Notice: Wiley Online Library will be unavailable on Saturday 27th February from 09:00-14:00 GMT / 04:00-09:00 EST / 17:00-22:00 SGT for essential maintenance. Apologies for the inconvenience.
 Site diversity is considered to be an effective technique to overcome potential large rain fade margins in satellite communication links. In addition, because of the expected orbital and frequency congestion, the aggravation of the signal leakage from an adjacent Earth-space system operating at the same frequency due to potential differential rain attenuation will be considered as a dominant source of interference. In this paper, an existing method to predict the differential rain attenuation statistics with respect to a double-diversity system is properly modified after assuming a more realistic model for the rain height. Some simple regression-derived formulas appropriate for use by the system designer and concerning both existing and proposed analysis are also presented. Further, the difference between existing results and the ones deduced from the use of the modified procedure for various availabilities and frequencies is examined, and some useful conclusions are derived.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 Modern satellite communication systems employ frequencies above 10GHz, thus allowing for higher antenna directivities and larger communication capacities. On the other hand, in this band of frequencies, the propagation is adversely effected by rain, which is a basic limiting factor concerning the outage performance of radio link. In heavy rain climatic regions, the multiple site diversity has been introduced as an adaptive method to reduce the large rain fade margins. As a result of the spatial inhomogeneity in the precipitation medium the use of two (or more) interconnected Earth terminals properly spaced can significantly decrease the outage time or, equivalently, the required fade margins.
 Furthermore, the consideration of the interference effects is of current importance for the reliable design of a modern communication system. Because of the expected orbital and frequency congestion, we will adopt here as dominant source of interference the aggravation of the signal leakage from an adjacent Earth-space path operating at the same frequency due to differential rain attenuation. When examining the predictive analysis of the above statistical variable, we should point out the following: First, Rogers et al.  have proposed a semiempirical model for the differential rain attenuation statistics at the 1% conditional probability level. Later, Kanellopoulos et al.  have presented a more general predictive methodology, based on a model of convective raincells and the lognormal assumption for the point rainfall rate statistics. Some recent experimental results [Matricciani and Mauri, 1996; Matricciani, 1997] have shown the importance of the problem and the necessity of the interference analysis on the Earth-space systems. In a next step, the analysis has been extended to include double site interfered systems [Kanellopoulos and Ventouras, 1996]. As noted previously, this is an important subject for Earth-space stations located in regions characterized by heavy rain climatic conditions, where the use of the double site diversity scheme seems to be inevitable. One of the fundamental assumptions of the above analysis concerns Crane's simplified consideration for the vertical variation of the rainfall structure [Crane, 1978, 1980]. According to this consideration, a uniform rain structure from the ground up to a constant mean seasonal rain height H is being assumed. But this is a simplification and the more realistic model for the determination of the rain height has been experimentally predicted to take into account the correlation between surface rain rate and rain rate height H. For this reason, Stutzman and Dishman  have presented a more realistic model for the effective rain height, consisting of using the constant level for low rainrates and adding a rainrate dependent term for higher rainrates. This model has been employed by the same authors [Stutzman and Dishman, 1982] for the prediction of the slant path rain attenuation statistics. As a following step, a motivation exists for the proper modification of the differential rain attenuation predictive analysis to take into account the more realistic model description of the rain height. Most recently, Kanellopoulos and Margetis  have proposed a modified method, valid for the single-site interfered systems. The more complicated, but inevitable for heavy rain climatic regions, double site diversity is examined here. Also, there is a preference here for using the Stutzman and Dishman  formulation for the rain height, instead of the corresponding one suggested by the International Radio Consultative Committee (CCIR) , because the former is more consistent with the proposed calculation procedure for determining the differential rain attenuation statistics. Because of the complicated nature of the expressions that are derived, some simple pocket calculator regression-derived formulas, appropriate for use from the system designer, have been developed and concern both existing and modified results. Numerical results presented at the end indicate the influence of the more realistic assumptions for the rain height for various availabilities and frequencies.
