adjacent satellite interference effects on the outage performance of a dual polarized triple site...

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 7, JULY 2007 2043

Adjacent Satellite Interference Effects on theOutage Performance of a Dual Polarized Triple Site

Diversity SchemeJohn D. Kanellopoulos, Senior Member, IEEE, Theodor D. Kritikos, Member, IEEE, and

Athanasios D. Panagopoulos, Member, IEEE

Abstract—The design of reliable, modern satellite communica-tion networks, in which both frequency and orbital congestion areincreasing, requires modeling of interference effects. The domi-nant sources of aggravation of nominal interference due to propa-gation phenomena are assumed to be differential rain attenuationfrom an adjacent satellite communication network operating at thesame frequency and cross polarization due to rain and ice-crys-tals. A physical methodology to predict the statistics of the car-rier-to-noise-plus-total-interference (CNIDR), which has alreadybeen applied to single and double-site systems, is extended to in-clude triple-site diversity reception schemes. This method is basedon a model of convective raincells model and the lognormal as-sumption for both the point rainfall statistics and slant path rainattenuation. The statistical properties of spatial inhomogeneity ofrain attenuation over six satellite slant paths is firstly here pre-sented. A set of simple, approximate formulas are presented whichfollow from a regression analysis on the previous theoretical re-sults. The results serve to examine the influence of various param-eters upon the total availability performance.

Index Terms—Interference, outage analysis, rain attenuation,satellite communications, site diversity.

I. NOMENCLATURE

: Elevation angle of the slant paths pointingtowards satellite

: Differential angle between two satellites.

: Site separation between stations and.

: Distance between the projections of the slantradio paths and

: Rain attenuations of the wanted signals referringto earth-space paths

: Rain attenuations of the interfering signalsreferring to earth-space paths

: Carrier-to-noise ratio at the earth stationreceiver under clear sky conditions.

: Carrier-to-interference ratio at the earth stationreceiver under clear sky

conditions.: .

Manuscript received February 3, 2005; revised October 4, 2006.The authors are with the Wireless & Satellite Communications Group, Divi-

sion of Information Transmission Systems and Materials Technology, Schoolof Electrical and Computer Engineering, National Technical University ofAthens, Zografou-GR 15780, Athens, Greece (e-mail: [email protected];[email protected]; [email protected]).

Digital Object Identifier 10.1109/TAP.2007.900173

: Cross-polarization discrimination due to rainattenuation and ice crystals concerningsatellite path .

: Cross-polarization discrimination concerningsatellite path under clear sky

conditions.: Carrier-to-noise plus total interference ratio at

the earth station receiver .: Effective average length of the earth-satellite

paths corresponding to thewanted signals.

: Effective average length of the earth-satellitepaths corresponding to theinterfering signals.

: The projected and respectively.

: Rain attenuations calculated for the projectionsof the slant paths .

: Rain attenuations calculated for the projectionsof the slant paths .

: Constants of the specific rain attenuation Ao(dB/km).

: Polarization tilt angle.

: Lognormal statistical parameters of the pointrainfall distribution.

: Non exceedance level of the (CNIDR) ratio (dB).

: Lognormal statistical parameters of thedistributions concerning , andr.v.s.

: Lognormal statistical parameters of thedistributions concerning , and r.v.s.

: 2-dimensional normal joint density function.

: 3-dimensional normal joint density function.

:4-dimensional normal joint density functions.

:5-dimensional normal joint density function.

:

Statistical parameters ex-pressed in terms of the loga-rithmic correlation coefficients

and .

0018-926X/$25.00 © 2007 IEEE

2044 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 7, JULY 2007

Fig. 1. Configuration of the problem under consideration.

II. INTRODUCTION

WAVE attenuation on the Earth-space path through the tro-posphere is associated with various physical phenomena

such as atmospheric gases and clouds, but rain attenuation isconsidered to be the dominant factor at carrier frequencies above10 GHz [1]. Due to their large available bandwidths, the Ka(20/30 GHz) and V(40/50 GHz) frequency bands are becomingincreasingly attractive for modern commercial satellite servicesin comparison to the almost congested Ku (12/14 GHz) band.On the other hand, the former frequency bands suffer from rainfades to a larger extent. This is also more severe for systemsoperating in heavy rain climatic regions, e.g., subtropical andtropical regions.

