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

Digi . Signal P ocess. 165 (2025) 105310 A ailable online 14 May 2025 1051-2004/© 2025 The Au ho s. Published by Else ie Inc. This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by- nc-nd/4.0/). Con en s lis s a ailable a ScienceDi ec Digi al Signal P ocessing jou nal homepage: www.else ie .com/loca e/dsp Symbol cons ella ion p edis o ion o DCO-OFDM isible ligh communica ions sys em linea iza ion Ana Cin a O ia O ia a, ,∗, Juan A. Bece a b, Ma ía J. Made o-Ayo a b, Vicen e Baena Lecuye a, Ca los C espo-Cadenasb aDepa men o Elec onic Enginee ing, Uni e si y o Se ille, Escuela Técnica Supe io de Ingenie ía, Camino de los Descub imien os, Se ille, 41092, Spain bDepa men o Signal Theo y and Communica ions, Uni e si y o Se ille, Escuela Técnica Supe io de Ingenie ía, Camino de los Descub imien os, Se ille, 41092, Spain A R T I C L E I N F O A B S T R A C T Keywo ds: Digi al p edis o ion Linea iza ion Spa se Bayesian lea ning DCO-OFDM Visible ligh communica ions Visible Ligh Communica ion (VLC) sys ems ace significan challenges due o he inhe en nonlinea cha ac e - is ics o ligh emi ing diodes (LEDs), which cause dis o ion in he ansmi ed DCO-OFDM signals. This signal quali y deg ada ion can be lowe o cing a la ge inpu powe back-off (IBO), bu leading o an inefficien use o he LEDs. In his wo k, we p opose a p edis o ion echnique, e e ed o as cons ella ion p edis o e (CPD), based on a no el equency-domain algo i hm o he linea iza ion o OFDM signals in VLC sys ems. The CPD ope a es on he signal cons ella ion and is based on applying a Bayesian pu sui o ob ain a spa se memo y polynomial (MP) model ma ix. Fo compa ison pu poses, his me hod has been compa ed o he MP-based ime- domain digi al p edis o e (DPD). The linea iza ion pe o mance o he CPD is measu ed in e ms o he e o ec o magni ude (EVM) and illumina ion- o-communica ion con e sion efficiency (ICE) pa ame e s. Wi h he p oposed p edis o e , we achie e a significan IBO educ ion as la ge as 7.6 dB, enhancing he efficiency o VLC sys ems, o a nea ly 62% dec ease in he EVM o a fixed IBO, which ep esen s a subs an ial educ ion in signal dis o ion and an imp o emen in ICE. Con en s 1. In oduc ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Sys em model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1. LED model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2. DCO-OFDM sys em . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3. Me ics and figu es o me i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. P oposed digi al p edis o ion me hod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.1. Cons ella ion p edis o e (CPD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4. Simula ion esul s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4.1. EVM s IBO analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4.2. EVM s BR analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.3. ICE analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5. CPD compu a ional complexi y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Decla a ion o compe ing in e es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Da a a ailabili y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Re e ences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 *Co esponding au ho . E-mail add ess: [email p o ec ed] (A.C. O ia O ia). h ps://doi.o g/10.1016/j.dsp.2025.105310 Digi al Signal P ocessing 165 (2025) 105310 2 A.C. O ia O ia, J.A. Bece a, M.J. Made o-Ayo a e al. 1. In oduc ion The uns oppable g ow h o mobile da a affic in ecen yea s has d awn significan a en ion o he s udy, defini ion and de elopmen o new echnologies ha complemen cu en adio equency (RF) wi e- less communica ion sys ems. To add ess he spec um c unch p oblem in 6G ne wo ks, Op ical Wi eless Communica ions (OWC) a e eme ging as p omising solu ions o alle ia ing he s ingen demand o wi eless da a se ices [1]. This is because he combined isible ligh and in a ed (IR) spec um is o e 2,000 imes la ge han he adio spec um [2]. The e a e ou implemen a ions o OWC sys ems [3]: F ee Space Op ical (FSO) Communica ions [4], Visible Ligh Communica ions (VLC) [5], Ligh Fi- deli y (LiFi) [6], and Op ical Came a Communica ions (OCC) [7]. The s anda diza ion p ocess o ce ain OWC echnologies has been e y ac i e in ecen yea s [2]. Fo example, he ea lie s anda d o VLC, IEEE 