the aplitude of the LOS coponent is assued to be a lognoral rando variable. However, as discussed in [9], the application of the lognoral distribution for characterizing shadow fading ost often results in coplicated expressions for the key first and second-order channel statistics such as envelope probability density function (PDF) and envelope level crossing rate (LCR), respectively. Matheatical anipulation of those expressions is usually hard, as they cannot be written in ters of known atheatical functions. This, in turn, akes data fitting and paraeter estiation for the lognoral-based odels a coplex and tie-consuing task. Perforance analysis of counication systes, e.g., interference analysis or calculation of the average bit error rate (BER) for single and ultichannel reception, is even uch ore difficult for the lognoral-based odels, as soeties even the nuerical procedures for these odels fail to give the correct answer. On the other hand, as is conjectured in [9], application of the gaa distribution, as an alternative to the lognoral distribution, can result in sipler statistical odels with the sae perforance for practical cases of interest. For exaple, it is shown in [1] that for LMT channels, the gaa-based K distribution is uch sipler than the lognoral-based Suzuki distribution, with nearly the sae perforance. In this paper, we assue the power of the LOS coponent is a gaa rando variable. According to the siple relationship between the gaa and Nakagai distributions [11], this eans that we are odeling the aplitude of the LOS coponent with the Nakagai distribution. As we will see in the sequel, a Rice PDF with Nakagai-distributed LOS aplitude constitutes a versatile odel which not only agrees very well with easured LMS channel data, but also offers significant analytical and nuerical advantages for syste perforance prediction. The rest of this paper is organized as follows. In Section II the statistical properties of the odel, i.e., PDF, oents, and oent generating function (MGF) are derived. Section III is devoted to establishing a connection between the paraeters of Loo s odel and our odel. In Section IV, the new odel is copared with easured data. Different types of syste perforance assessents, such as outage probability of systes subject to interference and BER calculation of uncoded/coded odulations with/without diversity, are discussed in Section V. The paper concludes with a suary given in Section VI. II. STRUCTURE AND STATISTICS OF THE NEW MODEL Using the sae notation as [5], the shadowed Rice PDF for the signal envelope in a LMS channel can be written as r r + Z Z r pr ( r) = EZ exp I, r, (1) b b b where R is the envelope, E Z[.] is the atheatical expectation with respect to Z, b is the total power of ultipath coponents, Z is the aplitude of the LOS coponent, and I (.) is the zero-order odified Bessel function of the first kind. In this paper, we odel the rando variable Z with the Nakagai PDF, i.e. z p ( ) 1 Z z = z exp, z Γ( ) Ω Ω, () in which is the shape paraeter and Ω = E[ Z ] is the scale paraeter. The reader should notice that in the traditional Nakagai odel for ultipath fading [11], has the liited range of.5, while here we allow to vary over a wider range, i.e.,. This

enables the Nakagai PDF to odel different types of LOS in a variety of LMS channels [4]. For = we have pz ( z) = δ ( z), with δ (.) as the Dirac delta function, which corresponds to urban areas with coplete obstruction of the LOS. The case of < < is associated with suburban and rural areas with partial obstruction of the LOS. For = we have pz ( z) = δ ( z Ω ), which corresponds to open areas with no obstruction of the LOS. Of course the abstract cases of = and = cannot be et in practice, and in real-world situations we expect nonzero sall and finite but large values of for urban and open areas, respectively. The oderate values of corresponds to suburban and rural areas. By calculating the expectation in (1) using [1], and after soe algebraic anipulations, we obtain the new envelope PDF as b r r pr ( r) = exp b + Ω b b (3) Ω r 1F1, 1,, r, b ( b + Ω) where 1 F 1 (.,.,.) is the confluent hypergeoetric function [13]. For =, Eq. (3) siplifies to the Rayleigh PDF ( r b )exp( r b ), while for = it reduces to the Rice PDF ( r b )exp( ( r + Ω) b ) I( Ω r b ). In contrast with Loo s PDF, which is an infinite-range integral, Eq. (3) has a copact for in ters of the tabulated function 1 F 1 (.,.,.), also available in standard atheatical packages such as Matheatica and Maple for both nuerical and sybolic operations. Using [1], The oents of the proposed PDF can be shown to be b k = b k Γ + b E k R ( ) 1 + Ω (4) k Ω F1 + 1,, 1,, k =, 1,,..., b + Ω where Γ (.) is the gaa function, and F 1 (.,.,.,.) is the Gauss hypergeoetric function [13]. As is shown in [14], the MGF of the instantaneous power, defined by S = R, plays a key role in calculating the