To combat rain attenuation several fade mitigation tech-niques have been developed such as diversity protectionschemes, power control and adaptive processing techniques[2]. Among these techniques, the most efficient is site diversity(SD) [3]. SD engages two or more earth stations to ensure thatthe probability of a specified rain attenuation being simulta-neously exceeded on all alternative paths is significantly lessthan the probability of exceedance on each individual path. Itis important to note here that interest in SD has been recentlyrenewed (see for example the Iridium project [4]), since groundterminal antennas and other hardware sizes have been reducedto an extent where terminals can now be installed in customers’premises and the use of public terrestrial networks to carry outthe signalling between them seems to be possible. Multiplesite diversity schemes are recommended, in heavy rain cli-matic regions for advanced satellite based applications such astelemedicine and teleconferencing where the outage time needsto be minimized [5], [6]. These terminals are commonly knownas very or ultra small aperture terminals (VSAT or USAT) andnumerous relevant applications are emerging.

In addition, interference must be seriously taken into accountfor reliable system design. The dominant interference sourcesexamined here are: (a) Adjacent satellite interference, due totraffic transmitted to and from adjacent satellites and (b) Cross-polarization interference, caused by traffic transmitted to andfrom transponders on the other polarization of the same satellite.The later source, of course, is of consequence for systems em-ploying the frequency re-use technique. Under rain fade condi-tions, both sources may degrade significantly the carrier to inter-

ference ratio due to potential existing differential rain at-tenuation and rain and ice crystals depolarization, respectively.

As a first step, the outage performance analysis has beenbased on the assumption that the thermal noise is dominant,and, consequently, the contribution of the interference effectshas been taken into account by calculating the degradation ofthe ratio under the assumption that the rain attenuationinduced on the wanted path does not exceed an allowed rainfade margin M [7]–[10]. On the other hand, due to the contin-uing increase in frequency and orbital congestion and the subse-quent interference effects, the more complicated analysis of thedegradation of the total carrier-to-noise plus interference ratio(CNIDR), under the presence of rain fades, becomes imperative.Up to now, several models have been proposed dealing with theprediction of (CNIDR) considering both single and double sitediversity protected systems as well as systems using single anddual polarization [11]–[14].

The subject of the present paper is an extension of the latteranalysis to include interference systems operating with triple-site diversity protection, which will be indispensable in somecases in the near future, such as earth-space paths located inheavy rain climatic regions, as mentioned previously [5], [6].So far, a number of three-site diversity systems have alreadybeen operated such as the two years three site diversity experi-ment at 20 GHz with ACTS [15]. From another point of view,the present analysis can also be considered to be an extension ofthe corresponding problem concerning an interfered triple sitediversity system where only the degradation of the carrier to in-terference ratio has been examined [10]. Due to the math-ematical complexity of the presented results, a set of simple ap-proximate formulas is also presented, derived after employingan appropriate regression analysis on the previous results. Fi-nally, numerical results are presented dealing with the applica-tion of the predicted (CNIDR) statistics towards the reliable de-sign of an interfered triple-site diversity system under rain fadeconditions.

III. THE INTERFERENCE ANALYSIS

The configuration of the problem under consideration isshown in Fig. 1. Three earth stations , and are in com-munication with a satellite , forming a triple-site diversityprotection scheme. A second interfering satellite , operating

KANELLOPOULOS et al.: ADJACENT SATELLITE INTERFERENCE EFFECTS ON THE OUTAGE PERFORMANCE 2045

at the same frequency, is in orbit close to ; the two subtendingthe same angle to , and . The elevation angles of thevarious earth-space paths pointingtowards satellites and are considered to be different,denoted by for the , and paths, and forthe , and paths, respectively.

The main objective of the analysis is the evaluation of the timewhere the carrier-to-noise plus total interference ratio (CNIDR),under rain fade conditions, does not exceed a specified level r(in dB), for paths , and . Following their defi-nitions, the (CNIDR) (in dB) for the th path , cor-responding to respectively, can be expressed as

(1)

(2)

In the above expressions, is the rain attenuation of thereceived signal for the path, and the attenuationof the potential interfering signal concerning the path.Moreover, , and represent the nominalvalues (clear sky conditions) for (CNR) (carrier-to-noise ratio)and (CIR) (carrier-to-interference ratio). It is worthwhile tonotice here that the and are expressedin terms of the basic link parameters [16]. Details for theanalytical derivation of the expressions (1)–(2) can be found inAppendix A. is the cross-polarization discrimina-tion for the th path, due to rain fading and ice crystals. Moredetails for its expression can be found in [17].