802.15.7, was eleased in 2011, bu e ised in 2018 [8]. Cu - en ly, i is ocusing on OCC. Ano he new s anda d, IEEE 802.15.13, was app o ed in 2023 [9], which is ocused on indus ial applica ions bu is no compa ible wi h exis ing wi eless ne wo ks. To o e come his limi a ion, he IEEE 802.11bb s anda d was ecen ly app o ed (2023) [10], which defines he physical laye specifica ions and sys em a chi- ec u es o wi eless communica ion using ligh wa es in he ange o 800 nm o 1000 nm (nea -in a ed). The app o al o his las s anda d is an impo an miles one o he u u e deploymen o Li-Fi echnol- ogy [6]. Al hough OWC solu ions co e he en i e op ical spec um o ul a iole (UV), isible and in a ed (IR) [3], we ocus on he VLC ech- nology ha uses he isible ligh spec um band. Compa ed o con en- ional RF wi eless echnologies, VLC sys ems achie e illumina ion and wi eless communica ion simul aneously, eusing he LED-based in as- uc u e o communica ion pu poses. These sys ems p esen ema kable ad an ages, such as high-speed da a ansmission, enhanced secu i y, no elec omagne ic in e e ence, use o un egula ed spec um, and low-cos on -end de ices [3]. In o de o implemen he VLC echnology, LEDs a e equi ed in he ansmission chain. These de ices ha e a limi ed modula ion band- wid h, which es ic s he achie able ansmission a es in hese sys ems. To o e come his limi a ion and enhance i s capaci y, esea che s ha e adop ed high spec al efficiency modula ion schemes, such as di ec cu - en biased op ical o hogonal equency di ision (DCO-OFDM) [11], asymme ically clipped op ical OFDM (ACO-OFDM) [12], o Flip-OFDM [13], among o he s [14]. In his pape , ou ocus will be on DCO-OFDM sys ems due o hei easily configu able DC componen , essen ial o he ligh ing unc ion in VLC sys ems, and hei supe io spec al efficiency compa ed o he p e ious me hods. Despi e he choice o efficien modula ion echniques o VLC sys- ems, he e a e nonlinea componen s in hei on -ends, such as LEDs, digi al- o-analog con e e s (DACs), analog- o-digi al con e e s (ADCs), and pho odiodes (PDs) ha dis o he signal and deg ade sys- em pe o mance. Among all nonlinea de ices, LEDs a e he majo sou ces o nonlinea i ies, p oducing significan in-band dis o ion [15]. The e o e, compensa ing o nonlinea impai men s is one o he key challenges in VLC sys ems. To add ess he issue o nonlinea i y in hese sys ems, he mos common app oaches a e digi al p edis o ion (DPD) [16–18] o pos dis o ion-based linea iza ion echniques [19]. Recen ly, nonlinea adap i e algo i hms, also called in he li e a u e machine lea ning ech- niques based on neu al ne wo ks, a e becoming inc easingly popula o modeling and mi iga ing he nonlinea i y o LEDs in VLC sys ems [20–23]. A na u al app oach o DPD o OFDM sys ems is o exploi i s o mu- la ion in he equency domain (FD), which akes ad an age o a lowe compu a ional complexi y by p ocessing QAM symbols ins ead o an o e sampled signal in he ime domain. Se e al p edis o ion schemes, based on FD es ima ion wi h a memo yless polynomial model, we e p e- sen ed in [24]. In [25], he s uc u e was based on a wo-block model o he powe amplifie (PA), app oxima ed as a Hamme s ein model. The p oposal in [26] allowed o diffe en deg ees o linea iza ion in diffe en pa s o he spec um. In his pape , he linea iza ion o a VLC sys em in he cons ella- ion domain is explo ed. The o mula ion o a cons ella ion p edis o e (CPD) is defined in e ms o a memo y polynomial (MP) model in he FD wi h he abili y o spa si y he p edis o e coefficien s h ough Bayesian echniques. The es o his pape is o ganized as ollows. In Sec ion 2, LED mod- els and he DCO-OFDM VLC sys em a e desc ibed. The signal me ics and figu es o me i a e also defined. Sec ion 3de i es he p oposed spa se p edis o ion echnique in he cons ella ion domain. Sec ion 4p esen s simula ion esul s and discussions. Finally, conclusions a e summa ized in Sec ion 6. 2. Sys em model The mos common de ec ion me hod in VLC sys ems is in ensi y mod- ula ed/di ec de ec ion (IM/DD). This echnique in ol es modula ing only he signal in ensi y wi hou any phase in o ma ion. In IM/DD sys- ems, he signal mus be eal- alued and unipola (non-nega i e). Con- sequen ly, i OFDM is implemen ed in a VLC sys em, he ime-domain elec ical OFDM signal ha modula es he LED mus be eal (fi s con- di ion) and non-nega i e (second condi ion). To achie e he fi s con- di ion, he He mi ian symme y p ope y is equi ed in he equency domain. To ensu e he second condi ion, a DC bias mus be added o he ime-domain OFDM signal, ob aining a DCO-OFDM signal, one o he manda o y wa e o ms in he op ical wi eless s anda diza ion, such as in VLC sys ems [9,10]. 