510 pp. 510-516

Prediction of the outage performance of a microwave multiple-hop network

due to rain attenuation John D. KANELLOPOULOS *

Lampros GAKIS *

Abstract

In the design of tandem links using frequencies above 10 GHz, it is necessary to estimate outage time occur- rence probability due to rain attenuation. Subject of this paper is the theoretical analysis of simultaneous probability of rain attenuation for tandem links by studying the joint distribution of correlated lognormal variables. This analysis is appropriate to locations where the point rainrate distribution approximates the lognormal function. The theoretical predictions for the outage performance of the multiple-hop network have been compared with existing experimental data for tandem links located in France, USA and Japan. The agreement has been found to be encouraging.

Key words : Radio-relay link, Outage, Attenuation, Rain, Stochastic model, Log-normal law.

normale. Les prdvisions thdoriques pour les interruptions d'une liaison gl bonds multiples ont dtd compardes aux donndes expdrimentales obtenues par des liaisons situdes en France, aux Etats-Unis et au Japon. L'accord des rdsultats a dtd encourageant.

Mots cl~s : Faisceau hertzien, Interruption, Affaiblissement, Pluie, Mod61e stochastique, Loi lognormale.

Contents

1. Introduction.

2. The analysis.

3. Numerical results and discussion.

Annexe.

References (17 ref ).

PRI~VISION DES CARACTI~RISTIQUES DES I N T E R R U P T I O N S

D ' U N F A I S C E A U HERTZIEN A B O N D S MULTIPLES

D U E S A L'AFFAIBLISSEMENT PAR LA PLUIE

Analyse

L'dtude des liaisons en cascade utilisant des frd- quences supdrieures dt 10 GHz ndcessite d'estimer la probabilitd d'occurrence des interruptions dues d l'attdnuation par la pluie. L'objet de cet article est l'analyse thdorique de la probabilitg simultande de l'affaiblissement d~ ~ la pluie pour des liaisons en cascade, par dtude de la distribution jointe de variables lognormales corrdldes. Cette analyse est appliqude aux sites oi4 la distribution de l'intensitd de prdcipitation ponetuelle peut ~tre approximde par fonction log-

1. I N T R O D U C T I O N

A problem of continuing interest in millimeter wave communication system design is the establish- ment of reliable attenuation predictions for terrestrial links. Above 10 GHz, rain attenuation becomes one of the most important factors governing system reliability. In this way, the estimation of the outage time occurrence probability due to rain attenuation for a multirelay system is important for the appro- priate design of the system.

This outage time can be estimated if rain attenuation probability and the simultaneous probability of rain attenuation in multirelay links can be found. Over the past decade there has been a considerable interest in the analysis of correlated rain attenuation and its influence on microwave system design. But this interest has been almost concentrated in the applica-

* Department of Electrical Engineering, National Technical University of Athens, 42, Patission Street, Ath~nes - 147, Gr6ce, GR-106 82.

ANN. T~L~COMMtn'r 42, n ~ 9-10, 1987 1/7

J. D. K A N E L L O P O U L O S . - O U T A G E P E R F O R M A N C E OF A M I C R O W A V E M U L T I P L E - H O P N E T W O R K 511

tion of route diversity systems and the expected improvement over the single route systems [1-2]. On the other hand, the influence of spatial correlation of intense rainfall on the outage performance of a tandem system has not received a great deal of attention in the literature.

Morita and Higuti [3] have presented a theoretical analysis for the joint probability of rain attenuation for a multihop network by studying the joint distri- bution of correlated Gamma variables. This analysis has been mainly based on the assumed Gamma form for the point rainrate distribution in relation with the appropriate rainfall spatial structure model which has been proposed for the Japan area. On the other hand, Segal [4] has developed a model for multihop networks based on measured rainfall correlation without reference to the form of distribution.

