Symbol constellation predistortion for DCO-OFDM visible light communications system linearization

Digit. Signal Process. 165 (2025) 105310 Available online 14 May 2025 1051-2004/© 2025 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Contents lists available at ScienceDirect Digital Signal Processing journal homepage: www.elsevier.com/locate/dsp Symbol constellation predistortion for DCO-OFDM visible light communications system linearization Ana Cinta Oria Oria a, ,∗, Juan A. Becerra b, María J. Madero-Ayora b, Vicente Baena Lecuyera, Carlos Crespo-Cadenasb aDepartment of Electronic Engineering, University of Seville, Escuela Técnica Superior de Ingeniería, Camino de los Descubrimientos, Seville, 41092, Spain bDepartment of Signal Theory and Communications, University of Seville, Escuela Técnica Superior de Ingeniería, Camino de los Descubrimientos, Seville, 41092, Spain A R T I C L E I N F O A B S T R A C T Keywords: Digital predistortion Linearization Sparse Bayesian learning DCO-OFDM Visible light communications Visible Light Communication (VLC) systems face significant challenges due to the inherent nonlinear characteristics of light emitting diodes (LEDs), which cause distortion in the transmitted DCO-OFDM signals. This signal quality degradation can be lower forcing a large input power back-off (IBO), but leading to an inefficient use of the LEDs. In this work, we propose a predistortion technique, referred to as constellation predistorter (CPD), based on a novel frequency-domain algorithm for the linearization of OFDM signals in VLC systems. The CPD operates on the signal constellation and is based on applying a Bayesian pursuit to obtain a sparse memory polynomial (MP) model matrix. For comparison purposes, this method has been compared to the MP-based timedomain digital predistorter (DPD). The linearization performance of the CPD is measured in terms of the error vector magnitude (EVM) and illumination-to-communication conversion efficiency (ICE) parameters. With the proposed predistorter, we achieve a significant IBO reduction as large as 7.6 dB, enhancing the efficiency of VLC systems, or a nearly 62% decrease in the EVM for a fixed IBO, which represents a substantial reduction in signal distortion and an improvement in ICE. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1. LED model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2. DCO-OFDM system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3. Metrics and figures of merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. Proposed digital predistortion method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.1. Constellation predistorter (CPD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4. Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4.1. EVM vs IBO analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4.2. EVM vs BR analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.3. ICE analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5. CPD computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Declaration of competing interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Data availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 *Corresponding author. E-mail address: [email protected] (A.C. Oria Oria). https://doi.org/10.1016/j.dsp.2025.105310 Digital Signal Processing 165 (2025) 105310 2 A.C. Oria Oria, J.A. Becerra, M.J. Madero-Ayora et al. 1. Introduction The unstoppable growth of mobile data traffic in recent years has drawn significant attention to the study, definition and development of new technologies that complement current radio frequency (RF) wireless communication systems. To address the spectrum crunch problem in 6G networks, Optical Wireless Communications (OWC) are emerging as promising solutions for alleviating the stringent demand of wireless data services [1]. This is because the combined visible light and infrared (IR) spectrum is over 2,000 times larger than the radio spectrum [2]. There are four implementations of OWC systems [3]: Free Space Optical (FSO) Communications [4], Visible Light Communications (VLC) [5], Light Fidelity (LiFi) [6], and Optical Camera Communications (OCC) [7]. The standardization process of certain OWC technologies has been very active in recent years [2]. For example, the earlier standard for VLC, IEEE 802.15.7, was released in 2011, but revised in 2018 [8]. Currently, it is focusing on OCC. Another new standard, IEEE 802.15.13, was approved in 2023 [9], which is focused on industrial