On the use of digital predistortion for power line communications

On the Use of Digital Predistortion for Power Line Communications Pere L. Gilabert Dept. of Signal Theory and Communications, Universitat Politècnica de Catalunya. [email protected] Abstract This contribution pretends to highlight the benefits of including digital predistortion (DPD) linearization in the digital front-end of Power Line Communications (PLC). Introduction PLC networks encounter major challenges due to noise, attenuation, and interference from electrical appliances. A key approach to enhancing PLC performance is power amplification, which strengthens signal robustness against these disruptions. However, power amplifiers (PAs) must balance efficiency and linearity, as achieving both simultaneously is difficult. Typically, PA design prioritizes maximizing power efficiency at high back-off levels to accommodate the high peak-to-average power ratio (PAPR) of OFDM signals, while linearity is managed at the system level using linearization techniques. In addition to filtering and noise cancellation methods for interference mitigation, digital predistortion linearization can compensate for distortion introduced by PAs and address impedance mismatches that significantly degrade signal quality in PLC systems. Theoretical Background The principles of predistortion are depicted Fig. 1. A nonlinear system called a predistorter is placed before the PA to compensate for its nonlinear characteristics. Therefore, the predistorter's goal is to ideally replicate the inverse nonlinear behavior of the PA, thereby achieving linear amplification at the PA's output. Fig. 1. Principles of predistortion linearization of power amplifiers. DPD is implemented within the baseband processing of the digital front-end. To characterize and compensate for the nonlinear distortion and memory effects of PAs, mathematical descriptors are essential. Unlike physical models, which require detailed knowledge of the PA’s electronic components, their constitutive relations, and the theoretical principles governing their interactions, PA and DPD behavioral models are derived solely from input-output observations. As a result, their accuracy is highly dependent on the chosen model structure and parameter extraction method [Gil25]. Polynomial-based models, including simplified versions of the Volterra series such as the memory polynomial (MP) [Kim01] and the generalized memory polynomial (GMP) [Mor06], are among the most widely used behavioral models in the literature. Additionally, piecewise behavioral models like the decomposed vector rotation (DVR) model [Zhu15] and look-up table (LUT) implementations [Mol07], [Gil18] have been proposed to address stronger nonlinear PA behavior by leveraging the locality of piecewise functions. While these models are nonlinear, they remain linear in their parameters, allowing for efficient extraction using the least squares (LS) method. More recently, artificial neural networks (ANNs) have emerged as a promising approach for PA behavioral modeling and DPD linearization, particularly in the context of wideband B5G modulated signals and highly efficient yet nonlinear PAs [Fis23], [Lop22]. This approach aims to surpass the limitations of conventional single-stage DPD models in terms of linearization performance. However, the improved accuracy of ANNs comes at the cost of a significantly larger number of parameters, raising concerns about power consumption and resource utilization. As an alternative, cascaded (CC) behavioral models and composite architectures have been proposed for DPD applications, drawing inspiration from the multistage structure of ANNs. These models are