Machine Learning-Based Linearization Schemes for Radio Over Fiber Systems

This work proposes a novel machine learning (ML)-based linearization scheme for radio-over-fiber (RoF) systems with external modulation. The proposed approach has the advantage of not requiring new training campaigns in case the Mach-Zehnder modulator (MZM) parameters are changed over time. Our innovative digital pre-distortion (DPD) was designed to favor enhanced remote areas (eRAC) scenarios, in which the non-linearities introduced by the MZM become more severe. It employs a multi-layer perceptron (MLP) artificial neural network (ANN) to model the RoF system and estimate its post-inverse response, which is then applied to the DPD block. We investigate the ML-based DPD performance in terms of adjacent channel leakage ratio (ACLR), normalized mean square error (NMSE), resultant signal root mean square error vector magnitude error (EVM$_\mathrm{RMS}$), and complexity. Numerical results demonstrate that the intended DPD method is less complex and outperforms the orthogonal scalar feedback linearization (OSFL) scheme, which has been considered a state-of-the-art DPD technique. The proposal has the potential to effectively and efficiently compensate for the RoF nonlinear distortions, especially in a time-variant system, without needing new training campaigns.

ultra-reliable low-latency communication (URLLC), for reducing the network time response. Many enabling technologies are used to support the services expected in each scenario, including optical/wireless techniques employed in the fronthaul architecture [4]. Most of the applied technologies are prone to reduce cell coverage. Moreover, coverage is also limited in the URLLC and the mMTC scenarios, since the power limitation in the device and restriction on the symbol duration will limit the link budget and robustness against the long channel delay profile, respectively [5], [6].
One scenario that has been attracting attention, especially in continental-sized countries, is known as enhanced remote area communications (eRAC) [7]. In this scenario, new business models and cost-effective solutions are required to offer connectivity for subscribers in remote and rural areas [8]. However, this network operating model is usually economically unattractive, since there are few potential subscribers at these locations. Moreover, the capital expenditures (CAPEX) and frequency licenses have hindered the remote network deployment [7]. These restrictions must be overcome to finally offer connectivity in remote areas. Some initiatives have been proposed to address this problem. The Remote Area Access Network for the 5th Generation (5G-RANGE) project aims to exploit TV white space (TVWS) for 5G communications in remote areas [9]. The TVWS usage demands a low out-of-band emission (OOBE) waveform, which means the adjacent channel leakage ratio (ACLR) must be kept as low as possible for avoiding co-channel interference. The 5G Rural First project aims to allow local communities to exploit idle 3rd Generation Partnership Project (3GPP) bands in remote areas [10].
All the initiatives mentioned above show that reducing the cost of deploying the telecommunication infrastructure is key for successfully covering remote areas. In this case, analog RoF (A-RoF) technology [11] can play an important role in bringing connectivity to unserved or underserved regions since it might be efficiently used in the transport centralized radio access network (C-RAN) [12]. The centralization of the RAN functions allows sharing the processing resources among different services and applications, favoring the dynamic allocation and simplifying the network operation and maintenance. This approach reduces the CAPEX [1] and brings more flexibility for the Telecom operators since the radiofrequency (RF) power for one base station (BS) can be dynamically allocated according to the demand in each region. In C-RAN architectures, the core network is connected to the central office (CO) using a backhaul link. In contrast, the remote radio unit (RRU) is connected to the This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ CO using a fronthaul link. Microwave and optical technologies might be applied in these links. A-RoF is particularly interesting since it reduces the RRU complexity and occupied bandwidth in the optical link when compared to the digital RoF (D-RoF) [13].
Recent advances on 5G fronthaul solutions indicate to the replacement of D-RoF by A-RoF technology. Although D-RoF is currently widely employed, it presents scalability issues, highpower consumption and costly operation in the millimeter-wave frequency range. In [14], authors present a dual-band A-RoF system using a polarization control to switch the RF operating frequency and single-sideband modulation for reducing distortions caused by intermodulation products. A digital signal processing (DSP)-assisted A-RoF was proposed in [15], focused on reducing the sampling rate for a high-speed analog-to-digital converter (ADC) and digital-to-analog converter (DAC) in fronthaul links. Regarding fronthaul flexibility and latency demands, Datsika et al. demonstrate an A-RoF fronthaul assisted by a software-defined network (SDN), which enables dynamically performing network functional splits for minimizing the system latency.
