A Statistical Model for Land Mobile Satellite Channels and Its Application to Nongeostationary

A Statistical Model for Land Mobile Satellite Channels and Its Application to Nongeostationary

Orbit S y s terns

Giovanni E. Corazza, Member, IEEE, and Francesco Vatalaro, Senior Member, IEEE

Abstract-The paper introduces a statistical channel model which is a combination of Rice and lognormal statistics, and is suitable in principle to all types of environment (rural, subur- ban, urban) simply by tuning the model parameters. The model validity is confirmed by comparisons with measurement data collected in the literature. Empirical formulas are derived for the model parameters to fit measured data over a wide range of elevation angles. In particular, the model is applied to non- geostationary satellite channels, such as low-Earth orbit and medium-Earth orbit channels, in which for a given user located in a generic site the elevation angle changes continuously. Fi- nally, examples of average bit error probability evaluations in the channel are provided.

I. INTRODUCTION

S a consequence of the growing interest in land mo-

A

bile satellite (LMS) systems, much effort is being de- voted to the problem of modeling nonselective multipath fading and shadowing in the LMS communication chan- nel. Among the different viable approaches [l], a proba- bility distribution model, which is less complicated than a geometric analytic model and is more phenomenological than an empirical regression model, has the advantage to more easily allow performance predictions and system comparisons under different conditions of modulation, coding, and multiple access. Loo [2] proposed a model, suitable for rural environments, which assumes that the received signal is affected by nonselective Rice fading with lognormal shadowing on the direct component only, while the diffuse scattered component has constant aver- age power level. Lutz et al. [3] introduced a two-state model which is Rice under good propagation conditions and Rayleigh-lognormal otherwise.

In this paper we propose a probability distribution model which is a combination of Rice and lognormal sta- tistics, with shadowing affecting both direct and diffise components, and not only the former as in [2]. In prin- ciple, our model is suitable to all types of environment (rural, suburban, urban) simply by tuning the model pa-

Manuscript received September 31, 1993; revised March 20, 1994. This work was carried out within the framework of the “Progetto Finalizzato Telecomunicazioni” of the Italian CNR.

The authors are with the Universitk di Roma “Tor Vergata,” Diparti- mento di Ingegneria Elettronica, Via della Ricerca Scientifica, 00133 Roma, Italy.

IEEE Log Number 9403213.

rameters. In particular, in built-up areas it reduces to the well-known and well-tested Suzuki model (i.e., Rayleigh- lognormal) [4], that is widely accepted for terrestrial land mobile macrocellular channels [5]. In this paper the model is applied to nongeostationary satellite channels, such as low-Earth orbit (LEO) and medium-Earth orbit (MEO) channels.

LEO and M E 0 constellations are being considered in many proposals for future LMS systems [6], with the ob- jective to achieve a global coverage with acceptable ele- vation angle, CY. Being such orbits nongeostationary , the elevation angle at any site changes continuously over time;

as a consequence, the propagation conditions are also time varying, even if the mobile terminal is static as it is often the case in personal communications.

In order to be able to predict the performance of a sys- tem adopting such constellations, it is therefore necessary to possess a statistical model which fits measured data over a wide range of elevation angles. Here we adopt a novel approach, in which the parameters of the probability dis- tribution model are described by empirical formulas to fit measured data. The resulting hybrid empirical-probabil- ity distribution model fits measured data over a wide range of CY, and it allows the evaluation of performance char- acteristics such as the average bit error probability.

The paper is organized as follows. In Section I1 we present the statistical channel model, its validation against measured data and the empirical regression formulas for a rural tree-shadowed environment. In Section I11 we de- scribe a three step procedure to evaluate the probability of error for nongeostationary systems. In Section IV we discuss some important features of the nongeostationary (in particular LEO and MEO) satellite channels and then provide examples of bit error probability evaluations. In Section V we draw paper conclusions.