designed to handle complex nonlinearities and wideband operating conditions with strong memory effects, where traditional single-stage DPD models struggle to meet linearity requirements [Cri25]. Considering the PLC application, where the signal bandwidth remains within tens of MHz, the GMP model offers a balanced trade-off between computational complexity and linearization performance. Additionally, it provides the flexibility needed to compensate for unwanted mismatched impedance effects. The GMP model is an advanced extension of the MP, incorporating bi-dimensional kernels that account for cross-term interactions between the complex signal and both lagging and leading envelope terms. This enhancement improves modeling accuracy but comes at the cost of a higher number of parameters compared to the MP. The low-pass equivalent PA input-output relationship using the GMP behavioral model is expressed as follows, 𝑦𝑦�[𝑛𝑛]=� � 𝑎𝑎𝑘𝑘𝑘𝑘𝑥𝑥[𝑛𝑛−𝑙𝑙]|𝑥𝑥[𝑛𝑛−𝑙𝑙]|𝑘𝑘 𝐿𝐿 𝑎𝑎 −1 𝑘𝑘=0 𝐾𝐾 𝑎𝑎 −1 𝑘𝑘=0 + � � � 𝑏𝑏𝑘𝑘𝑘𝑘𝑘𝑘𝑥𝑥[𝑛𝑛−𝑙𝑙]|𝑥𝑥[𝑛𝑛−𝑙𝑙−𝑚𝑚]|𝑘𝑘 𝑀𝑀𝑏𝑏 𝑘𝑘=1 𝐿𝐿𝑏𝑏−1 𝑘𝑘=0 𝐾𝐾𝑏𝑏 𝑘𝑘=1 + � � � 𝑐𝑐𝑘𝑘𝑘𝑘𝑘𝑘𝑥𝑥[𝑛𝑛−𝑙𝑙]|𝑥𝑥[𝑛𝑛−𝑙𝑙+𝑚𝑚]|𝑘𝑘 𝑀𝑀𝑐𝑐 𝑘𝑘=1 𝐿𝐿𝑐𝑐−1 𝑘𝑘=0 𝐾𝐾𝑐𝑐 𝑘𝑘=1 (1) with 𝑦𝑦�[𝑛𝑛] and 𝑥𝑥[𝑛𝑛] being the estimated PA output and PA input complex baseband signals, respectively. The complex coefficients describing the PA model are 𝑎𝑎𝑘𝑘𝑘𝑘, 𝑏𝑏𝑘𝑘𝑘𝑘𝑘𝑘 and 𝑐𝑐𝑘𝑘𝑘𝑘𝑘𝑘. The nonlinear order of the polynomials at each of the branches is determined by Ka, Kb and Kc, while the memory depth is determined by La, Lb and Lc. Finally, the leading and lagging terms of the envelope defining the cross-products is given by Mb and Mc. The number of coefficients of the GMP model is 𝑂𝑂=𝐾𝐾𝑎𝑎𝐿𝐿𝑎𝑎+𝐾𝐾𝑏𝑏𝐿𝐿𝑏𝑏𝑀𝑀𝑏𝑏+𝐾𝐾𝑐𝑐𝐿𝐿𝑐𝑐𝑀𝑀𝑐𝑐. DPD coefficients adaptation Figure 2 presents the block diagram of the adaptive DPD subsystem, which follows a direct learning closed-loop approach [Gil25]. Fig. 2. Block diagram of a direct learning DPD adaptation architecture. In the forward path, the input-output relationship within the DPD block can be expressed using matrix notation as follows, 𝒙𝒙=𝒖𝒖−𝑼𝑼𝑼𝑼 (2) where u is the Nx1 input vector and N is the number of samples (i.e., n=0,1,···,N-1)), x is the Nx1 predistorted vector and w is the Ox1 vector of coefficients. The NxO data matrix U containing the GMP basis functions described in (1), is defined as 𝑼𝑼=(𝝋𝝋𝑢𝑢[0], 𝝋𝝋𝑢𝑢[1], … , 𝝋𝝋𝑢𝑢[𝑁𝑁−1])𝑇𝑇 (3) where [] u nφ is the Ox1 vector of basis functions [] u j n ϕ , for j = 1,···,O, at time n. 𝝋𝝋𝑢𝑢𝑇𝑇[𝑛𝑛] = (𝜑𝜑1 𝑢𝑢[𝑛𝑛], 𝜑𝜑2 𝑢𝑢[𝑛𝑛], ⋯,𝜑𝜑𝑂𝑂 𝑢𝑢[𝑛𝑛]) (4) These basis functions can be particularized by any nonlinear behavioral model that is linear in parameters. Then, following the direct learning approach depicted in 2, the Ox1 vector of DPD coefficients (w) are extracted iteratively finding the LS solution as follows, 𝑼𝑼𝑖𝑖+1 =𝑼𝑼𝑖𝑖+𝜇𝜇�𝑼𝑼𝑯𝑯𝑼𝑼�−1𝑼𝑼𝑯𝑯𝒆𝒆 (5) with 𝜇𝜇 (0 < 𝜇𝜇< 1) being a weighting factor and e being the Nx1 vector of the residual estimation error, defined as 𝒆𝒆=𝒚𝒚 𝐺𝐺0−𝒖𝒖 (6) where 𝐺𝐺0 determines the desired linear gain of the PA and where y and u are the Nx1 vectors of the PA output and the transmitted input, respectively. Simulation Environment Considering a two-port unmatched network (see Fig. 3), the outputs and the inputs are composed of a vector of two signals. Fig. 3. Block diagram of a general two-port unmatched network. The two outputs, y1 and y2, are independent from each other, but each output is the result of nonlinear combination and interaction of the two inputs, x1 and x2. Therefore, the output y1 can be defined as 𝒚𝒚𝟏𝟏=𝑓𝑓(𝒙𝒙𝟏𝟏,𝒙𝒙𝟐𝟐) (7) A GMP behavioral model, derived from real measurements, was implemented to replicate the PA’s nonlinear distortion and memory effects. Additionally, impedance mismatch was simulated by incorporating