For the following analysis we assume a balanced diversitysystem, something usual for adjacent satellite interference ef-fects, and consequently we have (3) and (4), shown at the bottomof the page. As described previously, the aim of the analysisis the calculation of the following joint non-exceedance proba-bility, shown in (5) at the bottom of the page. The analysis of(5) is then based on some fundamental assumptions such as:

1. The lognormal form is adopted for both the unconditional(including non raining time) point rainfall rate and at-

tenuation distributions. The lognormal statistical param-eters of the point rainfall distribution for each lo-cation are calculated by using available experimental dataprovided by ITU-R rainmaps [18].

2. The constants and of the specific rain attenuationcan be found in [19].

3. The convective raincell model proposed by Lin [20] for thedescription of the horizontal variation of the rainfall spatialstructure is employed.

4. Crane’s [21] consideration of the vertical variation of therainfall structure is also adopted.

As a result of the above considerations and following its def-inition, the joint outage probability can be written as:

(6)

where

(7)

(8)

(9)

(10)

(11)

and

(3)

(4)

(5)


(12)

(13)

(14)

and

(15)

where are the roots of the following transcendentalequations:

(16)

and

(17)

Moreover we have

(18)

In the above expressions

andare the joint probability density

functions between the statistical variablesand . For the definition of ,and see the Appendix A. Details for thederivation of the expressions (7)–(18) can also be found in

the same Appendix. It should be noted that the same kindof joint probabilities also appear elsewhere [14], where thecorresponding interference problem, but valid for a double sitediversity system, has been examined. In the present case, dueto the complex geometry, the encountered joint probabilitiesare much more complicated and consequently their reliableand accurate evaluation is one of the fundamental keys of thewhole analysis.

The next step is the calculation of the probabilities (7)–(14).This can be achieved by using the previous considerations [par-ticularly assumption (1)] and after following a straightforwardanalysis (given in Appendix A). The final results are1

(19)

(20)

(21)

(22)

(23)

(24)

(25)

1The programming code concerning the numerical calculation of the multidi-mensional integrals can be obtained by sending a request to: [email protected]


(26)

where

(27)

(28)

The analytical forms of the 2,3-dimensional normal densityfunctions and the parameters

and can be found in [13],where the corresponding problem concerning an interfereddouble site diversity system using single polarization isexamined.

Furthermore, the forms for theand , along with the parameters

,and are expressed in terms of

. is the correlation coefficientbetween between ,and between . In the same way,is the correlation coefficient between

between , and between. The same holds for the , andcorrelating the

with the variables respectively.Analytical expressions for these coefficients can be foundelsewhere [10]. Finally, is the correlation coefficientbetween between ,and between . Analogous formulaswith the can also be derived for thesecoefficients (see expressions (B-28)–(B-32) of [10]). Further,

are the lognormal statistical parameters of theattenuation oror . In the same way,are the lognormal statistical parameters of the attenuation

or or .The above parameters are expressed in terms of the pointrainfall parameters , the constants and of thespecific rain attenuation and the characteristicparameter depending on the rainfall spatial structure, asproposed by Lin [20]. The analytical expressions for ,and are presented elsewhere [12]. It should benoticed that the numerical calculation of the multidimensionalintegrals encountered in (19)–(26) is a problem itself andhas been achieved by using adaptive Monte-Carlo techniques[22]. The convergence of the above technique has been foundquite satisfactory for all the examined cases shown in the nextsection.

As a final step, due to the complexity of the proposed expres-sions, a simple and easily applicable model in a form compat-ible with the ITU-R suggestions is introduced. This has beenachieved by employing a regression analysis on the numericaldata derived from the analytical model. For this reason, the non-linear least square fitting Levenberg-Marquadt algorithms [23]has been used. The resulting expression is given by:

(29)

where

(30)

(31)

(32)

(33)

where the involved variables take values in the following ranges:

(34)

(35)

It is worthwhile to notice that for the derivation of relation(29), the following typical in current satellite communicationnetworks values have been assumed: dB,

dB and dB (see expression(36)). The numerical constants , and

depend on the frequency, the climatic conditionsof the location where the earth station is cited and the polariza-tion tilt angle .

Numerical values for these constants, corresponding toGHz, a heavy rain climatic region such as the P-climatic

zone, where the triple site diversity cannot be avoided andfor the double polarization case, are presented in Table I.