applications but is not compatible with existing wireless networks. To overcome this limitation, the IEEE 802.11bb standard was recently approved (2023) [10], which defines the physical layer specifications and system architectures for wireless communication using light waves in the range of 800 nm to 1000 nm (near-infrared). The approval of this last standard is an important milestone for the future deployment of Li-Fi technology [6]. Although OWC solutions cover the entire optical spectrum of ultraviolet (UV), visible and infrared (IR) [3], we focus on the VLC technology that uses the visible light spectrum band. Compared to conventional RF wireless technologies, VLC systems achieve illumination and wireless communication simultaneously, reusing the LED-based infrastructure for communication purposes. These systems present remarkable advantages, such as high-speed data transmission, enhanced security, no electromagnetic interference, use of unregulated spectrum, and low-cost front-end devices [3]. In order to implement the VLC technology, LEDs are required in the transmission chain. These devices have a limited modulation bandwidth, which restricts the achievable transmission rates in these systems. To overcome this limitation and enhance its capacity, researchers have adopted high spectral efficiency modulation schemes, such as direct current biased optical orthogonal frequency division (DCO-OFDM) [11], asymmetrically clipped optical OFDM (ACO-OFDM) [12], or Flip-OFDM [13], among others [14]. In this paper, our focus will be on DCO-OFDM systems due to their easily configurable DC component, essential for the lighting function in VLC systems, and their superior spectral efficiency compared to the previous methods. Despite the choice of efficient modulation techniques for VLC systems, there are nonlinear components in their front-ends, such as LEDs, digital-to-analog converters (DACs), analog-to-digital converters (ADCs), and photodiodes (PDs) that distort the signal and degrade system performance. Among all nonlinear devices, LEDs are the major sources of nonlinearities, producing significant in-band distortion [15]. Therefore, compensating for nonlinear impairments is one of the key challenges in VLC systems. To address the issue of nonlinearity in these systems, the most common approaches are digital predistortion (DPD) [16–18] or post distortion-based linearization techniques [19]. Recently, nonlinear adaptive algorithms, also called in the literature machine learning techniques based on neural networks, are becoming increasingly popular for modeling and mitigating the nonlinearity of LEDs in VLC systems [20–23]. A natural approach to DPD of OFDM systems is to exploit its formulation in the frequency domain (FD), which takes advantage of a lower computational complexity by processing QAM symbols instead of an oversampled signal in the time domain. Several predistortion schemes, based on FD estimation with a memoryless polynomial model, were presented in [24]. In [25], the structure was based on a two-block model of the power amplifier (PA), approximated as a Hammerstein model. The proposal in [26] allowed for different degrees of linearization in different parts of the spectrum. In this paper, the linearization of a VLC system in the constellation domain is explored. The formulation of a constellation predistorter (CPD) is defined in terms of a memory polynomial (MP) model in the FD with the ability to sparsity the predistorter coefficients through Bayesian techniques. The rest of this paper is organized as follows. In Section 2, LED models and the DCO-OFDM VLC system are described. The signal metrics and figures of merit are also defined. Section 3derives the proposed sparse predistortion technique in the constellation domain. Section 4presents simulation results and discussions. Finally, conclusions are summarized in Section 6. 2. System model The most common detection method in VLC systems is intensity modulated/direct detection (IM/DD). This technique involves modulating only the signal intensity without any phase information. In IM/DD systems, the signal must be real-valued and unipolar (non-negative). Consequently, if OFDM is implemented in a VLC system, the time-domain electrical OFDM signal that modulates the LED must be real (first condition) and non-negative (second condition). To achieve the first condition, the Hermitian symmetry property is required in the frequency domain. To ensure the second condition, a DC bias must be added to the time-domain OFDM signal, obtaining a DCO-OFDM signal, one of the mandatory waveforms in the optical wireless standardization, such as in VLC systems [9,10]. 