IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH 2016 1167
Hadamard-Coded Modulation for Visible LightCommunications
Mohammad Noshad, Member, IEEE, and Maïté Brandt-Pearce, Senior Member, IEEE
Abstract—Visible light communication (VLC) systems using theindoor lighting system to also provide downlink communicationsrequire high-average optical powers to satisfy the illuminationneeds. This can cause high-amplitude signals common in higher-order modulation schemes to be clipped by the peak power con-straint of the light emitting diode (LED) and lead to high-signaldistortion. In this paper, we introduce Hadamard coded modula-tion (HCM) to achieve low error probabilities in LED-based VLCsystems needing high-average optical powers. This technique usesa fast Walsh–Hadamard transform (FWHT) to modulate the dataas an alternative modulation technique to orthogonal frequencydivision multiplexing (OFDM). HCM achieves a better perfor-mance for high-illumination levels because of its small peak toaverage power ratio (PAPR). The power efficiency of HCM can beimproved by reducing the DC part of the transmitted signals. Theresulting so-called DC-reduced HCM (DCR-HCM) is well suitedto environments requiring dimmer lighting.
Index Terms—Visible light communications (VLC), Hadamardmatrix, Walsh-Hadamard transform, orthogonal frequency divi-sion multiplexing (OFDM), peak-to-average power ratio (PAPR),LED nonlinearity.
I. INTRODUCTION
V ISIBLE light communications (VLC) is an emergingtechnology for indoor wireless networking that can offer
energy efficient Gbps streaming through the lighting system.The idea is to transmit downlink data by modulating white lightemitting diodes (LED) that are already being used by energyefficient and cost effective lighting systems. The unregulatedbandwidth available in VLC technology can relieve the traf-fic on radio-frequency (RF) communications. However, usingLEDs as sources adds restrictions on the modulation schemesand codes that can be used. Main limitations of these LEDs aretheir limited peak optical power, nonlinear transfer function,and limited modulation bandwidth [1]. Therefore, modulationand coding schemes with high spectral efficiencies are requiredto provide a high data-rate connection. In this work we proposea new modulation technique to achieve reliable and high-speeddata transmission in nonlinear-LED-based VLC systems.
Manuscript received June 18, 2015; revised October 29, 2015; acceptedDecember 30, 2015. Date of publication January 22, 2016; date of currentversion March 15, 2016. This work was supported by the U.S. Departmentof Energy (DOE) through the SBIR program under Grant DE-SC0013195.The associate editor coordinating the review of this paper and approving it forpublication was H. Haas.
M. Noshad is with VLNComm LLC, Charlottesville, VA 22903 USA(e-mail: [email protected]).
M. Brandt-Pearce is with the Charles L. Brown Department of Electrical andComputer Engineering, University of Virginia, Charlottesville, VA 22904 USA(e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCOMM.2016.2520471
Orthogonal frequency-division multiplexing (OFDM) is anefficient modulation technique for high speed data commu-nication through bandlimited channels, and is being widelyused in modern systems because of its high spectral effi-ciency and robustness against narrow-band interference [2].OFDM signals are generated by applying an inverse fast Fouriertransform (IFFT) on the data stream at the transmitter anddecoded using a fast Fourier transform (FFT) at the receiver.OFDM has been adapted to work in energy efficient opticalcommunications because of its better average-power efficiencycompared to other schemes [3]. Asymmetrically clipped opti-cal OFDM (ACO-OFDM), DC-biased optical OFDM (DCO-OFDM) and asymmetrically clipped DC biased optical OFDM(ADO-OFDM) are modified forms of OFDM proposed forintensity-modulation direct-detection (IM/DD) optical com-munication systems [4]–[6]. These OFDM techniques useHermitian symmetry to generate real signals from the datasequence, trading-off loss in the encoding rate. Due to the lowrate of these techniques, high order quadratic amplitude mod-ulation (QAM) has to be employed to achieve high spectralefficiencies, which degrades the energy efficiency.
