Joint Digital Analogue DVB-S2(X) Link Optimization in Non-linear Channel

Future broadband satellite communication (SatCom) systems require a high throughput of data transmission, which calls for operation at higher frequency bands. Adaptive coding and modulation (ACM) technology has been considered as a means to obtain improved performance further at these frequencies. However, the ACM protocols in current DVB-S2 and DVB-S2(X) standards do not take the effects that arise from the non-linear link with memory into account, which is generated by the cascade of a high-power ampliﬁer (HPA) and digital root raised cosine (RRC) ﬁlters. This paper ﬁrst reveals the non-linear characteristics of the SatCom link, and transfers this into an equivalent full-linear link with an additive noise source thanks to the pre-distortion algorithm. Using this equivalent modeling, an approach of selecting/adapting the optimum coding, modulation scheme and associated system conﬁgurations for a given channel fading level is proposed. Both simulations and experimental test-bed performance are presented to validate the proposed method.


I. INTRODUCTION
Demands for the high-data-rate satellite communication (Sat-Com) are dramatically increasing as services evolve from broadcast to broadband [1], [2]. Currently, high throughput SatCom systems have a capacity of about 100 Gbps and next generation systems are targeting unprecedented levels of Terabit/s [3]. Meanwhile, a number of medium and lower Earth orbit constellations under development are targeting to deliver comparable capacities [4], [5]. The aforementioned changes in capacity are underpinned by the transition to higher frequencies (e.g. Ka-, Q/V-and beyond) [5], [6], where larger bandwidths are available.
Atmospheric phenomena associated with the higher frequencies leads to increased losses and more challenging radio signal amplifications. Consequently, traditional uplink power control (UPC) [7], which provides margin at the expense of an The associate editor coordinating the review of this manuscript and approving it for publication was Mohamed M. A. Moustafa . oversized RF front-end, is no longer suitable to maintain high spectrum efficiency links. Instead flexibility on the digital configurations was introduced in the 2nd generation digital video broadcasting satellite standard (DVB-S2) ratified in 2005 [8], as a means to optimize spectrum efficiency. These include a wide range of modulation and coding (MODCOD) rates, options for lower roll-off (ROF) factors for the spectrum shaping filters as well as an adaptive coding and modulation (ACM) functionality that enables optimizing the channel coding and modulation on a frame-by-frame basis [9], [10]. As spectrum efficiency is continuing to be a major drive for emerging satellite missions, the extended version of DVB-S2, DVB-S2(X) [11] published in 2014 introduced additional MODCODs for much finer signal to noise (SNR) granularity as well as three additional smaller ROF factors for the digital filters.
The DVB-S2 framework leads to the exploitation of ACM and the development of modem algorithms to mitigate the non-linearity introduced by high power amplifiers (HPAs) [12]. Specifically, the non-linear characteristics of HPAs depend on the input backoff (IBO). Higher IBOs result in less non-linear distortion to the signal waveform at the expense of reduced signal strength [13]. Recognizing the associated trade-off between non-linearity and signal strength has led to the optimization of IBO in light of different modulation schemes [12]. Existing publications have optimized the digital signal processing parameters but overlooked the interaction between the digital and analogue signal processing [11]. For links relying on a practical HPA, the effects of digital and analogue configurations are intertwined. Consequently, achieving the highest spectrum efficiency in practical link scenarios requires coordinated analogue-digital domain optimization. In particular, lower ROF factors of digital root raised cosine (RRC) filters lead to higher baud-rates for a given bandwidth at the cost of higher inter-symbol interference (ISI), such that the trade-off between ROF and baud-rate applies. On the other hand, the ISI associated with a given ROF is strongly linked with the HPA's IBO, as discussed in [14]. However, the simultaneous optimization for the HPA's IBO, filter's ROF and signal's MODCOD that can maximize the spectrum efficiency while guaranteeing the QoS requirement is yet to be developed. Even though machine learning techniques are a way to work around this, the training process is not compatible with real-time operation of ACM.
This paper fills this gap and proposes a deterministic methodology that enables optimizing the analogue and digital signal processing for emerging broadband SatCom missions, and the key contributions are summarized as follows: • The intertwined effects of the ROF of RRC filters, MODCOD and IBO of a HPA to the BER performance of the non-linear SatCom link are analyzed in Section II. Then, the non-linear SatCom link is transferred to an equivalent full-linear link with an additive noise source in Section III.B.
• Test-bed based experiments are designed and performed to validate the effectiveness and generality of the equivalent full-linear link in Section III.C.
• The adaptive operation of a DVB-S2(X) link for spectrum efficiency maximization is achieved in Section IV by computationally allocating the MODCOD scheme, ROF and IBO operation for a given HPA and the channel condition. The adaptive control schedule can maintain spectral performance close to optimum and guarantee the QoS requirement.
The paper is organized as follows. In Section II, we provide the background including the SatCom link descriptions, the adverse impacts from the non-linearity and ISI. In Section III, the equivalent full-linear link by translating the ISI into an additive Gaussian noise is established, the designed test-bed and the measured results are also provided. In Section IV, the adaptive control schedule specifying the selected combination of the MODCOD, ROF and IBO is proposed. Finally, conclusions are drawn in Section V.

