1

Complete Modeling of Nonlinear Distortion inOFDM-based Optical Wireless Communication

Dobroslav Tsonev, Sinan Sinanovic and Harald Haas

Abstract—This paper presents a complete analytical frame-work for modeling memoryless nonlinear effects in an intensitymodulation and direct detection optical wireless communicationsystem based on orthogonal frequency division multiplexing. Thetheory employs the Bussgang theorem, which is widely acceptedas a means to characterise the impact of nonlinear distortionson normally-distributed signals. The current work proposes anew method to generalise this approach, and it describes howa closed-form analytical expression for the system bit error ratecan be obtained for an arbitrary memoryless distortion. Majordistortion effects at the transmitter stage such as quantisationand nonlinearity from the light-emitting-diode are analysed. Fourknown orthogonal-frequency-division-multiplexing-based modu-lation schemes for optical communication are considered inthis study: direct-current-biased optical OFDM, asymmetricallyclipped optical OFDM, pulse-amplitude-modulated discrete mul-titone modulation, and unipolar orthogonal frequency divisionmultiplexing.

Index Terms—Wireless communication, nonlinear distortion,OFDM, optical modulation.

I. I NTRODUCTION

W IRELESS data rates have been growing exponentiallyin the past decade. According to some recent forecasts,

in 2015 more than 6 Exabytes of wireless data would berequired per month [1]. The continuously enhanced wirelesscommunication standards will not be able to fully satisfy thefuture demand for mobile data throughput because the avail-able radio frequency (RF) spectrum is very limited. Hence, anexpansion of the wireless spectrum into a new and largelyunexplored domain – the visible light spectrum – has thepotential to change the face of future wireless communications.The advantages of an optical wireless system include amongothers: 1) vast amount of unused bandwidth; 2) no licensingfees; 3) low-cost front end devices; and 4) no interference withsensitive electronic systems. In addition, the existing lightinginfrastructure can be used for the realisation of visible lightcommunication (VLC).

Optical wireless communication (OWC) using incoherentoff-the-shelf illumination devices, which are the foremostcandidates for mass-produced front-end elements, is realisable

Manuscript received September 01, 2012; revised February 01, 2013 andJune 04, 2013; accepted July 30, 2013.

The authors gratefully acknowledge support for this work from the UKEngineering and Physical Sciences Research Council (EPSRC) under grantEP/I013539/1.

D. Tsonev and H. Haas are with The University of Edinburgh, Edinburgh,EH9 3JL, UK, (e–mail:d.tsonev, h.haas@ed.ac.uk.).

S. Sinanovic is with Glasgow Caledonian University, Glasgow, G4 0BA,UK, (e–mail: sinan.sinanovic@gcu.ac.uk).

Copyright c© 2013 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

as an intensity modulation and direct detection (IM/DD)system. This means that only signal intensity can be detectedreliably. Hence, without modification it is not possible to useall digital modulation techniques known in RF communica-tion. Unipolar techniques like on-off keying (OOK), pulse-position modulation (PPM), and pulse-amplitude modulation(M -PAM) can be adopted in a relatively straightforwardway. As transmission rates increase, however, unwanted in-tersymbol interference (ISI) appears. Hence, more resilienttechniques such as orthogonal frequency division multiplexing(OFDM) are preferred. OFDM allows equalisation to beperformed with single-tap equalisers in the frequency domain,which reduces design complexity and equalisation cost. It alsoallows different frequency subcarriers to be adaptively loadedwith information according to the channel characteristics.This enables more optimal usage of the channel, especiallywhen attenuation or interference is significant in certain fre-quency bands [2]. Conventional OFDM signals are bipolar andcomplex-valued. However, they have to be both real and unipo-lar in IM/DD systems. It is possible to transform an OFDMsignal into a real signal by imposing Hermitian symmetryon the subcarriers in the frequency domain. Furthermore, anumber of possible approaches to deal with the issue ofbipolarity in OFDM signals have been proposed. The currentwork focuses on four of them, in particular: direct-current-biased optical OFDM (DCO-OFDM), asymmetrically clippedoptical OFDM (ACO-OFDM) [3], pulse-amplitude-modulateddiscrete multitone modulation (PAM-DMT) [4], and unipolarorthogonal frequency division multiplexing (U-OFDM) [5].Itanalytically characterises their performance in a nonlinear ad-ditive white Gaussian noise (AWGN) channel which is typicalfor an OWC system. Some of the schemes - DCO-OFDM andACO-OFDM - have already been analysed in the context ofcertain nonlinearities that are present in OWC [6]–[10]. Thecurrent work gives a more complete analysis encompassingthe joint effect of a number of different distortions which tothe best of the authors’ knowledge have not been analysedjointly in OWC and have never been analysed for PAM-DMTand U-OFDM. An interesting observation is that the conceptspresented for U-OFDM have been previously introduced inFlip-OFDM [11]. In addition, the four schemes, ACO-OFDM,PAM-DMT, U-OFDM and Flip-OFDM, perform equivalentlyin a simple AWGN channel [3]–[5], [11].

An information signal in an OWC system undergoes anumber of distortions, including nonlinear ones. Linear dis-tortions such as attenuation and ISI can be compensatedwith amplifiers, equalisers and signal processing. Nonlineardistortions, however, often make irreversible changes to thesignal. Therefore, it is necessary to be able to characterise

2

Fig. 1. Optical Wireless Communication System

and evaluate distortion effects as fully as possible. Suchexamples include quantisation effects in the digital-to-analogconverters (DACs) and analog-to-digital converters (ADCs), aswell as the effects caused by the nonlinear output characteristicof a light emitting diode (LED). A number of works havebeen published on distortion in OFDM-based modulationschemes [6]–[10], [12]–[16]. A significant number of themfocus specifically on the nonlinear distortions present in anOWC system [6]–[10]. The analysis of nonlinear distortionis not straightforward. Even though general procedures havebeen introduced for obtaining an analytical solution [12],aclosed-form solution is not always available. The current workdescribes a complete general procedure whichalways leads toa closed-form solution. It can be used to solve the problemspresented in [6]–[10] as well as to analyse any other arbitrarymemoryless nonlinear distortion, which can be part of an OWCsystem based on OFDM. The specific case study in this paperinvolves a complete set of the significant nonlinear effectsatthe transmitter combining distortion due to quantisation at theDAC as well as distortion due to the nonlinear characteristicof the LED. Pulse shaping has also been considered unlike inprevious works on the subject [6]–[10].

The rest of this paper is organised as follows. SectionII provides a description of the OWC system. Section IIIdescribes the modulation schemes under investigation. Sec-tion IV discusses the issues incurred by pulse shaping tech-niques in IM/DD transmission systems. Section V introducesthe expected nonlinearities in an OWC system. Section VIpresents the theoretical approach for obtaining a closed-formassessment of the performance. Section VII confirms analyticalsolutions with numerical simulations. Finally, section VIIIprovides concluding remarks.

II. OWC SYSTEM

The diagram of an OWC system is presented in Fig. 1. Theincoming bits are divided into data chunks and mapped tosymbols from a known modulation scheme such as quadra-ture amplitude modulation (M -QAM) or M -PAM. The M -QAM/M -PAM symbols are modulated onto different fre-quency subcarriers according to one of the following schemes:DCO-OFDM, ACO-OFDM, PAM-DMT, U-OFDM. Then, theresulting time domain signal is subjected to a number ofpredistortion techniques, which condition it for transmission.This block includes oversampling, pulse shaping as well as

clipping any values below the allowed minimum or above theallowed maximum. Clipping is performed because a DAC,an amplifier, and an LED can only operate in a limitedrange, specified by their electrical properties. The conditionedsignal is fed to a DAC which outputs an analog signal. Thisstage of the system consists of a zero-order-hold element orother type of interpolator followed by a low pass filter. Theoutput signal from the zero-order hold is continuous in time.However, because the signal has discrete amplitude levels,corresponding to the samples of the oversampled pulse-shapedand clipped signals′[t], it is analysed in terms of the discretetime-domain signals′[t]. It is assumed that the oversamplingis sufficient, and the pulse shaping operation is such that thelow-pass filter after the zero-order hold outputs a continuous-time signal which is equivalent to the signal at its input forall practical considerations. Hence, in the analysis, nonlineartransformations of the signals′[t] are investigated. The analogoutput of the DAC is encoded into a current signal by avoltage-to-current transducer with appropriate bias and sup-plied to the LED. OFDM-based OWC with incoherent off-the-shelf illumination devices can only be realised as a basebandcommunication technique. Therefore, frequency upconversionis not required and, thus, has not been considered in thepresented analysis. Light intensity at the diode varies withthe current. At the receiver side, a photo diode transforms thevariations in the intensity of the received light into variationsof a current signal, which is turned into a voltage signal bya transimpedance amplifier. The resulting signal is discretisedat an ADC and passed on to the processing circuitry, whichincludes a matched filter, an OFDM demodulator with anequaliser, as well as a bit demodulator.