As in the original OFDM, these optical communicationstechniques generate signals with large peaks, which can endup clipped by the peak optical power constraint of the opticalsources. This clipping causes a distortion of the OFDM sig-nals that becomes larger by increasing the transmitted averagepower. Consequently, in VLC systems where high average opti-cal powers are required for illumination, some forms of OFDMcan suffer from signal clipping. This problem can be alleviatedby using peak to average power ratio (PAPR) reduction tech-niques that trade-off complexity and energy inefficiency [7].For example, Hadamard matrices can be used as precoders inOFDM systems to decrease the PAPR [8], [9], reduce the BER[10] and increase the resistance of the signals against frequencyselective fading [11]. The challenge of supporting a wide rangeof dimming levels is another big drawback in the application ofthese modified forms of OFDM to VLC systems. There havebeen significant efforts to address this problem [12]. Reversepolarity optical-OFDM (RPO-OFDM) combines pulse widthmodulation (PWM) with OFDM to change the dimming levelof the transmitted signals [13].
Pulsed modulation techniques are another solution to achievereliable high-speed communication at high optical averagepower levels in VLC systems [14]. Among these techniques,those that use the optical sources in their on/off mode arepreferred since they avoid the nonlinear effects of the LEDs[15]. Although using sources only in their on/off mode lim-its the spectral efficiency of the system in single-LED systems
1168 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH 2016
[16], systems with multiple LED sources have the potential touse multilevel signaling, which can be employed to use theavailable bandwidth more efficiently [17]–[19].
In [20] we introduce a multilevel modulation techniquenamed HCM that uses the Hadamard matrices as a modulationtechnique (rather than a precoder). In this technique, the datais modulated using a fast Walsh-Hadamard transform (FWHT)and the receiver uses an inverse fast Walsh-Hadamard trans-form (IFWHT) to decode the received signals. Therefore, it hasa low complexity and can exploit the bandwidth effectively. Wealso propose a modification referred to as the DC-reduced HCM(DCR-HCM) technique. Because of their low PAPR, HCM andDCR-HCM can provide high illumination levels in VLC sys-tems without being too affected by LED-induced distortion.The technique is reminiscent of the approach in [21] to use thebipolar Hadamard transform for channel orthogonalization incode division multiple access (CDMA) systems because of theirlow PAPR.
In this paper we analyze these modulations and study theirperformance in LED-based VLC systems. We propose usingsinc pulses for efficient use of the available LED bandwidthand for fair comparison with OFDM. We show that using DCR-HCM the energy efficiency of HCM can be improved. Thisimprovement becomes more significant by increasing the sizeof the FWHT. DCR-HCM is also able to achieve lower BERlevels compared to HCM and OFDM due to its reduced DClevel, which decreases the amplitudes of the transmitted signalsand makes them less likely to be clipped by the peak-powerlimit of the LEDs. As in OFDM systems, a cyclic prefix is usedto avoid interference between adjacent symbols in bandlimitedchannels, and then symbol-length interleaving is applied on theHCM signals, as was proposed in [22], to decrease the effect ofthe intra-symbol interference. This approach is shown to lowerthe error probability of high data-rate transmissions throughVLC channels.
The rest of the paper is organized as follows. Section IIdescribes the VLC system model including the LED nonlineartransfer function and VLC channel characteristics. Section IIIintroduces the principles of HCM and presents modified formsof HCM to increase its energy efficiency and make it reni-tent against inter-symbol interference (ISI) in dispersive VLCchannels. Numerical results are presented in Section IV thatcompare the performance of the HCM and its modified formsto OFDM in VLC systems. Finally, conclusions are drawn inSection V.
II. PROBLEM DESCRIPTION
This section describes the principles of a VLC system.Models for the optical sources and VLC channel are discussed.
In this work we represent vectors with boldfaced lower-caseletters, and boldfaced upper-case letters are reserved for matri-ces. The identity matrix is represented by I. The notation AT
denotes the transpose of the matrix A, and x∗ indicates theconjugate of the complex number x . The notations A − x anda − x are respectively used to show the matrix and vector thatare obtained by subtracting a scalar x from all elements of thematrix A and vector a. We define the complement of the vector
a as a := (1 − a), and that of the matrix A as A := (1 − A). Inthis paper, a(�) and A(�) represent the �th right cyclic-shifts ofvector a and matrix A, respectively.