In order to achieve high-speed modulation, the Mach-Zehnder modulator (MZM) is typically used to externally modulate the optical carrier with the RF signal. One challenge in this approach is to deal with the non-linearities introduced by the MZM as the input level of the RF signal increases. The MZM nonlinear response leads to a spectral regrowth, which is produced by the intermodulation products of the RF signal and is of major concern in TVWS application. Therefore, a limited input power must be applied to the MZM input or a linearization technique must be employed in order to minimize the signal distortion [16].
The digital pre-distortion (DPD) has been considered the most prominent solution for the linearization of nonlinear devices [17], [18], [19], [20]. Recent advances in computational capacity and the increase in the available data sets have enabled new machine learning (ML) approaches, which might be applied in distinct network layers, aiming to deal with the unprecedented growth in the complexity of the communication systems [21], [22]. For instance, in the sixth-generation of mobile network (6G) conception, ML-based solutions are considered an enabling technology for increasing the efficiency of distinct levels of the mobile networks [23]. Najarro et al. have described a model based on an artificial neural network (ANN) for distortion compensation by estimating the inverse response of a radio over fiber (RoF) system [24]. The received root mean square error vector magnitude (EVM RMS ) was the metric evaluated. Additionally, a DPD approach based on ANNs has been proposed by Hadi et al. [25]. Authors have shown a substantial reduction in EVM RMS when compared to a Volterrabased approach. An ANN for simultaneously equalizing and decoding RF signals was reported in [26]. The ANN solution has outperformed the conventional Volterra equalization followed by a hard decision detection. In [27], authors have demonstrated an RoF system equalization using a multi-level ANN equalizer for compensating the nonlinear signal compression due to the in-band distortions. Liu et al. have proposed an ANN equalizer to mitigate the interference between multiple users in uplink transmissions [28].
Although the above works have considerably reduced the RoF distortions, a natural component response variation with time, due to aging and temperature drift, will require new training campaigns to maintain the system's high-performance level. Moreover, the replacement of silicon components and/or fluctuations in the MZM polarization voltage might also demand a new training campaign, since these variations would imply in a different nonlinear function [29].
This work main contribution is the proposal and implementation of a novel ML-based scheme for linearizing RoF systems, which is able to generalize possible variations of the MZM parameters for enabling a non-recalibrated DPD. The proposed linearization scheme does not demand new training campaigns when the MZM parameters change over time, promoting robustness against time variations in the nonlinear RoF system response in eRAC applications. The manuscript is structured in six sections. Section II describes the nonlinear model used for modeling the RoF system. Section III presents the theoretical background on DPD techniques. Section IV introduces the proposed ML-based DPD concepts, whereas the Section V reports its results. Finally, the conclusions and future works are drawn in Section VI.

II. RADIO-OVER-FIBER MODEL
The scenario considered in this paper is depicted in Fig. 1. The Software-defined radio (SDR) approach is used to implement all the physical layer (PHY) and medium access control (MAC) functions of the base stations, which run at the CO server. It will be assumed that BSs will be used to provide connectivity to remote areas using eRAC, which means that low out-of-band emission is necessary [7]. The signal from BS is linearized by the proposed ML-based DPD approach and then applied to MZM. The fronthaul link is implemented using A-RoF due to its lower operational expenditure (OPEX), compared to the D-RoF solution. In case D-RoF is employed, OPEX is significantly increased for each additional kilometer between CO and remote base station, which might hinder its use in the eRAC scenario [30]. A single CO might serve multiple eRAC sites. A single-mode fiber (SMF) is used to distribute the signal to all eRAC sites, where a photodetector (PD) converts the signal from the optical to electrical domain and a RRU radiate it over the covered area, giving rise to a fiber-wireless (FiWi) system. The RRU also receives the signals from the mobile users and converts them to the optical domain to be transmitted to the respective BS.