Analogous formulas with expression (29), but with differentconstants can be obtained for every location of the world. Theagreement between the analytical and simplified model has beenfound very good, with the rms error being less than 1 dB in allexamined cases. Rms error of proposed simple model for var-ious levels of availability is shown in Table II.

IV. NUMERICAL RESULTS AND DISCUSSION

A. Numerical Application of the Proposed Model

We now proceed to the application of the preceding analysisto the prediction of the degradation of the total carrier-to-noiseplus interference ratio (CNIDR) for a triple site diversity systemoperating under the conditions indicated previously. As notedelsewhere [14], experimental data for this kind of problems exist


TABLE INUMERICAL VALUES FOR CONSTANTS a (i = 1; 4); b (i � 1; 2),

AND c (i = 1; 5)

TABLE IIRMS ERROR OF THE PROPOSED MODEL

only for the single site case, with respect to a noise-dominant op-eration [24], [25]. In a recent publication [26], an extensive com-parison between the above experimental data and results takenfrom a relative predictive model [7], [8] has been attempted,where a clear tendency of coincidence is obvious. A sufficientdegree of credibility for the present procedure is thus estab-lished after taking into account that the analysis of the single-sitecase is considered to be the basis of the proposed methodology.Moreover, it is also strengthened by the fact that the input of thepredictive analysis (point rainfall statistics) is consistent withthe recent ITU-R suggestions [18].

Furthermore, the numerical results presented hereafter areconcentrated on the examination of the optimum utilization ofthe geostationary orbit as well as the influence of various param-eters upon the total availability performance. For this reason, anexisting 20 GHz earth-space system located in Sao Paolo, Brazilis used [27]. We assume that the station is operating under a triplesite diversity protection scheme to overcome the very large fademargins due to the very large prescribed availability times. Thesystem is considered to be interfered by an hypothetical adjacent

TABLE IIIPARAMETERS OF THE PROBLEM UNDER CONSIDERATION

TABLE IVOUTAGE PROBABILITY VERSUS SERVICE FOR VARIOUS RECEPTION SCHEMES

satellite path using the same frequency. The implementation ofthe predictive procedure requires the knowledge of the param-eters , and with respect to theslant paths under consideration. The definition of the effectivelengths , and is given in the Appendix B. A list of appro-priate values for these parameters is presented in Table III. Thelognormal statistical parameters of the point rainfalldistribution are calculated by using available experimental dataprovided by ITU-R rainmaps [18], while the values for and ofthe specific attenuation have been estimated by using an appro-priate distribution for heavy rain climatic regions [19] (Table IV).We also present an example with the application of the proposedmodel in a digital transmission system (voice or video).

Before proceeding to the numerical results, we will calculatethe additional gain (according to Matricciani’s formula [28])using the parameters in Table III. The built-in-gain is

dB, while the triple site diversity gain usingconvective raincell model [10] is dB Then

. It is obvious from the above, thatthe additional gain is much larger than 0 dB, and that result in acost-effective SD system.

In Fig. 2, the non-exceeded levels (in dB) of the (CNIDR)versus angular separation (in degrees), between the wantedand interfering satellites, are presented. The non-exceededlevels have been calculated for outage time

and various interference levels under clearsky conditions. The interfered system has been examined byconsidering both double and triple site diversity configurationswith the same separation distance km. The determina-tion of the interference levels, under clear-sky conditions, canbe taken into account by means of the following expression:

(36)


Fig. 2. Carrier-to-noise plus total Interference ratio (CNIDR) (in dB) versusangular separation between adjacent satellites (� ) for an interfered double/triple site diversity system located in Sao Paolo, Brazil and using dual polariza-tion. The other parameters are: f = 20 GHz, (CNR) = 25 dB, outagetime = 30 min/year, (XPD) = 25 dB, � = 45 . (A): Clear-sky curve,(INR) = 20 dB (B): Double-site (S = 10 km), (INR) = 20 dB(C): Triple-site (S = 10 km), (INR) = 20 dB (D): Clear-sky curve,(INR) = 3 dB (E): Double-site (S = 10 km), (INR) = 3 dB (F):Triple-site (S = 10 km), (INR) = 3 dB.

