Implementation of Alternating Direction Method of Multipliers for Solving Power Amplifier Linearization Problem: Theoretical Foundations and Proof of Concept

Received 31 July 2024, accepted 18 August 2024, date of publication 26 August 2024, date of current version 16 September 2024. Digital Object Identifier 10.1109/ACCESS.2024.3450352 Implementation of Alternating Direction Method of Multipliers for Solving Power Amplifier Linearization Problem: Theoretical Foundations and Proof of Concept ELÍAS MARQUÉS-VALDERRAMA , JUAN A. BECERRA , (Senior Member, IEEE), AND MARÍA JOSÉ MADERO-AYORA , (Senior Member, IEEE) Departamento de Teorí a de la Se˜nal y Comunicaciones, Escuela Técnica Superior de Ingenierí a, Universidad de Seville, 41092 Seville, Spain Corresponding author: Elías Marqués-Valderrama ([email protected]) This work was supported by Grant PID2021-123090NB-I00 funded by MICIU/AEI/10.13039/501100011033 and by ‘‘ERDF/EU.’’ ABSTRACT In this work, a formulation of alternating direction method of multipliers (ADMM) for addressing power amplifiers (PA) modeling and linearization problems is presented. The proposal consists on leveraging the implicit redundancy of the equations in order to achieve a distributed architecture. A detailed theoretical formulation of the method is provided in order to get a better comprehension of its advantages and the approach to integrating the technique in the standard direct learning architecture scheme for digital predistortion. In addition, a discussion on how the implicit regularization helps to deal with numerical problems is presented. It is proven that the implicit regression for the modeling and linearization of PAs can be carried out in a distributed fashion with similar accuracy, enabling the use of resource-constrained devices for digital predistortion. Furthermore, a computational assessment is presented for measuring in terms of arithmetic operations how simpler the devices that implement the proposed ADMM can be, compared with the classical least squares (LS) data-centralized approach. A proof of concept with three scenarios is included: the first two scenarios feature a commercial PA driven by 5G-NR signals with a bandwidth of 30 MHz, and the third scenario involves a commercial Doherty PA with a 100-MHz signal. In a first scenario, experimental results show that an ADMM implementation of a digital predistorter can achieve the same performance as the classical LS solution. The benefits of the proposed regularization are examined by assessing ADMM in a second scenario characterized by significant numerical instabilities. Finally, the robustness of the technique is illustrated through the linearization of the Doherty PA with a 100-MHz OFDM signal. INDEX TERMS Alternating direction method of multipliers (ADMM), augmented Lagrangian, behavioral modeling, computational reduction, digital predistortion (DPD), distributed algorithms, least squares (LS), linear regression, nonlinear model identification, power amplifier (PA), regression conditioning, regularization, Volterra series. I. INTRODUCTION Digital services have been recently impulsed by the improvements deployed by mobile communication standards, including the fifth generation (5G), which, amongst others, The associate editor coordinating the review of this manuscript and approving it for publication was Tae Wook Kim . contains use case for enhanced mobile broadband (eMBB). This is committed to an increase in bandwidth, enabling a very high throughput data rate transmission. 5G new radio (5G-NR) is supported by orthogonal frequency division multiplexing (OFDM) signals that include cyclic prefix (CPOFDM) to mitigate multipath effects of the physical channel and a set of numerologies to provide flexibility for the 124682 2024 The Authors. This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ VOLUME 12, 2024 E. Marqués-Valderrama et al.: Implementation of ADMMs for Solving PA Linearization Problem adaptation to diverse scenarios in terms of bandwidth and latency. In spite of the advantages in the usage of this modulation, OFDM signals present a high peak-to-average power ratio (PAPR) [1],[2],[3]. Moreover, at the end of the transmitter chain, the power amplifier (PA) operates as close as possible to the saturation region to achieve maximum power efficiency. The high PAPR of OFDM is translated into amplitude excursions to the saturation zone of the PA, where nonlinear effects appear, distorting the signal and degrading power efficiency [4],[5]. Nonlinearities not only affect in the signal bandwidth, but also introduce spectral regrowth that violates spectral masks on adjacent bands. For dealing with these problems there exist several techniques, where digital predistortion (DPD) schemes are commonly employed [6], [7],[8],[9]. In general, this approximation requires a behavioral model, where Volterra-based models [10],[11] are widely extended due to their intrinsic physical meaning. Furthermore, although Volterra