2.1. LED model LED devices are the major sources of nonlinearity in VLC systems. This nonlinear behavior is mainly reflected in the relationship between output voltage (𝑉𝐿) and current (𝐼𝐿) of LED. As a result, signal distortion occurs, manifesting as amplitude clipping and harmonic distortion. The quality of the transmitted signal is degraded, leading to a higher Error Vector Magnitude (EVM) and increased Bit Error Rate (BER). Therefore, the inherent nonlinearity of LEDs is a challenge for the DCO-OFDM VLC systems due to its sensitivity to the resulting distortion. Behavior modeling of nonlinear LED devices is essentially important for the design of mitigation techniques. In general, nonlinear models for LEDs can be classified into two categories: memoryless and memory models [16]. Memoryless models are generally based on memoryless polynomial [27], which use a truncated version of Taylor series, or the Rapp’s model [28], that describes the I-V curve of an LED inspired by the nonlinear characteristics of a power amplifier in RF system and can be expressed as: 𝐼𝐿(𝑉𝐿)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑓(𝑉𝐿) (1+(𝑓(𝑉𝐿) 𝐼𝐿𝑚𝑎𝑥 )2𝑘)1 2𝑘 if 𝑉𝐿≥0 0if 𝑉𝐿<0 (1) where 𝑓(𝑉𝐿)represents the function derived from the data sheet IV curve of the LED, 𝐼𝐿𝑚𝑎𝑥 denotes the maximum alternating current flowing through the LED, and 𝑘is the knee factor that influences the smoothness of LED’s I-V curve. A higher 𝑘results in a less smooth curve [28]. Since memoryless models are only adequate for narrowband transmission, in this paper we will use the memory model proposed in [29]. This model describes the nonlinear behavior of LEDs by characterizing the nonlinear effects of their static and transient behaviors using the dynamic rate equation in the quantum well, which is based on the underlying physical mechanisms. In this paper, we will employ this model, identified as NL-LED. Digital Signal Processing 165 (2025) 105310 3 A.C. Oria Oria, J.A. Becerra, M.J. Madero-Ayora et al. Fig. 1. DCO-OFDM model in VLC systems with predistortion block. For comparison purposes, we will also employ the Rapp’s model with 𝑘=50to represent a LED model whose input-output characteristic is fully linearized within its dynamic range (𝐷𝑅), but exhibits saturation effects beyond this limited interval. The dynamic range of the LED is defined as 𝐷𝑅 =𝑉𝑆𝐴𝑇 −𝑉𝑇𝑂𝑉 , being 𝑉𝑇𝑂𝑉 the turn-on voltage, and 𝑉𝑆𝐴𝑇 the saturation input voltage. 2.2. DCO-OFDM system Fig. 1illustrates the blocks diagram of the DCO-OFDM system used in this paper. Assuming 𝑁FFT subcarriers in each OFDM symbol, and once the symbols are generated after the mapping process according to an 𝑀-QAM constellation, the Hermitian symmetry is implemented in the signal to guarantee a time-domain real-valued signal. Thus, the last 𝑁FFT∕2−1 subcarriers are conformed by the Hermitian symmetry of the subcarriers from 1 to 𝑁FFT∕2 − 1. In addition, subcarriers 1and 𝑁FFT∕2 are zero-valued. The resulting signal (U) is applied to the predistorter or the OFDM modulator, in the case of not implementing a predistortion block (X=U). The OFDM modulation is carried out with an inverse Fast Fourier Transform (IFFT) of 𝑁FFT points. The real-valued signal obtained from this block, 𝑥[𝑛], is converted from parallel to serial format. Next, a cyclic prefix (CP) is added. This signal is digitally converted to the analog domain, 𝑥(𝑡), which is then linearly scaled and biased to generate a unipolar (non-negative) OFDM signal suitable for the LED. The LED converts the magnitude of the electrical signal into optical intensity, which is then transmitted. At the receiver, direct detection is performed by using a photodiode, which transforms the received optical intensity into the amplitude of an electrical signal. The received constellation is distorted because of the nonlinear behavior of the LED, unless an inefficient high input back-off (IBO) transmission is employed or a predistorter is implemented at the transmitter. 2.3. Metrics and figures of merit In a DCO-OFDM system, the signal driving the LED is derived from 𝑥(𝑡)through a linear scaling and a biasing operation as [30] 𝑦(𝑡)=𝛼𝑥(𝑡)+𝐵𝐷𝐶 (2) where 𝛼and 𝐵𝐷𝐶 are both real values. 