Estimation of the rain attenuation outage time for a multirelay link at millimeter wavelengths

556 pp. 556-561

Estimation of the rain attenuation for a multirelay link at millimeter

outage time wavelengths

John D. KANELLOPOULOS *

Spyros VENTOURAS *

Abstract

In the design of long haul radio relay links using frequencies above 10 GHz, it is necessary to estimate outage time occurrence probability due to rain attenuation for multirelay systems. A theoretical predictive method is proposed in this paper which is involved with the analysis of joint attenuation exceedance probabilities using a very simple exponentially shaped spatial rainrate profile. This same rain structure model has been originally developed for the prediction of rain attenuation of microwave satellite links. Numerical results have been obtained and compared with experimental data taken from a multirelay system located in Japan.

Key words : Radio-relay link, Attenuation, Microwave, Rain, Outage, Statistical model, Spatial correlation.

numdriques ont dtd obtenus et compards aux donndes expdrimentales obtenues avec un systdme similaire au Japon.

Mots el6s : Faisceau hertzien, Affaiblissement, Hyperfr&luence Pluie, Interruption, Mod61e statistique, Corr61ation spatiale.

Contents

I. Introduction. 2. Analysis.

3. Numerical results and conclusions.

Appendix.

References (9 ref.).

ESTIMATION DU TEMPS D'INTERRUPTION DU A L'AFFAIBLISSEMENT PAR LA PLUIE

DANS LES FAISCEAUX HERTZIENS A PLUSIEURS BONDS

Analyse

Dans l'~tude des faisceaux hertziens ~ plusieurs bonds utilisant des frdquences supdrieures ~ 10 GHz, il est ndcessaire d'estimer la probabilit~ d'occurrence des interruptions dues d l'affaiblissement par la pluie. La mdthode thdorique proposde dans cet article porte sur l'analyse des probabilit~s conjointes de ddpassement de la marge de protection contre les dvanouissements dus ~ l'affaiblissement dans deux sections hertziennes consdcutives. Cette mdthode utilise un profil spatial trds simple d'allure exponentielle pour le moddle de pluie. Le m~me module de pluie a dtd d~velopp~ pour la pr~vision de l'affaiblissement d6 ~ la pluie dans les liaisons hyperfr~quences par satellite. Des r~sultats

1. INTRODUCTION

In the design of long haul radio relay links using frequencies above 10 GHz, it is necessary to estimate outage time occurrence probability due to rain attenuation for multirelay links with a high degree of precision. A very limited number of theoretical techniques have been proposed so far, dealing with the prediction of this outage probability [1, 2]. More particularly, the methodology is based on the calculation of the joint attenuation exceedance probabilities for two arbitrary adjacent single links of the system. The knowledge of the point rainfall distribution in the specific region is also required.

The evaluation of this joint attenuation statistics has been carried out by using a rigorous and compli- cated statistical model for the spatial structure of the rainfall medium. But as it has been pointed out elsewhere [3] this rain structure model is appropriate for the description of the intense rainfalls (the higher

* Department of Electrical Engineering, National Technical University of Athens, Athens-147, Greece.

ANN. Tt~L[COMMUN., 41, n ~ 11-12, 1986 5/6

J. D. KANELLOPOULOS. - RAIN ATTENUATION OUTAGE TIME FOR A MULTIRELAY LINK 557

portion of the rainfall distribution). On the other hand, the outage performance analysis of the near future 20/30 GHz communication systems requires the knowledge of both background and rain from intense cells. For this reason, an improved prediction technique is proposed here, which uses a very simple exponentially shaped spatial rainrate profile. This same rain structure model has been originally developed for the prediction of rain attenuation of microwave satellite links [4].

Numerical results for the outage performance have been obtained and they are compared with experimental data taken from a series of microwave multirelay links located in Japan.

2. ANALYSIS

The outage probability for a system of m multi- tandem links, which is the probability that the rain attenuation will exceed the fading margin in at least one arbitrary link, can be approximately given by [1]:

(l) P.,,~(x) ~ m P l ( x ) - mP2,o(X),

where mP~(x) is the sum of individual probabilities that rain attenuation will exceed x dB on each of the m single paths of the multirelay system and mP2,o is the sum of joint attenuations probabilities on two arbitrary adjacent links. Using the assumption that the m individual paths of the multirelay system have the same length L, then the previous expression is reduced to [1] :

(2) P~,x(x) = ,P~(x) - - (m - - 1) P2(x).