II. DVB-S2/DVB-S2(X) SATCOM LINK
In this section, we elaborate the end-to-end SatCom link for DVB-S2/DVB-S2(X) standards and investigate the nonlinear distortion. The system block diagram of the forward link is shown Fig. 1 [8], [11].
• At the Gateway transmitter (Tx), the mode adaptation is first applied on the input data, which provides input stream interfacing, synchronization, null-packet deletion, etc. Then, the stream adaptation is applied, i.e. padder and base-band (BB) scrambling, followed by the forward error correction (FEC) encoding, consisting of Bose-Chaudhuri-Hocquenghem (BCH) codes, lowdensity parity check (LDPC) inner codes and bit interleaver which is discussed in Section II.A. The available LDPC rates in DVB-S2 include {1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 5/6, 8/9, 9/10}. The resulting sequence is then modulated using one of the schemes including QPSK, 8PSK, 16APSK and 32APSK in the DVB-S2 standard [8], and 64APSK, 128APSK and 256APSK modulation schemes are added in the extended DVB-S2(X) standard [11]. In this paper, we focus on the 16APSK modulation elaborated in Section II.B. The physical layer frame is then applied to the modulated signal, synchronous with the FEC frames. Finally, an RRC filter is applied before the up-conversion for transmitting to satellite transponders through the uplink/feeder link.
• An HPA is equipped at the satellite transponder to amplify the signals received from the uplink and then the amplified signals are relayed to the receiver on the ground through the downlink/user link. The details of the HPA characteristic are discussed in Section II.C. As pointed out by the DVB-S2 working group, the downlink noise dominates the overall link thermal noise and can be assumed as additive white Gaussian noise (AWGN) [8], [11].
• At the receiver end (Rx), i.e. the ground users/terminals, a matched RRC filter is applied after frequency down-conversion and digital sampling, then followed by base-band signal demodulation and decoding. Output signals from the matched RRC, i.e. signals at Point B in Fig. 1, suffer from the distortion which is classified into warping and clustering [12]. The factors affecting the distortion are elaborated in Section II.B to Section II.E.

A. FEC CODE
The LDPC-based FEC concatenated with BCH codes are applied in the DVB-S2(X) system which is very powerful to improve the BER performance [11], [15]. The FEC and modulation order may be changed across different frames when an ACM is used, but remains constant within a frame. Fig. 2 indicates the mapping relations between the input and output BER of the BCH decoder with different LDPC coderate and the 16APSK modulation [16], [17]. For example, the FEC is able to improve BER performance from 0.02 to as low  as 10 −4 by applying LDPC-9/10. The LDPC code efficiency for error correction becomes greater with the decreasing of the code rate, while leads to lower spectrum efficiency.