III. OFDM M ODULATION SCHEMES

The modulation schemes presented in this paper are mod-ifications of conventional OFDM. The subcarriers in thefrequency domain are modulated withM -QAM symbols inthe case of DCO-OFDM, ACO-OFDM, and U-OFDM andwith M -PAM symbols in the case of PAM-DMT. A time-domain block of samples is obtained by taking the inversefast Fourier transform (IFFT) of a block ofNFFT complexM -QAM/M -PAM symbols. Hermitian symmetry is imposed,which, according to the properties of the Fourier transform,generates a real time-domain signal [17]. The subcarriers atpositionsk=0 andk=

NFFT

2 are set to zero in order to satisfy

3

the requirements of the Hermitian symmetry. The real time-domain signal is bipolar in nature. An LED can convey onlypositive signals when it is active. Hence, the following fourdifferent methods have been designed for the generation ofunipolar signals, suitable for OWC.

A. DCO-OFDM

DCO-OFDM generates a unipolar signal by introducing aDC bias. Fig. 2 illustrates the concept. The spectral efficiencyof the scheme is:

ηDCO

=log2(M)(N

FFT− 2)

2(NFFT

+ Ncp)bits/s/Hz (1)

provided that all available carriers are loaded withM -QAM.The factorsN

FFT−2 and 0.5 occur due to the Hermitian

symmetry requirement.Ncp is the length of the cyclic prefix.

0 2 4 6 8 10 12 14 16

−2

−1

0

1

2

n

s[n]

(a) Bipolar OFDM before biasing.

0 2 4 6 8 10 12 14 16−1

0

1

2

3

4

n

s[n] bias

(b) Biased DCO-OFDM.

Fig. 2. DCO-OFDM Generation. Cyclic prefix is not illustrated.

OFDM has a very high peak-to-average power ratio (PAPR).Following the calculations presented in [18], a lower boundforthe PAPR of real signals can be calculated as follows:

3Na(√

M − 1)

2√

M + 1(2)

whereNa is the number of modulated carriers in the frequencydomain. Therefore, it is impractical to introduce a biasinglevelwhich ensures that all possible time samples are positive. Inaddition, electronic elements have an operational range, whichis limited both in terms of a minimum and a maximum value.Hence, an OFDM signal would be clipped both from aboveand from below in order to fit within the required range. Atypical value of a few signal standard deviations is used inpractice for clipping on each side of the signal distribution.This distortion is easily modeled by the upper and lower limitof the DAC. This modeling approach is adopted in the currentstudy.

B. ACO-OFDM

Biasing in DCO-OFDM increases the dissipated electricaland optical energy at the transmitter by a substantial amount.The dissipated electrical energy is proportional toE[i2(t)] =E[(isignal(t) + ibias(t))

2], and the dissipated optical energyis proportional toE[i(t)] = E[isignal(t) + ibias(t)], whereE[·] denotes statistical expectation. ACO-OFDM, illustratedin Fig. 3, avoids the biasing requirement of DCO-OFDM byexploiting the properties of the Fourier transform so that aunipolar signal can be generated without biasing. As presentedin [3], only odd frequency subcarriers are modulated. This

creates a symmetry between samples in the time-domainOFDM frame. In general, ifs(k, n) is the contribution ofsubcarrierS[k] to the sample at timen, then [3]:

s(k, n) =1

√

NFFT

S[k]ej2πnkN

FFT

s(k, n +N

FFT

2)=

1√

NFFT

S[k]ej2π(n+N

FFT/2)k

NFFT =

=1

√

NFFT

S[k]ej2πnkN

FFT ejπk. (3)

For odd values ofk, s(k, n) = −s(k, n + NFFT

/2). For evenvalues ofk, s(k, n) = s(k, n + N

FFT/2). Hence, if only the

odd subcarriers in an OFDM frame are modulated, the time-domain signal,s[n], has the property:

s[n] = −s[n + NFFT

/2]. (4)

If only the even subcarriers in an OFDM frame are modulated,the time-domain signal has the property:

s[n] = s[n + NFFT

/2]. (5)

Because complex exponential functions are orthogonal to eachother, if a signal has the property in (4), this means that inthe frequency domain only odd samples contain information.Similarly, if a signal has the property in (5), then in thefrequency domain only its even samples contain information.Clipping the negative samples of an arbitrary time-domainsignal,s[n], can be represented as:

CLIP(s[n]) =1

2(s[n] + |s[n]|). (6)

0 2 4 6 8 10 12 14 16

−2

−1

0

1

2

n

s[n]

positivenegative

(a) ACO-OFDM before clipping.

0 2 4 6 8 10 12 14 16

−3

−2

−1

0

1

2

3

n

s[n]

positivenegative

(b) ACO-OFDM after clipping.

Fig. 3. ACO-OFDM Generation. Cyclic prefix is not illustrated.

In ACO-OFDM, only the odd subcarriers are modulated.Hence, (4) applies. Therefore,s[n]= − s[n + N

FFT/2]. Then

|s[n]|=|s[n + NFFT

/2]|. This symmetry allows all negativevalues to be removed. The clipping distortion|s[n]|, describedin (6), has the property stipulated in (5), and so it distortsonly the even subcarriers. The factor0.5 occurs from theclipping and is consistent with the analysis presented in[3]. An additional factor of

√2 is introduced to rescale the

unipolar signal and normalise the amount of dissipated energy,which would lead to an overall signal-to-noise ratio (SNR)performance penalty of3dB. This short proof describes theanalysis in [3] in a more concise manner. Not using theeven subcarriers sacrifices about half the spectral efficiency,compared to DCO-OFDM, and it becomes:

ηACO

=log2(M)N

FFT

4(NFFT

+ Ncp)bits/s/Hz. (7)

4

C. PAM-DMT

In PAM-DMT, illustrated in Fig. 4, the frequency subcarri-ers in an OFDM frame are modulated with imaginary symbolsfrom the M -PAM modulation scheme. Due to Hermitiansymmetry in the frequency domain, the PAM-DMT time-domain signal becomes [4]:

s[n] =1

√

NFFT

Nfft−1∑

k=0

S[k]ej2πknN

FFT = (8)

=1

√

NFFT

NFFT

−1∑

k=0

S[k]

(

cos2πkn

NFFT

+ j sin2πkn

NFFT

)

=

=1

√

NFFT

NFFT

−1∑

k=0

jS[k] sin2πkn

NFFT

. (9)

The time-domain structure of a PAM-DMT frame exhibitsan antisymmetry wheres[0]=0, s[N

FFT/2]=0 if N

FFTis

even, ands[n]= − s[NFFT

− n]. This means that|s[0]|=0,|s[NFFT/2]|=0 if NFFT is even, and|s[n]|=|s[NFFT − n]|.Therefore, if the negative values are removed, as describedin (6), the distortion term|s[n]| has Hermitian symmetry inthe time domain. This means that in the frequency domain,the distortion is transformed into a real-valued signal. Hence,it is completely orthogonal to the useful information. Thisproof has not been formally completed in [4], but it isstraightforward with the representation of clipping in (6). Thespectral efficiency of PAM-DMT is:

ηPAM−DMT

=log2(M)(N

FFT− 2)

2(NFFT

+ Ncp)bits/s/Hz (10)

whereM denotes the order ofM -PAM modulation. It shouldbe noted that

√M -PAM has roughly the same performance

as M -QAM in an AWGN channel. This makes PAM-DMTcomparable to ACO-OFDM in spectral efficiency for the samebit error rate (BER) performance.

0 2 4 6 8 10 12 14 16−4

−2

0

2

4

s[n]

n

positvenegative

(a) Bipolar PAM-DMT before clip-ping.

0 2 4 6 8 10 12 14 16−4

−2

0

2

4

n

s[n]

positvenegative

(b) Unipolar PAM-DMT after clip-ping and rescale.