𝐵𝐷𝐶 is the biasing level that is added to 𝑥(𝑡)to ensure a unipolar OFDM signal at the LED input, while 𝛼is the parameter to scale 𝑥(𝑡)within the dynamic range of the LED. The 𝛼and 𝐵𝐷𝐶 parameters determine another related parameters such as the Biasing Ratio (𝐵𝑅) [31] and the input power back-off (𝐼𝐵𝑂), which are defined mathematically as 𝐵𝑅 =(𝐵𝐷𝐶 −𝑉𝑇𝑂𝑉 )∕𝐷𝑅 (3) 𝐼𝐵𝑂 =𝐷𝑅2∕(𝛼2𝜎2 𝑥)(4) where 𝜎2 𝑥is the variance of 𝑥(𝑡). The scale factor 𝛼must be carefully selected to work with the dynamic range constraints of the LED. A low value of 𝛼(a high IBO) leads to an inefficient scheme concerning the LED’s dynamic range. Whereas a high value of 𝛼(a low IBO) may cause the optical signal to be clipped, thereby compromising communication performance. The figures of merit used to measure signal distortions include the EVM, the BER and the normalized mean square error (NMSE). Nonetheless, BER also depends on the system’s robustness, primarily determined by the type of forward error correction (FEC) [32]. For this reason, EVM will be the main figure of merit used in this article. In the context of VLC systems, which simultaneously transmit information and provide illumination, the brightness factor 𝐵𝐹 parameter [31], related to the illumination level, is usually employed and is mathematically defined as 𝐵𝐹 =(𝑉𝐴𝑉 𝐺 −𝑉𝑇𝑂𝑉 )∕𝐷𝑅 =𝑂𝐴𝑉 𝐺∕𝑂𝑆𝐴𝑇 (5) where 𝑉𝐴𝑉 𝐺 is the average input voltage associated with the average optical power of the LED (𝑂𝐴𝑉 𝐺), which represents its illumination level. 𝑂𝑆𝐴𝑇 represents the LED’s output optical power at the input voltage 𝑉𝑆𝐴𝑇 . Theoretically, 𝐵𝐹 ∈[0,1], but in practical scenarios, 𝐵𝐹 must be less than a maximum value 𝐵𝐹𝑚𝑎𝑥, which is constrained by the maximum permissible DC voltage of the LED. A lower 𝐵𝐹 results in a lower illumination level and reduced transmission capacity. In order to analyze the efficiency in VLC systems, the illumination to communication conversion efficiency (𝐼𝐶𝐸) parameter is defined as [31] 𝐼𝐶𝐸 =𝐷𝑜∕𝑂𝐴𝑉 𝐺 =𝐷𝑖∕(𝑉𝐴𝑉 𝐺 −𝑉𝑇𝑂𝑉 )(6) where 𝐷𝑜is the standard deviation of the output optical intensity, and 𝐷𝑖the standard deviation of the input electrical signal 𝑦(𝑡). It should be noted that a higher ICE indicates a higher dynamic range but a lower illumination level. Consequently, a trade-off between ICE and BR must be considered. 3. Proposed digital predistortion method As mentioned above, the intrinsic nonlinear behavior of a LED degrades significantly the performance of a VLC system, being DPD a commonly employed linearization method [16,17]. Although most DPD algorithms are implemented in the time domain, we proposed here a novel approach implementing predistortion in the frequency domain, called Constellation Predistorter (CPD), that benefits from lower computational complexity. 3.1. Constellation predistorter (CPD) We work with a DCO-OFDM VLC system with a new predistortion method, called in this paper CPD (see Fig. 1). The CPD block predistorts each 𝑀-QAM symbol 𝑈(𝑚)obtaining the 𝑋(𝑚)symbol before its transmission on the 𝑚th subcarrier, with the objective that the signal driving the LED generates an optical intensity with an undistorted constellation. In order to reduce the computational complexity of the predistorter, it is necessary to select a nonlinear model and an efficient algorithm to identify a sparse set of regressors with equivalent performance, like the Digital Signal Processing 165 (2025) 105310 4 A.C. Oria Oria, J.A. Becerra, M.J. Madero-Ayora et al. sparse Bayesian learning (SBL) published in [33] for a time-domain procedure. Here, we employ a direct learning architecture [34] to predistort the constellation in the frequency domain. First, an SBL algorithm is proposed to identify the active regressors and estimate the coefficients of the CPD, and then, the coefficients are employed to set the predistorted M-QAM constellation of the successive OFDM symbols. The estimation of coefficients of the CPD is carried out using the following procedure. For each OFDM symbol of a limited set (for example, 20 symbols), the coefficients of the CPD are initially identified. Then, these coefficients are applied to a DCO-OFDM VLC system calculating the EVM for the rest of symbols. In this iterative process, the set of coefficients of CPD is updated if a minimum is achieved in the EVM parameter. CPD identification with the novel FD-SBL algorithm is as follows. In the closed-loop linearization, the input-output relationship at the CPD block can be written as 𝐗=𝐔−𝚽𝐰,(7) where 𝚽is the frequency-domain observation matrix and 𝐰is the coefficients vector. Given a model structure with conventional time-domain regressors 𝝓𝑟, the Fourier-transformed regressors are  𝚽𝑟={𝝓𝑟}, and the observation matrix is