## 4 The Analytical Performance of Uplink CDM-Serving Satellites with a Saturated Amplifier  
As explained before, since it is convenient to drive the HPA at saturation, and since--in this condition--the signal is, in practice, hard-limited, this is equivalent to introducing a hard limiter (a sign function, as in Equation (2)) before the amplifier (see Figure 5). Such a transmitter was first proposed in [25]; however, to the best of our knowledge, no analytical result was provided for the CDM performance calculation with a non-linear amplifier working at saturation. Figure 11: HPA nonlinearity effects on the probability mass function of the instantaneous power of the input and output signals; case of average input power equal to \(-20\) dBm (nonlinear region), \(11\) satellites. Figure 10: HPA nonlinearity effects on the probability mass function of the instantaneous power of the input and output signals; case of average input power equal to \(-40\) dBm (linear region), \(11\) satellites. The baseband input of the HPA is as follows:  
\[s_{T}^{\prime}(t)=\sqrt{P_{in}}\text{ sign}\left[\sum_{i=0}^{N_{S}-1}\sum_{n}b_{i}(n)P_{T_{b}}(t-nT_{b}) \sum_{m}c_{i}(m)P_{T_{c}}(t-mT_{c})\right]\]  
and the transmitted signal is (a complex envelope) as follows:  
\[x(t)=\sqrt{2P_{out}}\text{ sign}\left[\sum_{i=0}^{N_{S}-1}\sum_{n}b_{i}(n)P_{T_{ b}}(t-nT_{b})\sum_{m}c_{i}(m)P_{T_{c}}(t-mT_{c})\right]\]  
where \(P_{out}\) is the transmitted signal power when the HPA input power is \(P_{in}\). The maximum value of \(P_{out}\) is \(P_{max}\) for a sufficiently large \(P_{in}\) and an odd number of served satellites. Note that, having introduced the hard limiter, the \(E_{b}/N_{0}\) for a desired BER, in general, will be larger for the system in Figure 5 than for that in Figure 2. However, when \(P_{in}\) is sufficiently large (i.e., HPA working in saturation, which as shown in the previous section is the best case), the two systems exhibit the same performance. The system depicted in Figure 5 has the advantage of allowing for a theoretical analysis, which is provided in the following theorem. Theorem 3.1: _The error probability for an uplink CDM system serving \(N_{S}\) satellites with random spreading codes and a saturated amplifier is as follows:_  
* _For_ \(N_{S}\) _even, the exact error probability is as follows:_ \[P_{b}(e)=\sum_{k_{0}=0}^{M}\sum_{k_{1}=0}^{M-k_{0}} \frac{1}{2}\text{ erfc }\left(\sqrt{\frac{N_{S}E_{b}}{M^{2}N_{0}}}(M-k_{0}-2k_{1})\right)\] \[\times\frac{M!}{k_{0}!k_{1}!(M-k_{0}-k_{1})!}\text{ }\pi_{0}^{k_{0}}\text{ }\pi_{m}^{k_{1}}\text{ }\pi_{p}^{M-k_{0}-k_{1}}\text{ }\text{ }\text{ }N_{S}\geq 4\] (3) \[P_{b}(e)=\sum_{k=0}^{M}\frac{1}{2}\text{ erfc }\left(k\sqrt{\frac{N_{S}E_{b}}{M^{2}N_{0}}}\right)\binom{M}{k}\left(\frac{1}{2} \right)^{M}\text{ }\text{ }\text{ }N_{S}=2\] (4) _and can be approximated by the following:_ \[P_{b}(e)\simeq\frac{1}{2}\text{erfc}\sqrt{\frac{p_{0}^{2}N_{S}(E_{b}/N_{0})} {1+2(1-p_{0}-p_{0}^{2})\,N_{S}(E_{c}/N_{0})}}\] (5) _where_ \[E_{c}=E_{b}/M,\text{ }\text{ }\text{ }\text{ }\text{ }p_{0}=\binom{N_{S}-1}{N_{S}/2-1}2^{-(N_{S}-1)}\] Figure 10:0  
Figure 10:1  
The proof of Theorem 1 is available in Appendix A. In the above equations, \(E_{b}/N_{0}\) is the signal-to-noise ratio for the received signal at the observed satellite. The \(N_{S}\) satellites might be at different distances from the ground station and, thus, have different path losses, but the signal-to-noise ratio and the effects of interfering components in the received CDM signal for satellite \(n\) do not depend on the positions of the other satellites, as we assume that the HPA output power is evenly distributed among all the served satellites. This theorem is highly significant because it enables the performance calculation analytically, as a function of the number of satellites, eliminating the need for simulations. First, we numerically validate the exact probability results of Theorem 1.