A Reduced-Complexity Direct Learning Architecture for Digital Predistortion Through Iterative Pseudoinverse Calculation

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by IEEE: J. A. Becerra, A. Pérez-Hérnandez, M. J. Madero-Ayora and C. Crespo- Cadenas, "A Reduced-Complexity Direct Learning Architecture for Digital Predistortion Through Iterative Pseudoinverse Calculation," in IEEE Microwave and Wireless Components Letters, vol. 31, no. 8, pp. 933-936, Aug. 2021, doi: 10.1109/LMWC.2021.3079839. “© 2021 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.” IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. XX, NO. X, MONTH YYYY 1 A Reduced-Complexity Direct Learning Architecture for Digital Predistortion Through Iterative Pseudoinverse Calculation J. A. Becerra, Senior Member, IEEE, Abraham P´ erez-H´ ernandez, Member, IEEE, M. J. Madero-Ayora, Senior Member, IEEE, and C. Crespo-Cadenas, Senior Member, IEEE Abstract—In this paper, a novel approach is proposed for digital predistortion (DPD) with direct learning architecture (DLA). Regression of a Volterra behavioral model requires the pseudoinverse of a matrix, which needs many resources due to the inverse operation when the Moore-Penrose pseudoinverse is used. This work substitutes the pseudoinverse calculation by a polynomial expansion (PE) method to obtain a PE-DLA, which attains a pseudoinverse in an iterative fashion avoiding the inverse operation and consequently reducing the algorithm computational complexity. Experimental results show that the number of iterations in the PE-DLA affects the convergence speed. The proposal is benchmarked against other state-of-the-art approaches such as the classic DLA and the covariance matrix DLA (CM-DLA) in the DPD of a commercial class AB power amplifier, concluding that the linearization performance of the current proposal is equivalent to others while featuring simple operations. Index Terms—digital predistortion, direct learning, power amplifier, Volterra series. I. INTRODUCTION MODERN communication standards set high efficiency and linearity requirements for power amplifiers (PAs). Digital predistortion (DPD) techniques arise as a solution to linearize the PA. Since direct learning architecture (DLA) was first reported as an adaptive nonlinear predistorter architecture [1], it has shown better linearization performance [2], [3], a greater robustness in presence of additive white Gaussian noise (AWGN) [4] and a lower impact of feedback noise [5] compared to indirect learning architecture (ILA). Several low computational complexity techniques have been proposed over DLA, mainly by means of the substitution of the inverse calculation by an alternative method. Considering the input signal is a stationary random process, the time-varying correlation matrix was substituted by the covariance matrix (CM-DLA) [6], [7]. In [8] a negative feedback structure (NFS) was presented, which was enhanced with model coefficient selection in [9]. Adaptive principal component analysis (APCA) Manuscript received December 23, 2020; revised February 21, 2021; March 23, 2021 and April 20, 2021. This work was supported by the Ministerio de Ciencia e Innovaci´ on - Agencia Estatal de Investigaci´ on of the Government of Spain under Grant TEC2017-82807-P, and by the European Regional Development Fund (ERDF) of the European Commission. The authors are with the Departamento de Teor´ ıa de la Se˜ nal y Comunicaciones, Escuela T´ ecnica Superior de Ingenier´ ıa, Universidad de Sevilla, Sevilla, 41092, Spain (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Forward Path Feedback Path DPD x=u−Uw PA DPD Adaptation wi+1 =wi+µU†e U† k+1 =U† k2I−U† k·U 1 g0 − u x y e=y g0−u Fig. 1. Block diagram of the proposed polynomial expansion direct learning architecture (PE-DLA). was proposed in [10] to take advantage of performing the regression in an orthogonal basis. The use of the Moore-Penrose solution to the pseudoinverse operation in DPD architectures is known to bring numerical issues due to the high correlation [11] and the dimensional nonuniformity of the basis waveforms. Techniques such as coefficient selection and orthogonal transformations reduce the condition number of the matrix alleviating this issue. Nevertheless, the inverse operation remains as a step with a high computational complexity associated. This work introduces in DLA an iterative formula