2. Interference Analysis
 The configuration of the problem under consideration is shown in Figure 1. Two Earth stations E1 and E2 are in communication with a satellite S1, forming a double-site diversity protection scheme. Another satellite S2 operating at the same frequency is in orbit close to S1, the two subtending an angle θd to E1 and E2, with elevation angles φ1 and φ2 respectively. For the following analysis, we denote , the attenuation levels of the wanted signal referring to Earth stations E1 and E2, while , are the corresponding ones of the potential interfering signals and M1, M2 the system margins available for rain attenuation. A balanced diversity system will be adopted here leading to the assumption of M1 = M2 = M. As a result of the site diversity operation, the selection of the receiving Earth terminal (E1 or E2) at each instant will depend on the carrier-to-noise plus interference ratio levels, respectively. The analysis of the outage time in the present case requires the calculation of the cumulative probability distribution that a given ratio is not achieved simultaneously for the two Earth stations. This is a quite complex problem to address and for this reason the analysis for a simplified version is presented here dealing with an interfered noise dominant system. In this case, the carrier-to-interference ratio is always at least 3dB greater than the carrier-to-noise ratio [Ha, 1991]. The above assumption is equivalent to the reasonable consideration that the selection of the receiving station (E1 or E2) at each instant will depend on the corresponding least carrier-to-noise ratio.
 In the light of the above considerations the major part of the nonavailable time concerning the total diversity system will occur whenever ≥ M and ≥ M. For a noise dominant system, an upper limit of the additive contribution of the interference effects on the total outage time can be taken into account by means of the following probability: this is the fraction of the time when the system suffers from the interference, as part of the time when the event ≥ M and ≥ M is not present, but the system is under rain fade conditions. In mathematical terms this conditional probability can be expressed as
where rM is an indicative threshold depending on the sensitivity of the attenuation measurements, used to describe the period of time when the system operates under rain fade conditions. The specific value “0.5” suggested by Rogers et al.  will be adopted here. Under rain fade conditions the are given by
 For the evaluation of the above probabilities the following considerations should be taken into account:
As noted in the Introduction, we adopt here the model proposed by Stutzman and Dishman  for the determination of the effective rain height. There the rain structure is assumed to be uniform from the ground up to an effective rain height He.
In the above expressions Λ is the latitude of the location of the Earth terminal and R is the value of the rainfall rate at the specific point. The above formulas concern mean seasonal values for He, as it is proper for long term rain attenuation statistics. The rain height He varies generally with rainrate and as a direct result the effective slant path length is also variable. Details for the determination of the appropriate effective lengths for both wanted and interfering paths are given by Kanellopoulos and Margetis .
According to the assumption of uniform vertical rain structure, the single and joint exceedance probabilities (equations (1)–(3)) can be obtained as
In these expressions, , and , are the attenuations calculated for hypothetical terrestrial links which are the projections of the corresponding slant paths affected by the rain medium.
All other assumptions concerning point rainrate statistics, the specific rain attenuation (Ao = aRb), and the horizontal structure of the rainfall medium are the same as those employed for the analysis of the respective problem [Kanellopoulos and Ventouras, 1996], where the constant rain height model is used.
2.2. Calculation of the Conditional Probability
 Following the previous considerations, the probabilities in expressions (7)–(10) can be expressed in terms of the lognormal parameters , and , concerning the attenuations , and , , as well as the correlation coefficients ρ12, ρ23, ρ13 and ρ′12, ρ′23, ρ′13 between them. The use of the model by Stutzman and Dishman  for the rain height affects only the calculation of the above parameters, not the analytical forms of the exceedance probabilities. These new values of the above parameters may be directly used in the formulas for the proposed interference analysis by Kanellopoulos and Ventouras .
 The evaluation of the , and , can proceed by expressing these parameters as [Papoulis, 1991]
in terms of , and , which are the mean values and standard deviations of the variables , and , respectively. Following now a straightforward analysis, the , can be calculated by means of the Rm, also, Sr concerning the point rainrate distribution, the constants a and b of the specific rain attenuation (Ao = aRb) as well as the parameters G and Dr appropriate for the description of the spatial rainfall inhomogeneity [Lin, 1975]. The final results are presented in Appendix A.