shaped as 𝚽=[ 𝚽1 𝚽2⋯ 𝚽𝑟]. Considering a real-valued MP structure in the time-domain for the CPD, the Fourier transformed regressors are easily calculated as phase rotations. For example, the frequency-domain regressor corresponding to the 𝑞-delayed linear regressor is given by 𝑢(𝑘−𝑞)→𝑈(𝑚)𝑒−𝑗(2𝜋∕𝑁)𝑚𝑞 . To avoid aliasing in the case of 𝑛th-order regressors, 𝐔is zero-padded before inverse Fourier transforming, acquiring the time-domain regressor, computing 𝑢𝑛(𝑘)and then Fourier transforming back to the frequency domain. If the system uses 𝑁FFT subcarriers, the 𝑛th-order model requires an FFT size of 𝑛𝑁FFT, i.e., the bandwidth is increased 𝑛times. In the direct learning architecture, the transmitter output signal is recovered mimicking the channel propagation and the output of the observation receiver is Fourier transformed to the frequency domain. The scaled output 𝐘is used to generate the error with respect to the input 𝐄=𝐘−𝐔.(8) At this point, we can remark that the regressors set is prohibitively large. A significant reduction is achieved by observing that the objective is the minimization of the error 𝐄only in the fundamental band. For example, for a size of the OFDM symbol 𝑁FFT = 1024 and 8th-order model, the FD regressors  𝚽𝑟are constrained to 𝑁FFT = 1024 points, resulting in a notable reduction of the observation matrix 𝚽with 8𝑁FFT = 8192 regressors and 𝑁FFT = 1024 frequency points each. Further pruning of 𝚽is attainable by implementing in the frequency domain the SBL. The sparsity and quality factors of all potential regressors in 𝚽are computed with the expressions 𝑠𝑖= 𝚽𝐻 𝑖𝐂−1 −𝑖 𝚽𝑖and 𝑞𝑖= 𝚽𝐻 𝑖𝐂−1 −𝑖𝐄,(9) respectively, 𝐂−𝑖is the covariance matrix of the measurement vector, 𝐂, without the contribution of the regressor 𝝓𝑖. Since the pursuit is initiated with an empty active set of regressors, a sensible value for the initialization can be 𝐂(0) −𝑖←𝛽−1 =𝜎2=10 −5, with 𝜎2denoting the variance of the additive noise. The algorithm runs entirely in the frequency domain and selects as active the regressor that maximizes the marginal likelihood, mirroring the equivalent time domain SBL proposed in [33]. Then, the posterior covariance and mean of the coefficients (which are scalars initially) are 𝚺=(𝛽𝚽𝐻𝚽+𝐀)−1 (10) 𝝁=(𝚽𝐻𝚽+𝛽−1𝐀)−1𝚽𝐻𝐄,(11) where the a priori precision 𝐀is a diagonal matrix. The procedure is repeated for all the candidate regressors until the potential set is empty with the result of a sparse active set which can be considered as the most likely reduced model. Fig. 2. Evolution of the identification NMSE with the number of active coefficients for 2nd-, 4thand 8th-order models. The evolution of NMSE as the active set is upgraded is shown in Fig. 2 for CPD structures of 2nd-, 4thand 8th-order. The transmitted signal is a 256-QAM constellation with IBO = 15 dB. It is remarkable that, in the case of the 8th-order structure, the potential set of 8192 regressors is reduced to a sparse set of S = 111 active regressors with NMSE =−35dB. Once the sparse set of active regressors has been identified, the estimated CPD coefficients are employed to predistort the successive OFDM symbols, providing the constellation shown in Fig. 3b with an average EVM of 5%. The robustness of the proposed method is illustrated by applying the identified coefficients to a new signal with a different constellation, in this case with a 64-QAM mapping. The constellation distorted by the LED nonlinearity is shown in Fig. 3c and the constellation after the application of the CPD is displayed in Fig. 3d. 4. Simulation results The effectiveness of the proposed predistorter is evaluated in a VLC system by means of Monte Carlo simulations. We have considered a DCO-OFDM signal. The nonlinear LED (NL-LED) model proposed in [29] has been assumed. For comparison purposes, we have also obtained results employing the Rapp’s model with 𝑘=50[28] and with the memory polynomial (MP)-based time-domain DPD method. Regarding the MPbased DPD, it was applied to the time-domain signal referred to 𝑥(𝑡) in Fig. 1, as conventionally [35–39]. The oversampling factor (OVS) assumed for the MP-based time-domain DPD is the same as that considered in the NL-LED model. Different values of memory depth were heuristically tested, being the best performance of the DPD provided by a memory depth of 5 samples. 