11. STATISTICAL CHANNEL MODELING A N D VALIDATION A. Probability Distribution Model

The probability density function (p.d.f.) of received signal envelope, r , is given by

m

P m =

so

^{P(rl S)Ps(S) dS. (1)}

In (1) p ( r l S ) is a Rice p.d.f. conditioned on a certain

0018-9545/94$04.00 0 1994 IEEE

CORAZZA A N D VATALARO: MODEL FOR LAND MOBILE SATELLITE CHANNELS 739

shadowing, S:

1

[

^{S r 2}

p ( r l S ) = 2 ( K + 1 ) G e x p - ( K _S

+

^{1 ) i - K}

- z0

2

IS ^JKO

^{(r 2}

⁰¹

^{( 2 )}

where Zo is the zero order modified Bessel function of the 2 20 - 2

B

( )

first kind, and K is the so called Rice factor. The shad- ^{.e 25 -}

owing, S, is lognormal with p.d.f.: 30

( 3 ) where h = (In 10)/20, p and (ha)2 are the mean and the variance of the associated normal variate, respectively; in terrestrial channels u is usually referred to as the "dB spread. "

When K = 0, (1)-(3) provide the Suzuki p.d.f. In the limit for ^{u -+} 0, ps(S) tends to a Dirac pulse located at the mean value of the distribution, i.e., it tends to 6 ( S - e"). Therefore p,(r)

-,

p (rle@) and the channel is Rice.

The signal envelope which meets the channel model (1)-(3) can be interpreted as the product of two indepen- dent processes, Le., r = RS, where R is a Rice process, and S is lognormal. Due to the independence between R and S we have [ 7 ] :

and by comparing (1) and (4):

which implies ^0; = 1 / 2 ( K

+

^{1j. Equation (4)} allows fur- ther observations: when K ^{+ 03,} pR(R) tends to a Dirac pulse located at R = 1 andp,(r) tends tops(r), i.e., the channel is lognormal. When K -+ 03 and u -+ 0 fading is absent. Therefore, depending on the combination of K , p , u the proposed channel model can be reduced to any one of the usual nonselective fading models.

From the moments of the lognormal process [8] and those of the Rice process [9], the nth order moment of r can be derived as

E ( r " } = E{R"}E{S"} = ( K

+ l)-'1'2cKr (

1

+

^-

9

e"@ exp (in'h'o') . IF1 ( 1

+ i,

^{1 ; K )}

-35

001 0 1 0 5 1 2 5 10 20 3 0 1 0 5 0 6 0 70 80 BO 95 BB 09 9 9 5 W.0 W W

Percent of time received signal is above ordinate

Fig. 1 . Comparison between measured c.d.f. data in light shadowing ( 0 ) and heavy shadowing ( x), the c.d.f. given by (7) (continuous lines), and the c.d.f. provided in [2] (dashed lines). Light shadowing: K = 4.0, p = 0.13, u = 1.0. Heavy shadowing: K = 0.6, p = -1.08, u = 2 . 5 .

where

r

is the gamma function and I F , is the confluent hypergeometric function [ 101.

Finally, the cumulative distribution function (c.d.f.) of the envelope is

Pr(ro) 2 Prob { r < ro}

= 1 - E s [ Q ( & , : m

)I

⁽⁷⁾

where Es { } stands for the average with respect to S and Q is the Marcum's Q-function [9].

B. Model Validation and Empirical Formulas

The proposed channel model was validated with respect to measurement data available in the literature. Fig. 1 col- lects the c.d.f. measured data provided in [ 2 ] for the cases referred to as "infrequent light shadowing" and "fre- quent heavy shadowing." In the same Fig. 1 we provide the fitting curves obtained by means of ( 7 ) with parame- ters p , u, K optimized by trial and error, and for compar- ison the curves calculated with the c.d.f. in [2]. Curves calculated with our model match very well both with mea- surement data and with Loo's c.d.f.

As stated in the "Introduction, ^" empirical formulas should be derived to fit measured data over a wide range of elevation angles. As an example, we used some data collected by ESA at L-band in a rural tree-shadowed en- vironment [ I 11. Data fitting was conditioned on the fol- lowing intuitive indications: the greater is ^a, the larger is K and the smaller is u. The resulting empirical formulas allow interpolation for any cy in the range 20" < cy <

80" :

K ( a ) = KO

+

Klcy

+

^K2cy2

P ( a > = Po + PIQ + P2cy2 + P P 3

u ( a ) = uo

+

u1cy. (81

& = 2.731

4 = -1.074 10’

4 = 2.774 10-9

TABLE I ENVIRONMENT)

COEFFICIENTS FOR EMPIRICAL FORMULAS (RURAL TREE-SHADOWED

pa = -2.331 a0 = 4.5

= 1.142 10-1 a1 = ^~ 0.05 pz = - ^{1.939 10-9}

= 1.094106

I I I I I 1

20 30 40 50 60 70 80

a

Fig. 2. Model parameters, p , u, and K as a function of the elevation angle, a, in a mal tree-shadowed environment.