Fig. 4. PAM-DMT Generation. Cyclic prefix is not illustrated.

D. U-OFDM

In U-OFDM, all possible subcarriers in the frequency do-main are modulated as in DCO-OFDM. After the time-domainsignal is obtained, it is divided into two blocks: a positiveand anegative one. The positive block is a copy of the original signalframe, where all negative samples are set to zero. The negativeblock is a copy of the original signal frame, where all samplesare multiplied by−1 to switch signs. After this operation, thenegative samples are set to zero. The principle of how bothblocks form the original OFDM frame can be observed in Fig.5(a). The two blocks are transmitted separately. This can be

seen in Fig. 5(b). The cyclic prefixes are omitted in the givenexamples for simplicity of illustration. The increased numberof samples in the time domain decreases the spectral efficiencyby a factor of0.5 compared to DCO-OFDM, and it becomes:

ηU

=log2(M)(N

FFT− 2)

4(NFFT

+ Ncp)bits/s/Hz. (11)

At the demodulator, the original OFDM frame is obtainedby subtracting the negative block from the positive one.This effectively doubles the noise at each resulting sample,and so the performance of U-OFDM becomes the same asthe performance of ACO-OFDM and PAM-DMT where theclipping introduces an SNR penalty of3 dB.

−1 0 1 2 3 4 5 6 7 8 9−2

−1

0

1

2

ns[n]

positivenegative

(a) Bipolar OFDM.

0 2 4 6 8 10 12 14 16−2

−1

0

1

2

n

s[n]

positivenegative

(b) Unipolar U-OFDM.

Fig. 5. U-OFDM Generation. Cyclic prefixes are not illustrated.

IV. PULSE SHAPING

An LED is modulated with a continuous electrical signal,and it emits a continuously varying light signal. A digitalimplementation of OFDM generates discrete values whichneed to be encoded into an analog signal, suitable to modulatethe LED. The pulse shaping operation allows digital samplesto be mapped to continuous pulse shapes. The selection of thepulse-shaping filters is important because the communicationchannel restricts the bandwidth of the signals which canbe successfully propagated to the receiver. The maximummodulation frequency of off-the-shelf white LEDs is in theorder of2 MHz and in the order of20 MHz when blue fileringis applied at the receiver [19]. The transmitted informationsignals must be tailored to fit in that frequency range in orderto avoid distortion. Different pulse shapes have differenttime-domain properties as well as different bandwidth requirements.An example of a pulse shape is a square pulse which corre-sponds to the zero-order hold function of a DAC [20]. Thisshape is easy to implement and has a time duration which isperfectly limited within a symbol period. However, it requiresan infinite bandwidth. Therefore, it is not possible to realiseit without distorting the received signal. In practice, if squarepulses are used as an interpolation technique, the signal islow-pass-filtered afterwards to incorporate only a desired portionof the frequencies, for example, until the first zero crossingin the frequency domain. This occurs at1/Ts if Ts is thesymbol period. A similar shape is the triangle pulse, whichcorresponds to a first-order interpolation of discrete samples[20]. This shape is characterised with an improved bandwidthprofile and a longer time-domain duration compared to thesquare pulse. Exact recovery of the transmitted signal, withoutISI, requires accurate sampling at the receiver. Theoretically,the most bandwidth efficient interpolation filter is the sinc

5

function [20]. Its bandwidth requirement is1/2Ts, so it is twotimes more efficient than a square pulse low-pass-filtered atthe first zero crossing. However, it spans an impulse responsewith an infinite duration in the time domain. This meansthat in practice the shape is truncated, and due to the longerimpulse response, it requires more processing. In addition,time jitter can introduce significant ISI. For this reason, theraised-cosine filter and its modified version, the root-raisedcosine filter, are used in many practical implementations. Theyallow the generation of an interpolation pulse with an arbitrarybandwidth requirement between1/2Ts and1/Ts dependent onan adjustable roll-off factor. The raised cosine filter gives thedesigner the freedom to choose between the length of the pulsein the time domain and the frequency requirement of the shape.In practice, these filters are implemented by oversampling thediscrete signal, interpolating it with a discrete pulse shape andthen supplying it to the DAC, which typically consists of azero-order hold and a low-pass filter as described in Fig. 1.

0 10 20 30 40 50−1

0

1

2

3

4

5

(a) T

s(t)

0 10 20 30 40 50−1

0

1

2

3

4

5

(b) T

s(t)

bias

Fig. 6. (a) ACO-OFDM pulse-shaped after removing negative values. (b)Addition of necessary bias to make the pulse-shaped signal unipolar.

−1.5 −1 −0.5 0 0.5 1 1.5

−1.5

−1

−0.5

0

0.5

1

1.5

(a)ReS[k]

ImagS[k]

−1.5 −1 −0.5 0 0.5 1 1.5

−1.5

−1

−0.5

0

0.5

1

1.5

(b)ReS[k]

ImagS[k]

Fig. 7. (a) Distortion in ACO-OFDM/U-OFDM after clipping negative valuesof the pulse-shaped unipolar signal. (b) Distortion in PAM-DMT after clippingnegative values of the pulse-shaped unipolar signal.

Signals in ACO-OFDM, PAM-DMT, and U-OFDM are madeunipolar after clipping of the negative values. Bipolar pulseshapes like the sinc and the raised-cosine filter turn aunipolarsignal before pulse shaping into abipolar signal after pulseshaping. An example is presented in Fig. 6. The issue canbe solved by introducing a bias to account for the negativevalues. This leads to an increase in the energy consumptionof approximately 3 dB. Alternatively, the negative values afterpulse shaping can be clipped at zero again, but this leads tothe distortion presented in Fig. 7. Therefore, it is important toimplement pulse shaping before the negative values in ACO-OFDM, PAM-DMT, and U-OFDM are removed. Then, as de-scribed in (6), the clipping operation leads to a unipolar signal

that consists of the original bipolar signal and a distortion termwhich is present both inside and outside the desired bandwidth.The useful signal, however, remains within the bandwidth limitand is not affected by the distortion term as described inSection III. The rest of this section provides a proof.

The sampling frequency is denoted byFs, Ts denotes thesymbol period,T=FsTs denotes the discrete-time symbolperiod, andp[t] denotes the digital pulse shape. Then, the partof the oversampled pulse-shaped discrete bipolar signal, whichcontains the information of a given frame, is expressed as:

s′[t] =

NFFT

−1∑

n=NFFT

−Ncp

s[n, 0]p[t−(n−NFFT

)T ]+

NFFT

−1∑

n=0

s[n, 0]p[t−nT ]

+

NFFT

−Ncp+Npcp−1

∑

n=NFFT

−Ncp

s[n, 1]p[t−(NFFT

+Ncp)T−(n − NFFT

)T ]

(12)

where Npcp is the length of the cyclic prefix sufficient to

remove the effects ofp[t], ands[n, 0] indicates thenth sampleof the OFDM frame at position0, i.e., the current frame. Thediscrete-time pulse-shaped bipolar signal relevant for samplingthe first

NFFT

2 points of the current frame can be expressedas:

s′1[t] =

NFFT

−1∑

n=NFFT

−Ncp

s[n, 0]p[t−(n−NFFT

)T ] +

NFFT

/2−1∑

n=0

s[n, 0]p[t−nT ]

+

NFFT

/2−1+Npcp

∑

n=NFFT

/2

s[n, 0]p[t−nT ]. (13)

The discrete-time pulse-shaped bipolar signal relevant forsampling the next

NFFT

2 points of the current frame can beexpressed as:

s′2[t] =

NFFT

/2−1∑

n=NFFT/2−Ncp

s[n, 0]p[t−nT ] +

NFFT

−1∑

n=NFFT

/2

s[n, 0]p[t−nT ]

+

NFFT

−Ncp+Npcp−1

∑

n=NFFT

−Ncp

s[n, 1]p[t−(NFFT+Ncp)T−(n − NFFT)T ].