series represent the inputoutput relation of a nonlinear function with memory, one of the convenient features of Volterra-based models is that they are linear with respect to the coefficients and can be viewed as regression models where the output is expressed as a linear combination of basis functions or regressors. The objective of regression is to calculate an estimator of the coefficients so that the error between the output of the model and the actual output signal is minimized. The direct mathematical minimization of the error leads to the classical least squares (LS) solution. The nonlinear behavior exhibited by efficient PAs driven by high bandwidth signals usually requires a rich Volterra structure for its modeling, worsening numerical issues implicit in the regression problem associated to the DPD- schemes. The high dimensionality not only increases the computational complexity of the problem, but also introduces instability in the solutions that, sometimes, are not even able to converge [12]. The parameter adaptation also becomes an issue in the direct learning architecture (DLA) [13] because of its real-time-processing requirements. Various preconditioning techniques have been employed to enhance the numerical properties of the regression and to deal with this implicit condition, such as normalization, whitening or orthogonalization, coefficient selection, coefficient exclusion, and regularization [14]. Among them, coefficient selection techniques rest in the assumption of the sparsity of solutions to find the best subset of regressors, producing a reduced-order model with enhanced conditions for its regression. Techniques such as doubly orthogonal matching pursuit (DOMP) [15], within the family of greedy algorithms [16], or the sparse Bayesian framework adopted in [17], can be employed for coefficient selection. On the other hand, regularization is a widely-used technique that seeks to recondition the regression and reformulate the minimization problem between the model output obtained with the estimated coefficients and the system output into a more suitable one for numerical performance. In general, regularization schemes add a penalty term to the cost function to prevent unwanted solutions [12], this way avoiding potential overfitting and reducing the condition number of the matrices involved. Ridge regularization [18],[19],[20] and least absolute shrinkage and selection operator (LASSO) [21],[22],[23] are the most commonly used regularization techniques [24]. Regardless of the employment of reduced-order models, there exist a high computational cost that has to be assumed at the time of performing the regression mainly due to the fact that the inversion of large size matrices is necessary. This scenario calls into question the suitability of machine learning with centralized computation since, generally, DPD-schemes have to be implemented in resource-constrained devices, typically in a field-programmable gate array (FPGA) [25]. The implementation cost can be measured by different quantities such as the number of multiply-accumulate operations, the memory usage, floating-point operations per second (flops) or power consumption [26]. These constrictions reinforce the need to reach a feasible and more efficient DPD implementation. For this reason, technologies that are enablers of the so-called edge computing [27], such as federated learning [28], take special interest. In this context, techniques as the alternating direction method of multipliers (ADMM) [29] appear as good candidates to adapt the classical centralized approach to a distributed architecture. Generally, ADMM is used to accomplish a minimization problem which relates two arbitrary functions that are subject to specific restrictions. This structure can be found in a variety of research fields and consequently ADMM has been employed with subtle modifications in other problems. For example, [30] uses ADMM to solve a marketlike distributed coordination of a production plant problem. Also, in [31] the distributed ADMM (D-ADMM) is presented to solve a basis pruning problem. Furthermore, in the field of communications, ADMM is interesting to provide support in different situations. For instance, in [32] three variations of ADMM are presented to solve a millimeter-wave imaging problem of a compressive reflector antenna. Additionally, a PAPR reduction problem is solved with linearized-ADMM (LADMM) in [33]. Considering devices with computational limitations, the possibility of distributing in simpler chunks the equations involved in the DPD problem is especially interesting, particularly in 5G, where a considerable increase in base station density is expected. Thus, it would not only be possible to reduce the costs implicit by the device requirements, but also create cooperative massive networks that help improve the adaptability of base stations. In this work, the ADMM technique is both used to directly predict the optimal DPD block and