2.1. LED model LED de ices a e he majo sou ces o nonlinea i y in VLC sys ems. This nonlinea beha io is mainly eflec ed in he ela ionship be ween ou pu ol age (𝑉𝐿) and cu en (𝐼𝐿) o LED. As a esul , signal dis o ion occu s, mani es ing as ampli ude clipping and ha monic dis o ion. The quali y o he ansmi ed signal is deg aded, leading o a highe E o Vec o Magni ude (EVM) and inc eased Bi E o Ra e (BER). The e o e, he inhe en nonlinea i y o LEDs is a challenge o he DCO-OFDM VLC sys ems due o i s sensi i i y o he esul ing dis o ion. Beha io modeling o nonlinea LED de ices is essen ially impo an o he design o mi iga ion echniques. In gene al, nonlinea models o LEDs can be classified in o wo ca ego ies: memo yless and memo y models [16]. Memo yless models a e gene ally based on memo yless polynomial [27], which use a unca ed e sion o Taylo se ies, o he Rapp’s model [28], ha desc ibes he I-V cu e o an LED inspi ed by he nonlinea cha ac e is ics o a powe amplifie in RF sys em and can be exp essed as: 𝐼𝐿(𝑉𝐿)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑓(𝑉𝐿) (1+(𝑓(𝑉𝐿) 𝐼𝐿𝑚𝑎𝑥 )2𝑘)1 2𝑘 i 𝑉𝐿≥0 0i 𝑉𝐿<0 (1) whe e 𝑓(𝑉𝐿) ep esen s he unc ion de i ed om he da a shee I- V cu e o he LED, 𝐼𝐿𝑚𝑎𝑥 deno es he maximum al e na ing cu en flowing h ough he LED, and 𝑘is he knee ac o ha influences he smoo hness o LED’s I-V cu e. A highe 𝑘 esul s in a less smoo h cu e [28]. Since memo yless models a e only adequa e o na owband ans- mission, in his pape we will use he memo y model p oposed in [29]. This model desc ibes he nonlinea beha io o LEDs by cha ac e izing he nonlinea effec s o hei s a ic and ansien beha io s using he dynamic a e equa ion in he quan um well, which is based on he un- de lying physical mechanisms. In his pape , we will employ his model, iden ified as NL-LED. Digi al Signal P ocessing 165 (2025) 105310 3 A.C. O ia O ia, J.A. Bece a, M.J. Made o-Ayo a e al. Fig. 1. DCO-OFDM model in VLC sys ems wi h p edis o ion block. Fo compa ison pu poses, we will also employ he Rapp’s model wi h 𝑘=50 o ep esen a LED model whose inpu -ou pu cha ac e is ic is ully linea ized wi hin i s dynamic ange (𝐷𝑅), bu exhibi s sa u a ion effec s beyond his limi ed in e al. The dynamic ange o he LED is defined as 𝐷𝑅 =𝑉𝑆𝐴𝑇 −𝑉𝑇𝑂𝑉 , being 𝑉𝑇𝑂𝑉 he u n-on ol age, and 𝑉𝑆𝐴𝑇 he sa u a ion inpu ol age. 2.2. DCO-OFDM sys em Fig. 1illus a es he blocks diag am o he DCO-OFDM sys em used in his pape . Assuming 𝑁FFT subca ie s in each OFDM symbol, and once he symbols a e gene a ed a e he mapping p ocess acco ding o an 𝑀-QAM cons ella ion, he He mi ian symme y is implemen ed in he signal o gua an ee a ime-domain eal- alued signal. Thus, he las 𝑁FFT∕2−1 subca ie s a e con o med by he He mi ian symme y o he subca ie s om 1 o 𝑁FFT∕2 − 1. In addi ion, subca ie s 1and 𝑁FFT∕2 a e ze o- alued. The esul ing signal (U) is applied o he p edis o e o he OFDM modula o , in he case o no implemen ing a p edis o - ion block (X=U). The OFDM modula ion is ca ied ou wi h an in e se Fas Fou ie T ans o m (IFFT) o 𝑁FFT poin s. The eal- alued signal ob- ained om his block, 𝑥[𝑛], is con e ed om pa allel o se ial o ma . Nex , a cyclic p efix (CP) is added. This signal is digi ally con e ed o he analog domain, 𝑥(𝑡), which is hen linea ly scaled and biased o gen- e a e a unipola (non-nega i e) OFDM signal sui able o he LED. The LED con e s he magni ude o he elec ical signal in o op ical in ensi y, which is hen ansmi ed. A he ecei e , di ec de ec ion is pe o med by using a pho odiode, which ans o ms he ecei ed op ical in ensi y in o he ampli ude o an elec ical signal. The ecei ed cons ella ion is dis o ed because o he nonlinea be- ha io o he LED, unless an inefficien high inpu back-off (IBO) ans- mission is employed o a p edis o e is implemen ed a he ansmi e . 2.3. Me ics and figu es o me i In a DCO-OFDM sys em, he signal d i ing he LED is de i ed om 𝑥(𝑡) h ough a linea scaling and a biasing ope a ion as [30] 𝑦(𝑡)=𝛼𝑥(𝑡)+𝐵𝐷𝐶 (2) whe e 𝛼and 𝐵𝐷𝐶 a e bo h eal alues. 