(14)

The discrete-time unipolar signal, ready for digital-to-analogconversion and transmission, can be expressed as:

s[t] =√

2CLIP (s′[t]) =1√2

(s′[t] + |s′[t]|) . (15)

At the receiver, the samples of a given frame after matchfiltering can be expressed as:

s[n] = (s[t] ∗ h[t] + n[t]) ∗ p[t]|t=nT

=

(

1√2

(s′[t] + |s′[t]|) ∗ h[t] + n[t]

)

∗ p[t]

∣

∣

∣

∣

t=nT

=

(

1√2

(s1′[t]+|s1

′[t]|) ∗h[t]+n[t])

∗p[t]∣

∣

∣

t=nT, n<

NFFT

2(

1√2

(s2′[t]+|s2

′[t]|) ∗h[t]+n[t])

∗p[t]∣

∣

∣

t=nT,

NFFT

2 ≤ n

(16)

6

where∗ denotes convolution,n[t] denotes AWGN, andh[t]denotes the impulse response of the channel in discrete time.The cyclic prefix is sufficient to remove ISI and to turn thecontinuous-time convolution with the channel,h(t), into acircular convolution indiscrete time with the channel,h[t].Assuming perfect channel knowledge, the equalised samplesat the receiver are expressed as:

sE[n] =

(

s[t] + n[t] ∗ h−1[t])

∗ p[t]∣

∣

t=nT

=

(

1√2

(s′[t] + |s′[t]|) + n[t] ∗ h−1[t]

)

∗ p[t]

∣

∣

∣

∣

t=nT

=

(

1√2

(s1′[t]+|s1

′[t]|)+n[t]∗h−1[t])

∗p[t]∣

∣

∣

t=nT, n<

NFFT

2(

1√2

(s2′[t]+|s2

′[t]|)+n[t]∗h−1[t])

∗p[t]∣

∣

∣

t=nT,

NFFT

2 ≤n.

(17)

A. ACO-OFDM

In ACO-OFDM, s[n] = −s[n + NFFT

/2]. Therefore,s′1[t] ≈ −s′2[t+

NFFT

2 T ] except for the third terms in (13) and(14). Differences appear because in the time domainp[t] spansbeyond a single symbol duration and beyond the boundaries ofthe OFDM frame. However, this effect is not significant whenNp

cp << NFFT

. As a consequence,|s′1[t]|≈∣

∣

∣s′2[t +

NFFT

2 T ]∣

∣

∣.

At the receiver, the distortion term in the firstN

FFT

2 points,1√2|s′1[t]| ∗ h[t] ∗ p[t]

∣

∣

∣

nT, is the same as the distortion term

in the secondN

FFT

2 points, 1√2|s′2[t]| ∗ h[t] ∗ p[t]

∣

∣

∣

nT, because

|s′1[t]|≈∣

∣

∣s′2

[

t +N

FFT

2 T]∣

∣

∣. Therefore, distortion falls on the

even subcarriers only as described in Section III-B.

B. PAM-DMT

In PAM-DMT, s[n] = −s[NFFT − n]. Therefore, with therepresentations in (13) and (14),s′1[t] ≈ −s′2[NFFT

T − t].Again, differences appear between the first term in (13) andthe third term in (14). The differences are caused by the time-domain span ofp[t] but are not significant forNp

cp << NFFT

.Hence, it can be concluded that|s′1[t]| ≈ |s′2[NFFT

T−t]|. Thepulse-shaping filter’s impulse response is an even function, sop[t] = p[−t]. From (17), the distortion term after equalisation

consists of 1√2|s′1[t]| ∗ p[t]

∣

∣

∣

t=nTand 1√

2|s′2[t]| ∗ p[t]

∣

∣

∣

t=nT.

It maintains Hermitian symmetry, and hence noise due todistortion is orthogonal to useful information as describedbefore in Section III-C.

C. U-OFDM

The U-OFDM bipolardiscrete signal is encoded in two con-secutive frame blocks. The oversampled discrete-time bipolarpart of s′[t] which contains the information of the positiveframe block can be expressed with the representation in (12)and denoted ass′p[t]. The oversampled discrete-time bipolarpart of s′[t] which contains the information of the negativeframe block can be expressed with the representation in (12)and denoted ass′n[t]. If sp[n] denotes the original bipolarsamples of the positive frame block, andsn[n] denotes theoriginal bipolar samples of the negative frame block, then by

designsp[n] = −sn[n]. Hence, a closer look at (12) shows thats′p[t] = −s′n[t] except for the third terms in the summation.The differences appear due to the time-domain span ofp[t] butare not significant whenNp

cp << NFFT

. Then, the samples ofthe positive frame block and the negative frame block aftermatch filtering at the receiver become respectively:

sp[n]=

(

1

2

(

s′p[t] + |s′p[t]|

)

∗ h[t] + n1[t]

)

∗p[t]

∣

∣

∣

∣

t=nT(18)

sn[n]=

(

1

2

(

s′n[t] + |s′n[t]|

)

∗ h[t] + n2[t]

)

∗p[t]

∣

∣

∣

∣

t=nT(19)

where n1[t] and n2[t] are two independent identically-distributed instances of the AWGN process. The bipolar sam-ples at the receiver can be reconstructed by subtracting thereceived samples of the negative frame block from the receivedsamples of the positive frame block:

sb[n] = sp[n] − sn[n]

=

(

1

2

(

s′p[t]−s′

n[t]

)

∗ h[t]+n1[t]−n2[t]

)

∗ p[t]

∣

∣

∣

∣

t=nT

.

(20)

The nonlinear distortion terms12 |s′p[t]| ∗ h[t] ∗ p[t]

∣

∣

t=nTand

12 |s′

n[t]| ∗ h[t] ∗ p[t]∣

∣

t=nTare equal and so are completely

removed by the subtraction operation. The noise doubles asdescribed in Section III-D.

An important implication of these proofs is that pulseshaping can also be incorporated in the analysis of the nonlin-ear distortions for ACO-OFDM, PAM-DMT, and U-OFDM.If pulse shaping is applied after clipping at zero, then thedistribution of samples in the time domain changes, and thatcompromises the accuracy of the analysis. OFDM samplesin the time domain follow a Gaussian distribution when thenumber of carriers is greater than64 [8], [10], [12]. Hence,the positive samples of ACO-OFDM, PAM-DMT, and U-OFDM follow the distribution of a Gaussian function, clippedat zero. Pulse shaping is a linear operation, which linearlycombines the discrete samples of an OFDM frame, scaledby the samples of a pulse-shaping filter. Linearly combiningsamples that follow a clipped Gaussian distribution resultsin samples that follow a different distribution. Hence, pulseshaping after clipping at zero produces a signal which cannotbe analysed with the Bussgang theorem. On the other hand,combining samples that follow a Gaussian distribution resultsin samples that again follow a Gaussian distribution. Hence, ifthe pulse shaping is done before the clipping at zero, the pulse-shaped samples follow a Gaussian distribution and after theclipping operation they follow a clipped Gaussian distribution.This enables the use of the Bussgang theorem and the analysispresented in this work.

V. NONLINEARITIES IN OWC

There are a number of possible sources of nonlinear dis-tortion in an OWC system. Electronic devices have limiteddynamic ranges and often nonlinear characteristics withinthe

7

dynamic range. Furthermore, transitions between the digitaland the analog domain lead to signal quantisation effects.

The processed digital time-domain signal needs to be passedthrough a DAC in order to obtain a signal, which can beused to drive the LED. An increase in the resolution of aDAC increases the design complexity and cost. Decreasing theresolution leads to signal quantisation. In addition, nonlineardistortion occurs from the limited range of the device whichleads to clipping. An accurate analysis of the distortion effectscaused to a signal allows for making informed choices betweencost, range and accuracy of the DAC.

LEDs and photodiodes (PDs) are another source of nonlin-earity. The voltage-current relationship at an LED is not linear.With the design of suitable V-to-I transducers, this transitionstep in the system can be almost completely linearised. How-ever, the relationship between the current through the LEDand the produced light intensity is also not linear. In addition,there is a minimum and a maximum allowed current levelso that the diode can operate properly. This means that theOFDM signal should be clipped at the processing step in orderto become suitable for transmission through the LED. Thesame effects due to the nonlinear relationships between lightintensity, current and voltage are present in the PD. However,this device operates in a much smaller range, which meansthat distortion is not as significant as in the LED.