B. MODULATION SCHEMES
APSK modulation schemes are used in DVB-S2 and DVB-S2(X) standards as their circular symmetry constellation diagrams are particularly suited to combat the non-linear effects of HPAs [18], [19]. The constellation of the 16APSK modulation focused in this paper is composed of two concentric rings of uniformly spaced 4 and 12 PSK points, respectively [8], as the reference points shown in Fig. 3. The modulated constellation symbol set is S := {s 0 , . . . , s M −1 } , M = 16. The ratio of the outer circle radius R 2 to the inner circle radius R 1 is denoted as λ = R 2 /R 1 . The average symbol energy of the 16APSK is calculated as To facilitate analysis in this paper, E s is normalized to be unity. We note that although the analysis presented in this paper is based on the 16APSK modulation, but it can be readily adapted for other APSK modulations.

C. HPA AT THE SATELLITE TRANSPONDER
The non-linear amplification characteristic of the satellite HPA defined in DVB-S2(X) is adopted and analyzed in this paper. The HPA is constructed using the Saleh's model [8], [20]. The compression of signal power level is characterized as AM/AM effect, see (2), and the relationship between input power level and output phase rotation is characterized by AM/PM effect, see (3).
Here, u denotes the magnitude of the input signal, and the coefficient values a A = 2.1322, b A = 1.0746, a = 1.70, b = 1.5072 are taken from [21]. The nonlinear characteristics of the HPA, i.e. AM/AM compression and AM/PM rotation, cause the constellation centroids of the received signals to no longer be lying on the top of the reference points. This effect is referred to warping distortion and is shown in Fig. 3. From Fig. 3, we can see that the warping distortion of the outer ring points, i.e. the points with larger amplitude, is more distinct than the inner ring points, i.e. the points with smaller amplitude. The IBO of a HPA (in dB) is defined as β = 10 log 10 (P avg /P sat ), indicating the deviation of the average input power P avg = |u| 2 from the input saturation power P sat . Smaller values of β indicate higher IBO. The output backoff (OBO) is a function of IBO. Fig. 4 illustrates the nonlinear OBO-IBO and phase rotation-IBO behaviors. Higher IBO results in reduced non-linear HPA distortions at the cost of lower output power and DC power efficiency.

D. RRC FILTERS
Raised-cosine (RC) filters are widely used in digital systems for pulse-shaping [22]. Usually the RC filter is split equally into a pair of RRC filters, with one at the Tx, and the other at the Rx. They perform matched filtering to optimize the signal to noise ratio (SNR) in the presence of additive channel noise [23], [24]. The impulse response of an RRC filter in time-domain is [22] h RRC where γ represents the ROF, f represents the frequency and T is the sampling period. The spectrum efficiency of a communication system is defined as the ratio between the bit rate R b and the transmission bandwidth [25]. Therefore, for a given R b , smaller γ gives higher spectrum efficiency due to narrower bandwidth. However, the capability of suppression in stop bands is reduced [22]. Note, RRC filters are theoretically infinite impulse response (IIR) filters. However, they are implemented as finite impulse response (FIR) filters in practice since FIR filters are intrinsically stable and use less hardware resources. The number of FIR filter taps is given as N tap = D × M sam , where D defines the number of symbols spanned by the impulse response of the filter and M sam is the amount of up-sampling, D and M sam are both integers. Therefore, the impulse response of IIR-RRC filter is written as (5) we can see that two cascaded FIR RRC filters, as shown in Fig. 5(a), cannot represent an ideal RC filter response. Specifically, the output signals from the Rx RRC, i.e. Point B in Fig. 5(a), are derived as Here, ⊗ represents the convolution operation, S is the modulated signals at the transmitter end, and the ISI introduced by the concatenation of two matched FIR RRCs is denoted as S ISI . The constellations of the output signals at Point B for different ROF values are shown in Fig. 5(b), which indicate that the resulting ISI is a nearly circularly symmetric complex Gaussian spreading and can be approximated by a complex Gaussian distribution with zero mean. Moreover, the simulated results of the complex Gaussian spreading variance are shown in Fig. 5(c), which indicate that the variance is dependent on γ , M sam and D. From this plot, we can observe that a larger D gives a better approximation to the ideal response, and the ISI is not monotonic versus γ but fluctuates with a decreasing trend. Furthermore, M sam has little impact on the filter performance other than to reduce the variation in stop band attenuation for small changes in D or γ . Following DVB-S2(X) standard and for the ease of comparison with some previous works, the RRCs are implemented with N tap = 80, M sam = 8 and D = 10 in our study [26].