𝐵𝐷𝐶 is he biasing le el ha is added o 𝑥(𝑡) o ensu e a unipola OFDM signal a he LED inpu , while 𝛼is he pa ame e o scale 𝑥(𝑡)wi hin he dynamic ange o he LED. The 𝛼and 𝐵𝐷𝐶 pa ame e s de e mine ano he ela ed pa ame e s such as he Biasing Ra io (𝐵𝑅) [31] and he inpu powe back-off (𝐼𝐵𝑂), which a e defined ma hema ically as 𝐵𝑅 =(𝐵𝐷𝐶 −𝑉𝑇𝑂𝑉 )∕𝐷𝑅 (3) 𝐼𝐵𝑂 =𝐷𝑅2∕(𝛼2𝜎2 𝑥)(4) whe e 𝜎2 𝑥is he a iance o 𝑥(𝑡). The scale ac o 𝛼mus be ca e ully selec ed o wo k wi h he dynamic ange cons ain s o he LED. A low alue o 𝛼(a high IBO) leads o an inefficien scheme conce ning he LED’s dynamic ange. Whe eas a high alue o 𝛼(a low IBO) may cause he op ical signal o be clipped, he eby comp omising communica ion pe o mance. The figu es o me i used o measu e signal dis o ions include he EVM, he BER and he no malized mean squa e e o (NMSE). None he- less, BER also depends on he sys em’s obus ness, p ima ily de e mined by he ype o o wa d e o co ec ion (FEC) [32]. Fo his eason, EVM will be he main figu e o me i used in his a icle. In he con ex o VLC sys ems, which simul aneously ansmi in o - ma ion and p o ide illumina ion, he b igh ness ac o 𝐵𝐹 pa ame e [31], ela ed o he illumina ion le el, is usually employed and is ma h- ema ically defined as 𝐵𝐹 =(𝑉𝐴𝑉 𝐺 −𝑉𝑇𝑂𝑉 )∕𝐷𝑅 =𝑂𝐴𝑉 𝐺∕𝑂𝑆𝐴𝑇 (5) whe e 𝑉𝐴𝑉 𝐺 is he a e age inpu ol age associa ed wi h he a e age op- ical powe o he LED (𝑂𝐴𝑉 𝐺), which ep esen s i s illumina ion le el. 𝑂𝑆𝐴𝑇 ep esen s he LED’s ou pu op ical powe a he inpu ol age 𝑉𝑆𝐴𝑇 . Theo e ically, 𝐵𝐹 ∈[0,1], bu in p ac ical scena ios, 𝐵𝐹 mus be less han a maximum alue 𝐵𝐹𝑚𝑎𝑥, which is cons ained by he max- imum pe missible DC ol age o he LED. A lowe 𝐵𝐹 esul s in a lowe illumina ion le el and educed ansmission capaci y. In o de o analyze he efficiency in VLC sys ems, he illumina ion o communica ion con e sion efficiency (𝐼𝐶𝐸) pa ame e is defined as [31] 𝐼𝐶𝐸 =𝐷𝑜∕𝑂𝐴𝑉 𝐺 =𝐷𝑖∕(𝑉𝐴𝑉 𝐺 −𝑉𝑇𝑂𝑉 )(6) whe e 𝐷𝑜is he s anda d de ia ion o he ou pu op ical in ensi y, and 𝐷𝑖 he s anda d de ia ion o he inpu elec ical signal 𝑦(𝑡). I should be no ed ha a highe ICE indica es a highe dynamic ange bu a lowe illumina ion le el. Consequen ly, a ade-off be ween ICE and BR mus be conside ed. 3. P oposed digi al p edis o ion me hod As men ioned abo e, he in insic nonlinea beha io o a LED de- g ades significan ly he pe o mance o a VLC sys em, being DPD a commonly employed linea iza ion me hod [16,17]. Al hough mos DPD algo i hms a e implemen ed in he ime domain, we p oposed he e a no el app oach implemen ing p edis o ion in he equency domain, called Cons ella ion P edis o e (CPD), ha benefi s om lowe com- pu a ional complexi y. 3.1. Cons ella ion p edis o e (CPD) We wo k wi h a DCO-OFDM VLC sys em wi h a new p edis o ion me hod, called in his pape CPD (see Fig. 1). The CPD block p edis o s each 𝑀-QAM symbol 𝑈(𝑚)ob aining he 𝑋(𝑚)symbol be o e i s ans- mission on he 𝑚 h subca ie , wi h he objec i e ha he signal d i ing he LED gene a es an op ical in ensi y wi h an undis o ed cons ella ion. In o de o educe he compu a ional complexi y o he p edis o e , i is necessa y o selec a nonlinea model and an efficien algo i hm o iden i y a spa se se o eg esso s wi h equi alen pe o mance, like he Digi al Signal P ocessing 165 (2025) 105310 4 A.C. O ia O ia, J.A. Bece a, M.J. Made o-Ayo a e al. spa se Bayesian lea ning (SBL) published in [33] o a ime-domain p o- cedu e. He e, we employ a di ec lea ning a chi ec u e [34] o p edis o he cons ella ion in he equency domain. Fi s , an SBL algo i hm is p o- posed o iden i y he ac i e eg esso s and es ima e he coefficien s o he CPD, and hen, he coefficien s a e employed o se he p edis o ed M-QAM cons ella ion o he successi e OFDM symbols. The es ima ion o coefficien s o he CPD is ca ied ou using he ollowing p ocedu e. Fo each OFDM symbol o a limi ed se ( o example, 20 symbols), he coefficien s o he CPD a e ini ially iden ified. Then, hese coefficien s a e applied o a DCO-OFDM VLC sys em calcula ing he EVM o he es o symbols. In his i e a i e p ocess, he se o coefficien s o CPD is upda ed i a minimum is achie ed in he EVM pa ame e . CPD iden ifica ion wi h he no el FD-SBL algo i hm is as ollows. In he closed-loop linea iza ion, he inpu -ou pu ela ionship a he CPD block can be w i en as 𝐗=𝐔−𝚽𝐰,(7) whe e 𝚽is he equency-domain obse a ion ma ix and 𝐰is he coe - ficien s ec o . Gi en a model s uc u e wi h con en ional ime-domain eg esso s 𝝓𝑟, he Fou ie - ans o med eg esso s a e  𝚽𝑟={𝝓𝑟}, and he obse a ion ma ix is shaped as 𝚽=[ 𝚽1 𝚽2⋯ 𝚽𝑟]. Conside ing a eal- alued MP s uc u e in he ime-domain o he CPD, he Fou ie ans o med eg esso s a e easily calcula ed as phase o a ions. Fo