reflected waveforms. The evaluation test signal consisted of a 16-QAM OFDM with a 10 MHz bandwidth, while impedance mismatch was emulated using a reflected wave with a voltage standing wave ratio (VSWR) of 3. For DPD purposes the GMP model has been considered. The most relevant basis functions of the GMP DPD model are selected using the doubly orthogonal matching pursuit (DOMP) algorithm described in [Bec18]. Therefore, from an original configuration of 563 basis, with 𝐾𝐾𝑎𝑎= 8, 𝐿𝐿𝑎𝑎=11,𝐾𝐾𝑏𝑏= 5, 𝐿𝐿𝑏𝑏= 8, 𝑀𝑀𝑏𝑏= 5, 𝐾𝐾𝑐𝑐= 5, 𝐿𝐿𝑐𝑐=11 𝑎𝑎𝑛𝑛𝑎𝑎 𝑀𝑀𝑐𝑐= 5 in (1), after the DOMP selection, only 200 coefficients were considered to perform DPD. Results Figures 4 and 5 illustrate the 16-QAM constellations and power spectra, respectively, at the PA output under both well-matched and mismatched impedance conditions. As seen in the power spectrum (red line) in Figure 5, impedance mismatch introduces a stronger frequency dependence. The out-of-band (OOB) spectral regrowth becomes more asymmetrical compared to the well-matched case, as reflected in the adjacent channel power ratio (ACPR) values in Table I. Additionally, this unwanted frequency response impacts the in-band signal, leading to a more significant degradation in error vector magnitude (EVM), as shown in Table I. In the well-matched scenario, the EVM is 5.2%, whereas the mismatched condition results in a much higher in-band distortion, with an EVM of 26.4%. Fig. 4. 16-QAM constellation considering: PA nonlinear distortion and memory effects (left); and PA nonlinear distortion, memory effects and impedance mismatched (right). Fig. 5. 16-QAM constellation considering: PA nonlinear distortion and memory effects (left); and PA nonlinear distortion, memory effects and impedance mismatched (right). Table I. ACPR and EVM values for different scenarios. ACPR (dB) Lower Band/Upper Band EVM (%) Nonlinear PA -29.7 / -28.8 5.2 Nonlinear and mismatched PA (VSWR=3) -35.6 / -28.0 26.4 NL and Mismatched w. DPD (VSWR=3) -49.2 / -45.6 1.1 Figures 6, 7, and 8 show the 16-QAM constellations, power spectra, and AM-AM characteristics, respectively, at the PA output under a mismatched impedance condition, both with and without DPD linearization. As seen in Fig. 7 and Table I, the out-of-band distortion is effectively compensated with DPD, resulting in an ACPR improvement of approximately 14 dB in the lower band and 17 dB in the upper band, even in the challenging case of mismatched nonlinear PA behavior. Additionally, as shown in Fig. 6 and Table I, the in-band distortion is also mitigated, with the EVM improving by more than 25 percentage points compared to the mismatched nonlinear PA. Finally, the AM-AM characteristic in Fig. 8 demonstrates the successful linearization and compensation of memory and impedance mismatched effects using the GMP DPD model. Fig. 6. 16-QAM constellations before (blue) and after (red) DPD under mismatched condition. Fig. 7. PA output power spectra before (blue) and after (red) DPD under mismatched condition. Fig. 8. PA AM-AM characteristic before (blue) and after (red) DPD under mismatched condition. Conclusion The results demonstrate the effectiveness of the GMP DPD model in mitigating nonlinear distortions and memory effects, even under challenging impedance mismatched conditions, which are particularly relevant in PLC. The selection of the most relevant basis functions using the DOMP approach enabled an efficient implementation of DPD, significantly reducing both in-band and out-of-band distortions. As shown in Table I, the proposed method achieved up to 17 dB improvement in ACPR and reduced the EVM from 26.4% to 1.1%, highlighting its capability to enhance signal integrity in PLC systems where