VI. N ONLINEARITY ANALYSIS

For a large number of subcarriers,NFFT

> 64, an OFDMtime-domain signal can be approximated by a set of indepen-dent identically distributed random variables with a continuousGaussian distribution [8], [10], [12]. According to [12], anon-linear distortion in an OFDM-based system can be describedwith a gain factor and an additional noise component, bothof which can be explained and quantified with the help ofthe Bussgang theorem. IfX is a zero-mean Gaussian randomvariable andz(X) is an arbitrary memoryless distortion onX , then following the Bussgang theorem in [21] and Rowe’ssubsequent work in [22],

z(X) = αX + Y (21)

E[XY ] = 0. (22)

In these equations,α is a constant,E[·] stands for statisticalexpectation, andY is a noise component not correlated withX . Using (21) and (22),α can be derived as:

α =E[Xz(X)]

σ2x

(23)

whereσx is the standard deviation ofX . The noise componentY can be quantified as follows:

E[Y 2] = E[z2(X)] − α2σ2x (24)

E[Y ] = E[z(X)] (25)

σ2Y

= E[Y 2] − E[Y ]2 (26)

whereσ2Y

denotes the variance ofY . When the fast Fouriertransform (FFT) is applied at the system receiver, in the

frequency domain the noiseY is transformed into additiveGaussian noise due to the central limit theorem (CLT). Thevariance ofY in the frequency domain is againσ2

Y, and its

time-domain average contributes only to the0th subcarrier.Therefore, at each modulated subcarrier an additional zero-mean additive Gaussian noise component with varianceσ2

Y

is present. Overall the system experiences an increase in theadditive Gaussian noise power byσ2

Yand a decrease in the

useful signal power by a factor ofα2. This approach hasbeen used in a number of works to analyse nonlinearitiesin an analytical or semi-analytical fashion [8], [10], [12].The analytical solution, however, is not guaranteed to be ina closed form. Whenever a closed-form solution is desired,an additional step is required as the derivation needs to betailored to the respective nonlinear distortion function.In thecurrent paper, a general derivation approach which leads toa closed-form analytical solution with arbitrary accuracyforan arbitrary memoryless distortion function is proposed. Itis applicable to the four OFDM-based modulation schemesinvestigated in this work. The rest of this section introducesthe modified technique and describes how it can be applied toDCO-OFDM, ACO-OFDM, PAM-DMT, and U-OFDM.

An arbitrary distortion functionz(X) can be expressedas a set of intervalsI with cardinality |I| and a set ofcontinuous polynomials which accurately approximatez(X)in those intervals. The polynomials can be generated throughinterpolation of empirical data, or with a polynomial expansionof a function. The polynomial degree sets the accuracy of theapproximation. Thenz(x) can be represented as:

z(x) =

|I|∑

l=1

nl∑

j=0

cl,j

xj (U(x − xmin,l) − U(x − xmax,l)) (27)

where l denotes thelth interval, nl denotes the order ofthe polynomial in intervall, c

l,jdenotes thejth polynomial

coefficient in intervall, and U(x) is the unit step function.Moreover, xmin,l and xmax,l denote the lower and upperboundaries of intervall. Thenα can be calculated as:

α =E[Xz(X)]

σ2x

=1

σ2x

∫ ∞

−∞xz(x)

1

σxφ

(

x

σx

)

dx

=1

σ2x

∫ ∞

−∞

|I|∑

l=1

nl∑

j=0

cl,j

xj+1 (U(x − xmin,l)

−U(x − xmax,l))

1

σxφ

(

x

σx

)

dx

=1

σ2x

|I|∑

l=1

nl∑

j=0

cl,j

∫ xmax,l

xmin,l

xj+1 1

σxφ

(

x

σx

)

dx

(46)=

1

σ2x

|I|∑

l=1

nl∑

j=0

cl,j

dj+1D(t, xmin,l, xmax,l, 0, σx)

dtj+1

∣

∣

∣

∣

t=0

(28)

where functionD(t, a, b, µ, σx) is defined in the Appendix.The variance of the time domain signal,σ2

x, can be calculatedwith the following formula for DCO-OFDM and U-OFDM:

8

σ2x =

1

NFFT

NFFT

−1∑

j=0

log2(Mj)Ebj . (29)

For ACO-OFDM and PAM-DMT, the time-domain variancecan be calculated as:

σ2x =

2

NFFT

NFFT

−1∑

j=0

log2(Mj)Ebj (30)

whereMj is the size of the signal constellation andEbj is theenergy per bit at thejth subcarrier. The factor of two for ACO-OFDM and PAM-DMT results from the power rescaling afterclipping of the negative samples. It should be noted that afterpulse shaping, the variance of the oversampled signal is notconstant over time. However, the authors of [12] have shownthat for commonly used pulse shapes this does not influencethe validity of the analysis. Results in this work confirm thisfinding. The calculations necessary to complete the statisticaldescription ofY can be expressed as:

E[Y ] = E[z(X)] =

∫ ∞

−∞z(x)

1

σxφ

(

x

σx

)

dx

=

∫ ∞

−∞

|I|∑

l=1

nl∑

j=0

cl,j

xj (U(x − xmin,l)

−U(x − xmax,l))

1

σxφ

(

x

σx

)

dx

=

|I|∑

l=1

nl∑

j=0

cl,j

∫ xmax,l

xmin,l

xj 1

σxφ

(

x

σx

)

dx

(46)=

|I|∑

l=1

nl∑

j=0

cl,j

djD(t, xmin,l, xmax,l, 0, σx)

dtj

∣

∣

∣

∣

t=0

(31)

E[z2(X)] =

∫ ∞

−∞z2(x)

1

σxφ

(

x

σx

)

dx

=

∫ ∞

−∞

|I|∑

l=1

nl∑

j=0

cl,j

xj (U(x − xmin,l)

−U(x − xmax,l))

2

1

σxφ

(

x

σx

)

dx

=

|I|∑

l=1

nl∑

j=0

nl∑

k=0

cl,j

cl,k

∫ xmax,l

xmin,l

xj+k 1

σxφ

(

x

σx

)

dx

(46)=

|I|∑

l=1

nl∑

j=0

nl∑

k=0

cl,j

cl,k

dj+kD(t, xmin,l, xmax,l, 0, σx)

dtj+k

∣

∣

∣

∣

t=0

(32)

With the help of the Appendix and the standard differentiationrules, it is straightforward to obtain closed-form expressionsfor (28), (31) and (32). The procedure can easily be pro-grammed on a computer. The resulting SNR at each frequencysubcarrier can be calculated according to the following rela-

tionship:

Enewbj

σ2NY

=α2Ebj

σ2N

+ σ2Y

(33)

whereEnewbj is the resulting energy per bit of thejth subcarrier,

Ebj is the initial energy per bit of thejth subcarrier,σ2N

is thevariance of the channel AWGN, andσ2

NYis the overall noise

variance. Closed-form analytical expressions for the BER inM -PAM andM -QAM as a function of the SNR exist in theliterature [20].

The Bussgang analysis presented so far is valid for zero-mean signals with Gaussian distribution. All four opticalmodulation schemes – DCO-OFDM, ACO-OFDM, PAM-DMT, U-OFDM – are modifications of the original zero-mean OFDM signal and need to be treated with care for acorrect assessment of the nonlinearity effects. The analysisshould take into account all effects on the signal up to thepoint where an FFT operation is performed by the OFDMdemodulator at the receiver. Let’s assume thatz1(x) is amemoryless distortion at one stage of the system, for examplecaused by clipping the signal within the allowed range;z2(x)is a memoryless distortion at another stage, for example dueto quantisation at the DAC;z3(x) is a third memorylessdistortion, for example the addition of a bias level. Then,the overall distortion after the three separate consecutivedistortions isz(s′[t]) = z3(z2(z1(s

′[t]))). It does not matterwhether a distortion is linear or nonlinear. It can always beincorporated in the analysis if it is memoryless. This workassumes that there is no distortion with memory or any suchdistortion can be completely equalised, for example ISI.

For comparison purposes, the average electrical energy perbit Eb,elec and the average optical energy per bitEb,opt

dissipated at the transmitter are defined as:

Eb,elec =P avg

elec

Bη=

E[z2elec(s

′[t])]

Bη(34)

Eb,opt =P avg

opt

Bη=

E[zopt(s′[t])]

Bη(35)

where P avgelec is the average electrical power of the signal,

proportional to the mean square of the electrical signal;P avgopt

is the average optical power of the signal, proportional tothe average intensity of the optical signal;η is the respectivespectral efficiency defined for the various schemes in (1), (7),(10) and (11);zelec(s

′[t]) is the current signal at the diode;zopt(s

′[t]) is the light intensity signal at the diode;B is thesignal bandwidth in Hz. For the examples which are presentedin this work,zelec(s

′[t]) includes clipping, quantisation effectsat the DAC and biasing whilezopt(s

′[t]) includes the current-to-light output characteristic of the diode in addition.