where represents the differenceof the interfered system under consideration for

. Two representative values for (3 dB and20 dB) have been chosen and the corresponding diagrams areshown in Fig. 2. As a consequence of the above selected values,the interfered systems with respect to dB be-have as interference-dominant for less than about 5 , whereasfor higher values of the dominates. On the otherhand, the systems characterized by dB demon-strate a noise dominant behavior for all values of greater thanabout 1 . The numerical values for aregiven in the figure captions. As can also be seen, the triple-sitediversity system has the same outage behavior as happens withthe double-site corresponding one for both dBand dB values. But the influence on the de-crease of the required rain fade margin, due to the employmentof the triple site diversity protection, is more significant for thenoise than the interference dominant systems. For example,for the use of triple site scheme reduces the rainfade margin from 7.35 dB to 2.85 dB for thedB system. For the dB system, however, therespective values are 12.61 dB (double-site system) and 6.62dB (triple-site configuration). It is also obvious that the triplesite mitigates the interference from the adjacent satellite net-work better than the double site one. If we have used an n-siteconfiguration for example with the same linearway as in Fig. 1, the corresponding curves of (CNIDR) curvesin 4-site scheme will tend more to the clear sky curve but thestep will be smaller than the one from the double to triple site

Fig. 3. Carrier-to-noise plus total Interference ratio (CNIDR) (in dB) versus� for an interfered triple-site diversity system located in Sao Paolo, Brazil andusing dual polarization. The other parameters are: f = 20GHz, (CNR) =25 dB, (INR) = 20 dB, outage time = 30 min/year, (XPD) = 25dB, � = 45 . (A): Clear-sky curve, (B): S = 15 km, (C): S = 10 km, (D):S = 7 km.

diversity and this is due to the existence of more correlatedpaths. All these will also be observed going from a triangularscheme to a quadrangle one. These comments are valid fromthe transition of an n-site to n+1-site diversity scheme.

In Fig. 3, (CNIDR) curves (in dB) versus are presentedfor the dB system as above, but for variousdiversity distances , and 15 km. As it is obvious, theincrease of the parameter has a significant impact on the re-quired separation angle between the satellites under consider-ation. More particularly, for a desired non-exceedance level of13 dB and , a minimum value of 2.5 isappropriate for the km system, whereas the above valueincreases to 7 for the km system.

In Figs. 4 and 5, the determination of the appropriate (inkm) in order to achieve a particular (CNIDR) level, is exam-ined for and various interference levels andoutage times. As can be seen, the influence of the increase of

on the achievable non-exceedance level (CNIDR) (in dB) ismore significant for a system characterized by noise dominantoperation in clear-sky conditions and small prescribed outagetimes. For example, an increase of from 5 to 25 km leads to a(CNIDR) improvement of about 6 dB for thedB system while the corresponding value is only 4 dB for the

dB one (see Fig. 4). In the same way (see Fig. 5),the improvement of (CNIDR) is about 4 dB for a system oper-ating under and 5.3 dB for

. These remarks are valuable for a system designerin order to determine the optimum value for the diversity dis-tance in a triple-site protection configuration.

Finally, the dependence of the system availability perfor-mance upon the joint variation of the various parameters of the


Fig. 4. The CNIDR threshold (in dB) as a function of the separation distance S(in km) for various interference levels. The other parameters are: Location: SaoPaolo, Brazil, f = 20 GHz, � = 3 ; (CNR) = 25 dB, outage time =30 min/year, (XPD) = 25 dB, � = 45 . (A): Triple-site, (INR) =�3 dB, (B): Triple-site, (INR) = 3 dB, (C): Triple-site, (INR) =9 dB.

Fig. 5. The (CNIDR) threshold (in dB) as a function of the separation distanceS (in km) for various outage times. The other parameters are: Location: SaoPaolo, Brazil, f = 20 GHz, � = 3 ; (CNR) = 25 dB, (INR) = 3dB, (XPD) = 25 dB, � = 45 . (A): Triple-site, Outage time : 90 min/year, (B): Triple-site, Outage time : 60 min/year, (C): Triple-site, Outage time :30 min/year.

problem is examined. For this reason, the simple arithmeticalmodel, [see expression (29)], provides an efficient and easilyapplicable tool. In Fig. 6, nomograms of non-exceeded levels(CNIDR) (dB) versus separation angle and diversity distance

(in km) are shown for the Earth-space system located in SaoPaolo, Brazil. The other parameters correspond are shown inTable III Diagrams of this kind are very useful because, bymeans of these, one is able to determine appropriate sets ofnumerical values for the parameters involved in the analysissuch that a particular non-exceeded level of (CNIDR) shouldbe established.