adapted to a DPD-DLA scheme for leveraging the distributed computing capabilities that relax the computational requirements of the devices. For that, the classical LS solution for the regression problem will be developed into an ADMM formulation, choosing suitable VOLUME 12, 2024 124683 E. Marqués-Valderrama et al.: Implementation of ADMMs for Solving PA Linearization Problem restrictions that not only serve to solve the problem in a distributed way, but also mitigate the numerical issues of the regression. To the best of the authors’ knowledge, this is the first time the ADMM technique is employed for the identification of a DPD. The technique is experimentally demonstrated against a state-of-the-art DPD technique and their performance is compared in terms of computational complexity and linearization figures of merit. As a proof of concept, experimental results for the linearization of both a class AB PA and a Doherty PA driven by a 5G-NR signal are provided. This work is organized as follows. First, Section II sets the theoretical framework. Next, Section III presents the logical implementation and delves deeper in aspects as the computational evaluation. Experimental results are shown in Section IV and Section Vsummarizes this contribution. II. VOLTERRA-BASED REGRESSION FRAMEWORK The regression problem inherent in Volterra-based models can be written in an algebraic form with the so-called measurement equation as y=Xh +e,(1) where X=[ϕ1,ϕ2,...,ϕNr]∈CNsa×Nris the regressor matrix, that weighted by the model coefficients h∈CNr×1, attempts to estimate the output samples y∈CNsa×1, assuming that there exist an error e∈CNsa×1. The vector notation y=[y[k],y[k−1],...,y[k−Nsa +1]]Trepresents the buffered output set of Nsa consecutive baseband complex envelope samples with discrete-time index k. Each regressor ϕris built by the multiplication of delayed and/or conjugate samples of the input baseband complex envelope vector x∈ CNsa×1. Obtaining an estimation of the output ˆ y=Xh consists in solving the minimization problem min hRSS =min h∥y−ˆ y∥2 2,(2) where the residual sum of squares (RSS) is equal to ∥e∥2 2. The minimization function can be rewritten as RSS = ∥y−ˆ y∥2 2=(y−ˆ y)H(y−ˆ y) =yHy−ˆ hHXHy−yHXˆ h+ˆ hHXHXˆ h,(3) where the conjugate and transpose operation is represented by the hermitic superscript (·)H. Given the Wirtinger calculus [34], it is possible to differentiate with respect to conjugate hand make it equal to 0 dRSS dh∗= −2XHy+2XHXˆ h=0,(4) getting the classical LS solution ˆ hLS of minimal error ˆ hLS =(XHX)−1XHy=X†y,(5) where X†stands the Moore-Penrose pseudoinverse. As it was explained in [14],(5) can be reinterpreted as a projection of yinto the model basis functions in RXy =XHyto be later decorrelated by the precision matrix PX=R−1 X, where RX=XHXcontains the covariance between the regressors. The ill-conditioning of RXproduces instabilities in the regression that make ˆ hLS to be very sensitive to variations in the input. This feature is intrinsic to the underling Volterra model and rests in the nature of the regressors. There are different means for dealing with this problem, where one possibility is to apply a coefficient selection technique. This kind of approach forces the matrix to be conditioned in a first step, to finally perform the regression in a second step. In other words, the result is a subset of basis functions that is better conditioned and, as a result, the numerical problem in the regression is avoided. Nevertheless, the instability of the regression operator is the same, considering that the regression equation (5) remains intact. Alternatively, relaxing the exigence of the minimization function rather than the numerical condition of RX, known as regularization, consist in imposing some kind of restriction to the cost function to achieve a less abrupt region in the minimization function, although deviating from the minimal error solution. That is, the reduction of the numerical problems is achieved by admitting a certain bias. Regularization techniques are also known as shrinkage methods, since they can be formulated as a constriction applied to the coefficients in some specific cases. In general, the selection of the penalty function is subject to a prior belief that the new penalized cost function presents a smoother behavior [35]. In the case of study, the prior conviction is the existence of columns of RXvery closed to 0, what among other things is related to the poor variance of the related basis functions. This leads to opt between the most common shrinkage methods, that are Ridge regression —or ℓ2-norm regularization— and the least absolute shrinkage and selection operator (LASSO) —or ℓ1- norm regularization— [24]. LASSO regularization restricts the amount of components, pruning the coefficients smaller than a threshold that is related with their variance. On the other hand, Ridge regularization penalizes the magnitude of the coefficients, what is translated in the introduction of a bias that mostly overshadows the insignificant coefficients. Despite the similarities between ℓ2- and ℓ1-norm constrains, the latter has not an algebraic solution. The penalized residual sum of squares (PRSS) with the LASSO application to the modeling problem can be described as follows PRSSLASSO =1 2∥y−Xh∥2 2+λ 2∥h∥1,(6) where, as mentioned, an ℓ1-norm penalty is added to the LS equation weighted by the regularization coefficient λ. In order to facilitate next operations, a 1/2 factor has been added. 