for computing the pseudoinverse called polynomial expansion (PE) method [12], [13], also found in literature as Newton iteration [14]–[16]. Newton iteration rapidly refines a crude initial approximation to the inverse or the Moore-Penrose generalized inverse of a structured matrix [17]. It is strongly stable numerically and is reduced essentially to performing matrix multiplication twice per step, in an iterative way. This level-three basic linear algebra subprograms (BLAS) is particularly effective on parallel computing. The remainder of this letter is organized as follows. First, Section II reviews the DLA framework followed by its modifications to introduce the PE-DLA in Section III. A computational complexity assessment is provided in Section IV. Section V shows experimental results proving the advantages of the proposal. Finally, conclusions are drawn in Section VI. Notation:(·)T,(·)H,(·)−1,(·)†,(·)+,k·k2denote transpose, conjugate transpose, inverse, pseudoinverse, Moore-Penrose pseudoinverse and Euclidean norm, respectively. OMdenotes the computational complexity of operation M. IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. XX, NO. X, MONTH YYYY 2 II. FRAMEWORK This section describes the general framework in a DLA DPD system. The PA input signal xis defined as the sum x=u−d,(1) where x[n]=[x[n], x[n−1],· · · , x[n−M+ 1]]T,u[n] = [u[n], u[n−1],· · · , u[n−M+ 1]]Tand d[n] = [d[n], d[n− 1],· · · , d[n−M+1]]Tare the predistorted, input and distortion signals, respectively, nis the index of samples and Mis the number of sample points. Dependence on the index of samples is dropped for notation simplicity. The distortion signal is generated by the DPD following d=Uwi,(2) where U= [ϕ1,· · · ,ϕN]is the matrix containing the N model basis functions ϕand wicontains the DPD coefficients at iteration i. The model basis functions are dependent on the model structure and generally consist in multiplications of some delayed versions of the input signal and its conjugate. Examples of these structures are the memory polynomial (MP) [18] and the generalized memory polynomial (GMP) [19]. Order reduction and standarization techniques can be applied to Uto enhance matrix conditioning. The DPD coefficients are updated in an iterative fashion wi+1 =wi+µU†e,(3) where µ∈(0,1) is the weighting factor, U†is a pseudoinverse matrix of Uand eis the error vector defined as e=y g0 −u,(4) where g0is the desired linear gain of the PA that, in general, takes the value of g0=kyk2/kuk2when no distortion signal is applied. III. POLYNOMIAL EXPANSION FOR MATRIX INVERSION A well-known procedure to obtain the pseudoinverse of a matrix Uis the Moore-Penrose pseudoinverse [13] U+= (UHU)−1UH.(5) Although this formation allows to obtain the pseudoinverse in one step, it includes a matrix inversion, that is characterized by having a high computational complexity. The polynomial expansion (PE) is a recursive method to calculate the pseudoinverse U†[12], [13], which states that the recursion U† k+1 =2I−U† kUU† k,(6) where Iis the identity matrix of size N, converges quadratically to U†as k→ ∞. Note that the term 2I−U† kUtends to the identity matrix and lim k→∞ U† kU=I.(7) The PE-DLA is depicted in Fig. 1. 2 4 6 8 10 12 14 16 18 20 −50 −45 −40 µ= 1/2 µ= 1/4 Iteration Worst ACPR (dBc) µ= 1/4, kmax = 5 µ= 1/2, kmax = 5 µ= 1/4, kmax = 10 µ= 1/2, kmax = 10 µ= 1/4, kmax = 15 µ= 1/2, kmax = 15 µ= 1/4, kmax = 20 µ= 1/2, kmax = 20 (a) 2 4 6 8 10 12 14 16 18 20 −50 −45 −40 Iteration Worst ACPR (dBc) µ= 1/2 µ= 1/3 µ= 1/4 (b) Fig. 2. Worst linearization ACPR of a 30-MHz 5G-NR signal and output power of 31 dBm for (a) the PE-DLA, with µ= 1/2and µ= 1/4in a sweep of number of PE iterations kmax and (b) the DLA, in a sweep of the weighting factor. A. Initialization Ben-Israel [12] and Ben-Israel and Cohen [20] proved that the iteration converges when the initial value is taken as U† 0=αUH(8) for a sufficiently small positive scalar αwith 0≤α≤2 λ1(UHU),(9) where λ1(·)is the maximum eigenvalue of its matricial argument. After a signal change, the initial value U† 0can be taken from the last iteration of the previous signal to speed up convergence. B. Number of Iterations The number of iterations in khas an effect on the DLA convergence speed, that is equivalent to a weighting factor change. IV. COMPUTATIONAL COMPLEXITY ASSESSMENT The computational complexity of the benchmarked