E. INTERACTION BETWEEN DIGITAL AND ANALOGUE SIGNAL PROCESSING
The insertion of the non-linear HPA between two RRC filters, as shown in Fig. 1, results in the mismatch of the receiver RRC filter to the Rx signal. Therefore, the uncertainty caused by the ISI doesn't follow Gaussian spreading any more, but in irregular shapes as shown in Fig. 3, which is called clustering. Note, the resulting clustering is the energy exchange between transmitted signals, which is related to both the amplitude of current signal and the amplitudes of neighboring signals, in a statistical sense, consist of all 16 symbols with equal probability. Due to the equal probability of the neighboring symbols, the clustering effect is independent of magnitude of symbols currently transmitted. Hence, there is no distinct differences of the spreading between the inner-ring points VOLUME 10, 2022 and outer-ring points in Fig. 3. Overall, the combination of the HPA with two matched RRC filters leads to a non-linear SatCom link with memory and results in the signals at Point B in Fig. 1 suffering from warping and clustering distortions. The distortions in turn lead to significant degradation of QoS performance, making the existing propagation impairments mitigation techniques (PIMTs) inefficient.

III. DIGITAL PRE-DISTORTION ALGORITHM AND EQUIVALENT FULL-LINEAR LINK
Digital pre-distortion algorithms commonly rely on iterative numerical optimization techniques [27], [28]. In this section, a digital pre-distortion algorithm is firstly introduced. With the pre-distortion applied before the transmitter RRC filter, see Point A in Fig. 1, the warping and clustering effects can be significantly suppressed and the reduced ISI can be approximated as an equivalent addictive Gaussian noise. Hence, the non-linear link can then be transferred to an equivalent full-linear link with an additive noise source. In order to validate the effectiveness of the pre-distortion algorithm and the equivalent linear link, a DVB-S2(X) test-bed at intermediate frequency (IF) is designed and implemented, and the experimental results are presented at the end of this section.

A. DIGITAL PRE-DISTORTION
The pre-distortion used in this paper consists of two main steps, coarse and refined pre-distortion [12], [28], [29]. The coarse pre-distortion is designed to overcome the wrapping distortion, i.e. minimizing the error signal using an interactive least mean square (LMS) method. The output constellation of the coarse pre-distortion is shown in Fig. 6(b). The refined pre-distortion is then followed to combat the clustering effects, i.e. conditioning the pre-distorted modulator constellation not only to the current transmitted symbol but also to the (L − 1)/2 preceding and (L − 1)/2 following symbols (L symbols in total and L is odd). Therefore, for the current received symbol, there are M L−1 combinations of neighboring symbols. In this paper, M = 16 for 16APSK, and L is set to be 3, i.e. one preceding and one following neighboring symbols are considered. Hence, one constellation cloud consists of M 2 sub-clouds and there are M 3 subclouds when all M constellations are considered. The goal of refined pre-distortion is to minimize the error signal between the centroid of each sub-cloud with the correspond reference point. The result achieved by the pre-distortion is a look-up table specifying the output constellation mappings which can be represented as a 3D matrix, as the green squares shown in