124684 VOLUME 12, 2024 E. Marqués-Valderrama et al.: Implementation of ADMMs for Solving PA Linearization Problem III. THE ALTERNATING DIRECTION METHOD OF MULTIPLIERS (ADMM) The alternating direction method of multipliers can be applied to a problem with this structure: min x,y(f(x)+g(y)),s.t.Ax +By =b,(7) where now f(x) and g(y) are two arbitrary functions of two arbitrary variables and A,Band bhave to be determined. In order to find a solution, the augmented Lagrangian function is defined as Lρ(x,y,d)=f(x)+g(y)+dT(Ax +By −b) +ρ 2∥Ax +By −b∥2 2,(8) where dTis the vector of Lagrange multipliers and ρ > 0 is the augmented parameter. This augmentation helps dealing with possible non-convexity of f(x) and g(y) and improves the convergence speed [29]. The restriction can be grouped in r=Ax +By −bto apply the following algebraic transformation [32]: dTr+ρ 2∥r∥2 2=ρ 2∥r+s∥2 2−ρ 2∥s∥2 2,(9) where s=d/ρ is the new augmented vector of Lagrange multipliers. The augmented Lagrangian function is rewritten as Lρ(x,y,s)=f(x)+g(y)+ρ 2∥Ax+By −b∥2 2+s−ρ 2∥s∥2 2. (10) Addressing the minimization problem of (7) with the formulation of the augmented Lagrangian function consists in solving iteratively x(k+1),y(k+1)=argmin x,y Lρ(x,y,s(k)),(11a) s(k+1) =s(k)+Ax(k+1) +By(k+1) −b. (11b) As mentioned in [36], in a minimization problem with two variables coupled for the restrictions, the actualization of x(k+1) and y(k+1) is separable, what enables the splitting of (11a) into x(k+1) =argmin x Lρ(x,y(k),s(k)),(12a) y(k+1) =argmin y Lρ(x(k+1),y,s(k)),(12b) s(k+1) =s(k)+Ax(k+1) +By(k+1) −b.(12c) This structure is what gives the method its name, since in (12a) the expression is minimized in xdirection and the method alternates to ydirection in the (12b) step. Finally, sis updated in (12c). A. ADMM AND LASSO The PRSS equation in (6) can be adapted to an ADMM problem duplicating the coefficients vector [32] min u,h 1 2∥y−Xu∥2 2+λ 2∥h∥1s.t.u−h=0,(13) where A=I,B= −Iand b=0ensure that the problem is equivalent to the LASSO minimization. It must be emphasized how now f(u)= ∥y−Xu∥2 2and g(h)= λ∥h∥1. That is, the PRSS function is not only interpreted as a pre-penalized function, but also the restriction has been included as a second function, splitting the PRSS in two different functions for leading the roll of restriction to the u−h=0imposition. Its augmented Lagrangian function can be expressed as Lρ(u,h,s)=1 2∥y−Xu∥2 2+λ 2∥h∥1 +ρ 2∥u−h+s∥2 2−ρ 2∥s∥2 2,(14) yielding a feasible minimization problem that can be solved as indicated in (12a),(12b) and (12c). IV. PROPOSAL OF A CENTRALIZED ADMM SOLUTION FOR PA LINEARIZATION Until now, the theoretical development has focused on the adaptation of LASSO regularization to ADMM in order to provide an iterative solution. In this way, an ℓ1-norm regularization can be applied to the PA modeling problem. From (13), it can be seen that the minimization variable is duplicated, that is, the solution is redundant and, in the last iteration, uand hhave to be equal. This fact gives freedom of choice when dealing with functions f(u) and g(h). Thus, (13) can be reformulated as a centralized optimization problem [29], by sectioning f(u) into a set of {1,· · · ,M} subproblems min {ui},h M X i=1 fi(ui)s.t.ui=h,∀i∈{1,· · · ,M},(15) where the superscript irepresents the i-th agent or computational module that only has access to the i-th function and the i-th variable. In the case of study, the set of Mfunctions is identical, that is, fi=RSS,∀iand uiare Minitial replicas of u. Thus, adding the LASSO penalization, the structure presented in (13) is reached min {ui},h 1 2 M X i=1 ∥yi−Xiui∥2 2+λ 2∥h∥1 s.t.ui−h=0,∀i∈ {1,...,M},(16) where the subscript irepresents the i-th segment of yand X. This new problem is known as row-wise division [32] or centralized ADMM [29] and rests in some considerations. The first one is the fact that the subfunctions fiare as similar as they can be considered so that the consensual solution his valid for the whole set. The second one is that the amount VOLUME 12, 2024 124685 E. Marqués-Valderrama et al.: Implementation of ADMMs for Solving PA Linearization Problem of information available in each agent is enough, being able to reach uiintermediate solutions that are not too disparate. Then, the augmented Lagrangian for the centralized ADMM can be formulated as Lρ(u1,· · · ,uM,h,s1,· · · ,sM) =1 2 M X i=1 ∥yi−Xiui∥2 2+λ 2∥h∥1+ M X i=1 ρ 2∥ui−h∥2 2 +si−ρ 2 M X i=1 ∥si∥2 2.