example, he equency-domain eg esso co esponding o he 𝑞-delayed linea eg esso is gi en by 𝑢(𝑘−𝑞)→𝑈(𝑚)𝑒−𝑗(2𝜋∕𝑁)𝑚𝑞 . To a oid aliasing in he case o 𝑛 h-o de eg esso s, 𝐔is ze o-padded be o e in e se Fou ie ans o ming, acqui ing he ime-domain eg es- so , compu ing 𝑢𝑛(𝑘)and hen Fou ie ans o ming back o he e- quency domain. I he sys em uses 𝑁FFT subca ie s, he 𝑛 h-o de model equi es an FFT size o 𝑛𝑁FFT, i.e., he bandwid h is inc eased 𝑛 imes. In he di ec lea ning a chi ec u e, he ansmi e ou pu signal is eco e ed mimicking he channel p opaga ion and he ou pu o he ob- se a ion ecei e is Fou ie ans o med o he equency domain. The scaled ou pu 𝐘is used o gene a e he e o wi h espec o he inpu 𝐄=𝐘−𝐔.(8) A his poin , we can ema k ha he eg esso s se is p ohibi i ely la ge. A significan educ ion is achie ed by obse ing ha he objec i e is he minimiza ion o he e o 𝐄only in he undamen al band. Fo example, o a size o he OFDM symbol 𝑁FFT = 1024 and 8 h-o de model, he FD eg esso s  𝚽𝑟a e cons ained o 𝑁FFT = 1024 poin s, esul ing in a no able educ ion o he obse a ion ma ix 𝚽wi h 8𝑁FFT = 8192 eg esso s and 𝑁FFT = 1024 equency poin s each. Fu he p uning o 𝚽is a ainable by implemen ing in he equency domain he SBL. The spa si y and quali y ac o s o all po en ial eg es- so s in 𝚽a e compu ed wi h he exp essions 𝑠𝑖= 𝚽𝐻 𝑖𝐂−1 −𝑖 𝚽𝑖and 𝑞𝑖= 𝚽𝐻 𝑖𝐂−1 −𝑖𝐄,(9) espec i ely, 𝐂−𝑖is he co a iance ma ix o he measu emen ec o , 𝐂, wi hou he con ibu ion o he eg esso 𝝓𝑖. Since he pu sui is ini- ia ed wi h an emp y ac i e se o eg esso s, a sensible alue o he ini ializa ion can be 𝐂(0) −𝑖←𝛽−1 =𝜎2=10 −5, wi h 𝜎2deno ing he a i- ance o he addi i e noise. The algo i hm uns en i ely in he equency domain and selec s as ac i e he eg esso ha maximizes he ma ginal likelihood, mi o ing he equi alen ime domain SBL p oposed in [33]. Then, he pos e io co a iance and mean o he coefficien s (which a e scala s ini ially) a e 𝚺=(𝛽𝚽𝐻𝚽+𝐀)−1 (10) 𝝁=(𝚽𝐻𝚽+𝛽−1𝐀)−1𝚽𝐻𝐄,(11) whe e he a p io i p ecision 𝐀is a diagonal ma ix. The p ocedu e is epea ed o all he candida e eg esso s un il he po en ial se is emp y wi h he esul o a spa se ac i e se which can be conside ed as he mos likely educed model. Fig. 2. E olu ion o he iden ifica ion NMSE wi h he numbe o ac i e coeffi- cien s o 2nd-, 4 h- and 8 h-o de models. The e olu ion o NMSE as he ac i e se is upg aded is shown in Fig. 2 o CPD s uc u es o 2nd-, 4 h- and 8 h-o de . The ansmi ed signal is a 256-QAM cons ella ion wi h IBO = 15 dB. I is ema kable ha , in he case o he 8 h-o de s uc u e, he po en ial se o 8192 eg es- so s is educed o a spa se se o S = 111 ac i e eg esso s wi h NMSE =−35dB. Once he spa se se o ac i e eg esso s has been iden ified, he es ima ed CPD coefficien s a e employed o p edis o he succes- si e OFDM symbols, p o iding he cons ella ion shown in Fig. 3b wi h an a e age EVM o 5%. The obus ness o he p oposed me hod is il- lus a ed by applying he iden ified coefficien s o a new signal wi h a diffe en cons ella ion, in his case wi h a 64-QAM mapping. The con- s ella ion dis o ed by he LED nonlinea i y is shown in Fig. 3c and he cons ella ion a e he applica ion o he CPD is displayed in Fig. 3d. 4. Simula ion esul s The effec i eness o he p oposed p edis o e is e alua ed in a VLC sys em by means o Mon e Ca lo simula ions. We ha e conside ed a DCO-OFDM signal. The nonlinea LED (NL-LED) model p oposed in [29] has been assumed. Fo compa ison pu poses, we ha e also ob ained e- sul s employing he Rapp’s model wi h 𝑘=50[28] and wi h he memo y polynomial (MP)-based ime-domain DPD me hod. Rega ding he MP- based DPD, i was applied o he ime-domain signal e e ed o 𝑥(𝑡) in Fig. 1, as con en ionally [35–39]. The o e sampling ac o (OVS) assumed o he MP-based ime-domain DPD is he same as ha con- side ed in he NL-LED model. Diffe en alues o memo y dep h we e heu is ically es ed, being he bes pe o mance o he DPD p o ided by a memo y dep h o 5 samples. 