A. Illumination Requirements Effect
According to the Illuminating Engineering Society of NorthAmerica (IESNA), the standard illumination level for mostindoor environments (classroom, conference-room, lecture hall,offices, etc.) is between 300 and 500 lux at 0.8 m height fromthe floor [23, Table 32.I]. In the daytime, a portion of the indoorillumination needs could be provided by daylight, and the lightscan then be dimmed to reduce the energy consumed. The dim-ming level has a nonlinear relation to the average optical power[24], and affects the performance of the VLC system.
In order to evaluate the performance of a VLC link, it isessential to determine the optical power level that correspondsto a desired illumination level. This can be done using lumi-nance efficiency of radiation (LER), which is defined as theluminous flux per unit optical power. Although the theoreticallimit of LER for white LEDs is 260–300 lm/W [25], commer-cially available white LEDs have an LER of 50–150 lm/W. Oneof the most efficient currently available white LEDs is Cree’sXLamp XT-E white LED with an LER of 148 lm/W. Since1 lux = 1 lm/m2, an illumination level of 500 lux from a lightsource with an LER of 148 lm/W corresponds to 33.8 μW aver-age optical power on a photo-detector with an effective area of0.1 cm2.
B. Nonlinearity in Optical Sources
The optical source is a key component of any optical commu-nication system, as it generates optical power as a function ofthe modulated input electrical signal and converts the informa-tion into an optical beam. Because of the structure of LEDs,the output optical power and the forward current are relatedby a nonlinear function. The maximum optical power in thesesources is limited to a peak-power, and this can result in theclipping of large peaks in the modulated signal. In opticalcommunication systems using multilevel or continuous valuedsignaling, the nonlinearity of the optical sources introduces adistortion on the transmitted optical signal.
Predistortion is a solution to linearize the relation betweenthe output optical power and the forward current over a range.This technique requires an accurate model for the design ofthe predistortion and linearization over the dynamic range ofthe optical source. A polynomial model is presented in [26] todescribe the nonlinear transfer function of the optical sources,through which a predistortion function can be designed tolinearize the output optical power in terms of the input current.
A problem with the predistortion technique is that the non-linear transfer function of optical sources can change due tomany factors, one of which is the temperature of the transmit-ter. LEDs and lasers tend to dissipate a portion of the inputenergy as heat, which increases the temperature of the deviceover time and changes the nonlinear relation between the outputpower and forward current. This means that one predistortion
NOSHAD AND BRANDT-PEARCE: HADAMARD CODED MODULATION FOR VISIBLE LIGHT COMMUNICATIONS 1169
Fig. 1. (a) Nonlinear transfer function of an LED (solid) and its linearizationafter pre-distortion (dashed), and (b) transfer function of and ideal LED with alimited peak-power.
function is not able to keep the device linear over time, anddynamic feedback is needed to modify the model of the instan-taneous nonlinear transfer function of the optical source [27]and actively match the predistortion to that model in order tokeep the relation between the optical power and input currentlinear. This makes the design of the transmitter more complex.In this paper we assume no predistortion is employed.
As discussed in [22], in VLC systems that employ arraysof LEDs as sources, pulsed modulation techniques can solvethe problem by using the LEDs in an on/off mode. In thesemodulation techniques, multilevel signals can be generated byindependently turning on and off each element of the LED-array. In this way multilevel signaling can be used withoutconcern for the effects of the LED nonlinearity, and the opticalsignal level remains proportional to the intended modulatingsignal1. Based on a similar idea, quantized OFDM is pro-posed to utilize the full dynamic range of LEDs by using LEDarrays and employing discrete power level stepping [28]. Belowwe show that HCM symbols can also be generated using anLED-array without being affected by the LED nonlinearity.
In this paper we model the LED as an ideal peak-power lim-ited source, i.e., a hard limiter, which generates a power rangingfrom 0 to the LED peak power, Pmax, proportional to the for-ward input current (Fig. 1-(b)). Ignoring the bandwidth limitof the LED, the only distortion on the transmitted signals isassumed to be caused by clipping the transmitted signals at 0and the peak-power of the LED. Based on the central limit theo-rem, we model the clipping induced distortion by an attenuationand an additive Gaussian noise with variance [29]
σ 2clip =
∫ 0
−∞x2 f (x)dx +
∫ ∞
Pmax
(x − Pmax)2 f (x)dx, (1)
where f (·) is the probability density function (pdf) of theamplitude of the signal sent to the LED.