(17) Finally, the minimization of the augmented Lagrangian is performed, yielding the splitted iterative solution of the centralized ADMM ui,(k+1) =XH iXi+ρI−1XH iyi+ρh(k)−si,(k), (18a) h(k+1) =Sλ Mρ¯ u(k+1) +¯ s(k),(18b) si,(k+1) =si,(k)+ui,(k+1) −h(k+1),(18c) where the mean operation is represented by ¯ uand ¯ sin equation (18b), and the ℓ1-norm regularization is carried out by the soft-thresholding operator Sκ(a), defined as: Sκ(a)=(a−κsign(a),|a|> κ 0,|a|≤κ. (19) FIGURE 1. Centralized ADMM topology for solving the problem posed. The details for obtaining this solution are shown in Appendix A, and an schematic representation of the proposed procedure is shown in Fig. 1. There are some aspects than can be interpreted in the achieved solution. Firstly, comparing (18a) with the classical LS solution (5), it can be seen that the equations are very similar but a modification in the projection operation can be observed. The ρIterm introduced into the inverse operation of (18a) recalls the Ridge regularization solution [18]. This is due to the ℓ2-norm form of the unification restriction, that is, the term (ρ/2)∥ui−h+si∥2 2. This has a relevant interpretation, because each agent only has a small portion of the data and the introduction of this bias effect avoids overfitting in the solution of each subproblem. It should be highlighted that this Ridge regularization is intrinsic to the proposed ADMM formulation and benefits its robustness in terms of numerical instabilities. Then, minimization was performed in hdirection, yielding the (18b) step. As the restriction at the ℓ1norm only affects the hvariable, the minimization results in a first step of consensus (performed by the mean operation) to satisfy the ui−h=0 imposition and afterwards, the minimization of ∥h∥1by the application of the soft threshold operator. This consensed solution for all agents ¯ uis previously modified by the mean of the vectors si,(k), which can be interpreted not only as a Lagrange multiplier vector update, but also as an accumulated consensus error. Thus, the more similar u is to h, the smaller the increase in the accumulated error s. Finally, the loop is closed again with (18a), where the second term ρh(k)−si,(k)can be seen as a correction to the regression solution ui,(k+1) from the inclusion of the accumulated error in the ui,(k)calculation. This helps the agent to perform solution corrections progressively and favors a higher convergence speed. In summary, the proposed adaptation of the PRSS minimization in (6) to an ADMM-based procedure not only involves the ℓ1-norm regularization due to LASSO, but also incorporates effects of ℓ2-norm regularization in order to satisfy the ADMM intrinsic constraints. With the combination of these two techniques, the method is expected to present enhanced robustness against numerical problems. A. ADAPTATION TO CLOSED-LOOP DPD DESIGN The previous explanation about the proposed ADMM solution was focused on the nonlinear modeling problem presented in (1), since the regression problem to be solved is more intuitive in this context and relevant aspects are easier to interpret. However, the practical scenario for the application of the proposed method is the design of DPDs for PA linearization. Now, the notation has to be changed since, in the context of closed-loop digital predistortion solutions such as the DLA scheme [13], it is common to employ x for referring to the PA input and uto denote the original generated signal, as shown in Fig. 2. Hence, x=u−d represents the predistorted signal. This signal is essentially the original signal, u, modified by the distortion signal, d. The distortion signal accounts for the difference between the actual output of the PA, y, and the desired linear output, g0u, which is a scaled version of the original signal based on the objective gain, g0. Additionally, the coefficient vector his now denoted by wdue to its weight connotation in the machine learning architectures. The DLA DPD scheme aims at iteratively finding the distortion signal dn=Uwn so that the error e=y/g0−uis minimized. For that purpose, a ‘‘DPD Adaptation’’ block iteratively actualizes the coefficients wn+1=wn+µdwn. In the conventional DLA 124686 VOLUME 12, 2024 E. Marqués-Valderrama et al.: Implementation of ADMMs for Solving PA Linearization Problem DPD, the actualization is performed with the Moore-Penrose pseudoinverse associated to the classical LS solution, that is, dwn=U†e. Although now, under this structure, the origin of the regression is not so evident, the pseudoinverse operation implicit in the DPD Adaptation block can be replaced by the proposed centralized ADMM implementation, as illustrated in Fig. 2. Notice that also the notation in the ADMM procedure has been changed accordingly. Thus, dw is the unified solution of all the agents, where each one calculates an individual solution ti. FIGURE 2. ADMM implementation for direct learning architecture scheme of digital predistortion. B. COMPUTATIONAL