In recent publications [5-6] the application of a theoretical joint statistical analysis to a case where the experimental rainrate distribution can be well represented by a lognormal function, has been also examined. The strong motivation for the presentation of this particular joint statistical analysis has been due to the very wide use of the lognormal form as the appropriate rainfall cumulative distribution, although recently other distribution forms have been seriously suggested [7-8]. First, in an earlier form [5], some results of this analysis have appeared. A rigorous model [9] has been considered there as the most appropriate for representing the convectivity of the intense rainfall. Most recently, the study has been extended to the analysis of the outage performance of the near future 20-30 GHz communication systems [6] by employing for this reason art appropriate exponentially shaped spatial rainrate profile. The purpose of this paper is the presentation of the complete joint statistical analysis in an expanded form which employs the rigorous model for the rainfall spatial inhomogeneity [9], and particularly, the exact and detailed derivation of the correlation attenuation coefficient for two adjacent links. The latter analysis for the correlation coefficient can be considered to be crucial for the development of predictive techniques dealing with the outage performance of route diversity systems in this communication band, as it will be shown in future papers.

As a result of the analysis, the outage time of the multihop system has been evaluated. The theoretical predictions have been compared with existing experi- mental data concerning the outage performance of tandem links located in Japan [10], France [11] and USA [12].

2. THE ANALYSIS

The outage probability for a system of tandem links can be defined as the probability that the rain attenuation will exceed the fading margin in at least

one arbitrary link of the system. In the following analysis, the probability will be expressed as Pm, I(X) where m is the number of links in tandem and x the fading margin in dB. From a theoretical point of view, the probability P,,,l(x) is evaluated as [3] :

(1) P,.,,(x) 1 - - f i ( 1 - P,(x)) i = 1

and m--i

(2) mP,(x) = Y~ mPi4(x) J=O

= k (- - l) i+x mPi(x), i = 1

for i =>2.

Here Pi(x) is the probability that the rain attenuation exceeds x dB at the ith link, and ,,Pi(x) is the sum of probabilities that rain attenuation simultaneously exceeds x dB on each of i links, which contain the all possible combinations of i links from the m relay links. Finally, mPi,j is the sum of joint attenuation probabilities on i links arbitrarily skipping j links.

Further analysis of the expressions (1) and (2) is a very complicated task, but Morita and Higuti [3] using experimental data of this kind from several links in tandem located in Japan, have indicated that the probability Pm,l(x) can be approximately given by :

(3) Pro, l(x) = ,.Pl(x) - - mPz,o(X).

It should be noted that Morita and Higuti have obtained the approximation for P~,l(x) by neglecting the probability when simultaneous rain attenuation occurs in more than 4 paths. Following the previous definitions, mPI(x) is the sum of the individual proba- bilities that rain attenuation will exceed simultaneously x dB on each of the m single paths of the multirelay link and is given by :

(4) ,,Pl(X) = ~] Pi(x), i = 1

P~(x) is the exceedance attenuation probability for the /-single path. On the other hand, ,,P2,o(X) is the sum of joint attenuation probabilities for two adjacent links. More specifically, when the individual paths have the same length L, the previous expression is reduced to :

(5) Pm,1(X) = mPl(x) - - (m - - 1) P2(x).

The next step is the analytical evaluation of the exceedance probabilities P~(x) and P2(x) using the assumption that the point rainfall rate follows a lognormal approximation.

We will adopt, moreover the Lirt [9] model for the spatial structure of the rainfall medium. According to this model, the single path exceedance probability P~(x) cart be given by :

(lnx ,nam (6) Pl(x) = Po(L) erfc ~/~-~-~- / '

where Po(L) is the path rain probability [9] and am, S~ are the statistical parameters of the lognormal

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512 J . D . K A N E L L O P O U L O S . - O U T A G E P E R F O R M A N C E O F A M I C R O W A V E M U L T I P L E - H O P N E T W O R K

distribution for the single path attenuation. The probability density function for this variable will be given by :

1 ( + ) [ (lnx--lnam) 2] (7) Px(x) -- ~ / ~ - exp - - 2 S~ "

The parameters am, S~ are expressed analytically in terms of the point rain probability Po(0), the corresponding parameters Rm, Sr of the lognormal form for the point rainfall distribution, and the spatial correlation coefficient -r u between the variables R(z) and R(z') along the path [9].