Typically, the MZM transfer function has a nonlinear periodical behavior, more specifically a cosine-shape function [31]. Although the MZM transfer function might be considered approximately linear for small signals [32], in this work we are interested in maximizing the electrical-to-optical (EO) conversion efficiency by operating with flexible RF power for covering distinct remote regions. The EO conversion efficiency is directly proportional to the RF input power, which will produce high levels of signal distortion, which consists of intermodulation products and harmonics of the carrier frequency. This distortion increases the OOBE parameter and reduces the EVM RMS , affecting the overall performance of the system. Depending on the levels of the OOBE, it can even hinder the exploitation of TVWS [7].
The aforementioned operating scenario claims for a linearization technique to make possible increasing the RF input power and simultaneously reduce the signal distortion. One of the main approaches to achieve linearization consists of using a DPD technique before transmitting the RF signal to RRU. The DPD power response is the reciprocal of the MZM power response. Therefore, the combined power response of DPD and MZM is linear. The behavior of the classic DPD is defined by a model based on a set of coefficients, which are defined by a linearization algorithm.
A simplified version of the polynomial model from [17], [33] has been applied to digitally linearize a RoF system. In our investigation, we have not considered the laser diode (LD) nonlinearities, which might introduce a memory effect. However, as demonstrated in [33], the memory effect can be neglected in RoF systems, since it will not produce significant performance degradation. We have assumed a memoryless polynomial model with five coefficients, i.e. K = 5, which represents a trade-off between performance and complexity [33]. This non-linearity order was assumed in all evaluated scenarios. The memoryless polynomial model is given by where K is the polynomial order, h k are the model coefficients, with k = 0, 1, ..., K − 1, x n is the orthogonal frequency division multiplexing (OFDM) signal given by with n representing the nth sample with n = 0, 1, ..., N − 1 from the total N samples, d m is the quadrature amplitude modulation (QAM) data symbols, which are split in M orthogonal subcarriers. y n is the output discrete signal.

III. DIGITAL PREDISTORTION
The DPD has been widely used for mitigating signal distortions caused by nonlinear devices. The main DPD application is the linearization of high power amplifier (HPA), which typically presents a nonlinear system response. Similarly, the RoF systems also present a nonlinear response, imposed mainly by the MZM modulator. The DPD operating principle consists of predistorting the input signal by a function that is complementary to the nonlinear system. Therefore, the distortions introduced by the nonlinear device will cancel those added by DPD, linearizing the system response. The DPD distortions are ruled by a set of coefficients obtained according to a linearization algorithm. One of the most popular approach to extract the DPD coefficients is the Least Mean Square (LMS) algorithm. Fig. 2 illustrates the block diagram of a conventional DPD system. In this diagram, x n is the discrete input signal, v n is the pre-distorted signal with shape ruled by the coefficients h, and e n is the error signal used to estimate the system response. The input signal x n and output signal y n feed the linearization algorithm, which calculates the coefficient h that produces an output signal y n as similar to x n as possible.
The scalar feedback linearization (SFL) is an alternative solution to linear-regression DPDs [34]. The basic idea of SFL is to optimize a scalar cost function (C(h k )) for obtaining the DPD coefficients. The scalar feedback does not require time synchronization and magnitude normalization, as the linear regression-based approaches. Distinct cost functions can be selected for reducing the OOBE, in-band distortions, or even both interferences. The main metrics used by the SFL DPD are the ACLR, EVM RMS , and mean-squared error (MSE). While the ACLR is used to minimize the OOBE, the EVM RMS is used to minimize the in-band interference. The MSE is employed to reduce both interferences. It is important to notice that by improving one metric, the other one will be also improved. For instance, a DPD optimized for the ACLR will also improve the EVM RMS (and vice-versa). Since C(h k ) changes at each iteration, any generic mathematical linearization algorithm might be employed to optimize the cost function. The main drawback of the SFL is the inter-dependency among the coefficients. In other words, the evaluation of the kth coefficient h k leads to the reevaluation of all previous coefficients h 1 , h 2 , . . . , h k−1 , which must be readjusted. This procedure increases significantly the overall convergence time.