REDUCTION In this subsection, a computational complexity comparison is performed in order to judge the benefit of this modification in the regression equation. Furthermore, the assessment is made mainly between the classical LS solution of (5) and the first step of the proposed ADMM solution (18a), since that is the point where the regression is performed. As previously mentioned, ADMM computes the solution dividing the whole problem in chunks and each agent computes a part of the solution. This partitioning provokes that each agent only works with a reduced set of the data, having to process less information. This contributes to reducing the implementation cost, specifically decreasing multiply-accumulate operations [26]. On the other hand, if the problem were to be tackled through a centralized computational paradigm—wherein all the data is evaluated by a single device—the amount of information needing processing would be significantly larger. In this way, the proposal of replacing the least squares regression with a rowwise division ADMM enables a parallel computation of the measurement matrix, that is, each agent processes at the same time a matrix of dimensional order (Nsa/M)×Nr. The assessment incorporates the Bachman-Landau notation for analysis, as referenced in [37]. Notice that only the multiplications are counted, because they have more impact on the overall complexity, and that the terms with order less than two are neglected. Then, computing the inverse operation as indicated in [38], the following approximations can be reached: OLS =2N2 rNsa +2 3N3 r+2NrNsa +5 2N2 r,(20) OADMM =Niter 2N2 rNsa M+2 3N3 r+2NrNsa M+5 2N2 r, (21) where Orepresents the total number of multiplications. It can be seen that, the greater the number of agents, the lower the computation that each one has to perform. An evaluation of (20) and (21) is represented in Fig. 3for the values of Nsa = 368640 and Nr=205 corresponding to the second scenario of Section V-B, where it can be seen how incrementing the number of parallel agents Mdecreases the computational cost of each agent and the total number of operations increases proportionally with Niter . Therefore, this additional degree of freedom in the distribution of the solution is not only in favor of achieving parallel computing but also enables the implementation in more computationally constrained devices. Moreover, since Nsa ≫Nr, a partitioning on sample dimension can reduce significantly the device exigences, not only in terms of computation cost but also in memory storage capabilities. Thus, it is also possible to implement a modeling or linearization regression problem in a single device with poor specifications, simply by taking the role of each agent sequentially. FIGURE 3. Theoretical computational cost of performing the regression against the number of agents for Nsa =368640 and Nr=205. The dash-dotted line represent the number of operations consumed by the LS method and the lowest solid curve with markers is the computational cost of one iteration of ADMM. V. EXPERIMENTAL RESULTS In this section, ADMM technique will be employed in three different experimental scenarios in order to illustrate its benefits and provide a better understanding of its implications. VOLUME 12, 2024 124687 E. Marqués-Valderrama et al.: Implementation of ADMMs for Solving PA Linearization Problem A. MODELING PROCEDURE To demonstrate how the proposed technique works, several aspects need consideration. Firstly, choosing the parameters λand ρis typically arbitrary and has various consequences. The higher the chosen value of λ, the greater the expected thresholding; likewise, a higher value of ρis expected to increase the convergence speed while reducing the threshold. The coupled behavior of these two variables makes the initialization difficult, and constitutes an added problem to the model structure identification. In order to find the adequate basis functions and ADMM parameters, the modeling stage was performed as follows. Initially, an iterative learning control (ILC) [39] experiment was carried out to obtain the optimal predistorted signal, and thus, the input to output relationship of the ideal DPD block, as conducted in [40]. This approach can be used to directly estimate the DPD block, as well as to provide a preliminary assessment of the feasibility of the regressors in the DLA DPD. Additionally, it serves as a preliminary step for calculating the optimal λ and ρparameters in both cases. The generalized memory polinomial (GMP) model [41]: xGMP[n]=X k∈KaX l∈La aklu[n−l]|u[n−l]|k +X k∈KbX l∈LbX m∈Mb bklmu[n−l]|u[n−l−m]|k +X k∈KcX l∈LcX m∈Mc cklmu[n−l]|u[n−l+m]|k, (22) is taken to shape the structure of the regressors matrix, and in this stage, the order sets Ka,Kband Kc, the memory depth sets La,Lband Lcand the memory shift sets Mb and Mchave to be selected. The initial configuration sets are empirically selected by varying nonlinearity order and memory depth until a model with