4.1. EVM vs IBO analysis Fig. 4shows the EVM curves versus IBO for a DCO-OFDM signal with an 𝑁FFT of 1024 subcarriers, a 64-QAM constellation and a biasing point ratio of 0.5employing the NL-LED model for the transmitter in different scenarios: without a predistorter, with our proposed predistorter (CPD), and with the MP-based time-domain DPD method, using different orders (4th, 5th, and 6th). Results employing the Rapp’s model with 𝑘=50for the LED are also shown for comparison purposes. Each point on all the curves is obtained by transmitting 1,000 OFDM symbols. To compensate for the effects of the static and linear components of the system’s transfer function, an ideal one-tap OFDM equalizer is used in all the curves. From Fig. 4, it is evident that the implementation of the proposed predistorter significantly enhances the EVM compared to the system Digital Signal Processing 165 (2025) 105310 5 A.C. Oria Oria, J.A. Becerra, M.J. Madero-Ayora et al. Fig. 3. Constellation in reception in different scenarios for an IBO of 12 dB: (a) 256-QAM, VLC system without predistortion. (b) 256-QAM, VLC system with the proposed predistortion block. (c) 64-QAM, VLC system without predistorter. (d) 64-QAM, VLC system with the proposed distortion block. without a predistorter, or with the MP DPD. For example, for an EVM of approximately 7%, the 5th-order CPD or higher only requires an IBO of 10 dB, while the system without a predistorter requires at least 14.7 dB. This represents improvements in the power efficiency of the system of 4.7dB. In the case of implementing an MP DPD, the required IBO for an EVM of 7% is 12.5 dB, which implies that our predistorter offers a 2.5dB improvement in the power efficiency regarding to the MP. Note that 5th-order CPD presents the same performance than CPDs of higher orders. Even the 4th-order CPD, which performs worse than higher orders for IBO values lower than 16 dB, presents an improvement of 2dB for a 7% EVM compared to the case without a predistorter. Fig. 4shows that the MP predistorter performs considerably worse across the entire range of IBO values examined. Only the 4th-order CPD exhibits worse performance than the MP for IBO values below 13 dB. It is important to note that increasing the order of the MP algorithm does not enhance its performance; in fact, higher orders result in a detrimental effect. This is due to overfitting and numerical regression issues caused by the increase of the number of coefficients with the order of the MP. Compared to the results obtained with the Rapp’s model (𝑘=50), the fifth order CPD performs better for low IBO, that is, the proposed predistorter handles the saturation effects of LED better. For IBO values higher than 15 dB, the DCO-OFDM signal is mainly within the DR of the LED, where the Rapp’s model is perfectly linear, and therefore the Rapp’s model presents a lower EVM. Additionally, the curve labeled as “NL-LED + CPD (5𝑡ℎ)(256-QAM)” from Fig. 4represents the EVM vs IBO for a 256-QAM DCO-OFDM sigFig. 4. EVM curves versus IBO for a DCO-OFDM signal with 𝑁FFT = 1024 and a BR of 0.5. nal, but with a CPD whose coefficients have been obtained for a signal employing a 64-QAM constellation. Similarly, the approach was applied to the MP (see curve labeled as “NL-LED + MP (5𝑡ℎ)(256-QAM)”). These Digital Signal Processing 165 (2025) 105310 6 A.C. Oria Oria, J.A. Becerra, M.J. Madero-Ayora et al. Fig. 5. EVM curves versus IBO for a DCO-OFDM signal with 𝑁FFT = 1024 and a BR of 0.5 for a 5th-order CPD with different oversampling rates. curves demonstrate that the performance of these predistortion methods is not dependent on the order of the constellation used in the transmitted signal. Similar results have been obtained for QPSK, 16-QAM, and even 1024-QAM, which are defined in the main VLC standards [8–10]. Fig. 5shows the results of the analysis of EVM vs IBO in the same scenario but with different oversampling factors (OVS) for our method. Although theoretically the oversampling factor used in the CPD algorithm must be at least equal to the order of the CPD, Fig. 5indicates that an OVS of 4 is sufficient without any performance loss. This reduces the required hardware resources for implementing the 5th-order CPD. Considering an oversampling factor of 2 could also be beneficial if a slight increase in the EVM can be assumed. 