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

(8) P2(x) = P[X, _-> x, X2 > x]

---- P(1, 2) P ( X 1 ~ x, X 2 ~ x},

where P(I, 2) is the probability that rain will exist in two adjacent links of the system simultaneously and X~, )(2 are the corresponding attenuations. Using simple statistical ideas, one is able to express P(I, 2) as :

(9) P(1, 2) = 2 Po(L) - - Po(2L),

in terms of the path rain probabilities Po(L) and Po(2L) [9].

As far as the second term of the expression (8) is concerned, we have :

(10) P[fX, _-> x, X2 > x}]

= (xl , x2) dxl dx2. Xl

Using the fact that the variables X3 ---- In X~ and )(4 = In )(2 are separately two normal distributed variables, we make the reasonable assumption that they are also jointly normal distributed.

The subsequent use of the Jacobian transformations gives the joint density function Px~x~ as [13].

1 = X (11) Px,x2(X,,X2) 2 ~ S 2 4 1 - - ' r 2 xtx2

, [(taxi,ham)2 exp [ 2(1 "r 2)

2"r(lnxl--lnam)(lnx2--lnam) (lnxz~lnam~2] l s~ + & / j '

where -r is the correlation coefficient between the normal variables X3 and X4 under the condition of rain. The next step is the analytical evaluation of this parameter v in terms of the spatial correlation coefficient vu and the other parameters of the rain- fall medium. The final results are :

1 (12) v = -6-2 X

S~

t ((exp(S2b)/Po(0)) 1) p2(0) (Rib) 2 exp(S]b) 1 In P(1, 2) (R~b exp(S2)) L'-.- ~ I +

po2(L) - - P(1, 2) ) P(1, 2) + 1 1 '

where : Rmb---- a,,/aL, (13) Rmb = R~,

S,b = bSr, and :

(14) I = G2 l ~- [sinh-1 (-~-) -- sinh-1 (Z)] --

r + (~-~)2 +2V/1 +(L)2--11.

The constants a and b, appearing in the previous expressions, depend on the frequency and incident polarization and their values are tabulated by Nowland et al. [14]. It should be noted that the following semi- empirical formulation has been employed for the spatial correlation coefficient Zu, as a function of the distance d = ]z - - z'[ between two points z, z' along the path [9] :

G (15) "ru[R(z), R(z')] = (G 2 + d2)1/2,

G is a parameter characterizing the inhomogeneity of the rainfall medium. As it is obvious from the previous formula, G can be considered to be a distance between two points in the medium at which the correlation coefficient % for their point rainfall rates becomes 1[~2.

Details for the analytical derivation of the expres- sions (12)-(14) can be found in the Appendix.

Using now the set of expressions (11)-(14) into (10) and following a tedius but straightforward analysis, one is able to obtain the conditional joint exceedance probability as :

(16) P{X, >~x,)(2 ~>x)

1 e-a~ m ~o e-Z dz,

d ! - - In x - - In am

G

where :

(17)

and :

(18) F(~J2-z-)-- e-a,41-~-\ ~~ f' 1 e -z2/2 d• ,,)a4

= e -d~'/~-" - erfc ~-~ ,

(19) & = #~ + d~ (~/2-~ + dx),

In x - - In am (20) d2 -- G ~ / i ~ ~ ,

(21) d3 = 4 ] ~ - �9

The final step is the calculation of the integral in expression (16), which can be carried out by using efficient numerical techniques, such as modified Gauss-Laguerre quadratures.

ANN. TI~LI~COMMUN., 42, n ~ 9-10, 1987 3/7

.

10-

NUMERICAL RESULTS AND DISCUSSION

correlation factor of the attenuations calculated by the set of expressions (12)-(14) is not valid for a convergent arrangement of the adjacent paths.

