## 2 The Method to Download Clear-Sky Data Volume during Rainfall
### Channel Efficiency  
To deliver a fixed signal-to-noise ratio \(\rho_{o}\) in BPSK, QPSK and in the more general QAM schemes-mostly used in satellite communications-standard calculations show that the instantaneous symbol rate \(R_{s}(t)\) (symbols per second) at time \(t\), with rain attenuation \(A(t)\), is given by the following expression [6]:  
\[R_{s}(t)\geq\frac{P_{r}\times 10^{-A_{G}/10}\times 10^{-A(t)/10}}{k(T_{m}+T_{ rec})\rho_{o}}=R_{o}\times 10^{-A(t)/10} \tag{1}\]  
\[R_{o}=\frac{P_{r}\times 10^{-A_{G}/10}}{k(T_{m}+T_{rec})\rho_{o}} \tag{2}\]  
In Equations (1) and (2), \(k\) is [PERSON]'s constant, \(T_{m}\) (K) is the mean physical raindrops temperature (which is supposed to be constant), \(T_{rec}\) (K) is the receiver equivalent noise temperature; \(A_{G}\) (dB) is the total attenuation due to water vapor, oxygen, and clouds, which is supposed to be constant during rainfall (this restriction can be removed; see below); \(\rho_{o}=E_{b}/N_{o}\) is the signal-to-noise ratio that is tolerated, \(E_{b}\) is the received energy per symbol, and \(N_{o}=k(T_{m}+T_{rec})\) is the one-sided noise power spectral density.  
The inequality (1) arises because the antenna noise temperature \(T_{sky}\leq T_{m}\) (K) is, in fact, upper bounded by \(T_{m}\), a conservative hypothesis that models rain fade--for noise calculation--as due to a passive attenuator at physical temperature \(T_{m}\) with \(A\rightarrow\infty\)[23]. We do not consider scintillation because this phenomenon does not have an impact on the implementation of the method [6].  
In [6], we studied Equation (1) under the hypothesis of providing the minimum equivalent and tolerable \(\rho_{o}\) (fixed by the tolerated maximum probability of symbol error) by reducing \(R_{s}(t)\). In other words, the channel must dynamically match \(R_{s}(t)\) to the slow (compared to \(R_{s}(t)\)) time-varying \(A(t)\). Therefore, the volume of symbols down- or uploaded during the rain attenuation time \(T_{A}\) is given by:  
\[D=\int_{0}^{T_{A}}R_{s}(t)dt=R_{o}\!\int_{0}^{T_{A}}10^{-A(t)/10}dt=R_{o}\tau \tag{3}\]  
\[\tau=\int_{0}^{T_{A}}10^{-A(t)/10}dt \tag{4}\]  
The integral in Equation (4) gives the equivalent time \(\tau\) during which the constant rate \(R_{o}\) delivers the same total data volume \(D\). In other words, the perfect matching (1) in the interval \(T_{A}\) is equivalent to transmitting the symbol rate \(R_{o}\) in an ideal channel with \(A(t)=0\) dB, but for the shorter equivalent interval \(\tau\). In addition to QAM, this finding also applies M--PSK modulation and [PERSON]'s capacity formula [6].  
The mean channel efficiency \(\eta\) is defined as  
\[\eta=\frac{1}{T_{A}}\int_{0}^{T_{A}}10^{-A(t)/10}dt=\frac{\tau}{T_{A}} \tag{5}\]  
Of course, \(0\leq\eta\leq 1\) because if \(A\rightarrow\infty\) then \(\tau=0\), \(\eta=0\); if \(A=0\) (no rainfall) then \(\tau=T_{A}\), \(\eta=1\).  
Notice that it is not necessary to consider rain-attenuation time series \(A(t)\) but only the conditional complementary long-term probability distribution \(P_{c}(A)=P(A)/P_{o}\), with\(P_{o}=P(A>0)=T_{A}/T_{obs}\) where \(T_{obs}\geq T_{A}\) is the observation time (a year, a moth, etc.). It can be shown [10] that  
\[\eta=1-\frac{\ln(10)}{10}\int_{0}^{\infty}10^{-A/10}P_{c}(A)dA \tag{6}\]  
Notice that a fixed power margin \(S>0\) (dB)--see Section 4 below--gain (site diversity, time diversity, etc.) or a constant fade can be introduced in Equation (1) by only changing \(R_{o}\), not the mean efficiency (5) and (6). This argument applies also to a constant interference power, if we consider it another source of Gaussian noise (worst case, according to [PERSON]). In other words, all constant parameters do change \(R_{o}\) but not \(\eta\).  
The mean efficiency (6) is a random variable bounded in a range calculated by applying the Cauchy-Schwarz inequality [6]:  
\[\left[1-\frac{\ln(10)}{20}\int_{0}^{\infty}10^{-A/20}P_{c}(A)dA\right]^{2} \leq\eta\leq\sqrt{1-\frac{\ln(10)}{5}\int_{0}^{\infty}10^{-A/5}P_{c}(A)dA} \tag{7}\]