adequate DPD performance in terms of standard linearization metrics, that will be explained below, is found using the conventional LS method, as in [42]. Both modeling and linearization involve two stages. The first one is the training or identification step, which consists in calculating the coefficients with a probe signal. Then, the robustness of the model is validated by applying new input signals with the same OFDM format but generated with different random symbols. B. PROOF OF CONCEPT WITH 30 MHz BANDWIDTH The experimental setup in this first stage comprised a Rohde & Schwarz SMU200A signal generator, followed by two cascaded Minicircuits TVA-4W-422A+preamplifiers working in a linear zone. The output signal was measured using a Keysight Technologies PXA-N9030A vector signal analyzer (VSA), with a power supply driving the PA. In order to protect the VSA, the output signal was fed into the measurement equipment through the coupled port of a directional coupler, terminating its output port with an attenuator followed by a matched load. The PA under test was a SKY66394-11 from Skyworks with wide instantaneous bandwidth, operating at a frequency of 2.14 GHz. The probe signal was an OFDM following the 5G-NR waveform format with a bandwidth of 30 MHz and a numerology µ=1, corresponding to a subcarrier spacing of 30 kHz. The OFDM signal had a PAPR of approximately 11 dB and was generated with a sampling rate of 92.16 MHz. Subsequently, the signal analyzer acquired samples of the complex envelope at the output of the PA. Each measurement was repeated 10 times, time-aligned with the input signal, and the average between the repetitions was considered to improve the resulting signal- to-noise ratio. The entire setup is presented in Fig. 4. FIGURE 4. Laboratory setup employed to perform the experiments. The workstation that holds the algorithms was equipped with 64 GB of RAM and an Intel i7 processor with 8 CPU cores. For the computation of the ADMM, 64 computational agents were deployed, distributing the computational load across 8 CPU cores in parallel. Consequently, the ADMM algorithm was executed in a sequence of 8 turns, with 8 agents operating simultaneously. The limitations of the available hardware forces to opt for this implementation that, although is not a purely distributed scheme, serves to present a proof of concept. Notice that this fact only affects to the computational cost assessment, remaining the rest of the performance intact. In a first experiment, the operation level of the PA was characterized by a gain compression of 1.5 dB. The behavior of a class AB amplifier with moderate compression does not require a DPD with many regressors for compensation, since it is also a nonlinear system that does not need a high nonlinear order or large memory depth. Therefore, to model the ideal DPD for this PA under these conditions, a GMP model with only the 41 regressors indicated in Model A Table 1was enough. The conventional LS method achieved an identification normalized mean squared error (NMSE) of −44.8 dB and a validation NMSE of −44.2 dB, and ADMM reached an NMSE of −44.7 dB in identification and −44.2 dB in validation after tuning λ=1.28 ·10−8and ρ=3.45 ·10−9. In the regression for this case, the numerical problem is non-existent and consequently the LASSO term of 124688 VOLUME 12, 2024 E. Marqués-Valderrama et al.: Implementation of ADMMs for Solving PA Linearization Problem TABLE 1. GMP model configurations employed in the three experimental scenarios. ADMM implementation is small enough and little influential to consider the two solutions very close. Then, a DLA DPD experiment was performed to assess the capability of both approaches to achieve linearization. The DPD training was executed for 10 iterations, with a learning rate µ=1/3. Afterwards, the coefficients were fixed and the robustness of the identified DPD was tested over 3 more iterations by changing the random symbols, that is, generating a new validation signal for each iteration. This way, Fig. 5shows the evolution of the linearization metrics for both the training and robustness stages. See [43] for more information about the linearization figures of merit employed in the general practice of DPD design, namely, the linearization NMSE, the adjacent channel power ratio (ACPR) and the error vector magnitude (EVM). Specifically, in Fig. 5(a) the linearizarion NMSE is computed by comparing the scaled output against the input on each iteration. On the other hand, in Fig. 5(b) the evolution of the ACPR is illustrated. This metric is an indicator of the mitigation of the spectral regrowth in the adjacent channels. It can be seen that, although the DLA ADMM approach starts the progression with slightly worse values, once training has been finished, it reaches a similar performance to the classical LS. The DLA convergence is evidenced by the decrease of these metrics until reaching a steady state in both figures. The reduction of the out-of-band