In this paper we have compared our ML-based DPD solution with a state-of-the-art DPD technique known as orthogonal scalar feedback linearization (OSFL) [35], which represents a pruned version of the SFL. Basically, the OSFL principle consists of the orthogonalization of the coefficients h k , followed by numerical optimization of the scalar cost function. The main goal of the OSFL is to remove the inter-dependency among the coefficients, which means that the evaluation of one coefficient does not affect the previous one. The orthogonalization process is based on minimizing the difference (residual vector) when a new coefficient order is being adjusted. In other words, the orthogonalization produces a new coefficient set (h r ) that is incorporated in the DPD coefficients for promoting orthogonality. The coefficients h r are obtained using a least squares estimation algorithm. It is important to highlight this process might be repeated for enhancing the pre-distortion accuracy. The OSFL approach provides faster convergence when compared with the SFL technique, since the orthogonalization of the coefficients simplifies the optimization phase.
The DPD algorithm complexity is an important parameter to be taken into account. The complexity analysis was carried out using the float-point operations (flops) counter method, where one flop is defined as one multiplication followed by an addition of two float-point numbers [36]. The orthogonalization algorithm applied to the SFL DPD reduces the complexity, since only K matrix inversion operations are required (once for each order) regardless the iterations of the optimization algorithm [35]. The complexity of an n × n matrix inversion is O(n 3 ). Since this process is going to be repeated K times, we can infer a computation cost of OSFL will be which has its complexity mainly governed by the least square estimation of h r . Considering the dominant term of O, we can infer that the OSFL DPD computation complexity will cubically increase with the polynomial model non-linearity order.

IV. ML-BASED DPD FOR ROF SYSTEMS
The ANN is a subclass of the ML algorithms that has been widely employed to solve complex computational tasks. A supervised ANN learns the relation among the input samples and the target labeled samples by using a set of training instances (N TR ), with the purpose of creating an approximated function to map the input and target data samples. An ANN has been considered an interesting tool to deal with nonlinear systems since it is capable of learning complex nonlinear behaviors.
This section presents two distinct RoF system DPD schemes based on machine learning. The first one was designed to linearize time-invariant systems, while the second one is used for time-variant systems. The second approach is interesting for optical/electronic components since their nonlinear responses can change over time due to aging and temperature drift. In a conventional DPD, the previously calculated coefficients must be recalculated to avoid signal distortion, while the solution for time-variant system proposed in this paper will accommodate these changes, adding robustness to the pre-distortion process and minimizing the need for periodical maintenance for the coefficients recalculation.

A. MLP ANN-Based DPD for Time-Invariant Systems
Many nonlinear regression techniques have been widely adopted in recent works, such as Least Absolute Shrinkage and Selection Operator (LASSO), Random Forest, Support-vector Regressor (SVR) and multi-layer perceptron (MLP) [37], [38], [39]. The MZM non-linearities can be entirely modeled by a memoryless polynomial model. This statement is supported by [33], in which authors have proved that modeling the RoF system nonlinearities considering the memory effect will only unnecessarily increase the overall system complexity. For this reason, ANNs-based pre-distortion schemes that consider the memory effect were classified as high complexity solutions in our analysis. On the other hand, our proposed schemes employ a memoryless polynomial model, which is more appropriate to tackle the non-linearities introduced by the MZM. Once the memory effect might be neglected, a simpler ANN structure could be employed. Therefore, we have chosen the MLP for developing the ANN-based DPD, since it can be successfully deployed to represent continuous variables with nonlinear behavior. Although its simplicity, the MLP can be efficiently used for solving complex tasks.