The outage probability for two links in series obtained in Paris, France [11] is shown in Figure 2

In this section, the predicted results are compared with experimental data from some parts of the world. First, the experimental results for the excess simulta- neous attenuation on two 3-mile converging paths located near Palmetto, Georgia and operated in the 18 GHz band are compared with the numerical ones using the set of expressions (16)-(21). The data is referred to a 7-month period from November 18, 1970 to June 15, 1971 [12]. As has been pointed out in the previous section, the numerical values for the para- meters a and b appearing in the predictive procedure, have been taken by Nowland [14]. The experimental data for the point rainfall rate distribution in Palmetto, Georgia has been taken by Lin [9]. The theoretical predictions in Figure 1 have been obtained by using G = 1.5 km. It should be noted that this numerical value for the characteristic distance has been shown by Lin [9] to be the most appropriate for the represen- tation of the convectivity of intense rainfalls in various places in the United States of America. The slight under estimation of the experimental data by the numerical results is mainly due to the fact that the

Fraction of total time attenuation abscissa exceeded

•

10 -

x x

lO

I I 10 20 30 410 50

Attenuation in dB

FIG. 1. - - Analys is of the excess s imul taneous a t tenua t ion probability for two 18-GHz adjacent converging hops, in Pal-

metto Georgia. • : Experimental data.

- - �9 Predicted results, G = 1.5 km.

Analyse de la probabilitd de l'affaiblissement simultand en excds pour des bonds convergents et adjacents d Palmetto en Gdorgie.

• : donndes expdrimentales, : prdviMons thdoriques, G = 1,5 km.

10

g

10

Percent of time attenuation is exceeded

J. D. KANELLOPOULOS. - OUTAGE PERFORMANCE OF A MICROWAVE MULTIPLE-HOP NETWORK 513

x

I i I ! 10 20 2~5 3~0 3~5 40 415 50

Attenuation in dB

FIG. 2. - - Analys is of the outage probab i l i ty for two l inks in series, located in Paris. Frequency �9 13 GHz.

• : Exper imenta l data.

- - : Predic ted results, G = 1.5 km.

Analyze de la probabilitd d'interruption pour 2 liaisons en sdries, situdes d Paris, France. Frdquence : 13 GHz.

• : donndes expdrimentales,

- - : prdvisions thdoriques, G = 1,5 km.

in comparison with the results derived by using the approximation (5) and the predictive procedure presented in the previous section. The frequency was 13 GHz and the polarization was horizontal. The experimental results are based on measurements for the period 26 March 1975 to 3 March 1977. The experimental data for the point rainrate distribution in Paris, France has been taken by Fedi [15]. The theoretical predictions shown in Figure 2 are referred to G ---- 1.5 km and the agreement with the experimental results is quite good. The only discrepancy exists at the higher values of attenuation where model predictions overestimate the real situation. This is quite expectable, because the Lin [9] model concerning the single attenuation distribution overestimates the experimental data for high values of attenuation for locations in France [15].

Finally, numerical results concerning the outage probability (Fig. 3-4) for various fade margins have

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514 J. D. KANELLOPOULOS. -- OUTAGE PERFORMANCE OF A MICROWAVE MULTIPLE-HOP NETWORK

0.1

k Outage Probability Pro, ~(•

(A)

( A : ~

0.01 I I I I I I I f I I I : 2 3 4 5 6 7 8 9 10 11 12 13

m (Number of single links)

FIG. 3. - - Outage probability for a series of tandem links of 4 km single path at a frequency of 20 GHz.

(A) : Fade margin 15 dB : Experimental data for links in Japan [10].

(A1) : Predicted results, G = 1.5 km. (A2) : Predicted results, G = 3 km. (A3) : Predicted results, using the Gamma model [3]. (B) : Fade margin 20 d B : Experimental data for links in

Japan [10]. (B1) : Predicted results, G = 1.5 kin. (B2) : Predicted results, G = 3 km. (B3) : Predicted results, using the Gamma model [3].