emissions is also endorsed by the observation of the power spectral density (PSD) in Fig. 6. It can be seen how the output for the optimal predistorted signal obtained by ILC technique produces the highest possible reduction of the spectral regrowth. It is worth noticing that the designed Volterra-base predistorter achieves almost the same performance, whether conventional LS or ADMM is executed. In Table 2the validation metrics are collected, where the EVM is also stored. Even though the performance limit established by the optimal predistorted signal provided by ILC technique is able to avoid virtually all distortion effects, the linearization achieved by both DLADPD approaches is satisfactorily below the typical standards requirements for out-of-band emissions. And what is more important, it can be verified that the distributed ADMM technique can achieve the same performance than the data centralized LS approach. Finally, as mentioned above, in this first scenario it was expected, not only that the accuracy would be comparable, but that the coefficients achieved by the solution of both TABLE 2. Linearization performance when the PA is working with a gain compression of 1.5 dB. methods would be practically equal. In order to confirm this hypothesis, the magnitude and phase of the coefficients wobtained with each technique are shown in Fig. 7. This observation confirms that partitioning the problem in statistically representative chunks and reaching a consensus is a valid solution for the whole problem. Regarding computational cost, recall Fig. 3showing the number of operations associated to the LS and the ADMM approaches versus the number of agents. For this scenario, it can be appreciated that the total amount of operations performed by each agent is only the 31.3 % of the required by data centralized LS. Moreover, the most demanding operation, that is the pseudoinverse performed for each of the 64 agents at each iteration, presents the 1.6 % of the computational cost involved in performing the pseudoinverse of the whole matrix. This means that the LS estimation has been successfully adapted to a distributed architecture, enabling a deployment into resource-constrained devices. For the sake of a better comprehension of the internal consequences of the ADMM integration in the DLA architecture, a second scenario is studied. The operation level in this case produced a gain compression of 3.1 dB, that requires a DPD needing a higher nonlinear order and greater memory depth, and therefore, a greater number of regressors. In this way, a GMP structure of 205 regressors was selected with the configuration indicated in Model B of Table 1, but it was not possible to identify a model by the LS method, due to numerical problems. Then, an ADMM was tuned with λ=6.0·10−7and ρ=7.0·10−8, achieving an identification NMSE of −40.2 dB and a validation NMSE of −39.9 dB. In this kind of scenario, the advantages of ADMM regression are emphasized due to the convergence of the solution it provides, in contrast to the LS method. Once the parameters have been found, a DLA DPD was designed in the same way as for the first scenario. VOLUME 12, 2024 124689 E. Marqués-Valderrama et al.: Implementation of ADMMs for Solving PA Linearization Problem FIGURE 5. Linearization performance achieved by the LS and ADMM approaches in a DLA scheme for the scenario with a gain compression of 1.5 dB: (a) NMSE, and (b) ACPR. The training and robustness stages are separated by the dashed line. FIGURE 6. Spectrum of the output signal, with and without DPD, for a gain compression of 1.5 dB. As observed in Fig. 8, the spectral regrowth was mitigated up to very close values to the ideal limit imposed by ILC linearization. Considering that this second scenario is more complicated, DLA requires more iterations to achieve convergence and, therefore, a slower evolution of the linearization NMSE and ACPR is observed in Fig. 9 compared with the previous case. The validation results contained in Table 3demonstrate that the proposed ADMM FIGURE 7. Weight vector solution after the DLA loop achieved with LS and ADMM. Differences between both cases are minimal. TABLE 3. Linearization performance when the PA is working with a gain compression of 3.1 dB. implementation is suitable for dealing with the intrinsic numerical instabilities in those cases where a GMP model with a large number of regressors is necessary. FIGURE 8. Spectrum of the output signal, with and without DPD, for a gain compression of 3.1 dB. The integration of the proposed ADMM approach in the adaptation DLA block seems to bring some benefits in terms of stability of the whole DLA scheme. As explained in Section IV-A with the help of Fig. 2, ADMM intends to substitute the Moore-Penrose pseudoinverse operation associated to the LS method with an adaptive estimation of the coefficients that produce dwn. This way, the LASSO term introduced in the algorithm has the ability to modify 124690 VOLUME 12, 2024