B. Application of the Proposed Model in a Basic TransmissionSystem

The proposed physical model can be applied straightfor-ward to digital satellite communication networks for all themodern satellite services, including satellite voice and videoapplications, which have a growing share in today’s telecom-munication market (satellite VoIP, DVB-S). Firstly, we definethe bit-error-rate (BER) requirements of those services:for voice transmission and for Video transmission. Moreover adopting the 3/4 forward error correction (FEC) ratio,which is the most common one in current VSAT applications,the following thresholds must be achieved: 6.3 dBfor Voice transmission and 9.8 dB for Video transmission [3]in order to satisfy the quality of service requirements. Theminimum (CNIDR) (dB) at the input of the receiver, necessaryto achieve the above BER requirements, is given

(37)

where the usage of frequency re-use technique has been as-sumed, and for QPSK modulation. In thisway, we obtain the (CNIDR) requirements: 8.05 dB for voicetransmission and 11.55 dB for video transmission. Using theprevious analysis, we draw the outage probability versus theachieved (CNIDR) threshold for a dual polarized Earth terminalsuffering from adjacent satellite interference, under rain fades,located in Sao-Paolo, considering single, double and triple sitediversity schemes: , and have been con-sidered at the levels of 20 dB and 6 dB, respectively. FromFig. 7, it is obvious that the above BER requirements for voiceand video are satisfied with the lowest outage probability levelsby means of the triple site diversity scheme. In Table IV, thecorresponding values of outage probabilities in (min/year) forvoice and video transmission and various reception schemesare given.

V. CONCLUSION

The prediction of the degradation of the total carrier-to-noiseplus interference ratio (CNIDR) has been examined, for a dual-polarized system located in a heavy rainy climatic region andoperating under very large availability times. In this case, dueto the very large induced rain fade margins, the triple-site diver-sity technique is inevitably imposed, and the present analysisconcerns the calculation of the total outage time of a site diver-sity system suffering from both differential rain attenuation andrain depolarization. Numerical results taken from the proposedprocedure examine the outage performance for a system located


Fig. 6. Nomograms of non-exceeded levels (CNIDR) (dB) versus separation angle � and diversity distance S (in km) for the Earth-space system located in SaoPaolo, Brazil. The other parameters are the same as in Fig. 2, (INR) = 20 dB.

Fig. 7. Outage probability versus carrier-to-noise plus total interference ratio(CNIDR) (in dB) for an interfered double/triple site diversity system located inSao Paolo, Brazil and using dual polarization. The other parameters are: f =20 GHz, (CNR) = 20 dB, (INR) = 6 dB, (XPD) = 25 dB,� = 45 . Double-site (S = 10 km), triple-site (S = 10 km).

in Sao Paolo, Brazil. The conclusions that can be deduced aresummarized as follows.

a) The impact on the decrease of the required rain fademargin due to the employment of the triple-site diversityconfiguration is more remarkable for a system operatingunder noise dominant clear-sky conditions.

b) The influence of the increase of the separation distanceupon the achievable non-exceedance level (CNIDR) (indB) is more significant for a noise dominant operationsystem under clear sky conditions and small outage times.

c) Finally, diagrams are drawn where the dependence of thesystem availability upon the joint variation of the var-ious parameters involved is demonstrated. By means ofthese diagrams, the system designer is able to determineappropriate sets of numerical values for the above pa-rameters such that a particular (CNIDR) level should beestablished.

APPENDIX ADERIVATION OF THE EXPRESSIONS (1), (2)

Each earth station is receiving the followingsignals.

Receiving power of wanted signal under clear skyconditions.Receiving power of wanted signal during rain effects.

Receiving power of interfering signal under clear skyconditions.Receiving power of interfering signal during rain effects.

Receiving power, due to depolarization, of theorthogonal to the wanted signal, under clear skyconditions.Receiving power, due to depolarization, of theorthogonal to the wanted signal, during rain effects.Receiving power of white Gaussian noise.

The rain attenuation of the wanted signal can be expressed by

(A-1)

the rain attenuation of the interfering signal by

(A-2)

the cross polarization discrimination expressed by

(A-3)

and also the interference-to-noise ratio

(A-4)


Assuming that the signals are statistically independent,the (dB) under rain fades is given

(A-5)

Equation (A-4) can now be expressed as

(A-6)

which is (1) of the main text.