A. DCO-OFDM

The proposed analysis can be applied to DCO-OFDM ina straightforward manner. The biasing of the signal can beconsidered as part of the nonlinear transform. It does not needto be added separately like it has been done in other works[8], [10].

9

B. ACO-OFDM

The modulation process of ACO-OFDM includes clippingat zero. This operation is a nonlinear transform, but does notaffect the odd carriers of the system. As a consequence, aslight modification needs to be made to the analysis in orderto account for this effect. The modified approach is describedin the rest of this section. In ACO-OFDM, the bipolar OFDMsignal before clipping consists of a set of positive samplesand a set of negative samples. The two sets have identicalcontribution to each modulated value in the frequency domain[3]. Therefore, setting one set to zero does not distort theuseful signal except for a factor of0.5. This means that thenonlinearity analysis can be conducted only on the positivesamples. The result is the same as if the effect of a symmetricaldistortion function on the bipolar OFDM signal is analysed.Hence, for the calculations, the intervals of the nonlineartransform can be specified from0 to ∞. Then (28), (31) and(32) can be calculated in the interval[0;∞] and scaled by 2to account for the negative half of the signal distribution.Theequations become:

α =2

σ2x

|I|∑

l=1

nl∑

j=0

cl,j

dj+1D(t, xmin,l, xmax,l, 0, σx)

dtj+1

∣

∣

∣

∣

t=0

(36)

E[Y ] = 2

|I|∑

l=1

nl∑

j=0

cl,j

djD(t, xmin,l, xmax,l, 0, σx)

dtj

∣

∣

∣

∣

t=0

(37)

E[z2(X)] =

= 2

|I|∑

l=1

nl∑

j=0

nl∑

k=0

cl,j

cl,k

dj+kD(t, xmin,l, xmax,l, 0, σx)

dtj+k

∣

∣

∣

∣

t=0

.

(38)

The removal of the negative samples does not influence theodd subcarriers, as previously explained in Section III. Thecalculated noise variance,σ2

Y, needs to be halved before

addition to the AWGN variance because the noise is evenlydistributed on both odd and even subcarriers. There is a0.5factor to the SNR, stemming from the removal of the negativesamples. Then for ACO-OFDM, (33) becomes:

Enewbj

σ2NY

=α2Ebj

2(σ2N

+σ2Y

2 ). (39)

It should be noted that the zeroes from clipped negativesamples need to preserve their value in order not to influencethe modulated subcarriers. The overall distortion of the signalup to the demodulator iszd(s

′[t]); this distortion does notinclude the addition of AWGN. Ifzd(0) 6= 0, then the zerosobtained from clipping the negative samples are distorted.Thedistortion on the clipped zero values in the time domain addsdistortion on the odd subcarriers in the frequency domain. Thiseffect is avoided when the clipped values are zero. Ifzd(0) isinterpreted as a DC shift and subtracted fromzd(s′[t]), thenthe modified Bussgang analysis described in this section canbe applied without having to additionally model the distortionresulting from the distorted clipped samples. Hence, the overall

distortion, experienced by the nonzero ACO-OFDM samplesis:

z(s′[t])s′[t]≥0 = zd(s′[t])s′[t]≥0 − zd(0). (40)

C. PAM-DMT

The time-domain signal of PAM-DMT has the same sta-tistical properties as the ACO-OFDM signal. In addition, thepower of the additive Gaussian noise from the nonlinearityis equally split between real and imaginary components inthe frequency domain. Therefore, the nonlinearity analysis ofPAM-DMT is exactly the same as for ACO-OFDM. It shouldbe kept in mind that PAM-DMT employsM -PAM while ACO-OFDM employsM2-QAM for the same spectral efficiency.In an additive Gaussian noise environment,M -PAM andM2-QAM perform identically in terms of BER.

D. U-OFDM

In U-OFDM, the signs of negative samples are switched,they are transmitted as positive samples and switched back atthe demodulator. As a result, the nonlinear distortion on thebipolar OFDM signal is symmetric around zero,i.e., z(s′[t]) =−z(−s′[t]). Any two bipolar samples with the same absolutevalue experience exactly the same nonlinear distortion. Hence,z(s′[t])s′ [t]≤0 is formed as a mirrored version ofz(s′[t])s′[t]≥0

according toz(s′[t]) = −z(−s′[t]). Alternatively, it can bestated that due to the symmetry of the Gaussian probabilitydensity function (PDF),z(s′[t]) can be specified only in theinterval[0;∞] for the calculations of (28), (31) and (32). Eachof the equations, however, needs to be doubled to account forthe negative half of the distribution as is the case for ACO-OFDM and PAM-DMT. The subtraction of the negative blockfrom the positive block effectively doubles the system AWGN.Hence (33) is modified as:

Enewbj

σ2NY

=α2Ebj

2σ2N

+ σ2Y

. (41)

It should be noted that in the demodulator the bipolar signalis obtained by subtracting the negative block from the positiveone. If all zeroes corresponding to clipped negative samplesfrom the original signal are transformed tozd(0)6=0, then thesubtraction operation shifts all positive samples by−zd(0)and all negative sampleszd(0). The functionzd(s

′[t]) is theoverall nonlinearity distortion which the signal experiences onits path to the demodulator. The subtraction operation in thedemodulation process effectively adds additional distortion,which needs to be accounted for by subtractingzd(0) fromzd(s′[t]), i.e., the resulting overall nonlinearity becomes:

z(s′[t])s′[t]≥0 = zd(s′[t])s′[t]≥0 − zd(0) (42)

as in ACO-OFDM and PAM-DMT.The analyses of ACO-OFDM, PAM-DMT, and U-OFDM

show that all three schemes experience exactly the sameSNR deterioration for the same nonlinearity effects. Thisoccurs because the three schemes exhibit the same statisticalproperties in the time domain when their spectral efficienciesare equivalent. This finding is also supported by the results

10

in the next section. Therefore, in addition to the equivalentperformance in a simple AWGN channel, the three schemesexhibit the same performance in a nonlinear AWGN channel.

VII. N UMERICAL RESULTS

In this section, the joint effect of three nonlinear distortionsat the OWC transmitter is illustrated. These include clippingto account for the limited dynamic range of the electronicdevices, quantisation from a low-resolution DAC, and thecurrent-to-light output characteristic of the LED. Nonlinear-ities at the receiver are not examined since they are negligiblein comparison to the ones present at the transmitter. Theycould be analysed in an analogous manner using the conceptspresented in Section VI. In the following case study, theoversampled pulse-shaped signals′[t] has an oversamplingratio of 10.

The zero-order hold has a limited range of output amplitudesbetweens′min and s′max, and a finite number of bitsq. Thedifferent levels of the device are equally spaced in the interval[s′min; s

′max] at a distance of:

dq =s′max − s′min

2q − 1. (43)

Each quantisation threshold is set in the middle between twoconsecutive quantisation levels. For example, the thresholdbetweens′min ands′min+dq is set ats′min+dq/2. The valuesof s′min and s′max are determined by the desired accuracyof the DAC and by the allowed operational range of theLED, [imin; imax]. Therefore, they account for the clippingeffects introduced by the DAC and the LED. In the conductedsimulations, the resolution of the DAC is set toq=8 bits.

After the signal values are quantised by the DAC outputfunction, they are biased by a constant and are passed throughthe LED. The LED output function is a continuous function,specified in the interval[imin; imax]. According to the devicedatasheet [23],imin=0.1 A and imax=1 A. The LED outputfunction corresponds to a transition from a current signal toan optical signal. The output characteristic of the LED hasbeen obtained through interpolation of data from the devicedatasheet. A third degree polynomial has been used in theinterpolation, and its coefficients are presented in Table I.The relationship between radiant flux (power) and luminousflux (power) is linear. Therefore, since only the relationshipbetween current through the device and luminous flux isavailable in the datasheet, it has been adopted as an accuraterepresentation of the relationship between the current andtheradiation power.