4.1. EVM s IBO analysis Fig. 4shows he EVM cu es e sus IBO o a DCO-OFDM signal wi h an 𝑁FFT o 1024 subca ie s, a 64-QAM cons ella ion and a biasing poin a io o 0.5employing he NL-LED model o he ansmi e in diffe en scena ios: wi hou a p edis o e , wi h ou p oposed p edis o e (CPD), and wi h he MP-based ime-domain DPD me hod, using diffe en o de s (4 h, 5 h, and 6 h). Resul s employing he Rapp’s model wi h 𝑘=50 o he LED a e also shown o compa ison pu poses. Each poin on all he cu es is ob ained by ansmi ing 1,000 OFDM symbols. To compensa e o he effec s o he s a ic and linea componen s o he sys em’s ans e unc ion, an ideal one- ap OFDM equalize is used in all he cu es. F om Fig. 4, i is e iden ha he implemen a ion o he p oposed p edis o e significan ly enhances he EVM compa ed o he sys em Digi al Signal P ocessing 165 (2025) 105310 5 A.C. O ia O ia, J.A. Bece a, M.J. Made o-Ayo a e al. Fig. 3. Cons ella ion in ecep ion in diffe en scena ios o an IBO o 12 dB: (a) 256-QAM, VLC sys em wi hou p edis o ion. (b) 256-QAM, VLC sys em wi h he p oposed p edis o ion block. (c) 64-QAM, VLC sys em wi hou p edis o e . (d) 64-QAM, VLC sys em wi h he p oposed dis o ion block. wi hou a p edis o e , o wi h he MP DPD. Fo example, o an EVM o app oxima ely 7%, he 5 h-o de CPD o highe only equi es an IBO o 10 dB, while he sys em wi hou a p edis o e equi es a leas 14.7 dB. This ep esen s imp o emen s in he powe efficiency o he sys em o 4.7dB. In he case o implemen ing an MP DPD, he equi ed IBO o an EVM o 7% is 12.5 dB, which implies ha ou p edis o e offe s a 2.5dB imp o emen in he powe efficiency ega ding o he MP. No e ha 5 h-o de CPD p esen s he same pe o mance han CPDs o highe o de s. E en he 4 h-o de CPD, which pe o ms wo se han highe o de s o IBO alues lowe han 16 dB, p esen s an imp o emen o 2dB o a 7% EVM compa ed o he case wi hou a p edis o e . Fig. 4shows ha he MP p edis o e pe o ms conside ably wo se ac oss he en i e ange o IBO alues examined. Only he 4 h-o de CPD exhibi s wo se pe o mance han he MP o IBO alues below 13 dB. I is impo an o no e ha inc easing he o de o he MP algo i hm does no enhance i s pe o mance; in ac , highe o de s esul in a de imen al effec . This is due o o e fi ing and nume ical eg ession issues caused by he inc ease o he numbe o coefficien s wi h he o de o he MP. Compa ed o he esul s ob ained wi h he Rapp’s model (𝑘=50), he fi h o de CPD pe o ms be e o low IBO, ha is, he p oposed p edis o e handles he sa u a ion effec s o LED be e . Fo IBO alues highe han 15 dB, he DCO-OFDM signal is mainly wi hin he DR o he LED, whe e he Rapp’s model is pe ec ly linea , and he e o e he Rapp’s model p esen s a lowe EVM. Addi ionally, he cu e labeled as “NL-LED + CPD (5𝑡ℎ)(256-QAM)” om Fig. 4 ep esen s he EVM s IBO o a 256-QAM DCO-OFDM sig- Fig. 4. EVM cu es e sus IBO o a DCO-OFDM signal wi h 𝑁FFT = 1024 and a BR o 0.5. nal, bu wi h a CPD whose coefficien s ha e been ob ained o a signal employing a 64-QAM cons ella ion. Simila ly, he app oach was applied o he MP (see cu e labeled as “NL-LED + MP (5𝑡ℎ)(256-QAM)”). These Digi al Signal P ocessing 165 (2025) 105310 6 A.C. O ia O ia, J.A. Bece a, M.J. Made o-Ayo a e al. Fig. 5. EVM cu es e sus IBO o a DCO-OFDM signal wi h 𝑁FFT = 1024 and a BR o 0.5 o a 5 h-o de CPD wi h diffe en o e sampling a es. cu es demons a e ha he pe o mance o hese p edis o ion me hods is no dependen on he o de o he cons ella ion used in he ansmi - ed signal. Simila esul s ha e been ob ained o QPSK, 16-QAM, and e en 1024-QAM, which a e defined in he main VLC s anda ds [8–10]. Fig. 5shows he esul s o he analysis o EVM s IBO in he same scena io bu wi h diffe en o e sampling ac o s (OVS) o ou me hod. Al hough heo e ically he o e sampling ac o used in he CPD algo- i hm mus be a leas equal o he o de o he CPD, Fig. 5indica es ha an OVS o 4 is sufficien wi hou any pe o mance loss. This e- duces he equi ed ha dwa e esou ces o implemen ing he 5 h-o de CPD. Conside ing an o e sampling ac o o 2 could also be beneficial i a sligh inc ease in he EVM can be assumed. 4.2. EVM s BR analysis In he p e ious subsec ion, he EVM s IBO cu es we e analyzed o a fixed BR o 50% (BR=0.5). Nex , we will s udy he pe o mance o he CPD when diffe en alues o BR a e applied in he same sce- na ios. The BR pa ame e is closely ela ed o he biasing le el (𝐵𝐷𝐶 ) added o 𝑥(𝑡) o ob ain an unipola signal o he LED inpu . In o de o ob ain hese esul s, he IBO was fixed o 10 dB. Simila esul s we e also ob ained o he MP algo i hm. F om Fig. 6, i can be seen ha he minimum EVM is achie ed when he BR is 0.5in all scena ios. In his case, CPD educes he EVM by nea ly 62% compa ed o a DCO- OFDM signal wi hou p edis o e , whe eas MP DPD algo i hm only achie es a educ ion o app oxima ely 51%. Simila esul s a e ob ained o he emaining BR alues when using ou p edis o e . Fo example, o BR = 0.1, he pe cen age dec ease in EVM is 56.5%, and nea ly 49% o BR = 0.9. This analysis has also