B. EQUIVALENT GAUSSIAN NOISE OF ISI WITH PRE-DISTORTION
By applying the pre-distortion algorithm, the warping effect caused by the nonlinear characteristic of the HPA is largely overcome, and the clustering effect due to the ISI is reduced and can be approximated by an equivalent Gaussian noise. Then, the non-linear link, see Fig. 7(a), is equivalent to a full-linear link with an additive noise source, see Fig. 7(b). The mean of the additive noise is zero, and its variance σ 2 ISI is obtained with a Monte-Carlo simulation and calculated as where x n represents n-th modulated signal, x n ∈ S. N is the length of modulated signals. δ(x n , s m ) = 1 if and only if x n = s m , otherwise, δ(x n , s m ) = 0. y n represents the signals after the Rx RRC with pre-distortion. The obtained variance values are summarized in Table 1 as functions of the ROF γ and IBO β. From this table, we can see that the equivalent variance reduces as β decreases, as the non-linearity of the HPA becomes less distinct. However, there is a floor owing to the imperfect RRC characteristic as shown in Fig. 5. The achieved equivalent noise in Table 1 is validated by comparing the BER performance under the two links in Fig. 7. The numerical results are shown in Fig. 8, in which the BER curves under these two links show good agreement. We can also see the significant improvement of system performance relying on the pre-distortion algorithm. For example, for a higher IBO value, i.e. β = −3dB, the input BER can be suppressed down to approximately 10 −5 for moderate SNRs, while an SNR gain of 1.5 dB can be achieved by the pre-distortion method for a lower IBO value, i.e. β = −8dB, when the BER = 10 −4 .

C. TEST-BED BASED EXPERIMENTS FOR PRE-DISTORTION AND EQUIVALENT FULL-LINEAR LINK VALIDATION
The equivalent full-linear link is validated in this section experimentally. The amplifier ZJL-4HG+ from Mini-Circuits operating at 2.4GHz operation is used in the test-bed. The AM/AM behavior is modeled as the Bessel-Fourier considering the accuracy in all operating regions. The Bessel-Fourier model consists on a particular kind of generalized Fourier series based on Bessel functions of the first kind J 1 (·) [30], [31] where µ denotes the magnitude of the input signal, M J is the number of terms of the Bessel series with assigned coefficients {c m } M J m=1 and v 1 is another coefficient. The main advantage of the Bessel-Fourier model is the extensibility, in the sense that the number of Bessel series can be increased for improved model accuracy, and in this paper, the number is set to be M J = 23 and the coefficient v 1 = 16.006. The measured AM/AM conversion of the selected HPA and the fitted AM/AM conversion using the Bessel-Fourier behavioral model are depicted and compared in Fig. 9. We note that the selected solid state amplifier has negligible AM/PM non-linearity and hence this is not plotted.
Based on the two-step pre-distortion algorithm, the pre-distortion look-up table for amplifier ZJL-4HG+ is generated through Monte-Carlo simulation. Then, the experimental test-bed is set up as Fig. 10, which measures the BER performance of un-coded 16APSK signal for different  IBO levels and SNRs at the presence of the HPA and a pair of RRC filters. Specifically, the modulated source signals with the pre-distortion are firstly generated in MATLAB. Then, in order to generate the signals for transmission, the filtered symbols at the output of the transmit RRC filter are converted into I16 format and recorded into a binary file, before being translated into waveforms in the vector signal generator (VSG). A driver is connected between the VSG and the device under test (DUT) to keep the HPA input signal as linear as possible [30]. The output of the HPA is connected with a fixed attenuator and a variable attenuator to adjust the SNR value of the link, as shown in Fig. 10(b). The received data is recorded as a binary file by the vector signal analyzer (VSA). Finally, the received data is processed by the matched RRC filter and then demodulated in MATLAB.
The parameters and the corresponding values used in this section are listed in Table 2. Based on the digital pre-distortion in Section III.A, the variance of the additive Gaussian noise for the equivalent linear link approximating the non-linear link depicted in Fig. 10(a) is obtained with Monte-Carlo simulations, and listed in Table 3. Experimental setup with HPA devices whose AM/AM conversion is depicted in Fig. 9.
The comparisons of the simulated BER performance between the non-linear link with pre-distortion and the equivalent linear link with additive noise source are shown in Fig. 11, and exhibit good agreement. The comparisons between the measured BER and the simulated BER performance are shown  in Fig. 12. From this figure, we can see that for higher IBO values, e.g. β = 0dB, the application of the pre-distortion algorithm can improve the BER performance significantly, while for lower IBO value, i.e. β = −8dB, the performance without pre-distortion comes closest to the performance with pre-distortion, which implies that the operation is in the HPA linear region. Furthermore, it can also be observed that the simulated and measured BER values are in good overall agreement. The reason for the measured BER curves being slightly higher than the simulated curves relies on the fact that extra noise is inevitable in the test-bed, and also in simulations, perfect synchronization is assumed while this is not the case in the test-bed measurement.