The specific point of the paper is the calculation of the exceedance probabilities Pt and particularly P2(x) which is referred to two arbitrary adjacent links, under the assumption that the point rainfall rate in the region of the multirelay system follows a lognormal distribution [3].

The evaluation of these probabilities requires the knowledge of the rainrate profile R(I) along the extent of the propagation path, or in other words, the spatial rainfall distribution. In a previous publi- cation [2], the same subject has been analysed, adopting the convective raincell model for the spatial structure of the rain medium. But generally, precipi- tation systems are combinations of both stratiform and convective rain structures.

Radar measurements indicate that most precipi- tation is characterized by large areas of low rates with a number of smaller regions of high rainrates [5]. Combining these ideas, Stutzman and Dishman [4] have proposed the following exponential shaped effective path profile for rainfall :

(3) R(z)---- Ro, Ro ~< 10ram/h,

R(z )=Roexp ( - - y ln (~ -~ ) z ) , Ro > lOmm/h,

where Ro is the point rainrate at the l = 0 end of the single path or Ro ---- R(I = 0), z is the horizontal distance along the path and y is a parameter controll- ing the rate of decay of the profile.

The total attenuation for the single path can be now easily computed using the effective rain profile from (3) as :

(4) X(Ro) = aR~L, Ro ~ 10 mm/h,

X(Ro) = aRbo 1 - - exp(-- y b ln(Ro[ 10) L) yb In(Roll0)

Ro > 10 mm/h.

The constants a and b in the previous expression depend upon frequency and the microstructure of rain [6]. Using now a regression analysis, we can express the second of (4), as :

(5) X(Ro) = aRg 1 - - exp ( - -yb In(Roll0) L) yb In(Roll0)

= W + URo (Ro > 10 mm/h),

where the constants W and U depend upon the single path length L and the frequency of operation. Some more details about this regression analysis are also included. The numerical values for the X(Ro) which are required for the analysis have been taken in the rainrate region of 10.5 to 150 mm/h with step of 0.5 mm/h. For the characteristics of the Japanese experiment (see Section 3) the values of W and U are given by :

(6) W = --0.42939, U = 0.3287, s ---- 8.6.10 -2,

where s is the residual standard deviation. It should be noted here that some other choices of the upper limit for the rainrate region or the step used do not affect significantly the numerical values of W, U and s.

Further, using simple ideas of statistical analysis one is able to formulate the outage probability Pl(x) for the single path of the system, as follows :

(7) Px(x) = Po(0) [P[X(Ro) >/xl(Ro > 10)] •

P(Ro > 10) + P[(X(Ro) ~> x)l(Ro <~ 10)] P(Ro ~< 10)],

where Po(0) is the rain probability at the l = 0 end of the single path. The evaluation of the exceedance probabilities in expression (6) can be achieved by using the assumption that the point rainfall rate Ro follows the lognormal distribution together with the formulas (4), (5). The final results aie :

fin x - - In am) (8) P , ( x ) = P o ( 0 ) ( ~ e r f c \ ~ / ) '

for x ~ Xc,

[1 ~ [ln[(x--W)lU]--lnRm)] P~(x) ---- Po(0) [~ e r t c \ - ~ - ~ ,

for x > Xc, where :

(9) Xc = a 10 a L,

am = aR~ L, sa = b S , ,

2/6 ANN. TI~LI~COMMUN., 41, n o 11-12, 1986

558 J. D. KANELLOPOULOS. - RAIN ATTENUATION OUTAGE TIME FOR A MULTIRELAY LINK

and Rm, S, are the statistical parameters of the assumed lognormal form for the point rainfall distribution [3].

We come now to the most interesting and original part of the analysis, that is the evaluation of the joint probability P2(x). Following its definition, this joint probability can be expressed mathematically as follows :

(10) P2(x)= P(1, 2) [ P[(X1 ~ x, )(2 ~ x)I(Ro < 10)] •

P (Ro< 10)+ P[(X1 >~x, ,Y2 ~> x)[(Ro ~> 10)] P(Ro > 10)],

where P(1, 2) is the probability that rain will exist in two adjacent links of the system simultaneously or equivalently rain will fall at the 1 ----- 0 ends of the two adjacent links. Freeny and Gabbe [7] have publi- shed some experimental data concerning this rain probability parameter for a number of paths located in the USA, but further data are required for its analytical formulation. In this analysis, we will use the fact that in the 20/30 GHz band, the single path length is usually less than 5 kin, and hence we have approximately :

(11) P(1, 2) ~ Po(0).