I 1 I l l I

10 6Q 70 80 90 95 98 99 99.5 99.9 99.99

Pacent of h e received signal level is above ordinate

Fig. 3. Comparison between measured c.d.f. data in a rural tree-shadowed environment ( 0 ) as a function of the elevation angle, a, and the proposed c.d.f. with parameters given by (8) (continuous lines).

The coefficients for the specific example are reported in Table I and the resulting curves for K, p and ^(T are shown in Fig. 2 , while measured data fitting is reported in Fig.

2 J.

111. PROBABILITY OF ERROR ^{IN THE} NONGEOSTATIONARY LMS CHANNEL

The symbol error probability for transmission in chan- nels affected by time and frequency nonselective fading can be written as

P”

pe =

Io

P(eIr)pr(r) dr (9) where P ( e

I

r) is the symbol error probability conditioned on a certain value of r and p r ( r ) is given by (4). By sub- stituting (4) _into (9) and interchanging the order of inte-

gration we have

pe =

lo”

^~S(S)

^{[ j”}

p ( e l R ~ ) p R ( R ) dR d~ (10)

0

I

where R = r/S. The inner integral represents the average error probability in the presence of Rice fading only (Le., for a given value of S):

cn

E R { P ( e l R S ) } =

so

P ( e ( R S ) p R ( R ) dR (11) while the outer integral in (10) is the average o f f @ ) in the presence of lognormal shadowing. In conclusion, we have:

(12) pe = ES{f(S)) = ES{ER[P(elr)l).

The error probability provided by (12) depends on the model parameters p , ^(T and K , which for a given site are functions of the elevation angle, a.

While for geostationary systems (12) itself provides the local average probability of error, in the case of nongeo- stationary orbit systems P , must be additionally averaged with respect to the angle p.d.f.:

I“,

= E a { P e ) . (13) The three step procedure given by (1 1)-( 13) for evaluat- ing the error probability is applied in the following section to some specific examples.

IV. NUMERICAL RESULTS FOR SOME NONGEOSTATIONARY ORBIT SYSTEMS With the aim of characterizing nongeostationary orbit systems, a software package was developed. The package simulates the satellite constellation motion around the Earth, and evaluates system parameters and functions such as instantaneous coverage, satellite handovers, link un- availabilty, etc. [ 1 2 ] . In particular, the c.d.f. of the ele- vation angle can be achieved under minimum, average, and maximum global coverage conditions. As an exam- ple, Fig. 4 shows the minimum worldwide coverage, C,,, (%), as a function of a for some LEO and M E 0 con- stellations [ 6 ] . We note that the fractional area covered with high-elevation angles, e.g., a 2 60° , is generally small. This means that to achieve adequate protection from shadowing nongeostationary communication sys- tems must be provided with large power margins. From a channel modeling point of view, we must resort to local statistics for the elevation angle. For instance, setting Roma as the location in which the performance needs to be evaluated, the corresponding discretized p.d.f. for a is reported in Table I1 for the different constellations con- sidered.

Making use of the proposed model, the bit error prob- ability for binary DPSK modulation has been evaluated at different elevation angles (see Fig. 5 ) . For high elevation angles (greater than 6 0 ° ) , when shadowing is negligible, it is evident an error floor in the bit error probability curves

CORAZZA A N D VATALARO: MODEL FOR LAND MOBILE SATELLITE CHANNELS 74 1

,..

E

UE e, 60

8 .5 2

$

40

20

0

0 10 20 30 40 50 60 7 0 80 90

a (&flees)

Fig. 4 . Minimum global coverage (polar regions excluded) of some LEO and M E 0 systems as a function of the elevation angle, a (the number fol- lowing the system name provides the constellation size).

TABLE I1

% (SITE: ROMA; 41.9 N LAT., 12.5 E LONG)

DISCRETIZED PROBABILITY DENSITY FUNCTION FOR THE ELEVATION ANGLE,

100 ~ I I I I I 1 1

0 5 I O 15 20 25 30 35

Eb/No (dB)

Fig. 5 . Bit error probability, P,, for DPSK modulation as a function of the bit energyhoise power spectral density, E,,/N,, for several values of the elevation angle, a.

due to the presence of the diffuse component. Averaging the bit error probability over the discretized range of el- evation angles produces the average bit error probability for the different constellations as reported in Fig. 6 . Ob- viously, these results cannot be used to compare the ac- tual systems for several reasons, among which: DPSK is not the adopted modulation scheme; the systems have dif- ferent orbital heights, so that free-space losses are much different; shadowing and multipath fading are not the only impairments in the channel (in particular, co-channel in- terference has to be considered). Furthermore, the results strongly depend on latitude.