FIG. 3. - - Probabilitd d'interruption pour une sdrie de liaisons en cascade de trajet de 4 k ~ ~ une frdquence de 20 GHz.

( A ) : Donndes expdrimentales pour des liaisons au Japon [10] : marge de protection contre le~ ~vanouissements 15 dB.

(A1) : Pr#visions thdoriques, G = 1,5 km. (A2) : Prdvisions th~oriques, G = 3 km.

(A3) : Prdvisions thdoriques, utilisant le meddle Gamma [3]. (B) : Donn~es expdrimentales pour des liaisons au Japon [10] :

marge de protection contre les #vanouissements 20 dB. (B1) : Pr~visions thdoriques, G = 1,5 km.

(B2) : Prdvisions thdoriques, G = 3 km.

(B3) : Pr~visions thdoriques, utilisant le maddle Gamma [3].

been drawn as a funct ion o f the number m o f the single paths and have been compared with appropriate experimental data. The data in the appropria te form has been taken by Sasaki et al. [10] and is referred to a series o f tandem links operated in the 20 G H z band and located in the Tokyo bay area.

As has been reported by Sasaki et al. [10], the s i ng l epa th ranged f rom 2.7 to 6.5 km but due to a normalizat ion procedure the experimental results can be referred to a system o f tandem links with 4 km each. It should be noted that the single path o f 4 km considered is o f the order o f magnitude o f the rain cell in Japan. On the other hand, a l though the links were normalized to 4 km there is still the influence o f the locality on the experimental data as pointed out by Sasaki et al. [10]. The experimental data for the point rainfall rate distribution in the Tokyo bay area has been taken by Mori ta [16]. The theoretical pre-

I Outage Probability P~ 1 (~)

l ~ (A3 (A2) (A

0.01

0'001; 12 13 14 15 I I ; I I I ', : ~- 6 7 9 10 11 12 13

m (Number of single links)

FIG. 4. - - Outage probability for a series of tandem links of 4 km single path at a frequency of 20 GHz.

(A) : Fade margin 30 d B : Experimental data for links in Japan [10].

(AI) : Predicted results, G = 1.5 kin. (A2) : Predicted results, G = 3 km. (A3) :Predicted results, using the Gamma model [3]. (B) : Fade margin 40 d B : Experimental data for links

in Japan [10]. (B1) : Predicted results, G = 1.5 km. (B2) : Predicted results, G = 3 km. (B3) : Predicted results, using the Gamma model [3].

FIG. 4. - - Probabilitd d'interruption pour une sdrie de liaisons en cascade de trajet de 4 km dune frdqaence de 20 GHz.

( A ) : Donndes expdrimentales pour des liaisons au Japon [10] : marge de protection centre les #vanouissements 30 dB.

(A1) : Prdvisions thdoriques, G = 1,5 km.

(A2) : Prdvisions thdoriques, G = 3 kin. (A3) : Prdvisions thdoriques, utilisant le maddle Gamma [3]. (B) : Donndes expdrimentales pour des liaisons au Japon [10] :

marge de protection centre les dvanouissements 40 dB. (B1) : Prdvisions thdoriques, G = 1,5 kin. (B2) : Prdvisioas thdoriques, G = 3 kin. (B3) : Prdvisions thdoriques, utilisant le meddle Gamma [3].

dictions in Figures 3 and 4 have been obtained by using various reasonable values for the characteristic distance G (G ---- 1.5 and 3 km). As it is obvious f rom these figures for fade margins o f 15 dB, 20 dB and 30 dB, the experimental outage probabili ty is higher than the predicted values, whilst for a fade margin o f 40 dB the reverse is the case. This slight discrepancy is mainly due to the fact that the lognormal function is not the curve fitting in the best possible way to the experimental rainfall rate distribution in the Japan area. The whole Japan is characterized by strong showers (with high rain rates ) due to typhoons and thunderstorms. As a matter o f fact, in Japan G a m m a model provides better approximat ion than log normal model for the rainfall cumulative distribution, patti-

