Digi . Signal P ocess. 165 (2025) 105310
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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
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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.
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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.