The MLP is composed of several units or neurons densely connected in sequential layers. At least three layers are required for designing a MLP network, namely input layer, hidden layer and output layer. It has already been demonstrated in the specialized literature that a single-layer network can approximate any continuous function if there is enough data to train the neural network [40]. This means that a very simple neural network can already present high non-linear representation capacity, as long as we employ non-linear activation functions, have enough examples and computational resources to train the neural network. In this work, two hidden layers are enough to represent the nonlinearities imposed by MZM. Each connection between neurons is represented by a weights matrix W and a bias vector b, which are calculated accordingly to the backpropagation algorithm to minimize a cost function. In summary, the backpropagation algorithm jointly with the solver compute the gradient of the loss function for the ANN current weights. Sequentially, the algorithm backpropagates the error for identifying its parcel related to each connection. Finally, the weights are updated aiming for minimizing the cost function. Fig. 3 illustrates the proposed ML-based DPD for timeinvariant systems. It is based on the indirect learning architecture, in which a post-inversion system response is obtained and the matrix of weights W is copied to the DPD block. In our approach, x n is the original OFDM signal, whereas v n is the pre-distorted signal at MZM input and y n is the photodetected signal, which is obtained by applying (1). In our work, an ideal feedback loop was considered for training the DPD coefficients. However, in real RoF systems, the required output feedback signal is typically a few kilometers far from CO. Our goal is to perform the training in an offline fashion rather than on-the-fly. This strategy implies in transmitting the labeled data from CO to the remote base station, where signals are collected for the ANN training; once trained, ANN is not updated anymore. As a consequence, although costly, the training phase only takes place once, since our solution is able to generalize variations in the RoF system time-response.
The MLP ANN is composed of L + 1 layers, where each one of them contains O neurons per layer with ∈ {0, . . ., L + 1}. The post-inversion estimation is obtained by the ANN, as described in Fig. 4. The ANN is fed by a data set containing two features, which represent the real R(y) and imaginary I(y) parts of an OFDM signal at the RoF system output. The target labels are composed of the real R(x) and the imaginary I(x) parts of a non-distorted OFDM signal at the MZM input.
Mathematically, an MLP layer performs the following operation for a given input vector y where W is a matrix containing the MLP weights, y is the input vector, b is the biases vector, and f (.) is the nonlinear activation function. Several activation functions, including sigmoid, rectified linear unit (ReLU), hyperbolic tangent function (tanh), scaled exponential linear unit (SELU) might be used [41]. Finally, the loss function to be minimized, which depends on the estimated instancex n and desired instance x n , is described as follows where N TR is the size of the training set and L(.) is the loss function to be minimized by tuning W and b.
The neural network performs nonlinear transformations in the input signal to compensate the A-RoF system distortions. It means that both input and output layers of the neural network must contain two features i.e., O 0 = O L+1 = 2, which represent the real and imaginary part of the OFDM symbol. The training data set relied on a hundred OFDM symbols; each of them with a signal-to-noise ratio (SNR) equal to 45 dB and 1024 samples, resulting in 102400 instances. The other system hyperparameters are summarized in Table I. The hyperparameters were configured using a heuristic methodology, which means without a sophisticated random or grid search. Initially, we started with a few neurons per hidden layer. Sequentially, these hyperparameters were gradually increased accordingly with the DPD performance. According to our observations, 32 neurons in the first hidden layer and 16 neurons in the second one are the smallest dimensions for the hidden layers that outperform the conventional DPD. Additionally, we have noted that incrementing the number of neurons of each layer will only unnecessarily increase the algorithm complexity, with no additional performance gain. Regarding the activation function, we have tested the ReLU, SELU and tanh. However, the activation function has not considerably impacted on the results. Therefore, we have chosen the ReLU for performance evaluation. The number of hidden layers depends on the application however, typically one to five hidden layers are employed in regression problems [42]. We have chosen two hidden layers in our ML-based DPD, which have enabled a remarkable performance and employing more hidden layers will only unnecessarily increase the computational complexity.