Using then, the lognormal distribution for the point rainfall rate together with the formulas (3), (5), the joint exceedance probability P2(x) can be expressed as :

(12 ./lnx--lna~))~--~a (12) P (x) = P( l , 2) ,

for x <~ Xc,

(f Pdx) = P(1, 2) p(Ro, Ro) x d(x- W)ltt .)(x- w)ltd

dRodRo \ , for x > X~, /

where p(Ro, Ro) is the joint density probability function, for the rainrates Ro and Ro which are referred to the l = 0 ends of the two adjacent links respectively, and X~, am, S, are given by the expression (9). It should be noted here that the hypothesis of a joint probability function p(Ro , Ro) is not consistent with the equation (3). This is a small difficulty and introduces a discrepancy in the exact determination of P2(x) for x = X~. On the other hand, we believe that the approximation proposed by the second of (12) is very closed to the reality, as far as x becomes larger from X~.

The next step is the evaluation of the joint exceedance probability for x > X~ which can be done as follows. First, taking into account the lognormal forms for the p(Ro), p(Rg) and using the Jacobian transformations, we can express the joint density function as :

1 (13) p(Ro, Ro) = 2r:~1~ z R o R ; ~ / ~ X

1 [(,O .o - - exp t 2(1 _ _ ,.r 0.1

' (In R~__-- m2/2] t 2 "r (In Ro - - ml) (In Ro - - m 2 ) +

with :

(14) ~rl = ~r2 = S, ,

ml ---- m 2 = In Rm,

and "r in the previous expression (13) is the correlation coefficient between the normal variables lnRo and lnR'o. This parameter "r can be analytically expressed in terms of the unconditional spatial correlation coefficient ru between the point rainrates Ro, R'o and the other parameters of the rainfall medium. The final results are :

(15) "r = C11n(C2ru(Ro, Ro) + Ca},

1 (16) C 1 - ~ ,

( es'~ l ) po2(0) Po(0)

C 2 ~ P(1, 2)

p2(0) - - P(1, 2) Ca = + 1.

P(1, 2)

More details for the analytical derivation of the expressions (15), (16) can be found in the appendix. As far as the spatial correlation coefficient ru is concerned, two semi-empirical formulations are emplo- yed in the foregoing analysis. The one proposed by Lin (3) uses the following postulation :

G (17) ru(Ro, Ro) -- (G 2 + L2)1,2,

1 where G is a characteristic distance at which ru = ~/~-.

On the other hand, a negative exponential form for the G has been proposed by Morita and Higuti (1), and it has been also used here :

(18) ru(Ro, Ro) = e -~L''2,

is a constant (in km - I n ) ranging from 0.1 to 0.25. Using now the set of expressions (13), (18) into (12)

and following a straightforward algebraic analysis, one is able to obtain the conditional joint exceedance probability for x > Xc, as :

1 e_a~n(~e - 'O( t ) (19) P[{XI>~x, X2>~x}]----24 ~ 20 4~- d t ,

where : In [ (x- - W)[U]- -ml

dl = ffl

(20)

and :

(21) O(t) = e_adT/-1 (d2 - -da (~ /~ - [ -+dx) ) erfc ~/~- ,

In [(x ~ W)IU]- -m2 "r dz = t~2 41 __ "r z , d3 -- 4 1 ~ .

The last step is the calculation of the integral in expression (19), which can be carried out by using efficient numerical techniques, such as modified Gauss- Laguerre quadratures.

ANN. TI~LI~COMMUN., 41, n ~ 11-12, 1986 3/6


3. N U M E R I C A L R E S U L T S

A N D C O N C L U S I O N S

In this section, theoretical results for the outage probability Pm,x(x) are given for a series of tandem links operated in the 20 GHz band and located in

the Tokyo bay area. These results are given as a function of the number m of the single paths for various fade margins and they are compared with experimental ones for the same multirelay system taken from Sasaki et al. [8]. The main characteristics of the Japanese experiment are included here. Rain attenuation measurements on 13 tandem links have been made on about a 55 km path between Musashino Electrical Communication Laboratory (Musashino

lPm 1 (x)

1. */*~ ' ~

o

~ ~

/ 0.001 / 0.001-

3

' ' ' -~ ' ' ' o ' ' ' ~ 3 4 5 6 8 9 1 11 12 13 NUMBER OF SINGLE PATHS

Pm,1 (x)

I % A ~

NUMBER OF SINGLE PATHS

FIG. 1. - - Comparison of theoretical predictions with experimental data taken from a multirelay system located in Japan.