It is assumed that the modulating signal is contained in thecurrent signal through the LED. The initial average energyper bit of the original bipolar OFDM signals[n] is Eb. Theactual dissipated electrical energy per bit at the LED assumingthat the resistance is normalised to1Ω and including thequantisation effects and biasing isEb,elec = E[i2(t)]/(Bη) =E[(z3(z2(z1(s

′[t])))+ ibias)2]/(Bη). In this formula,z1(s

′[t])is the clipping effect in the preprocessing step applied on theoversampled pulse-shaped signals′[t]; z2(x) is the quantisa-tion effect of the DAC;z3(x) is the conversion from voltage tocurrent, which is assumed to be linear with gain1. In order to

evaluate the optical efficiency of the system, a third quantityis defined asEb,opt = E[z4(z3(z2(z1(s

′[t]))) + ibias)]/(Bη)wherez4(x) expresses the transition from current to opticalsignal in the LED, specified by the polynomial in Table I.

The modulation bandwidth of white-light LEDs is2 MHz[19]. The coherence bandwidth of the optical channel is around90 MHz [19], which is much higher than the modulationfrequency of the LED. Hence, ISI does not need to beconsidered. In an alternative scenario, where ISI is an issue, thepresented nonlinearity framework is still applicable as long aschannel knowledge is available at the receiver, and the signalcan be equalised. In the assumed system configuration, thereceived current signal can be expressed as:

i(t) = z4(i(t))hGγ (44)

wherehG

is the channel gain due to dispersion of light, andγ is the responsivity of the PD. Using the profile of the LEDlight spectrum and the PD responsivity to different opticalwavelengths [24], the receiver responsivity to white lightiscalculated asγ = 0.52. The channel gain depends on a numberof factors - distance, receiver area, angle with respect to thetransmitter. In the current work,h

Gis selected depending on

the M -QAM modulation order that is used. The aim is tooperate the LED in the full range of its active region becausethen the improvements of ACO-OFDM, PAM-DMT and U-OFDM over DCO-OFDM are demonstrated. In the cases whenthe optical signal varies only slightly around the biasing point,the energy consumption depends almost entirely on the biasinglevel, which makes the energy dissipation almost constant forthe four schemes.

The main source of AWGN in an OWC system is shot noiseat the photo detector caused by background light. The powerspectral density of shot noise isNo=10−21 W/Hz accordingto [19]. The variance of the AWGN is calculated asσ2

N=BNo.

Fig. 8 presents a working example which compares4-QAMDCO-OFDM,16-QAM ACO-OFDM,4-PAM PAM-DMT, and16-QAM U-OFDM – all with the same spectral efficiencyof 1 bit/s/Hz. The simulated channel gain ish

G=4×10−6

since this value allows all the presented schemes to reachBER values in the order of10−3 and 10−4 – required forsuccessfull communication [10], [19]. At the same time, thisvalue of the channel gain requires almost full utilisation ofthe LED active range. The bias levels for ACO-OFDM, PAM-DMT, and U-OFDM are set toibias=0.1 A since this isthe minimum biasing requirement of the LED. The selectedsimulation parameters set the minimum bias level for DCO-OFDM at ibias=0.19 A. The distribution region[−3σ; 3σ] isquantised for DCO-OFDM, and the region[0; 3σ] is quantisedfor the other three schemes. Anything outside those regionsis clipped. These clipping levels are chosen to be the defaultclipping levels as they introduce negligible nonlinear distortionaccording to simulations. Since the AWGN power is constant,higher SNR values can be achieved by amplifying the in-formation signal through the LED. However, if that signalfalls outside the allowed operational range of the device whenamplified, it is further clipped at the predistortion step inorderto satisfy the electrical properties of the LED. Hence, the

11

TABLE IPOLYNOMIAL COEFFICIENTScl,k IN INTERVAL l AND OF DEGREEk

Interval cl,3 cl,2 cl,1 cl,0

l = 1 ⇔ i(t) < imin 0 0 0 0.1947l = 2 ⇔ imin < i(t) < imax 0.2855 -1.0886 2.0565 -0.0003

l = 3 ⇔ imax < i(t) 0 0 0 1.2531

100 105 110 115 120 125 130 135 140

10−5

10−4

10−3

10−2

10−1

Eb

No

BE

R

DCO−OFDM Theory

DCO−OFDM Raised−cosine

DCO−OFDM Boxcar

ACO−OFDM Theory

ACO−OFDM Raised−cosine

ACO−OFDM Boxcar

PAM−DMT Theory

PAM−DMT Raised−cosine

PAM−DMT Boxcar

U−OFDM Theory

U−OFDM Raised−cosine

U−OFDM Boxcar

[dB]

(a) Comparison of bipolar signals. Bias not taken into account.

124 126 128 130 132 134 136 138 140 142

10−5

10−4

10−3

10−2

10−1

Eb,elec

No

BE

R

DCO−OFDM Theory

DCO−OFDM Raised−cosine

DCO−OFDM Boxcar

ACO−OFDM Theory

ACO−OFDM Raised−cosine

ACO−OFDM Boxcar

PAM−DMT Theory

PAM−DMT Raised−cosine

PAM−DMT Boxcar

U−OFDM Theory

U−OFDM Raised−cosine

U−OFDM Boxcar

[dB]

(b) Comparison of unipolar signals including contributionof bias to electricalpower dissipation.

82 83 84 85 86 87 88

10−5

10−4

10−3

10−2

10−1

Eb,opt

No

BE

R

DCO−OFDM Theory

DCO−OFDM Raised−cosine

DCO−OFDM Boxcar

ACO−OFDM Theory

ACO−OFDM Raised−cosine

ACO−OFDM Boxcar

PAM−DMT Theory

PAM−DMT Raised−cosine

PAM−DMT Boxcar

U−OFDM Theory

U−OFDM Raised−cosine

U−OFDM Boxcar

[dB]

(c) Comparison of unipolar signals in terms of optical powerrequirements.Bias contribution to power dissipation included.

Fig. 8. Comparison between4-QAM DCO-OFDM, 16-QAM ACO-OFDM,4-PAM PAM-DMT, and16-QAM U-OFDM in terms of electrical and opticalSNR.

BER curves presented in this section are V-shaped. The dip inthe plots emerges because after a certain point the distortiondue to clipping outweighs the improvement due to the SNRincrease. Fig. 8(a) shows the electrical energy efficiency ofthe four schemes when the energy dissipation due to biasingis neglected. As expected for bipolar signals, DCO-OFDMperforms better than the other three schemes. The benefitsof the latter, however, are due to the biasing requirement,

and this can be observed in Fig. 8(b). In that figure, ACO-OFDM, PAM-DMT, and U-OFDM require about1 dB lessenergy per bit than DCO-OFDM forBER=10−3 and about thesame energy forBER=10−4. According to Fig. 8(c), DCO-OFDM is about1 dB less optically efficient than the otherthree schemes forBER=10−3 and about0.3 dB less efficientfor BER=10−4. It is interesting to note that, as expected, theBER values decrease with the increase in SNR up to a certainpoint. Afterwards, the active region of the LED is exhausted,and the increase in power leads to increase in clipping andhence to more nonlinear distortion. It should also be noted thatthe theoretical analysis coincides well with the Monte Carlosimulation results for square pulse shapes (Boxcar filter).TheMonte Carlo results for a root-raised-cosine filter are slightlybetter than the other results because the match filter at thereceiver eliminates some of the distortion noise which fallsoutside the desired signal bandwidth. Nonetheless, the root-raised-cosine filter results follow closely the theoretical resultsand confirm validity of the new analytical framework. Forpractical purposes, the presented analysis can serve as a goodlower bound approximation for the performance of systemswhich employ pulse shapes with limited bandwidth. To thebest of the authors’ knowledge, other available techniquesfor analysis of nonlinear distortion in OWC do not considerbandlimited pulse shapes. They assume simple square pulses(Boxcar filter) [6]–[10] for which an exact analysis is providedin this paper. The curves for ACO-OFDM, PAM-DMT andU-OFDM fall almost on top of each other as predictedby the theoretical analysis. This highlights the very similarperformance of the three schemes.

ACO-OFDM, PAM-DMT, and U-OFDM exhibit better en-ergy efficiency than DCO-OFDM. However, they require abiggerM -QAM constellation size for the same spectral effi-ciency. Larger constellations are more vulnerable to distortion.This vulnerability can put the three schemes at a disadvantagein certain scenarios. Such an example occurs if the channelgain is decreased toh

G=2×10−6. In this case, the minimum

biasing requirement of DCO-OFDM becomesibias=0.5 A,and it is kept atibias=0.1 A for the other three schemes. Inspite of the worse performance in the previous simulation, inthis scenario DCO-OFDM becomes the better choice. It is stillable to achieve a BER of10−4, while all three other schemescannot even reach a BER of10−3. This example shows thata minor change of the system parameters such as a factor oftwo in the channel gain can be decisive in the selection of amodulation scheme.