4.2. EVM vs BR analysis In the previous subsection, the EVM vs IBO curves were analyzed for a fixed BR of 50% (BR=0.5). Next, we will study the performance of the CPD when different values of BR are applied in the same scenarios. The BR parameter is closely related to the biasing level (𝐵𝐷𝐶 ) added to 𝑥(𝑡)to obtain an unipolar signal for the LED input. In order to obtain these results, the IBO was fixed to 10 dB. Similar results were also obtained for the MP algorithm. From Fig. 6, it can be seen that the minimum EVM is achieved when the BR is 0.5in all scenarios. In this case, CPD reduces the EVM by nearly 62% compared to a DCOOFDM signal without predistorter, whereas MP DPD algorithm only achieves a reduction of approximately 51%. Similar results are obtained for the remaining BR values when using our predistorter. For example, for BR = 0.1, the percentage decrease in EVM is 56.5%, and nearly 49% for BR = 0.9. This analysis has also been carried out for other 𝑁FFT values (64,128,256,512,2048, and 4096) obtaining similar results to 𝑁FFT = 1024, with a percentage decrease between 40% and 58%. Additionally, note that for a BR further away from 0.5, the effects of signal saturation are more pronounced; however, the CPD resolves these effects better than the Rapp’s model of the LED. As expected, from Fig. 6, it can be seen that the 5th-order CPD is also sufficient. Fig. 6also illustrates that the MP DPD algorithm drastically reduces performance when a BR close to the extremes is used compared to our method. For example, for BR=0.1 and BR=0.9, the MP reduction percentages are only 14% and 12%, respectively. It is important to note that the MP algorithm does not effectively address the effects of LED saturation due to the limited richness of the mathematical expressions of its basis functions in the time domain, unlike our method. Fig. 6. EVM versus BR for a DCO-OFDM signal with 64-QAM constellation and 𝑁FFT = 1024, and for IBO =10dB. Fig. 7. IBO versus BR and IBO reduction for EVM = 10%. Finally, Fig. 7shows the required IBO values for an EVM value of 10% versus BR, for all analyzed scenarios. Note that the implementation of the proposed predistorter in a VLC DCO-OFDM transmitter reduces significantly the required IBO to achieve an EVM of 10% in the full range of BR. It can be observed that our proposed method achieves an IBO reduction of up to 7.6dB. This reduction allows that the LED can operate closer to its maximum output power without incurring significant distortions and increasing the efficient use of the LEDs. 4.3. ICE analysis The evaluation of the ICE parameter is interesting to know VLC systems efficiency, where illumination and communications take place simultaneously. The results are depicted in Fig. 8, where ICE versus BF curves for a VLC system with a 5th-order CPD and a VLC system without predistortion block are shown. To obtain these curves, the required IBO for each BF point has been configured such as it provides an EVM = 10%. It can be observed that an improvement regardless the BF is achieved with the proposed CPD over the signal without predistortion technique. The improvement is greater the smaller the value of the BF parameter. Digital Signal Processing 165 (2025) 105310 7 A.C. Oria Oria, J.A. Becerra, M.J. Madero-Ayora et al. Fig. 8. ICE versus brightness factor for EVM = 10%. 64-QAM constellation and 𝑁FFT = 1024. 5. CPD computational complexity Next, the computational complexity of the proposal is compared to that of the MP following the Bachmann–Landau notation operation, which measures the number of complex multiplications in the steps involved in the algorithms, providing a clear performance evaluation based on the computational cost of each operation. On the one hand, the total computational complexity for the CPD simplifies to 𝐶CPD =𝑂(OVS ⋅𝑛⋅log(𝑁FFT ⋅OVS)+(𝑛−1)⋅(OVS +𝑆 2 )),(12) where 𝑛stands for the nonlinear order, 𝑁FFT is the number of samples in the frequency domain, OVS is the oversampling factor and 𝑆is the number of selected coefficients in the model. On the other hand, the computational complexity of the MP model is 𝐶MP =𝑂(OVS(𝑆⋅OVS +1 )⋅log(𝑁FFT ⋅OVS)+(𝑛−1)⋅OVS).(13) The comparison between the computational complexities of the CPD and MP models reveals key differences. Both models share terms involving the oversampling factor and the logarithmic complexity associated with the FFT. However, the complexity of the CPD includes a term which grows linearly with the oversampling factor, while the complexity of the MP includes a term which grows quadratically with this variable. This makes the MP model more computationally expensive when the number of coefficients and oversampling factor are large. Additionally, while both models depend on the order, the CPD involves a logarithmic term that grows with it, whereas the MP has a more direct linear dependence. Overall, the CPD is more efficient when the number of coefficients is small, while the MP is more sensitive to increases in this term and the oversampling factor. 