A : Fade margin 5 dB. B : Fade margin 15 dB. C : Fade margin 30 dB. D : Fade margin 40 dB. E : Fade margin 50 dB. 1 :Theoret ical prediction for Pm,~ with G = 1.5 km. 2 :Theoret ical prediction for Pm,~ with G = 3.0 km. 3 : Experimental data for Pm,~.

Comparaison entre prdvisions thdoriques et donndes expdrimentales obtenues dans un faisceau hertzien d plusieurs sections situd

au Japon. A : Marge de protection contre les dvanouissements 5 dB. B : Marge de protection contre les dvanouissements 15 dB. C : Marge de protection contre les dvanouissements 30 dB. D : Marge de protection contre les dvanouissements 40 dB. E : Marge de protection contre les dvanouissements 50 dB. 1 :Prdvision thdorique pour Pro,1 avec G = 1,5 km. 2 : Pr~vision thdorique pour Pm,x avec G = 3,0 kin. 3 : Donn~es exp~rimentales pour Pro,1.

FIG. 2. - - Comparison of theoretical predictions with experimental data taken from a multirelay system located in Japan.

A : Fade margin 5 dB. B : Fade margin 15 dB. C : Fade margin 30 dB. D : Fade margin 40 d.B. E �9 Fade margin 50 dB. 1 : Theoretical prediction for P,n 1 with ~ ---- 0.25 km -lt2 2 Theoretical prediction for fim',l with ~t = 0.1 km-ltzl 3 :Exper imenta l data for Pm,~.

Comparaison entre prdvisions thdoriques et donndes expdrimentales obtenues dans un faisceau hertzien d plusieurs sections situ~

au Japon. A : Marge de protection contre les ~vanouissements 5 dB. B : Marge de protection contre les ~vanouissements 15 dB. C : Marge de protection contre les dvanouissements 30 dB. D : Marge de protection contre les dvanouissements 40 dB. E : Marge de protection contre les ~vanouissements 50 dB. 1 : Prdvision th~orique pour Prn,1 avec ct = 0,25 km -ltz. 2 : Prdvision thdorique pour Pm,x avee ~ = 0,1 km -ltz. 3 : Donndes expdrimentales pour Pro,1.

4/6 Aim. T~LI~COMMUN., 41, n ~ 11-12, 1986

560 J. D. KANELLOPOULOS. - RAIN ATTENUATION OUTAGE TIME FOR A MULTIRELAY LINK

ECL) and Yokosuko ECL. These measurements are referred to a period of about four years (1975-1979). The longest and the shortest path lengths are 6.5 km and 2.8 km, respectively. The average is about 4.6 km and the sum of path length is about 60 km. As the branching angles at all relay points are nearly 180 ~ , this tandem link is considered to be essentially a straight line. The lengths of the individual paths of the Japanese multirelay system range from 2.8 up to 6.5 km, but due to a normalization procedure, the experimental results for the rain attenuation accumulation can be referred to a system of tandem links with 4 km each [8]. The last statement is consistent with the assumption which has been used for the derivation of equation (2). The same occurs with the assumption which is underlying the approximation (11) for the probability P(1, 2). The experimental results for the point rainfall rate distribution in the Tokyo bay area have been taken by Morita and Higuti [1]. The theoretical predictions in Figures 1 and 2 have been obtained by using various reasonable values for the characteristic parameters G and of the expressions (17), (18) for the rn.

It should be noted that the particular range of reasonable values for the G (1.5 to 3 km) has been shown by Lin [3] to be the most appropriate for the representation of the convectivity of intense rainfalls in various places in the United States of America. On the other hand, Morita and Higuti [1] have shown that the constant ~ in the alternative negative exponential form for the r~ varies from 0.1 to 0.25 (in km -112) for various places in Japan.

As it is obvious from these figures, the agreement between theoretical and experimental results is quite encouraging, although the lognormal function is not the curve fitting best the experimental rainfall rate distribution in the Japan area, especially for the high rainfall rate values.

As a final conclusion, it can be stated that more experimental data of this kind is needed for tandem links located in the USA and the other places of the world where the lognormal function best approxi- mates the point rainrate distribution, for the esta- blishment of the prediction method.