The presented approach to the estimation of the average bit error probability provides a tool that enhances the link

0 5 IO 15 20 25 30 35

Eb/No (dB)

Fig. 6. Average bit error probability, Fc, for DPSK modulation as a func- tion of the bit energy/noise power spectral density, E,,/N,, associated to the probability density functions for the elevation angle given in Table 11 (site:

Roma) .

budget accuracy of any system adopting nongeostationary orbits.

V . CONCLUSIONS

The paper described a statistical model for land mobile satellite communications that is suitable to different prop- agation environments and a wide range of elevation an- gles. Therefore, the model is proposed for the statistical characterization of LEO and M E 0 satellite communica- tions. The model allows to predict communication per- formance under different conditions of modulation, cod- ing, and multiple access. The paper showed this by evaluating the average probability of error in few simple cases.

REFERENCES

J . Goldhirsh and W . I . Vogel, “Propagation effects for land mobile satellite systems: Overview of experimental and modeling re- sults,”NASA Ref. Publ. 1274, 1992.

C. Loo, “A statistical model for land mobile satellite link,” IEEE Trans. Veh. Technol., vol. VT-34, pp. 122-127, Aug. 1985.

E. Lutz, D. Cygan, M. Dippold, F . Dolainsky, and W. Papke, “The land mobile satellite communication channel-recording, statistics and channel model,” IEEE Trans. Veh. Technol., vol. 40, pp. 375-385, May 1991.

H. Suzuki, “A statistical model for urban radio propagation,” ZEEE Trans. Commun., vol. COM-25, pp. 673-680, July 1977.

D. Parsons, The Mobile Radio Propagation Channel. London: Pen- tech Press, 1992.

MITRE, “Study of the mobile satellite communication industry: In- terim report,” ESA Contract 9563/91/NL/RE, Nov. 1991.

A. Papoulis, Probabilities, Random Variables and Stochastic Pro- cesses. New York: McGraw Hill, 1991, 3rd ed.

L. F . Fenton, “The sum of log-normal probability distributions in scatter transmission systems,” IRE Trans. Comm. Sysr., pp. 57-67, Mar. 1960.

1. G. Proakis, Digital Communications. New York: McGraw Hill, 1983.

M. Abramowitz and 1. A. Stegun, Handbook of Mathematical Func- tions. New York: Dover, 1970.

M. Sforza and S . Buonomo, “Characterisation of the propagation channel for nongeostationary LMS systems at L- and S-bands: Nar- row band experimental data and channel modelling,” in Proc. XVII NAPEX Conf.. Pasadena, CA, June 14-15, 1993.

G. E. Corazza and F. Vatalaro, “Comparison of low and medium orbit systems for future satellite personal communications,” pre- sented at IEEE Pacijic Rim Con5 Comm., Compur. Signal Proc., IEEE 93CH32-88, May 19-21, 1993, pp. 678-681.

Giovanni E. Corazza (M’92) was born in Tri- este, Italy, in 1964. He received the Dr.Ing. de- gree in electronic engineering in 1988 from the University of Bologna, Bologna, Italy.

From 1989 to 1990 he was with the Canadian aerospace company COM DEV (Cambridge, Ont., Canada), where he was an advanced member of technical staff, working in the millimeter-wave subsystems group.

Since 1991 he has been with the Dipartimento di Elettronica of the Universiti di Roma “Tor Vergata,” where he is presently a research associate. His main interests are in the areas of terrestrial and satellite personal communication systems, and spread spectrum multiple access techniques.

Francesco Vatalaro (M’88-SM’91) was born in Vibo Valentia, Italy, in 1953. He received the Dr.Ing. degree in electronic engineering from the University of Bologna, Bologna, Italy, in 1977.

From 1977 to 1980 he was with Fondazione Ugo Bordoni at Pontecchio Marconi, Italy. Then he was with FACE Standard Central Laboratory, Po- mezia, Italy, from 1980 to 1985. While with Se- lenia Spazio, Roma, Italy, he was Group Leader of satellite ground segment radiosystems engi- neering. In 1987. he became Associate Professor Y

of Radio Systems at the University of Roma Tor Vergata. From 1987 to 1989 he was Project Manager of the ground segment of the European Data Relay System (ESA-DRS). He was co-winner of the 1990 “Piero Fanti”

INTELSAT/Telespazio international prize. His research interests include mobile and personal communication systems and spread spectrum systems.

Prof. Vatalaro is a member of AEI.