6. Conclusions In this paper, a novel FD-SBL algorithm has been proposed to linearize OFDM signals distorted by LEDs used in the transmitters of VLC systems. The proposed predistortion technique, CPD, acts on the signal constellation. For this, a Bayesian pursuit in the frequency domain has been applied to a MP model structure to select a sparse active set of regressors, following a similar procedure as that of the SBL in the time domain. The linearization performance of the CPD has been demonstrated for different constellation formats and different numbers of subcarriers in a VLC DCO-OFDM system. Firstly, we have shown that the CPD reduces signal distortions, improving EVM performance. It has been demonstrated that a fifth-order CPD is sufficient, allowing for up to a 7.6 dB reduction in the IBO of the DCO-OFDM signal for the same EVM compared to not using a predistorter. Secondly, for a fixed IBO value in both scenarios, it has been shown that the VLC system integrating the CPD achieves a percentage reduction in EVM of nearly 62% compared to the EVM obtained with the system without predistorter. 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Zhou, Adaptive postdistortion for nonlinear LEDs in visible light communications, IEEE Photonics J. 6 (4) (2014) 1–8, https://doi.org/ 10.1109/JPHOT.2014.2331242. [38] W. Zhao, Q. Guo, J. Tong, J. Xi, Y. Yu, P. Niu, X. Sun, Orthogonal polynomial-based nonlinearity modeling and mitigation for LED communications, IEEE Photonics J. 8 (4) (2016) 1–12, https://doi.org/10.1109/JPHOT.2016.2581485. [39] Z. Du, et al., Enhanced performance of an indoor non-line-of-sight VLC system utilizing a multi-pixel photon counter and interleaved single-carrier FDM scheme, Opt. Commun. 554 (2024) 130179, https://doi.org/10.1016/j.optcom.2023.130179. Ana Cinta Oria Oria was born in Huelva, Spain. She received his Master and Ph.D. degrees in Telecommunication Engineering in 2005 and 2010, both from the University of Seville. Since 2005, she has been with the Department of Electronic Engineering, High School of Engineering, University of Seville. Her current research interest is in multicarrier systems and digital signal processing. Juan Antonio Becerra (STM’12-M’18-SM’19) obtained his B.Sc. and M.Sc. degrees in Telecommunication Engineering from the Universidad de Sevilla, Seville, Spain, in 2009 and 2012, respectively. He further pursued his Ph.D. in Electrical and Computer Engineering at the University of Delaware, Newark, DE, USA, completing it in 2017. Simultaneously, he earned a Ph.D. in Telecommunication Engineering from the Universidad de Sevilla in 2019. Since 2017, he has been with the Department of Signal Theory and Communications, Universidad de Sevilla, currently serving as an Associate Professor. His main research areas include behavioral modeling and linearization of power amplifiers, and compressed-sensing signal processing. María J. Madero-Ayora (STM’06-M’09-SM’20) received the M.Sc. and Ph.D. degrees in Telecommunication Engineering in 2002 and 2008, respectively, from Universidad de Sevilla, Seville, Spain. Since 2003, she has been with the Department of Signal Theory and Communications, Universidad de Sevilla, where she is currently an Associate Professor. Her main research areas include compensation of impairments in modulators and power amplifiers, and measurement techniques of nonlinear communication systems. Vicente Baena-Lecuyer was born in Athis-Mons, France. He received his Telecommunication Engineering and Ph.D. degrees from the University of Seville (Spain) in 1997 and 2001, respectively. Since 1997, he has been with the Department of Electronic Engineering, High School of Engineering, University of Seville. His current research interests include digital signal processing and multicarrier communications systems. Carlos Crespo-Cadenas (M’93-SM’15-LM’22) was born in Madrid, Spain. He received the degree in physics from the University of Havana, Havana, Cuba, in 1973, and the Ph.D. degree from the Polytechnique University of Madrid, Madrid, in 1995. Since 1995, he has been with the Department of Signal Theory and Communications, Universidad de Sevilla, where he is currently a retired Professor and Honorary Researcher. His research interests include the communication systems, nonlinear analysis of active microwave devices, power amplifier behavioral modeling, and linearization techniques.