C. VLC Channel Model
The impulse-response of a VLC channel consists of line-of-sight (LOS) and non-line-of-sight (NLOS) parts. In VLCsystems, the NLOS part of the impulse response is due to reflec-tions of the light from the walls and other objects and usuallycauses inter-symbol interference at symbol-rates higher than
1Note that the LED nonlinearity could still affect the pulse-shape, an effectthat is ignored in this paper.
Fig. 2. VLC system front end.
50 Msps. The results of [30] and [31] can be used to find theimpulse-response of a VLC channel with a given room geom-etry. Given the sampling period, which is assumed to be thesame as the length of the time-slots in this work, an equiva-lent discrete impulse-response of the VLC channel, α = {α�},can be calculated from the continuous impulse-response. Formathematical simplicity, in this work we assume the normalizeddiscrete impulse-response of a VLC channel for which we have
∞∑�=−∞
α� = 1, and the channel loss is ignored for notational
convenience. We model the noise in the system as an additivewhite Gaussian noise (AWGN) source, which is a good approx-imation for high background light scenarios. The front-end ofour VLC system model is depicted in Fig. 2.
III. HADAMARD CODED MODULATION
Hadamard coded modulation (HCM), which uses a binaryHadamard matrix to modulate the input data in blocks oflength N , is introduced in [20] as an alternative to OFDM. Asdescribed in [20], the HCM signal x = [x0, x1, · · · , xN−1]T isgenerated from the data sequence u = [0, u1, · · · , uN−1]T as
x =(
HN u + HN u)
, (2)
where HN is the binary Hadamard matrix of order N [32], HN
is the complement of HN , and the matrix (HN − HN ) is thebipolar Hadamard matrix.
The components of u are assumed to be M-ary pulse ampli-
tude modulated (PAM), where un ∈{
0, 1M−1 , 2
M−1 , . . . , 1}
for
n = 0, 1, . . . , N − 1. The complexity of HCM is the sameas OFDM since an N -size FWHT also has a computationalcomplexity of order N log2 N .
Similar to [17], two structures can be used for the HCMtransmitter. In the first structure, shown in Fig. 3-(a), the HCMsymbols generated are sent to an amplitude modulator that thenmodulates the optical source. This structure, which we call thesingle-source structure, can be used with power-line communi-cation (PLC) integrated VLC networks, where the data is sendto the LED bulbs via the power lines and each component of theLED array cannot be modulated separately. In the single-sourcestructure, as mentioned earlier in Section II-B, the nonlineartransfer function of the optical source causes unequal spacingbetween the transmitted power levels, which makes the sym-bols more susceptible to noise, and therefore, a predistorter isrequired to make the power levels equal. A control circuit is alsoneeded to compensate for the drift due to the thermal changes,which leads to an increased complexity of the transmitter.
In the second structure, the nonlinearity problem of the opti-cal sources is solved by using each LED in an array in its on
1170 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH 2016
Fig. 3. Block diagram of the HCM transmitter using FWHT: (a) single-sourcestructure and (b) LED array structure.
Fig. 4. Block diagram of the HCM receiver using IFWHT.
or off mode. This structure, which is referred to as the LED-array structure, directly modulates a set of LEDs with oneHadamard code as shown in Fig. 3-(b). Given that the un’sare M-PAM modulated, the LED-array structure uses a total ofN × (M − 1) LEDs to modulate the data, i.e., (M − 1) LEDsfor each Hadamard code. This structure can be used in VLCsystems in which each LED can be modulated independently.The LED-array structure guarantees equal spacing between theoutput optical power levels, and avoids the effect of the LEDnonlinearity on the transmitted optical signal. Using HCM,this structure is able to transmit only average powers less thanPmax/2, where Pmax is the peak optical power of the LED array.
In either case, the decoded vector v is obtained from thereceived vector y as
v = 1
N
(HT
N y − HTN y)
+ P
2[1 − N , 1, 1 . . . , 1]T, (3)
which can be realized by an inverse FWHT (IFWHT) as shownin Fig. 4. The noise due to the channel, n, is assumed tobe an AWGN vector with auto-covariance matrix σ 2
N I, andhence, in the absence of nonlinearity, the output signal ofa non-dispersive channel, i.e., α� = 0 for � �= 0, is given byy = (P/N ) x + n, where P is the unclipped peak transmitted
Fig. 5. (a) An HCM signal, and (b) the transmitted pulses using sinc pulse-shaping.
power. Assuming a photo-detector with responsivity equal to 1,the decoded data can be rewritten as
v = P
Nu + n, (4)
where n = 1N
(HT
N − HTN
)n is a N × 1 noise vector with
independent components.