We have also estimated the complexity of the ML-based DPD using the flops counting method. Since each layer output is obtained by (4), we can express the MLP complexity by Assuming now that dim(y) which shows that the complexity of the proposed DPD grows linearly with the dimension of the input vector, the number of neurons of the input layer O 0 and the number of hidden layers L. Additionally, the complexity grows quadratically with the number of neurons from the hidden layers O .

B. MLP Dual-ANN-based DPD for Time-Variant Systems
Two MLP ANNs were employed for dealing with the variability of the nonlinear system over time, since the nonlinear behavior of the MZM is not constant over time. Aging, temperature variation and fluctuations of the MZM polarization voltage can cause the nonlinear behavior of the system to vary over time. In order to analyze the proposed scheme capacity of accommodating these variations, a Gaussian distributed random variation of approximately 10% of the coefficient value has been added to each coefficient of the memoryless polynomial model, with the purpose of representing the time-variable non-linear behavior of the MZM. Our strategy for this case was to apply the ANN for modeling the RoF system and obtaining the post-inversion system estimation using another independent MLP ANN, as illustrated in Fig. 5. We have named this scheme as dual-ANNbased DPD. This enhanced approach brings robustness against time-variations of the RoF system nonlinear response.
The post-inversion estimation from the time-variant system was obtained by applying the same approach from the timeinvariant systems, generating the weights matrix W 1 , which is copied to DPD. To model the RoF system, the ANN must be fed with the non-distorted samples and the label samples are the distorted signal at the RoF output. The RoF ML-based modeling produces the second matrix of weights W 2 , which might be used for emulating the RoF system. Modeling the RoF system with an ANN has an important advantage that comes from the high capacity of ANN to mimic nonlinear systems. That means, the ANN is able to capture nonlinear behaviors that a polynomial model can not represent. Additionally, such modeling might contribute to simultaneously compensate other distortions beyond those from MZM. Table I summarizes the main hyperparameters employed by the dual-ANN DPD for time-variant systems. Assuming that both ANNs were trained offline, the DPD process applied to the RoF system will have the same complexity derived in Section IV-A. Hence, considering the two ANN will run independently, the dual-ANN DPD scheme will present twice complexity when compared with the MLP ANN-based DPD presented in Section IV-A.

V. PERFORMANCE EVALUATION
This Section presents the proposed ML-based DPD results applied to the eRAC scenarios, aiming the exploitation of TVWS opportunities. All simulations presented in this work were performed for base-band discrete-time signals, since the pre-distortion processing is accomplished in the digital domain. In this case, we consider the OFDM signal at the reception side is in its base-band format and has already been converted from analog to digital. One important motivation for applying a DPD technique in this scenario is the possibility of dynamically allocating RF power to cover distinct regions, without producing prohibitive adjacent channels interference.
The novel ML-based DPD was implemented in Python using the high-level Tensorflow application programming interface (API) Keras. Its performance was investigated using three figures of merit, namely: EVM RMS , ACLR and normalized mean square error (NMSE). The NMSE was used for estimating the time-domain error between the MZM input signal and linearized signal at the photodetector output. Moreover, we have compared our ML-based DPD with the OSFL scheme for two distinct cases, i.e. time-invariant and time-variant systems, which were described in Sections IV-A and IV-B, respectively. The same data set have been used for both cases, which was generated for an OFDM signals at 20-dBm electrical power, which produces considerable distortion in the RoF system output, in case no linearization technique is applied. The performance evaluation of the proposed DPD scheme for both scenarios, i.e. time-invariant and time-variant systems, was carried out using 16-QAM. The data set was partitioned into 70% of training instances, leading to N TR = 71680 and 30% for validating the ANN. The MSE loss function, which is one of the metrics for evaluating regressors [43], was used for training the ANNs and is defined as Fig. 6(a) depicts the MSE between the predicted and real target labels, as a function of the training epoch for the time-invariant system. This result might be extended for the time-variant system post-inverse estimation, once the same hyperparameters and data set were used. Once no overfitting or underfitting behaviors were observed, we can infer that all configured hyperparameters have produced an appropriate generalization capability. Such conclusion is ratified by Fig. 6(b), which is regarding the ML-based RoF modeling.   ML-based linearization performance compared to the OSFL and long short-term memory (LSTM) schemes. It is important to highlight the ANN DPD training was realized only for a specific power for the entire tested RF power range. Additionally, we have performed new training campaigns to investigate the best training RF power in terms of performance. As shown in Fig. 7, using OFDM training symbols at 20 dBm leads to the best linearization result, for almost the entire analyzed RF power range, in comparison with the cases when the 15 and 28 dBm RF power levels are employed. The LSTM has only increased the complexity without improving the system performance, when compared with MLP and OSFL, since it was designed to process temporal data, which does not necessarily help to model the A-RoF system non-linearities. We are currently using ReLU activation function. Nonetheless, preliminary studies have demonstrated SELU can result in faster convergence time, without degrading the linearization performance. For the time-invariant system, we can note both DPD scheme has considerably reduced the EVM RMS in comparison to the results without DPD. Furthermore, it is possible to observe that our proposed scheme has performed slightly better than the OSFL method for almost the entire analyzed power range. In addition, our approach is less complex, as already discussed in the previous Sections, making it promising for PHY layer optimization using machine learning.