Fig. 9 presents a case for a significantly higher channel gain,h

G=4×10−4. In this scenario, the information signal energy

requirement is small. Hence, the main contributor to energyconsumption is the biasing level. The bias of DCO-OFDM is

12

set toibias = 0.1018 A, which is the minimum required biasthat is able to accommodate the information signal withoutsevere clipping distortion. The bias of the other three schemesis kept at the minimum,ibias = 0.1 A. The electrical energyadvantage of ACO-OFDM, PAM-DMT, and U-OFDM overDCO-OFDM is in the order of0.06 dB while the difference inthe optical energy requirement is in the order of0.04 dB. Thefour schemes are almost equivalent in performance because themain contributor to energy dissipation is the bias of the LED.It should be noted that the electrical energy efficiency of ACO-OFDM, PAM-DMT, and U-OFDM is lower than presented inthe papers which originally introduced these concepts – [3],[4], [5]. The reason for this is that all clipped values whichareset to zero at the modulator actually cannot be lower than theminimum required current at the LED and so also contributeto the power dissipation.

127 127.1 127.2

10−4

10−3

10−2

10−1

100

Eb,elec

No

BER

DCO−OFDM

ACO−OFDM

PAM−DMT

U−OFDM

100 100.1 100.2

10−4

10−3

10−2

10−1

100

Eb,opt

No

BER

(b)(a) [dB] [dB]

Fig. 9. Comparison of4-QAM DCO-OFDM, 16-QAM ACO-OFDM, 4-QAM PAM-DMT, and 16-QAM U-OFDM for a high channel gain,hG=4×10−4, scenario: (a) Electrical power efficiency including bias current; (b)Optical power efficiency including bias level.

The analysed schemes achieve higher spectral efficiencywhen theM -QAM/M -PAM modulation order is increased.This leads to more variance in the time-domain signal asdescribed in (29) and (30), which for the same system pa-rameters leads to more distortion. At the same time, biggerconstellations are more sensitive to noise. Therefore, forhigherspectral efficiencies and assuming the same OWC systemparameters, the BER performance suffers from additionaldeterioration.

VIII. C ONCLUSION

A complete analytical framework has been presented forthe analysis of memoryless nonlinear distortion in an OWCsystem. It allows for the analysis of an arbitrary distortionfunction and guarantees closed-form solutions. The concepthas been successfully applied to four separate OFDM-basedmodulation schemes proposed for IM/DD systems: DCO-OFDM, ACO-OFDM, PAM-DMT and U-OFDM. Exampleshave been given for the joint distortion effects from quanti-sation at a DAC element, as well as for distortion from thenonlinear relationship between electrical current and emittedlight in an LED, which are the major sources of nonlinearityin an OWC system. Monte Carlo simulations show very goodagreement with the proposed theory, thus confirming validityof the approach.

Analytical derivations as well as numerical results exhibitequivalent performance of ACO-OFDM, PAM-DMT, and U-OFDM in a nonlinear AWGN channel. The findings suggestthat these schemes are three separate and equally valid ap-proaches with respect to spectrum efficiency and energy effi-ciency. A brief analysis demonstrates that the optimal choiceof a modulation scheme depends on the operating conditionsand can change with variations in the system parameters. Thepresented framework provides a quick and accurate way toestimate system performance without computationally expen-sive Monte Carlo simulations and numerical integration. Thus,it enables system optimisation as the influence of a largerange of system parameters can be evaluated exhaustively withreasonable computational complexity.

APPENDIX

This section presents the necessary formulas for derivationof a closed-form solution in the proposed novel analyticalframework. The formulas are defined as follows:

D(t, a, b, µ, σx) =

∫ b

a

ext 1√

2πσ2x

e− (x−µ)2

2σ2x dx

=

∫ b

a

1√

2πσ2x

e−−2xtσ2

x+x2−2µx+µ2

2σ2x dx

=

∫ b

a

1√

2πσ2x

e−x2

−2(tσ2x+µ)x+µ2

2σ2x dx

=

∫ b

a

1√

2πσ2x

e−x2

−2(tσ2x+µ)x+t2σ4

x+2tσ2xµ+µ2

−t2σ4x−2tσ2

xµ

2σ2x dx

= et2σ2

x2 +tµ

∫ b

a

1√

2πσ2x

e− x2

−2(tσ2x+µ)x+(tσ2

x+µ)2

2σ2x dx

= et2σ2

x2 +tµ

∫ b

a

1√

2πσ2x

e− (x−µ−tσ2

x)2

2σ2x dx

= et2σ2

x2 +tµ

(

Q

(

a − µ − tσ2x

σx

)

− Q

(

b − µ − tσ2x

σx

))

(45)

where Q(x) is the tail probability of the standard normaldistribution, andφ(x) is its PDF.

dnD(t, a, b, µ, σx)

dtn

∣

∣

∣

∣

t=0

(45)=

=dn

dtn

∫ b

a

ext 1√

2πσ2x

e− (x−µ)2

2σ2x dx

∣

∣

∣

∣

∣

t=0

=

∫ b

a

dn

dtnext 1

√

2πσ2x

e− (x−µ)2

2σ2x dx

∣

∣

∣

∣

∣

t=0

=

∫ b

a

xnext 1√

2πσ2x

e− (x−µ)2

2σ2x dx

∣

∣

∣

∣

∣

t=0

=

∫ b

a

xn 1√

2πσ2x

e− (x−µ)2

2σ2x dx (46)

13

dQ(

x−µ−tσ2x

σx

)

dt= φ

(

x − µ − tσ2x

σx

)

σx (47)

dφ(

x−µ−tσ2x

σx

)

dt= φ

(

x − µ − tσ2x

σx

) (

x − µ − tσ2x

σx

)

σx

(48)

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Dobroslav Tsonev(S’11) received the BSc degree in electrical engineeringand computer science in 2008 from Jacobs University Bremen,Bremen,Germany and the MSc degree in communication engineering with a spe-cialisation in electronics in 2010 from the Munich Institute of Technology,Munich, Germany. Currently, he is pursuing a PhD degree in electricalengineering at the University of Edinburgh. His main research interests liein the area of optical wireless communication with an emphasis on visiblelight communication.

Sinan Sinanovic(S’98-M’07) is a lecturer at Glasgow Caledonian University.He has obtained his Ph.D. in electrical and computer engineering from RiceUniversity, Houston, Texas, in 2006. In the same year, he joined Jacobs Uni-versity Bremen in Germany as a post doctoral fellow. In 2007,he joined theUniversity of Edinburgh in the UK where he has worked as a research fellowin the Institute for Digital Communications. While workingwith HalliburtonEnergy Services, he has developed acoustic telemetry receiver which waspatented. He has also worked for Texas Instruments on development of ASICtesting. He is a member of the Tau Beta Pi engineering honor society anda member of Eta Kappa Nu electrical engineering honor society. He won anhonourable mention at the International Math Olympiad in 1994.

Prof. Harald Haas (S’98-A’00-M’03) holds the Chair of Mobile Communi-cations in the Institute for Digital Communications (IDCOM) at the Universityof Edinburgh, Edinburgh, U.K., and he currently is the CSO ofa universityspin-out company pureVLC Ltd. His main research interests are in interferencecoordination in wireless networks, spatial modulation, and optical wirelesscommunication. He holds more than 23 patents. He has published more than50 journal papers including a Science article and more than 170 peer-reviewedconference papers. Nine of his papers are invited papers. Hehas co-authoreda book entitled Next Generation Mobile Access Technologies: ImplementingTDD (Cambridge, U.K.: Cambridge Univ. Press, 2008). Since 2007, he hasbeen a Regular High Level Visiting Scientist supported by the Chinese 111program at Beijing University of Posts and Telecommunications (BUPT). Prof.Haas was an invited speaker at the TED Global conference 2011, and his workon optical wireless communication was listed among the 50 best inventionsin 2011 in Time Magazine. In 2011 he has received a prestigious Fellowshipof the Engineering and Physical Sciences Research Council (EPSRC) in theUK.