A P P E N D I X

Evaluation of the correlation coefficient v

The following definitions are given here :

cov(Ro, Re) r(Ro, R~) ~ (var(Ro) var(R~)) 1'2'

(correlation coefficient between Re, Re),

cov,(Ro, Re) r,(Ro, Re) zx (varu(Ro) var~(R~)) 1'2'

(unconditional correlation coefficient between Re, Re),

where the subscript u denotes the unconditional statistical parameter including raining and non- raining time periods. Combining the above definitions along with the obvious relations.

(A-l) Eu(Ro, Re) = P(I, 2) E(Ro, Re), r

Eu(Ro) = Eu(Ro) ---- Po(0) E(Ro),

one is able to express the r(Ro, Re) as follows :

vary(Re) (A-2) r(Ro, Re) ---- P(1, 2) var(Ro) G(Ro , Re) +

(p2(0) - - P(1, 2)) (E(Ro)) 2

P(I, 2) var(Ro)

At this point, the lognormal form for the variable Re will be used. Employing well known properties of the lognormal distribution [9], one is able to obtain :

) (A-3) varu(Ro) = \ p - ~ ) 1 E~(Ro),

var(Ro) = (e s~, - - 1) E2(Ro),

E(Ro) = Rm exp (-~-) .

Using further the expressions (A-3) in (A-2) one gets :

(A-4) r(Ro RE)= \P--~) 1 Po2(0) ' P( l , 2) (e s~, - - 1) r . (Ro , Re) +

p2(0) - - P(1, 2)

P(1, 2) (e s~, - - 1)"

The final step is the evaluation of the correlation coefficient z(ln Re, In Rg) in terms of the r(Ro, Re). Using the fact that the variables In Re and In R~ are separately two normal distributed variables, we make lhe reasonable assumption that they are also jointly normal distributed. After a straight forward analysis we can express the E(Ro, Re) as :

(A-5) E(Ro, Re) = R 2 e s~ e ~r176 lnRps2,

Using further the definitions for the statistical parameters r(Ro, Re) and cov(Ro, Re) together with the expressions (A-3), one is able to obtain :

1 (A-6) "r --- z(ln Re, In R e ) = ~ •

In ((e s~, - - 1) r(Ro, Re) + 1},

and using expression (A-4) ends-up with the formulas (15), (16) of the main text.

Manuscrit refu le 13 f~vrier 1986,

acceptd le 26 juin 1986.

A~r~. TI~LI~COMMUN., 41, n ~ 11-12, 1986 5/6


R E F E R E N C E S

[1] MOR1TA (K.), HIGUTI (1.). Theoretical studies on simulta- neous probability of rain attenuation in microwave and millimeter wave multi radio relay links. Rev. of the Elee. Comm. Labs, Jap. (1977), 25, pp. 329-335.

[2] KASELLOPOULOS (L D.), GAINS (L.). Analysis of the rain attenuation outage time for a microwave multirelay link, 3th Inter. Conf. on Ant. and Prop. lEE Conf. Proc., GB (1983), n ~ 219, part 2 : Propagation, pp. 219-223.

[3] LIN (S. H.). A method for calculating rain attenuation distributions on microwave paths. Bell Syst. Teeh. J., USA (1975), 54, pp. 1051-1086.

[4] STUTZMAN (W. L.), DISHMAN (W. K.). A simple model for the estimation of rain induced attenuation along earth- space paths at millimeter wavelengths. Radio Sci., USA (1982), 6, pp. 1465-1476.

[5] CRANE (R. K.). A global model for rain attenuation prediction. IEEE Eastcon Record, USA (1978), pp. 391-395.

[6] NOWLAND (W. L.), OLSEN (R. L.), SHKAROFSKY ([. P.). Theoretical relationship between rain depolarization and attenuation. Electron. Letters, GB (oct. 1977), 13, n o 22, pp. 676-678.

[7] FREENY (A. E.), (3ABBE (J. D.). A statistical description on intense rainfall. Bell Syst. Tech. J., USA (1969), 48, n ~ 6, p. 1789.

[8] SASAKI (O.), NAGAMUNE (][.), SATO (K.), HOSOYA (Y.). Rain attenuation characteristics on 20 GHz band multirelay links. IEEE Trans. AP, USA (1981), 7, pp. 585-594.

[9] ArrcmsoN (J.), BROWN (J. A. C.). The lognormal distribution. Cambridge Univ. Press, London (1957).

6/6 ANN. T~LI~COMMON., 41, n ~ 11-12, 1986

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Estimation of the rain attenuation outage time for a multirelay link at millimeter wavelengths