been ca ied ou o o he 𝑁FFT alues (64,128,256,512,2048, and 4096) ob aining simila esul s o 𝑁FFT = 1024, wi h a pe cen age dec ease be ween 40% and 58%. Addi ionally, no e ha o a BR u he away om 0.5, he effec s o signal sa u a ion a e mo e p onounced; howe e , he CPD esol es hese effec s be e han he Rapp’s model o he LED. As expec ed, om Fig. 6, i can be seen ha he 5 h-o de CPD is also sufficien . Fig. 6also illus a es ha he MP DPD algo i hm d as ically educes pe o mance when a BR close o he ex emes is used compa ed o ou me hod. Fo example, o BR=0.1 and BR=0.9, he MP educ ion pe - cen ages a e only 14% and 12%, espec i ely. I is impo an o no e ha he MP algo i hm does no effec i ely add ess he effec s o LED sa u a ion due o he limi ed ichness o he ma hema ical exp essions o i s basis unc ions in he ime domain, unlike ou me hod. Fig. 6. EVM e sus BR o a DCO-OFDM signal wi h 64-QAM cons ella ion and 𝑁FFT = 1024, and o IBO =10dB. Fig. 7. IBO e sus BR and IBO educ ion o EVM = 10%. Finally, Fig. 7shows he equi ed IBO alues o an EVM alue o 10% e sus BR, o all analyzed scena ios. No e ha he implemen a ion o he p oposed p edis o e in a VLC DCO-OFDM ansmi e educes significan ly he equi ed IBO o achie e an EVM o 10% in he ull ange o BR. I can be obse ed ha ou p oposed me hod achie es an IBO e- duc ion o up o 7.6dB. This educ ion allows ha he LED can ope a e close o i s maximum ou pu powe wi hou incu ing significan dis- o ions and inc easing he efficien use o he LEDs. 4.3. ICE analysis The e alua ion o he ICE pa ame e is in e es ing o know VLC sys ems efficiency, whe e illumina ion and communica ions ake place simul aneously. The esul s a e depic ed in Fig. 8, whe e ICE e sus BF cu es o a VLC sys em wi h a 5 h-o de CPD and a VLC sys em wi hou p edis o ion block a e shown. To ob ain hese cu es, he equi ed IBO o each BF poin has been configu ed such as i p o ides an EVM = 10%. I can be obse ed ha an imp o emen ega dless he BF is achie ed wi h he p oposed CPD o e he signal wi hou p edis o ion echnique. The imp o emen is g ea e he smalle he alue o he BF pa ame e . Digi al Signal P ocessing 165 (2025) 105310 7 A.C. O ia O ia, J.A. Bece a, M.J. Made o-Ayo a e al. Fig. 8. ICE e sus b igh ness ac o o EVM = 10%. 64-QAM cons ella ion and 𝑁FFT = 1024. 5. CPD compu a ional complexi y Nex , he compu a ional complexi y o he p oposal is compa ed o ha o he MP ollowing he Bachmann–Landau no a ion ope a ion, which measu es he numbe o complex mul iplica ions in he s eps in ol ed in he algo i hms, p o iding a clea pe o mance e alua ion based on he compu a ional cos o each ope a ion. On he one hand, he o al compu a ional complexi y o he CPD simplifies o 𝐶CPD =𝑂(OVS ⋅𝑛⋅log(𝑁FFT ⋅OVS)+(𝑛−1)⋅(OVS +𝑆 2 )),(12) whe e 𝑛s ands o he nonlinea o de , 𝑁FFT is he numbe o samples in he equency domain, OVS is he o e sampling ac o and 𝑆is he numbe o selec ed coefficien s in he model. On he o he hand, he compu a ional complexi y o he MP model is 𝐶MP =𝑂(OVS(𝑆⋅OVS +1 )⋅log(𝑁FFT ⋅OVS)+(𝑛−1)⋅OVS).(13) The compa ison be ween he compu a ional complexi ies o he CPD and MP models e eals key diffe ences. Bo h models sha e e ms in ol ing he o e sampling ac o and he loga i hmic complexi y associa ed wi h he FFT. Howe e , he complexi y o he CPD includes a e m which g ows linea ly wi h he o e sampling ac o , while he complexi y o he MP includes a e m which g ows quad a ically wi h his a iable. This makes he MP model mo e compu a ionally expensi e when he numbe o coefficien s and o e sampling ac o a e la ge. Addi ionally, while bo h models depend on he o de , he CPD in ol es a loga i hmic e m ha g ows wi h i , whe eas he MP has a mo e di ec linea dependence. O e all, he CPD is mo e efficien when he numbe o coefficien s is small, while he MP is mo e sensi i e o inc eases in his e m and he o e sampling ac o . 