 As far as the correlation coefficients, because of the symmetry of the configuration, we have
Further, the correlation coefficients ρ12, ρ13, ρ23 and ρ23′ can be expressed as [Papoulis, 1991]
and, consequently, as a result one could calculate the expected values E[ ], E[ ], E[ ] and E[ ]. In a similar manner, as before for and , one gets
More particularly, the calculation of the factors K12, K1d, K2d and Kdd is presented elsewhere (see expressions (36)–(52) of Kanellopoulos and Livieratos ). Because of the complexity of the expressions, there is no need to present these formulas here, but one is again referred to Kanellopoulos and Livieratos . In the same way, the analytical expressions for Λ12, Λ1d, Λ2d and Λdd have been calculated and are given by expressions (25)–(31) of Kanellopoulos and Margetis . As far as the factors in expressions (20)–(21) are concerned, their evaluation follows similar steps as before for Λ12, Λ1d, Λ2d and Λdd. Due to the complexity of the expressions, we prefer to give on request the PASCAL/C code concerning the numerical evaluation of the joint exceedance probabilities (07)–(10). (The PASCAL/C code under consideration can be obtained by sending a request to the last author by e-mail at firstname.lastname@example.org.)
 In the next section, we further present some simple pocket calculator expressions derived after using an appropriate regression fitting analysis. These expressions are very useful to system designer and concern the variation of the nonexceeded (C/I) levels versus the angular separation θd.
3. Simulated Results
3.1. Numerical Evaluation
 Here we apply the previous analysis to the prediction of the interference induced by an adjacent path on a satellite system using double site diversity reception. We show the results obtained from the single site (e.g. Figure 2) that give support to the use of the variable rain height [Matricciani and Mauri, 1996; Matricciani, 1997; Kanellopoulos et al., 2000]. Double and triple site diversity experimental results have been compared with the corresponding ones of the theoretical analysis; the latter also uses the model of convective raincells and the lognormal assumption for the point rainfall rate statistics [Kanellopoulos et al., 1990; Kanellopoulos and Koukoulas, 1990]. The above remarks provide a sufficient degree of credibility for the proposed methodology, because the theoretical analysis concerning the single site interfered case and the multiple site diversity problem constitute the basis of the present procedure.
 In addition our examination is mainly concentrated on studying the reliable and economic design of the interfered communication system. For this reason, we consider a hypothetical communication system operating both in the Ku (12GHz) and Kα (20GHz) bands, as defined by Ha . The climatic conditions of the Earth-space station are examined by considering the L climatic zone which is representative of the Mediterranean areas, although the prediction model is applicable to any region where the lognormal assumption for the point rainfall rate is valid. Considering now that heavy rain climatic conditions have occurred along with prescribed large availability times, a double diversity scheme with S = D = 10km (see Figure 1), is employed, in order to have the lower possible fading margins (LFD operation ) . For all cases, the potential interfering satellite is taken such that the separation angle θd is always 3°. On the other hand, the common elevation angle for both wanted and interfering signals is taken to be 40°. Implementation of the procedure requires knowledge of the parameters H, Ho, a, b, G, Dr, Rm, Sr, as defined previously. A list of appropriate values for these parameters is presented in Table 1. Some comments concerning their estimation are given here. The 0o C isotherm rain height H is given by expression (6) using the geographical latitude of the location under consideration. The numerical values for the parameters a and b (for both 12 and 20GHz and elevation angle 40°) are estimated by using the appropriate CCIR  report. As far as the parameter G is concerned, the value G = 1.5km has been selected as a representative one suggested by Lin . For the other parameter characterizing the spatial structure of the rain medium, namely the raincell diameter Dr, the value Dr = 30km has been selected in order to be compatible with respective experimental data [Lin, 1975]. As a final step, the lognormal parameters Rm, Sr are derived by means of a regression fitting technique implemented on the available tabulated data for the L-zone [CCIR, 1997]. In Figure 3, the conditional probability is drawn versus the variable r′(r′ = (C/I)nom − r), considering 30min/year as an appropriate time for the interfered communication system under examination. In the same figure, the results taken from the only available procedure [Kanellopoulos and Ventouras, 1996] are also drawn. At this point, the significance of choosing the 30min/year as an appropriate nonavailable time should be clarified. This is of course associated with the need of operating under low fading margins (LFD operation) and leads inevitably to the employment of double diversity protection, as the subject of the paper requires. (See also Table 2, where the rain fading margins for the single-site and double diversity schemes for both Ku and Kα bands are presented.) As shown in Figure 3, the operating frequency plays a significant role with respect to the novel considerations for the rain height. More particularly, the increase of the operating frequency leads to the most significant differences between present and existing results. For example, let us examine what happens with the nonexceedance conditional probability p = 10−4. The difference is about 2dB at f = 20GHz, whereas the difference for the same p and f = 12GHz is only about 0.5dB.
