### Motivation and Contribution  
In uplink communications, ground stations employ power amplifiers to boost the transmitted power levels, ensuring that the signal can reach multiple satellites in an LEO constellation. However, a significant challenge arises due to the non-linear nature of these amplifiers, which introduce distortion into the transmitted signal. This distortion can degrade the overall system performance, especially when multiple satellites are being served simultaneously using CDM. This could suggest operating in linearity by applying a back-off and introducing a loss in transmitted power. Nevertheless, a key characteristic of LEO constellations is that the number of visible satellites continuously changes, as the satellites move rapidly and remain visible for only short intervals. As a result, it is not feasible to design the ground station to serve a constant number of satellites at all times. This fact, combined with the tendency of a CDM signal to approximate a Gaussian distribution in amplitude when multiple signals are summed together, makes it extremely challenging to operate in linearity with a fixed back-off. The challenge is related to the peak-to-average power ratio (PAPR) of the CDM signal, which depends on the number of satellites. As the number of LEO satellites changes very rapidly, so does the PAPR, making it difficult to select an optimal back-off that ensures linearity across all conditions. Moreover, operating with a fixed back-off reduces the efficiency of the amplifier compared to when it operates in saturation. As a result, many ground stations operate directly in saturation. While the performance of satellite communications with an amplifier operating in saturation has been well investigated for the downlink, it has been less studied for the uplink, particularly when using CDMA.  
In this paper, we build on the concept of transmitting the sign of the sum of the signals when the amplifier operates in saturation, a technique first introduced in [25]. The performance of the CDMA satellite uplink was analyzed through accurate simulations in [26]. The objective of this paper is to analytically study the performance of a CDM-based uplink to multiple satellites in the presence of a non-linear power amplifier operating in saturation. The main contributions of this paper are as follows:  
* The study of the impact of input power level on system performance across different numbers of satellites.
* A first theorem for the exact calculation of error probability when the amplifier operates in saturation as a function of the number of satellites, including a simplified formula.
* The validation of exact and simplified analytical results through simulations.
* A second theorem for the analytical computation of loss based on the number of satellites served.
* A comprehensive analysis of the impact of amplifier non-linearities on system performance, along with guidelines for simplifying satellite uplink operation under power amplifier constraints.  
This paper is organized as follows: In Section 2, we present the problem. In Section 3, we discuss the impact of the power amplifier in terms of the output power and bit error rate across different numbers of satellites. In Section 4, we present the theorem for analytically computing the system performance of a CDM system operating with a saturated amplifier, as a function of the number of satellites. Both exact and simplified expressions are presented. The analytic results are validated through simulation and discussed. In Section 5, we present a theorem for analytically computing the loss for a given number of satellites. These results are discussed, and practical considerations for the operation of the ground station uplink are provided. Section 6 concludes the paper, while the proofs of the theorems are provided in Appendixes A and B.