IV. ADAPTIVE OPERATION OF DVB-S2X LINK
Based on the equivalent full-linear link with additive noise source, an adaptive operation which can jointly optimize the IBO, ROF, MODCOD is proposed in this section. The target of the link optimization is to maximize the spectrum efficiency of SatCom links while maintaining the required QoS performance, e.g. below a specific maximum allowable BER. This is a multi-dimensional optimization problem, involving three system parameters, i.e. MODCOD, RRC ROF, and IBO of the given HPA, and is also influenced by the channel noise power. The optimization problem is mathematically formulated in (9) and (10), Here, E represents the spectrum efficiency determined by the MODCOD scheme α and ROF γ . B represents the BER determined by α, γ , IBO value β and channel noise power N 0 . τ denotes the input BER threshold of the required detection performance. Note, β has impact only on the BER but no effect on the spectral efficiency. Therefore, the optimum IBO value which can minimize the input BER is firstly obtained.

A. OPTIMUM IBO FOR DIFFERENT ROF VALUES
The optimum IBO for the minimal input BER value can be determined by minimizing the total link degradation denoted  as D tot , see the derivation in Appendix. D tot is defined as the sum of the link degradation and the HPA OBO [12]  are the average symbol energy over noise density required to achieve the target BER in the non-linear and linear channels, respectively. ξ denotes the value of OBO in dB which is a negative constant as shown in Fig. 4. By approximating the ISI as additive noise, the relation between the required average symbol energy for non-linear and linear links can be written as Then, the required energy over noise density in a non-linear channel can be presented as

And in dB format, it is
. (14) Therefore, D tot can be rewritten as As mentioned previously, the equivalent variance σ 2 ISI and ξ are both functions of IBO, i.e. σ 2 ISI = f (β) and ξ = g(β). Then, the optimum IBO denoted as β † for the best BER performance can be obtained by minimizing D tot as below The numerical results of the optimum IBO values with the associate BER for different ROF γ and channel AWGN power N 0 are shown in Fig. 13. From this figure, the optimum IBO β † decreases as the link noise power reduces. This is because the signal distortion caused by HPA dominates the detection error when N 0 is small. Therefore, β should be reduced to make the HPA to work in a more linear region. It can be concluded from Fig. 2 that the impact from FEC codes to the BER performance is monotonic, i.e. the output BER reduces when decreasing the input BER. Therefore, the optimum IBO obtained for the minimal input BER in Fig. 13 is still valid for the output BER with FEC coding.