Table 1. Parameters of the Problem Under Consideration
(f = 12GHz) (0.0166, 1.2071)
(f = 20GHz) (0.0678, 1.0793)
Table 2. Fade Margins for the Single/Double Site Diversity System
f = 12GHz 30min/year
f = 20GHz
Ms = 17.537dB
Ms = 41.810dB
Ms = 12.962dB
Md = 5.017dB
Md = 11.167dB
Md = 3.743dB
 Some other sets of curves which may interest the system designer concern the variation of the nonexceeded (C/I) levels versus the angular separation θd. Following expression (4), one can express the (C/I) under rain fading conditions for a balanced diversity system as
where (C/I)nomo represents the (C/I)nom of the interfered Earth-space system for θd = 1°. The last term of equation (23) specifies a sidelobe envelope relative to the normalized peak gain (0dB) according to the U.S. Federal Communication Commission [Ha, 1991]. In Figures 4–6 curves of (C/I) nonexceeded levels versus θd are presented for the interfered communication system considered before. Two outage-times (30 and 200min/year) as well as the Ku (12GHz) and Kα (20GHz) frequency bands are also examined. In all the diagrams, the clear sky curve (C/I)nom is drawn for reference reasons, along with numerical results taken from the existing procedure [Kanellopoulos and Ventouras, 1996]. It is important to notice here that the curves for (C/I) should always satisfy the following inequality:
imposed by the noise domination conditions, as discussed in section 2. For the present case, we have taken (C/N)nom = 20dB as a quite reasonable value along with rain fade margins Md as presented in Table 2, corresponding to the specified outage times 30 and 200min/year for both 12 and 20GHz. Taking into consideration the above numerical values the validity of inequality (24) is certainly insured. As a general remark, the increase of frequency combined with the decrease of the prescribed outage time lead inevitably to appreciable differences between existing and modified results (see, for example, Figure 6). Furthermore, in order to have a more detailed view, the 30min/year results (Figures 4 and 6) are presented for various percentage nonexceedance conditional probabilities p%. As may be seen, the influence of the new assumptions for the rain height is more significant for the lower conditional probability levels. This becomes more obvious by examining the minimum angular allowed threshold θth, less than this, the (C/I) ratio takes invalid values. Assuming, therefore, an allowed level of 29dB for the (C/I) ratio, the differences of θth's between existing and modified results are 0.8o for p% = 0.05% and 1.3° for p% = 0.01% (as can be seen in Figure 6). On the other hand, taking f = 12GHz, outage time = 30min/year, and p% = 0.05% (Figure 4) the situation is totally different. The aforementioned difference of θth's is now 0.3°. As a direct consequence, the demand for the design of Earth-space systems operating in the Kα band with minimum specified outage times, along with the least possible interference by other adjacent satellite links, lead inevitably to the consideration of more accurate description for the rain height.