A. Performance of the MLP ANN DPD for Time-Invariant Systems
The second investigation was conducted by analyzing the frequency domain intermodulation using ACLR and the timedomain metric NMSE for calculating the error between the linearized and RoF output signals. Fig. 8 shows a normalized power spectrum density (PSD) for a 25-dBm OFDM signal. The ACLR metric defines the ratio between the undesired OOBE mean  II  DPD PERFORMANCE FOR THE TIME-INVARIANT SYSTEM AT 25 DBM   TABLE III  DPD PERFORMANCE FOR THE TIME-VARIANT SYSTEMS AT 25 DBM power and assigned channel frequency mean power. When DPD is not applied, the OOBE might become prohibitive, hindering the exploitation of TVWS. As can be seen, both DPD techniques have substantially reduced OOBE. We have also investigated the impact of variations in the RoF system nonlinear model, as a result of fluctuations in the MZM polarization voltage, in the ML-based DPD results. We refer to this case as a time-variant system. Table III shows a comparison of the DPD performance between the OSFL scheme and the proposed ML-based ANN DPD approach. The variation in the RoF system model coefficients has degraded the signal quality, which can be noted by comparing each column from Table II with the correspondent from Table III. As a conclusion, a new training campaign must be conducted in both cases for enhancing the DPD distortion reduction. In the next section, we present a second approach (dual-ANN DPD) to deal with this challenging feature in a dynamic scenario.

B. Performance of the MLP dual-ANN DPD for Time-Variant Systems
The dual-ANN DPD has been idealized to operate in cases that the nonlinear system response changes over time. In any case, it might also be applied to the static operating condition, in spite of presenting higher complexity, when compared to our first ML-based DPD approach. We have assumed that a time-variant nonlinear response has short and long terms components. The temperature drift over time results in a short-term instability to the system response. For instance, the environment temperature variation along the day might produce fluctuations in the MZM polarization voltage, which could imply in the MZM operating point variation and, consequently, performance fluctuation. The long-term instabilities will be observed as the components ages. In this case, a new training session needs to be performed. Alternatively, we can employ the MLP dual-ANN DPD for time variant-systems, that is able to generalize the variations of the polynomial model coefficients, outperforming the OSFL scheme.
