## 1 Introduction  
With the rapid rise of fifth-generation (5G) millimeter-wave (mmWave) communications and broadband low Earth orbit (LEO) satellites [1, 2], active phased array (APA) antennas have seen unprecedented development in recent years [3, 4, 5]. Compared with conventional antennas, APA antennas do not require bulky servo tracking systems. They can change the radiation pattern by adjusting the excitation phases of the antenna elements in the array, thus bringing advantages such as flexible beam control and high reliability [6, 7]. Higher power efficiency is required to support high data rates operation while driving power amplifiers (PAs) close to their saturation state will introduce severe nonlinear distortion. Nonlinear distortion leads to non-negligible problems such as: interference to other satellite and terrestrial communication systems due to the increased adjacent channel power ratio (ACPR), and deterioration of the error vector magnitude (EVM) leading to higher bit error rates. To address this problem, power backoff techniques and digital pre-distortion (DPD) [8, 9, 10, 11, 12, 13, 14] techniques are often adopted, of which DPD techniques have drawn widespread attention because they can better improve power efficiency to fully utilize the space-borne power resources. As shown in Figure 1a,b, the DPD techniques with 4-by-4 array antennas as an example can be further divided into two categories: one is individual DPD [8, 9, 11],and the other is overall DPD [12; 13; 14]. The impact of PA nonlinear distortion of a massive multiple-input multiple-output (MIMO) generalized frequency division multiplexing (GFDM) (MIMO-GFDM) system on the spectral shape, the peak to average power ratio (PAPR), and the bit error rate (BER) in the additive white Gaussian noise (AWGN) channel was researched and analyzed in [8]. It proposes a linearization technique for applying each PA pre-distorter separately at the transmitter side of the MIMO-GFDM system. Ref. [9] offers a method to simultaneously compensate for PA nonlinearity, antenna crosstalk, and mismatch in a broadband multi-antenna transmitter. Each transmit path is provided with a dual input DPD, which matches a linear model of the crosstalk and mismatch features of the antenna array shared by all transmitting paths. The DPD technique proposed by [11] includes overall multibeam compensation and separate dual-band compensation. The multibeam compensation deals with multibeam interference, while the dual-band compensation is designed to eliminate the nonlinear distortion of the PA. Nevertheless, these one-to-one DPD techniques are unsuitable for large-scale APAs because of the high complexity and the mutual disturbance between channels. For large-scale APA linearization, approaches with overall DPD of PAs for all array cells at the transmitter give more generic and effective solutions. The fundamental coefficients of the single-input single-output (SISO) array model include beamforming weights, antenna cross-coupling, channel coefficients, etc. As a result, many sets of trained DPD coefficients are sufficient to reduce the nonlinear distortion of mm-wave radio frequency beamforming array [12]. Ref. [13] assumed that the PAs are similar in all branches, the main beam is generated by collecting the output signal of one PA, and the memory polynomial (MP) model is used to calculate the overall DPD from the main beam and the input of the sub-array. Ref. [14] implements full-angle DPD for 5G mmWave large-scale MIMO transmitters by first compensating for differences of PAs in different transmitter chains and then linearizing the entire subarray using a common digital block. Unfortunately, since the DPD is done within the digital domain before the front-end, these approaches necessitate an RF feedback loop, and the resulting hardware resource consumption is almost equivalent to an space-borne receiver, which is unacceptably high for LEO satellite communications.