B. OPTIMUM DVB-S2(X) LINK CONFIGURATIONS
The adaptive operation is implemented by adjusting the MODCOD scheme and ROF value subject to a certain channel noise power, which is summarized as Algorithm I in Table 4. The improved spectrum efficiency of the proposed algorithm is validated by comparing the spectrum efficiency of the ACM with fixed IBO value β = −5dB. For example, the output BER threshold (i.e. with FEC) is set to be 10 −5 , the input BER thresholds for different MODCOD schemes denoted as τ are obtained from the curves in Fig. 2 are:   • { α = 16APSK-9/10, τ = 0.020 }. When the channel noise power, for example, is set to be N 0 = 10 −2 , the optimum IBO β † and the associated minimal BER BER m for each ROF γ are obtained from Fig. 13, and the BER comparisons between the proposed algorithm and the ACM with = −5dB are shown in Tabel 5.
Then, taking γ = 0.05 for example, only three MODCOD schemes for the proposed algorithm can be chosen, i.e. 16APSK-2/3, 16APSK-3/4 and 16APSK-4/5. This is because for other MODCOD schemes, the BER requirement cannot be met, i.e. BER m > τ . Finally, 16APSK-4/5 is selected under γ = 0.05 for the highest spectrum efficiency among the three optional MODCOD schemes. Accordingly, we can achieve the optimum combinations of MODCOD and IBO for each ROF value of the proposed algorithm under N 0 = 10 −2 , and the optimum MODCOD for each ROF value of the ACM under N 0 = 10 −2 , as listed in Table 6. Based on Table 5 and  Table 6, the maximum spectrum efficiency of the proposed algorithm under N 0 = 10 −2 is 3.1020 with the input BER BER = 0.0182, and the optimal combination of link parameters is { α = 16APSK-9/10, γ = 0.15, β † = −3.03dB }. By contrast, the maximum spectral efficiency of the ACM with fixed IBO β = −5dB under N 0 = 10 −2 is 3.002 with the input BER BER = 0.0348, and the optimal combination of link parameters is { α = 16APSK-5/6, γ = 0.10 }.
The scheme proposed in this paper aims to optimize the SatCom link throughput by joint designing the IBO of nonlinear PA, roll-off factor of RRC filters, as well as MODCOD conventionally used in current systems. This joint optimization is able to improve the link throughput. Here more information can be conveyed with the same bit rate since low order coding can be used. In this sense raw bit rate and bandwidth of the channel are not affected in our study, but the spectrum efficiency is improved.
It is worth mentioning that we study 16APSK modulation schemes as defined in the relevant the DVB-S2(X) standard as our paper is focused on SatCom systems. These systems presently rely on APSK simlations due to the favorable PAPR. When applying orthogonal frequency division multiplexing (OFDM), the signal PAPR will be determined by the modulation in each subcarrier and the number of subcarriers. The non-linear property is likely to have more profound effects not only in each subcarrier, but also across subcarriers. The investigation for the settings of this link will become more complicated, and this is a topic for future research, and will be reported separately.

V. CONCLUSION
This paper first reported an equivalent full-linear link with additive noise, which can approximate the DVBS2(X) nonlinear link with an HPA and RRC filters for BER analysis. Numerical and experimental results validated the equivalent linear link with additive noise source. Then, optimum HPA IBOs for different ROFs and channel noise power were obtained to minimize the system BER. With further consideration of the effects of different LDPC code rates to the BER performance and spectral efficiency, as well as the effects from different ROFs to the spectral efficiency, an adaptive operation algorithm to maximize the system spectrum efficiency while still guaranteeing the detection performance was formulated and validated.

APPENDICES
The output signal power from the HPA is denoted as P out . The BER is a Q-function related to the SNIR which is calculated as SNIR = P out /(σ 2 ISI + N 0 ), the increasing SNIR can enhance the BER performance. Therefore, the target of the BER minimization can be transferred to the SNIR maximization, i.e.
The definition of OBO is ξ = P out − P sat out [dB]. Therefore, the target T 1 could be rewritten as T 2 : . (18) As the saturation output power P sat out is a fixed value for a given HPA, P sat out can be removed as Next, the maximization is formulated as T 4 With the consideration of ISI, the required SNRs for nonlinear and linear systems are written as E s /(N 0 + σ 2 ISI ) req NL and (E s /N 0 ) req AWGN , respectively, and in dB format, the relation between the two required SNRs is rewritten as and in dB format: Note, the required (E s ) req NL only considers the adverse impact from the ISI. Hence, the (E s ) req NL is treated as the same value of P out . Substituting (23) into (21) yields the optimization target as As N 0 is independent of IBO, the final target is, which is the definition of D tot in Section IV.A. VOLUME 10, 2022