Any required change in the system will generate considerable expenses, since the skilled team would be necessarily sent to field, implying in travel and accommodation costs. This scenario could be even worse if the remote site is placed at hard-to-reach areas. In addition, the communication system must be periodically turned off for re-training the DPD algorithm whenever necessary, leaving customers without coverage. Therefore, although the time-variant response of the A-RoF components is typically slow, implementing a system that does not require changes will significantly reduce network operational expenditure. Fig. 9 presents the EVM RMS performance as a function of the electrical power at the MZM input. We have used a set of 100 OFDM symbols at 20 dBm for training the OSFL, LSTM and dual-ANN DPDs schemes. Our goal was to evaluate the generalization capacity of the DPD schemes when distinct RF powers are applied to the MZM input. Considering the timevariant system, the OSFL and LSTM schemes have not been shown efficient in terms of performance, since both will require a new training campaign because its polynomial model does not consider any changes in the coefficients model. Additionally, for low RF input power, using a DPD trained at 20 dBm might produce an overestimated pre-distortion that is not correspondent to the real observed distortion. In this case, the DPD block might produce a signal distortion that is even higher than not applying any DPD scheme. Therefore, the coefficients of the DPD must be re-estimated for low RF powers or DPD could even be turned off if the distortion level is acceptable. On the other hand, the dual-ANN DPD approach provided a remarkable distortion reduction, as a result of applying a second ANN for modeling the RoF system, which allowed to take advantage of the high-capacity of ANN to generalize and represent a nonlinear system. Therefore, in contrast to the OSFL and LSTM models, we can infer that our dual-ANN DPD scheme is capable to absorb variations in the model that represents the nonlinear system.
Similarly to the time-invariant system, we have also investigated the ACLR and NMSE metrics for the time-variant systems considering a 25-dBm OFDM signal. Fig. 10 reports the normalized PSD for the time-variant system. As expected, the OSFL scheme has presented higher intermodulation levels than the dual-ANN DPD approach, since it had not been designed for   this operating condition. This result endorses the importance of a dynamic approach for linearizing signals in TVWS, by reason of high levels of spectral regrowth might severely interfere with adjacent channels. Table IV summarizes the statistics of the performance results for the time-variant system based on 10,000 runs. Our dual-ANN DPD solution enabled significantly reduce the mean EVM RMS from 9.52 to 0.92%. The obtained mean ACLR = −36.54 dB and NMSE = −41.31 dB have also demonstrated its potential for distortion compensation. Furthermore, the dual-ANN DPD also has less computational complexity (6.144 k flops) when compared with OSFL (205.280 k flops).
Finally, we have evaluated our proposed dual-ANN DPD performance in terms of bit error rate (BER). Fig. 11 presents the results of our proposed scheme compared with the theoretical OFDM-additive white Gaussian noise (AWGN) system, as a function of SNR. In this analysis, we have considered 15, 20 and 28 dBm of RF power at MZM input. It can be observed that for 20 dBm, our proposed dual-ANN DPD has produced a BER performance similar to the theoretical OFDM-AWGN. This achievement proves the excellent linearization performance of our DPD scheme, in case nonlinear distortions are introduced by the MZM. Additionally, we believe that the dual-ANN can play a distinguished role on the future mobile communication system, since ANN can learn complex nonlinear behaviors, such as the interactions between the linear and nonlinear effects of RoF systems.

VI. CONCLUSION
This paper presented, for the first time in literature, a digital pre-distortion scheme based on machine learning for radio over fiber systems considering a time-variant response. It employs an MLP artificial neural network to model the RoF system and estimate its post-inverse response, which is then applied to the DPD block. The main advantage of our dual-ANN DPD approach relies on not requiring new training campaigns when the RoF system parameters change over time, due to temperature variations, aging or fluctuation on the electro-optic modulator polarization voltage. Its performance has been compared to that of the OSFL scheme, considered a state-of-the-art solution, as a function of the three following figures of merit: ACLR, NMSE and EVM RMS . It outperformed and has been shown simpler than OSFL. Particularly, the proposed ML-based linearizer allowed to significantly reduce mean EVM RMS from 9.52 to 0.92% with mean ACLR = −36.54 dB and NMSE = −41.31 dB.
Moreover, our DPD solution is potential for reducing CAPEX and increase flexibility to the mobile network operators for dynamically allocating RF power for serving different regions. Furthermore, it can be efficiently applied to the future 5G networks in the eRAC scenario, exploiting TV white space. As future works, we envisage utilizing an even more realistic nonlinear model and testing other artificial neural network architectures. Additionally, we aim to experimentally validate our approach in a real 5G network.