impedance mismatches are common due to varying line conditions. Furthermore, the AM-AM characteristic evidences the successful compensation of memory and mismatch effects, ensuring improved linearity and robustness in signal transmission. These findings validate the importance of adaptive DPD techniques in maintaining spectral efficiency and reliable communication performance in nonlinear and impedancemismatched PLC environments. References [Gil25] Gilabert, P.L. and Montoro, G. (2025). Digital Predistortion for Power Amplifiers. In Encyclopedia of RF and Microwave Engineering, K. Chang (Ed.). https://doi.org/10.1002/0471654507.erfme422 [Kim01] J. Kim and K. Konstantinou, Digital predistortion of wideband signals based on power amplifier model with memory, Electron. Lett. 37: 1417–1418 (Nov. 2001). [Mor06] D. R. Morgan, Z. Ma et al., A generalized memory polynomial model for digital predistortion of RF power amplifiers, IEEE Trans. Signal Process. 54(10): 3852–3860 (Oct. 2006). [Zhu15] A. Zhu, “Decomposed vector rotation-based behavioral modeling for digital predistortion of RF power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 63, no. 2, pp. 737–744, Feb. 2015. [Mol17] A. Molina, K. Rajamani, and K. Azadet, “Digital predistortion using lookup tables with linear interpolation and extrapolation: Direct least squares coefficient adaptation,” IEEE Trans. Microw. Theory Techn., vol. 65, no. 3, pp. 980–987, Mar. 2017. [Gil08] P. L. Gilabert, A. Cesari, G. Montoro, E. Bertran, and J.-M. Dilhac, “Multi-lookup table FPGA implementation of an adaptive digital predistorter for linearizing RF power amplifiers with memory effects,” IEEE Trans. Microw. Theory and Techn., vol. 56, no. 2, pp. 372–384, 2008. [Fis23] A. Fischer-Bühner, L. Anttila, M. D. Gomony, and M. Valkama, “Phase-normalized neural network for linearization of RF power amplifiers,” IEEE Microw. Wireless Tech. Lett., vol. 33, no. 9, pp. 1357–1360, 2023. [Lop22] D. López-Bueno, G. Montoro and P. L. Gilabert, "Training Data Selection and Dimensionality Reduction for Polynomial and Artificial Neural Network MIMO Adaptive Digital Predistortion," in IEEE Transactions on Microwave Theory and Techniques, vol. 70, no. 11, pp. 4940-4954, Nov. 2022 [Cri25] R. Criado, W. Li, W. Thompson, G. Montoro, K. Chuang and P. L. Gilabert, "Model-Order Reduction of Multistage Cascaded Models for Digital Predistortion," in IEEE Journal of Microwaves, vol. 5, no. 1, pp. 137-149, Jan. 2025. [Bec18] J. A. Becerra, M. J. Madero-Ayora, J. Reina-Tosina, C. Crespo-Cadenas, J. Garcia-Frias, and G. Arce, “A doubly orthogonal matching pursuit algorithm for sparse predistortion of power amplifiers,” IEEE Microw. Wireless Compon. Lett., vol. 28, no. 8, pp. 726–728, Aug. 2018. Add files to your IEEE Standards submission Delete Collapse all  CANCEL  Title * Item type * Report  Authors * Pere L. Gilabert  Add all authors by name, full email or ORCID. Hit enter after each. Description * This contribution pretends to highlight the benefits of including digital predistortion (DPD) linearization in the digital front-end of Power Line Communications. Use this form to edit all information related to your research. Please be as descriptive as possible. The file upload is independent from the rest of the form, so you don't need to save an upload. This message will be replaced with helpful tips and suggestions as you begin interacting with the form. On the Use of Digital Predistortion for Power Line Communications  Add file(s) DPD for PLC_v3.pdf (539.8 kB)  On the Use of Digital Predistortion for Power Line Communications IEEE Standards https://standardscollection.ieee.org/submit#/s/7556a11b3cc1e552c7f2b... 1 de 3 25/03/2025, 23:14