BER and sybol error rate (SER) of different odulation schees over fading channels. In our odel, the MGF of S, defined by M S ( η) = E[exp( η S)], η, can be found with the help of [1] as M ( b ) (1 + b η) 1 S ( η) =, η ( b + Ω )(1 + bη ) Ω. (5) The siple atheatical for of the above MGF in our odel entails very straightforward perforance evaluation procedures even for ultichannel reception of coplicated odulation/coding schees in LMS systes. On the other hand, the MGF of S in Loo s odel can be expressed at ost in ters of an infinite-range integral or a double infinite su [15]. Since the nuerical anipulation of both representations of the Loo s MGF is tie consuing, several approxiate expressions are proposed for different cases [15]. III. CONNECTIONS BETWEEN THE NEW & LOO S MODEL In Loo s odel, the LOS aplitude has a lognoral PDF [5] ( ln z µ ) 1 pz ( z) = exp, z. (6) π d z d With t as a positive real nuber, it is easy to show that for Nakagai and lognoral PDFs in () and (6) we have E[ Z t ] given respectively by Eqs. (7) and (8) as 1 Ω Ψ ( ) ln E t Z = ln + Ψ ( ) t + t 8 Ψ ( ) Ψ ( ) + t3 + t 4+..., 48 384 d ln E t Z = µ t + t, (8) where Ψ(.), Ψ (.), Ψ (.), and Ψ (.) are the psi function and its derivatives, respectively [13]. The absolute values of the psi function and its derivatives converge to zero very fast, as increases. By second-order atching of the two expressions in (7) and (8), the following relationship between the two sets of paraeters (, Ω ) and ( µ, d) can be established 1 Ω µ = ln + Ψ ( ), (9) Ψ ( ) d =. (1) 4 For a given d, the corresponding can be easily obtained by solving the equation in (1), nuerically. The value of Ω can be calculated by inverting the equation in (9), which yields Ω = exp[ µ Ψ ( )]. As we will see later, Loo s distribution and our distribution closely atch, if one coputes our paraeter set ( b,, Ω ) fro the Loo s paraeter set ( b, µ, d), using the relations given in (9) and (1). This is particularly useful when we wish to apply the new odel with unknown paraeters to a set of data collected previously, but the easured data is not available for paraeter estiation, or we ay not want to go through the tie-consuing procedures of paraeter estiation. In these cases, the paraeters of our odel can be siply obtained fro the estiated Loo s paraeters, using (9) and (1). IV. COMPARISON WITH PUBLISHED MEASUREMENTS In this section we consider two sets of published Loo s paraeters ( b, µ, d) [5] [16]. These paraeter values are listed in Table I and Table II, respectively, together with the paraeters ( b,, Ω ) of the proposed odel, coputed using (9) and (1). Although the entries for light and heavy in Table I and Table II are identical, we have listed Table II copletely, as Loo s paraeters, given in the table, have been used in several studies such as [15] [16] [17] [18] [19], for syste analysis and perforance prediction purposes. As we expect fro the theory, values in Table II decrease as the aount of increases fro light to average, and then to heavy. This epirical observation verifies the key role of the Nakagai paraeter in our odel, discussed at the beginning of Section II, in odeling different types of shadow fading conditions. In Fig. 1 and Fig., we have plotted the envelope copleentary cuulative distribution functions (CCDFs), r pr ( x) dx, for Loo s PDF and our PDF, given in (3). Interestingly, all of Loo s curves and our curves are alost indistinguishable and both are close enough to the easured data, for different cases and channel conditions. These epirical results indicate the utility of our odel for LMS channels. Also note the usefulness of the paraeter transforation rules given in (9) and (1), which gives alost perfect atch between Loo s CCDFs and ours. (7)

As is discussed in [], a LMS channel odel should be applicable for a wide range of elevation angles, under which the satellite is observed. One way of incorporating the effect of the elevation angle in a statistical LMS channel odel is to derive epirical expressions for the paraeters of the envelope PDF in ters of the elevation angle []. To deonstrate this procedure for our odel, we have considered the experiental data published in [1], also used in [], and have derived the following relationships o o by fitting polynoials over the range < θ < 8 b ( ) = 4 7943 1 + 5 5784 1 1344 1 + 3. 