3.2. Practical System Design Approximation
 In addition due to the complicated nature of the expressions (see section 2.2), some simple pocket calculator regression-derived formulas, derived with the use of the well-known Levenberg-Marquardt algorithm for nonlinear least squares fitting [Jacobs, 1977], are presented. These formulas are useful to the system designer and concern the calculation of the nonexceeded (C/I) levels in terms of the nonexceedance percentage probability (p%), the angular separation θd, the outage time, the common elevation angle φ and the separation distance S. More particularly, for a double site diversity interfered system the variable r′ in expression (22) is given by
for both the existing method, as well as the model taking into account the modified considerations for the rain height. The parameters k1, k2, k3 and k4 in (25) depend on the climatic zone, frequency and the latitude of the location. The numerical values for these parameters corresponding to L, N climatic zones for both 12 and 20 GHz frequency bands are presented in Tables 3 and 4. (The general PASCAL/C code concerning the derivation of the appropriate values for k1,k2,k3 and k4 applicable to any frequency, geographical latitude and local statistics for the rainfall rate can be obtained by sending a request to the last author e-mail at email@example.com.) It is also important to notice that expression (25) is valid under the assumption that the variables φ (φ1 = φ2 = φ) in (degrees), S in (Km), outage time in (min/yr) and θd (in degrees) take values in the following restricted ranges:
which are very common in current satellite systems.
Table 3. Parameters k1, k2, k3, k4 for the L Climatic Zone and 12/20GHz
Constant Rain Height Model
Modified Rain Height Model
f = 12GHz
f = 20GHz
Table 4. Parameters k1, k2, k3, k4 for the N Climatic Zone and 12/20GHz
Constant Rain Height Model
Modified Rain Height Model
f = 12GHz
f = 20GHz
 The aggravation of the signal leakage due to differential rain attenuation is considered to be one of the main propagation effects between adjacent Earth-space paths. The subject of the present work is the modification of an existing systematic procedure for the prediction of the differential rain attenuation induced by an adjacent satellite path on a double site diversity system. The modified procedure takes into account a more complicated but realistic model for the description of the rain height. Available experimental data exist only for the single–site interfered case as well as the multiple site-diversity problem without interference. For this reason, our effort has been concentrated on the examination of the coordination studies and the influence of the modified rain height assumptions upon this problem. As a general conclusion, the need for the design of Earth-space systems operating in the Kα band with the maximum availability conditions leads inevitably to the use of more realistic conditions for the rain height. As a final remark, although a thorough experimental verification of the proposed procedure is still needed, we believe that the simplification of the regression formula (25) makes it a useful tool, at least for a first examination of the coordination conditions related to an interfered site diversity system.
Appendix A:: Parameters , (i = 1,2)
 Following a cumbersome but straightforward analysis, one is able to express the , (i = 1,2) by means of the lognormal statistical parameters Rm and Sr of the point rainfall distribution, the constants a and b and the characteristic distance G.
Further, we have
In the above expressions:
Elevation angle of the slant path pointing toward satellite Si (i = 1,2).
Differential angle between two satellites.
Effective rain height.
Rain attenuations of the wanted signal referring to Earth-space slant path EiS1 (i = 1,2).
Rain attenuation of the interfering signal referring to Earth-space slant path EiS2 (i = 1,2).
System margin available for rain attenuation.
Carrier-to-interference ratio at the Earth station Ei receiver (i = 1,2).
Carrier-to-interference ratio as above but under clear-sky conditions (i = 1,2).
Rain attenuation concerning the projection of the slant path EiS1 (i = 1,2).
Rain attenuation concerning the projection of the slant path EiS2 (i = 1,2).
Constants of the specific rain attenuation Ao in (decibels/kilometer).
Characteristic distance modeling the spatial inhomogeneity of the rainfall structure.
Mean Diameter of the rain cell size.
Lognormal parameters of the point rainfall distribution.
Nonexceedance level of the (C/I) (in decibels).
Lognormal statistical parameters of the distributions concerning , random variables.
Lognormal statistical parameters of the distributions concerning , , r.v.