6. Conclusions In his pape , a no el FD-SBL algo i hm has been p oposed o lin- ea ize OFDM signals dis o ed by LEDs used in he ansmi e s o VLC sys ems. The p oposed p edis o ion echnique, CPD, ac s on he sig- nal cons ella ion. Fo his, a Bayesian pu sui in he equency domain has been applied o a MP model s uc u e o selec a spa se ac i e se o eg esso s, ollowing a simila p ocedu e as ha o he SBL in he ime domain. The linea iza ion pe o mance o he CPD has been demons a ed o diffe en cons ella ion o ma s and diffe en numbe s o subca ie s in a VLC DCO-OFDM sys em. Fi s ly, we ha e shown ha he CPD educes signal dis o ions, imp o ing EVM pe o mance. I has been demons a ed ha a fi h-o de CPD is sufficien , allowing o up o a 7.6 dB educ ion in he IBO o he DCO-OFDM signal o he same EVM compa ed o no using a p edis o e . Secondly, o a fixed IBO alue in bo h scena ios, i has been shown ha he VLC sys em in eg a ing he CPD achie es a pe cen age educ ion in EVM o nea ly 62% compa ed o he EVM ob ained wi h he sys em wi hou p edis o e . These im- p o emen s a e also measu ed in e ms o he ICE pa ame e , showing ha CPD enhance he efficiency o VLC sys ems. Decla a ion o compe ing in e es The au ho s decla e ha hey ha e no known compe ing financial in e es s o pe sonal ela ionships ha could ha e appea ed o influence he wo k epo ed in his pape . Da a a ailabili y The da a ha has been used is confiden ial. Re e ences [1] H. Haas, J. Elmi ghani, I. Whi e, Op ical wi eless communica ion, Philos. T ans. R. Soc., A 378 (2020) 20200051, h ps://doi.o g/10.1098/ s a.2020.0051. [2] C.W. Chow, Recen ad ances and u u e pe spec i es in op ical wi eless communi- ca ion, ee space op ical communica ion and sensing o 6G, J. Ligh wa e Technol. 42 (11) (June, 2024) 3972–3980, h ps://doi.o g/10.1109/JLT.2024.3386630. [3] H. 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Zhou, Adap i e pos dis o ion o nonlinea LEDs in isible ligh communica ions, IEEE Pho onics J. 6 (4) (2014) 1–8, h ps://doi.o g/ 10.1109/JPHOT.2014.2331242. [38] W. Zhao, Q. Guo, J. Tong, J. Xi, Y. Yu, P. Niu, X. Sun, O hogonal polynomial-based nonlinea i y modeling and mi iga ion o LED communica ions, IEEE Pho onics J. 8 (4) (2016) 1–12, h ps://doi.o g/10.1109/JPHOT.2016.2581485. [39] Z. Du, e al., Enhanced pe o mance o an indoo non-line-o -sigh VLC sys em u i- lizing a mul i-pixel pho on coun e and in e lea ed single-ca ie FDM scheme, Op . Commun. 554 (2024) 130179, h ps://doi.o g/10.1016/j.op com.2023.130179. Ana Cin a O ia O ia was bo n in Huel a, Spain. She ecei ed his Mas e and Ph.D. deg ees in Telecommunica ion Enginee ing in 2005 and 2010, bo h om he Uni e si y o Se ille. Since 2005, she has been wi h he Depa men o Elec onic Enginee ing, High School o Enginee ing, Uni e si y o Se ille. He cu en esea ch in e es is in mul ica ie sys ems and digi al signal p ocessing. Juan An onio Bece a (STM’12-M’18-SM’19) ob ained his B.Sc. and M.Sc. deg ees in Telecommunica ion Enginee ing om he Uni e sidad de Se illa, Se ille, Spain, in 2009 and 2012, espec i ely. He u he pu sued his Ph.D. in Elec ical and Compu e Enginee ing a he Uni e si y o Delawa e, Newa k, DE, USA, comple ing i in 2017. Si- mul aneously, he ea ned a Ph.D. in Telecommunica ion Enginee ing om he Uni e sidad de Se illa in 2019. Since 2017, he has been wi h he Depa men o Signal Theo y and Communica ions, Uni e sidad de Se illa, cu en ly se ing as an Associa e P o esso . His main esea ch a eas include beha io al modeling and linea iza ion o powe amplifie s, and comp essed-sensing signal p ocessing. Ma ía J. Made o-Ayo a (STM’06-M’09-SM’20) ecei ed he M.Sc. and Ph.D. deg ees in Telecommunica ion Enginee ing in 2002 and 2008, espec i ely, om Uni e sidad de Se illa, Se ille, Spain. Since 2003, she has been wi h he Depa men o Signal Theo y and Communica ions, Uni e sidad de Se illa, whe e she is cu en ly an Associa e P o esso . He main esea ch a eas include compensa ion o impai men s in modula o s and powe amplifie s, and measu emen echniques o nonlinea communica ion sys ems. Vicen e Baena-Lecuye was bo n in A his-Mons, F ance. He ecei ed his Telecom- munica ion Enginee ing and Ph.D. deg ees om he Uni e si y o Se ille (Spain) in 1997 and 2001, espec i ely. Since 1997, he has been wi h he Depa men o Elec onic Engi- nee ing, High School o Enginee ing, Uni e si y o Se ille. His cu en esea ch in e es s include digi al signal p ocessing and mul ica ie communica ions sys ems. Ca los C espo-Cadenas (M’93-SM’15-LM’22) was bo n in Mad id, Spain. He ecei ed he deg ee in physics om he Uni e si y o Ha ana, Ha ana, Cuba, in 1973, and he Ph.D. deg ee om he Poly- echnique Uni e si y o Mad id, Mad id, in 1995. Since 1995, he has been wi h he Depa men o Signal Theo y and Communica ions, Uni e sidad de Se illa, whe e he is cu en ly a e i ed P o esso and Hono a y Resea che . His esea ch in e - es s include he communica ion sys ems, nonlinea analysis o ac i e mic owa e de ices, powe amplifie beha io al modeling, and linea iza ion echniques.