71 1, 8 3 6 4 3 4 1 ( ) = 6 3739 1 5 8533 1 1 5973 1 3 5156, Ω = -5 3-3 -1 ( ) 1 448 1 3798 1 1 7 1 1 4864. The proposed PDF in (3), in conjunction with the above equations, copose a hybrid statistical/epirical odel. The epirical and the theoretical CCDFs of the new odel are plotted in Fig. 3 for different elevation angles. V. PERFORMANCE EVALUATION OF LMS SYSTEMS In this section we focus on two types of syste perforance evaluation: BER calculation of uncoded and coded odulations with diversity reception, and interference analysis of LMS systes. In both cases we clearly observe the utility and flexibility of the proposed odel for perforance evaluation purposes. A. BER of coded and uncoded odulations with diversity Define the instantaneous signal-to-noise ratio (SNR) per bit of the th branch as γ = ( Eb N) S, where E b is the energy per bit and N is the power spectral density of the additive white Gaussian noise. As an exaple of a coon diversity cobiner, consider a axial ratio cobiner (MRC) with L independent branches in a fading channel. With an MRC, the BER of QPSK and OQPSK, two coon odulations in LMS systes [] [3], is given by [14] π L γ = 1 1 1 Pb = M dθ π sin θ, (11) where Mγ ( η) = M S ( η Eb N). Note that the MGF of S in our odel, given by (5), is very convenient for the nuerical integration in (11), as the integrand is just a rational function, while the integrand in (11) for Loo s odel, according to the previous discussion, is either an L-fold infinite-range integral or a L-fold infinite-range su. The general scenario of independent branches with different fading statistics, as considered above, has practical iportance and is not just of theoretical interest. For exaple, in the application of spread spectru techniques such as DS-CDMA to LMS systes, to increase capacity, fading resistance, security, etc. [], the first arriving path of the wideband LMS channel can be odeled with a shadowed Rice distribution, while the reaining paths ay be odeled by Rayleigh distributions with different powers [18] [19]. To do soe nuerical exaples with (11), here we consider a wideband LMS channel with two paths, both distributed according to the new shadowed Rice PDF in (3). In one case we assue that the two paths are under light and average s, while in the second case the two paths are assued to be under average and heavy s. Based on (11) and the paraeters listed in Table II for the new odel, the BER curves of a two-finger MRC-RAKE receiver for these two cases are plotted in Fig. 4. As we expect, the BERs of the first case are saller because of the less severe conditions. For an MRC with L independent branches, all under the sae conditions and with the sae paraeters taken fro Table II, the BER curves are plotted in Fig. 5 according to (11). Now we consider two types of coding/odulation techniques for LMS channels: trellis-coded M-ary phase shift keying (MPSK) and Reed-Soloon (RS) coded MPSK. According to [15], we need the MGF of S to calculate the BER upper bounds for these two iportant schees. This again confirs the utility of our new odel with its siple MGF, given in (5), which akes the nuerical coputations for LMS channels very siple. B. Interference analysis The operation of two (or ore) satellites with the sae frequency and the siultaneous illuination of an area by two satellites with clear LOS ay be considered as two ajor sources of interference in LMS systes [3]. Let S d and S i represent the instantaneous power of the desired signal and the interfering signal, respectively. For a given protection ratio ζ, the probability of outage caused by the interfering signal is given by P = Pr[ S < ζ S ]. (1) outage d i It is reasonable to assue that the desired and interfering signals are independent. This yields sd ζ Poutage = 1 ps ( s ) ( ) d d ps s i i ds i dsd where based on (3), the PDF of S can be written as, (13) b 1 s S ( s) = exp b + Ω b b p (14) Ω s 1F1, 1,, s. b ( b + Ω) Nuerical calculation of the double integral in (13) is stable and straightforward, specially with a standard atheatical software such as Matheatica which has efficient built-in nuerical procedures for 1 F 1 (.,.,.) and any other special functions. On the other hand, upon the application of Loo s odel, (13) becoes a quadruple integral, which is definitely ore difficult to evaluate. As an exaple of the nuerical evaluation of (13), we consider four different interference scenarios in a LMS channel with an interfering satellite. In the first case we assue that both desired and interfering signals are under light s, while in the second case the interfering signal is under heavy. The other two cases can be defined as well. Using the paraeters of the new odel in Table II, the outage probability for these four cases are plotted in Fig. 6. As expected, the ost severe outage corresponds to the case where the desired and interfering signals are under heavy and light s, respectively. On the other hand, for the case in which the desired and interfering signals are under light and heavy s, respectively, we have the sallest outage probabilities. For the reaining two cases where the conditions are the sae for the desired and interfering signals, we observe an interesting result. For ζ > 1, the outage for these two cases are siilar, while for ζ < 1 the two curves diverge. VI. CONCLUSION In this paper a new Rice-based odel is proposed for land obile satellite channels, in which the aplitude of the line-of-sight is

