## 3 Impact of Power Amplifier Nonlinearities on the System Performance
### BER Performance  
Figure 6 shows the bit error rate (BER) performance for \(N_{S}=5\) and several HPA input powers \(P_{in}\). Note that we define \(E_{b}=P_{max}T_{b}\), where \(P_{max}=31\) dBW is the maximum power at the HPA output: \(E_{b}\) changes if we change the HPA power, whereas \(N_{0}\), which depends on the receiver, does not. We observe the following:  
* When the HPA average input power is very small (\(-60\) dBm), the amplifier works in its linear region, the output power is small and a much smaller \(N_{0}\) is necessary to obtain a desired BER.
* On the other hand, when the HPA input power is large (\([-20,-10]\) dBm), the amplifier works in the saturation region, and the output power is large, but interference arises among the CDM signal components, due to the hard limiting effect.
* When standard CDM without sign operation is transmitted, if the HPA introduces phase distortion (due to non-ideal AM/PM characteristics, which were not considered in the simulations), then the BER for the HPA linear region is higher than shown in Figure 6. On the contrary, when the sign is transmitted, the BER for \(P_{in}\) larger than \(-30\) dBm does not change since the output power is consistently equal to \(P_{max}\) and the HPA-induced phase shift is also constant, as assumed in the simulations. Note that the receiver's carrier phase recovery subsystem compensates for constant phase shifts.
* For desired BER values larger than \(5\times 10^{-4}\), the minimum \(E_{b}/N_{0}\) is obtained by driving the HPA at saturation (see Figure 7).  
In an LEO satellite constellation, the number of satellites being served continuously changes because the satellites move rapidly, and the window of visibility is very short. The results suggest that instead of trying to operate in linearity by constantly adjusting the back-off, which depends on the number of satellites, and considering that with CDM the transmitted signal still maintains a Gaussian distribution of amplitudes, it is more efficient to operate directly in saturation. For this reason, having an analytical result that enables  
Figure 5: Equivalent scheme with the sign function for HPA working at saturation.  
the calculation of BER performance for CDM with a saturated amplifier as a function of the number of satellites would be extremely useful. This is precisely the result presented in the next section.  
Figures 8 and 9 show the effects on the instantaneous power of the signals at the input and output of the HPA when the average input power \(P_{in}\) is equal to \(-40\) dBm (linear region of the AM/AM curve) and \(-20\) dBm (nonlinear region of the AM/AM curve), respectively. Clearly, if the HPA works in its nonlinear region, the output signal is compressed, and the instantaneous power is \(P_{max}\). The same conclusion can be drawn from Figures 10 and 11, showing the probability mass function of the input and output instantaneous power, for the HPA working in the linear and nonlinear regions, respectively.  
Figure 6: Simulated BER performance for \(N_{S}=5\), spreading factor \(M=100\), and different input powers \(P_{in}\) of the HPA. Best performance for \(P_{in}=-10\) dBm (saturation).  
Figure 7: Simulated \(E_{b}/N_{0}\) (dB) necessary to obtain the specified BER value versus the HPA input power \(P_{in}\), \(N_{S}=5\), spreading factor \(M=100\). Best performance for \(P_{in}=-10\) dBm (saturation).  
Figure 8: Example of HPA nonlinearity effects on the instantaneous power of the input and output signals; case of average input power equal to \(-40\) dBm (linear region), \(11\) satellites.  
Figure 9: Example of HPA nonlinearity effects on the instantaneous power of the input and output signals; case of average input power equal to \(-20\) dBm (nonlinear region), \(11\) satellites.