approaches are analyzed next. For this study, it has been considered that the multiplication of one n×mmatrix and one m×pmatrix has a computational complexity of O=nmp and the inversion of a matrix of size nresults O=n3. Let NiterDLA be the number of DPD iterations and kmax the number IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. XX, NO. X, MONTH YYYY 3 Iteration Worst ACPR (dBc) 0 2 4 6 8 10 12 14 16 18 20 −55 −50 −45 −40 DLA CM-DLA PE-DLA kmax = 10 Fig. 3. Worst adjacent channel power ratio (ACPR) of the algorithms in comparison. Identification with a 30-MHz 5G-NR signal with µ= 1/2. of PE iterations. Then, the computational complexity of the architectures in comparison follow ODLA =N3+ 2N2MNiterDLA, OCM-DLA =N3+N2M+N2MNiterDLA, OPE-DLA =2N2MkmaxNiterDLA. (10) The current proposal shows a computational complexity and running time that are equivalent to others and depends on the number of iterations while not using any matrix inverse operation. Regarding running time, it is dominated by kmax and identification size as well. The fact that the new technique requires a smaller subset of operations and avoids matrix inversion, i.e., only requires matrix sums and multiplications, is beneficial for working with low level operations and parallelization. V. EXPERIMENTAL DESIGN AND RESULTS To confirm the validity of the proposed framework, measurement experiments were performed. The testbench was composed by a SMU200A vector signal generator (VSG) from Rohde & Schwarz, a PXA-N9030A vector signal analyzer (VSA) from Keysight Technologies and a DC power supply. The device under test (DUT) was the cascade of two Mini-Circuits TVA-4W-422A+ preamplifiers and the Cree’s CGH40010F operating at a central carrier frequency of 3.6 GHz. The output power was set to +31 dBm characterized by 2.7 dB of gain compression. Two experiments with orthogonal frequency division multiplexing (OFDM) signals were performed: the illustration of the proposal convergence and a benchmark between several architectures. The model under test was a 13th order GMP with memory fading from 15 to memoryless in the MP part and 7th order with maximum memory depth of 1 in the nondiagonal parts. The convergence of the proposal was proven with a 5G New Radio (5G-NR) signal with 30 MHz bandwidth —acquired with oversampling of 3— and 10.5 dB of peak-to-average power ratio (PAPR). The DLA was executed for a set of weighting factors and the PE-DLA was run for µ= 1/2,1/4 and a sweep of number of PE iterations. Results of the worst adjacent channel power ratio (ACPR), i.e., the maximum ACPR of the two available adjacent channels, are shown in Fig. 2, where it can be observed that the number of PE 3560 3570 3580 3590 3600 3610 3620 3630 3640 −60 −50 −40 −30 −20 −10 0 case ACPR -1/+1 no dpd -37.2/-36.7 DLA -53.0/-52.5 CM-DLA -53.1/-52.6 PE-DLA -53.9/-53.3 Frequency (MHz) Normalized PSD (dB/Hz) no dpd DLA CM-DLA PE-DLA (a) 3560 3570 3580 3590 3600 3610 3620 3630 3640 −60 −50 −40 −30 −20 −10 0 case ACPR -1/+1 no dpd -37.0/-36.6 DLA -57.7/-57.2 CM-DLA -58.3/-57.8 PE-DLA -60.2/-59.6 Frequency (MHz) Normalized PSD (dB/Hz) no dpd DLA CM-DLA PE-DLA (b) Fig. 4. Power spectral densities (PSDs) of the PA without DPD and with DPD attained through the techniques in comparison validating with (a) 30-MHz and (b) 20-MHz signals. iterations affects the speed of convergence of the algorithm in a similar way that the weighting factor does. Note that Fig. 2b does not have dependence on kmax since it is set as a reference for comparison with the classical DLA approach. A benchmark between algorithms that include a regression in the learning process was performed. The DLA, CM-DLA and PE-DLA techniques were executed with a step size of µ= 1/2and kmax = 10 for PE-DLA. In order to highlight the capabilities of the algorithms in comparison, DPD adaptation was enabled for 20 iterations that were executed over a 30 MHz bandwidth without changing the probing signal. Once the algorithms converged, the DPDs were fixed and validated with different signal bandwidths of 30 and 20 MHz. The ACPR against the iteration number is shown in Fig. 3, that reveals the similarity of the evolution in the three techniques. Fig. 4 represents the normalized power spectral densities (PSDs) of the validation stage and highlights the proposal ACPR reduction outperformance. VI. 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