assued to follow the Nakagai odel. We have shown that this new odel has nice atheatical properties, its first-order statistics can be expressed in exact closed fors, and is very flexible for data fitting and perforance evaluation of narrowband and wideband land obile satellite systes. Moreover, We have shown that the proposed odel fits very well to the published data in the literature, collected at different locations and frequency bands. A connection is also established between the paraeters of the proposed odel and those of the Loo s odel, a odel widely used for satellite channels. Based on this connection, the paraeters of the new odel can be easily derived fro the estiated paraeters of Loo s odel, reported in the literature. This obviates the task of paraeter estiation for the new odel in cases where estiated Loo s paraeters are available. We have also shown, through an exaple based on easured data, that the paraeters of the proposed odel can be related to the elevation angle. This allows the application of the new odel over a wide range of elevation angles. Siilar to its first-order statistics, the second-order statistics of the new odel, such as level crossing rate and average fade duration, can also be expressed in closed fors. However, those theoretical results, together with the epirical justifications, are oitted here due to space liitations. They are copletely reported in [4], which is an extended version of this paper. ACKNOWLEDGMENT The work of the first, the third, and the fourth authors have been supported in part by the NSF, under the Wireless Initiative Progra, Grant #9979443. The authors appreciate the input provided by Mr. C. Loo at Counications Research Center, Ottawa, ON, Canada, with regard to reproducing the epirical curves in Fig. 1. REFERENCES [1] W. W. Wu, E. F. Miller, W. L. Pritchard, and R. L. Pickholtz, Mobile satellite counications, Proc. IEEE, vol. 8, pp. 1431-1448, 1994. [] W. W. Wu, Satellite counications, Proc. 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Table I LOO S PARAMETERS [5] AND THE CORRESPONDING PARAMETERS OF THE NEW MODEL, CALCULATED FROM (9) AND (1) Infrequent light Frequent heavy ln( ) Loo s odel 1 µ ln( d ) The new odel b Ω.115.115.158 19.4 1.9-3.914.86.63.739 8.97 1-4 Overall results -.69.3.51 5.1.78 Table II LOO S PARAMETERS [16] AND THE CORRESPONDING PARAMETERS OF THE NEW MODEL, CALCULATED FROM (9) AND (1) ln( ) Loo s odel 1 µ ln( d ) The new odel b Ω Light.115.115.158 19.4 1.9 Average Heavy -.115.161.16 1.1.835-3.914.86.63.739 8.97 1-4

Received signal relative to line of sight value (db) 1 5 1 3 35 Measured values, light Measured values, heavy Measured values, overall results Loo s odel The proposed odel.1.1.3 1 5 1 5 5 75 9 95 98 99 99.799.9 99.99 Percent of tie received signal is greater than ordinate 1 paths, Light and average paths, Average and heavy 1 1 1 1 3 1 4 1 1 6 5 1 15 5 3 Signal to noise ratio (SNR) per bit E b /N o [db] Fig. 1. Copleentary cuulative distribution function of the signal envelope in a land obile satellite channel under different conditions: Measured data [5], Loo s odel [5], and the proposed odel. Fig. 4. Bit error rate curves of a two-finger MRC-RAKE receiver in a wideband two-path land obile satellite channel with QPSK and OQPSK odulations, under different conditions. 1 Light Received signal relative to line of sight value (db) 5 Light Average 1 Heavy Loo s odel 3 The proposed odel.1.1.3 1 5 1 5 5 75 9 95 98 99 99.799.9 99.99 Percent of tie received signal is greater than ordinate 1 L=1 1 4 L= L=4 L=3 Proposed odel 1 6 1 3 4 5 6 7 8 9 1 Signal to noise ratio (SNR) per bit E b /N o [db] Heavy 1 1 1 L=1 L= 1 L=3 1 3 L=4 Proposed odel 1 4 5 1 15 Signal to noise ratio (SNR) per bit E /N [db] b o Fig.. Copleentary cuulative distribution function of the signal envelope in a land obile satellite channel under different conditions: Loo s odel [16] and the proposed odel. Fig. 5. Bit error rate curves of an L-branch MRC receiver in a land obile satellite channel with QPSK and OQPSK odulations, where all the branches are either under light or heavy. 1 Different outage scenarios Received signal relative to line of sight value (db) 1 8 o 6 o 4 o 3 o Experiental data o The proposed odel 75 9 95 98 99 99.7 Percent of tie received signal is greater than ordinate Outage probability 1 1 1 Int light,des light Int light,des heavy Int heavy,des heavy Int heavy,des light 1 3 1 8 6 4 4 6 8 1 Desired power to the interference power ratio ζ [db] Fig. 3. Copleentary cuulative distribution function of the signal envelope in a land obile satellite channel for different elevation angles: Measured data [1] and the proposed odel. Fig. 6. Outage probability curves for four different interference scenarios in a land obile satellite syste with an interfering satellite (Des. and Int. in the plot stand for the desired and interfering signals, respectively).