ANN. T~LgCOMMUN., 42, n ~ 9-10, 1987 5/7

J. D. KANELLOPOULOS. - OUTAGE PERFORMANCE OF A MICROWAVE MULTIPLE-HOP NETWORK 515

cularly for rainrates of over 10-20 mm/h [16]. Part of this discrepancy may be also due to the fact that the spatial correlation model appropriate to represent the intense rainfall convectivity in Japan has been shown to be a negative exponential function of the one half power of distance d [3]. For this reason, the model proposed by Morita and Higuti [3] for the outage probability of tandem links using the previous considerations has been also employed here. The numerical results for the same fade margins have been compared with those obtained with the lognormal model (Fig. 3-4). For fade margins of 20, 30 and 40 dB the two models give quite comparable results but if fade margins greater than 40 dB were considered, then Morita and Higuti model [3] predictions would be quite closer to the real situation.

In conclusion, as one step in the testing of the theoretical model, we have compared the theoretical predictions with a limited set of existing experimental data taken from multirelay systems located in USA, France and Japan. Generally, the agreement has been found to be encouraging. Finally, more data of this kind is needed for tandem links located in the USA and in the other places of the world, where the log- normal function is the best approximation for the point rainfall rate distribution.

ANNEXE

E v a l u a t i o n o f the c o r r e l a t i o n c o e f f i c i e n t "~

As mentioned in the main text, we have the random variables :

X1 = a .Rb(z) L1, (attenuation due to rain for the single path 1),

X2 - - a .Rb(z') L2, (attenuation due to rain for the adjacent single path 2),

Y 3 = In X l ,

)(4 In )(2.

The following definitions are given here :

cov[Xl, x d �9 [X,, x d =

(var[Xl] var[Xz]y/z '

(correlation coefficient between X~, X2),

cov,[X,, x~] "~'u[]~'l, X2] = (varu[Xd var.[X2]) i n "

(unconditional correlation coefficient between )(1, )(2). 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) E , [Xt , )(2] = e(1, 2) E[X, , )(21,

E.[X,] = E.[X2] -- Po(L)E[X1],

one is able to express the z[X~, )(2] as follows :

var,[Xd (A-2) -r[X~, X2] = P(1, 2) var[X~] zu[X~, X2] +

(p2(L) - - P(1, 2)) (E[X,]) 2

P(I, 2) var[X~]

But :

and : 1 N~

(A-3) Rb(z) = ~ Y~ R~(z), i=1

1 N2 = Z R,~(z'), ~(z ' ) ~ , = ,

(A-4) COV,[iRb(z), ,qb(z')] 1 NI N2

-- NxN2 i=Y~l J'~-~*l'= COVu[R~(z), R~(z')],

or, in the limit N~, N2 -+ 0%

(A-5) COVu[Rb(z), Rb(z')]

-- COVu[R~(z), Rb(z')] dz dz'. LIL2 o o

As a direct result, we have :

(A-6) "ru[X1, )(2] = %[_R~(z), Rb(z')]

var,[R~(z)] 1 --b !

varu[R (z)] L , L 2 I,

~[xl , x d = ~[R~(z), .~(z')],

and :

I L1 i L2 (A-7) I = %[Rb(z), R~(z')] dz dz'.

0 0

Substituting the semi-empirical expression (15) for the correlation coefficient %[Rb(z), Rb(z')] and using the same length for the single paths of the multirelay system, one gets the formula (14) of the main text.

At this point, the lognormal forms for the variables Rb(z) and Rb(z) will be used. As a direct consequence of the assumption that the point rainfall rate R(z) follows the lognormal distribution, the variables Rb(z) and Rb(z) will be, also, lognormal distributed with parameters :

Rmb = R~ (median value of the Rb(z)),

Srb = b S,. (standard deviation of the In Rb(z)),

and :

Rmb am a L

(median value of the Kb(z)),

Srb = S~ (standard deviation of the In Kb(z)).