APPENDIX BDERIVATION OF THE EXPRESSIONS (7)–(28)

As pointed out in the main text the probability under consid-eration can be calculated as in (B-1), shown at the bottom of thepage. Following Crane’s simplified considerations for the ver-tical variation of the rainfall structure [21] we may write

(B-2)

Fig. B-1. AAA � AAA plane.

where , and are the attenuationscalculated for the hypothetical terrestrial links with path lengths

for , andfor respectively. Moreover, and are the ef-fective average lengths of the Earth-satellite paths with respectto the wanted and interfering signals [21]. Substituting (B-2)into (B-1) we have (B-3), shown at the bottom of the page. Con-sidering now the planes and(see Fig. B-1) we draw the curves

(B-4)

(B-5)

(B-1)

(B-3)


(B-6)

By setting 0, for into (B-4)–(B-6) we can deter-mine the respective roots , and

. We have obviously,

(B-7)

Using now the definition of the double integral and the confir-mation that the functions , and are pos-itive increasing functions of , the following non-ex-ceedance probabilities can be expressed as:

(B-8)

(B-9)

(B-10)

and the levels are given by means of theexpressions (15)–(17) of the main text. The levels

are the roots of the (B-4)(B-6) for and respectively (seealso expression (18) of the main text).

It is now obvious that the joint non-exceedance probabilitygiven by (B-1) can be expressed as:

(B-11)

where the probabilities are givenby means of the expressions (7)–(14) of the main text.

Adopting now assumption (1) (Section III) and using the fol-lowing definition for the normal variables

(B-12)

we have

(B-13)

(B-14)

(B-15)

(B-16)

(B-17)

(B-18)

(B-19)

(B-20)

The definition of and is given in themain text [see expressions (27) and (28)]. Using now the Bayestheorem [29] the above joint density functions can be expressedas


(B-21)

Substituting (B-21) into (B-13) (B-20), and using the fol-lowing result from the theory of normal variables

(B-22)

where is a normal density function with parameters ,one gets expressions (19)–(26) of the main text.

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John D. Kanellopoulos (SM’90) was born inAthens, Greece, on December 12, 1948. He re-ceived the Diploma of Mechanical and ElectricalEngineering and the Dr.Eng. degree from NationalTechnical University of Athens (NTUA), in 1971and 1979, respectively. He also received the DIC andPh.D. degrees from the Imperial College of Science,Technology and Medicine, University of London,London, U.K., in 1979.

In November 1979, he joined the Department ofElectrical Engineering, NTUA, where he is now a

Professor. He has authored and coauthored more than 250 papers in interna-tional journals, transactions and conference proceedings. His areas of interestare microwave propagation through rain media, satellite communication sys-tems, non-linear optics, rough surface scattering and waveguide propagation.


Theodor D. Kritikos (M’99) was born in Athens,Greece in 1974. He received the Diploma Degreein electrical engineering and computer technologyfrom the Polytechnic Department of the Universityof Patras, Greece, in October 1998 and is currentlyworking toward the Ph.D. at the National TechnicalUniversity of Athens (NTUA).

He is a Research Assistant at the Wireless Commu-nications Laboratory of NTUA. His research interestsinclude satellite communications system design andthe effect of propagation impairments on microwave

links.Mr. Kritikos is a member of the Technical Chamber of Greece.

Athanasios D. Panagopoulos (M’98) was born inAthens, Greece, on January 26, 1975. He receivedthe Diploma degree in electrical and computer engi-neering (summa cum laude) and the Dr. Engineeringdegree from the National Technical Universityof Athens (NTUA) in July 1997 and April 2002,respectively.

From May 2002 to July 2003, he had served theTechnical Corps of Hellenic Army. In September2003, he joined the School of Pedagogical andTechnological Education, as an Assistant Professor.

He is also a Research Assistant with the Wireless and Satellite CommunicationsGroup of NTUA. He has authored and coauthored more than 100 papers ininternational journals, transactions and conference proceedings. His researchinterests include radio communication systems design, wireless and satellitecommunications networks and the propagation effects on multiple accesssystems and on communication protocols.

Dr. Panagopoulos is the recipient of URSI General Assembly Young ScientistAward in 2002 and 2005, respectively. He is a member of the Technical Chamberof Greece.

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