Using now well known properties of the lognormal distribution [17] one is able to obtain :

(A-8) var.[R~(z)]

and :

(A-9)

_ (cxp(S]b) ) \ Po(0) 1 po2(0) (R2b) exp(S~),

var[_Rb(z)] = (exp(S 2) - - 1 ) (E[Rb(z)]) 2 = (exp(S 2) - - 1) (R2 b exp(S2)).

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516 J. D. KANELLOPOULOS. -- OUTAGE PERFORMANCE OF A MICROWAVE MULTIPLE-HOP NETWORK

E m p l o y i n g fu r t he r the exp res s ions (A-6) , (A-8) a n d (A-9) in (A-2) one gets :

(A-IO) "r[X1, X21 -----

? x p ( S 2 ) ) Po(0) 1 p2(0) (R2mb) exp(S2b) 1

P(1, 2) ( e x p ( S 2 ) - - 1) (-~m2b e x p ( S 2 ) ) L - '~ I +

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

P ( l , 2) ( e x p ( S 2 ) - - 1)"

T h e f inal s tep is the e v a l u a t i o n o f the c o r r e l a t i o n coeff ic ient "r[Xa, X , ] in t e r m s o f the "r[X1, X2].

U s i n g the def in i t ions o f the va r i ab l e s Xa a n d X , a n d f o l l o w i n g a s t r a i g h t - f o r w a r d j o i n t s t a t i s t i ca l ana lys i s , one is ab le to o b t a i n :

( A - I 1) "r[X3, X4]

1 = ~ l n ( ( e x p ( S 2 ) - - 1) "r[X1 )(21 + 1}, S,~

and , us ing exp re s s ion (A-10) e n d s - u p wi th the f o r m u l a (12) o f the m a i n text .

Manuscri t recu le 1 er ddcembre 1986,

acceptd le 30 mars 1987.

R E F E R E N C E S

[1] DRUrUCA (G.), TORLASCHI (E.). Rain outage performance of tandem and route diversity systems at 11 GHz. Radio Sci., USA (1977), 12, n ~ 1, pp. 61-74.

[2] OSBORNE (T. L.). Rain outage performance of tandem and path diversity 18 GHz short hop radio systems. Bell Syst. tech. J., USA (1971), 50, n ~ 1, pp. 59-79.

[3] MORITA (K.), HIGUTI (L). Theoretical studies on simulta- neous probability of rain attenuation in microwave and millimeter wave multi radio relay links. Rev. electr. Commun. Labs, Japan (1977), 25, pp. 329-335.

[4] SEGAL (B.). Spatial correlation of intense precipitation with reference to the design of terrestrial microwave radio relay networks. IEE Conf. Publ., GB (1983), 219, pp. 209- 213.

[5] KANELLOPOULOS (J. D.), GAKIS (L.). Analysis of the rain attenuation outage time for a microwave multirelay link. IEE Conf. Publ., GB (1983), 219, pp. 219-223.

[6] KANELLOPOtfLOS (J. D.), V~NTOtmAS (S. A.). Estimation of the rain attenuation outage time for multi-relay link system at millimetre wavelengths. Ann. Telecommun., Fr. (1986), 41, n ~ 11-12, pp. 556-561.

[7] SEGAL (B.). An analytical examination of mathematical models for the rainfall rate distribution function. Ann. Telecommun., Fr. (1980), 35, n ~ 11-12, pp. 434-438.

[8] MOUPFOOMA (F.). Model of rainfall-rate distribution for radio system design, lEE Proc. pt. H., GB (1985), 132, n ~ 1, pp. 39-43.

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[101 SASAKI (O.), NAOAMt~NE (I3, SArO (K.), HOSOVA (Y.). Rain attenuation characteristics on 20 GHz Band Multi- relay links. IEEE Trans. AP, USA (1981), 29, pp. 587- 594.

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