A. Pulse Shaping to Increase Spectral Efficiency
In practice, transmitting rectangular pulses requires a largebandwidth and is not spectrally efficient. In order to overcomethis problem, we use sinc pulses instead of rectangular ones totransmit data. But since negative signals cannot be sent overthe optical link, we add a DC bias to the signals to make thempositive. Fig. 5-(b) illustrates the transmitted pulses for the threerectangular pulses shown in Fig. 5-(a). Replacing rectangularpulses with sinc pulses reduces the SNR by 0.83 dB.
B. BER Calculation
Through (4), the BER of M-PAM HCM for non-dispersiveAWGN channels can be calculated from [33] as
BERHCM ≈ (M − 1)
M log2 MQ
⎛⎝√√√√ 3
γ (M2 − 1)
P2/N
σ 2N + σ 2
clip
⎞⎠ , (5)
where γ represents the penalty in SNR due to the pulse-shaping,which is 1.21 in this work, σ 2
N is the variance of the additive
NOSHAD AND BRANDT-PEARCE: HADAMARD CODED MODULATION FOR VISIBLE LIGHT COMMUNICATIONS 1171
Gaussian noise at the receiver and σ 2clip is the variance of the
clipping noise.For the LED-array transmitter structure in Fig. 3-(b), σ 2
clip =0 and no further analysis is needed. Since each of the N − 1columns of HN that are used to modulate the data has anequal number of zeros and ones, HCM signals have a PAPRof 2, and therefore, for the single-source transmitter structurein Fig. 3-(a), its signals are not clipped by LEDs for averagepower levels less than Pmax/2 and the clipping noise is zero,i.e., σ 2
clip = 0. For average powers larger than Pmax/2, we use
(1) to find σ 2clip. In order to find the pdf of x, we first consider u
to be a binary vector, and then we generalize the results to thecase when the components of u are M-PAM.
For the binary case, xn ∈ {0, 1, 2, . . . , N − 1, N } for n =0, 1, . . . , N − 1, and the probability that xn = k is equal to
Pr (xn = k) =(
N
k
)(1
2
)N
, k = 0, 1, . . . , N . (6)
Through (1), the clipping noise for binary HCM is
σ 2clip =
(1
2
)N N∑k=�N Pmax
P �
(k
P
N− Pmax
)2 (N
k
), (7)
where �x� is the smallest integer larger than x .For M-PAM HCM, the components of x take values
from a larger set as xn ∈ {0, 1M−1 , 2
M−1 , . . . , N } for n =0, 1, . . . , N − 1, and the probability of xn = k
M−1 is
Pr
(xn = k
m − 1
)= C(m, N , k)
(1
m
)N
, k = 0, 1, . . . , N ,
(8)
where C(m, N , k)’s are the extended binomial coefficientsdefined as the coefficients of xk in the expansion of (1 + x +x2 + . . . , x (m−1))N [34].
C. Increasing Energy Efficiency Using DC-Reduced HCM
As shown in [20], the DC part of HCM signals (before thepulse-shaping) can be reduced without losing any information,making HCM more average power efficient. This is importantwhen for illumination reasons the light should be dimmed, i.e.,not operated at its brightest level. Let the first component ofu be set to zero and only N − 1 codewords of the Hadamardmatrix be modulated, as proposed in [20]. In this scheme, whichis called DC-reduced HCM (DCR-HCM), the average trans-mitted power is reduced by sending (x − min x) instead of x.The reduced DC level is per HCM symbol and its value canbe different for each symbol. The same receiver structure asin Fig. 4 can be used to decode the received signals. Fig. 6shows an example of DC reduction in an HCM symbol of sizeN = 8, where the transmitted energy of the HCM symbol inFig. 6-(a) is reduced by a factor of 3/7 in its correspondingDCR-HCM symbol in Fig. 6-(b). This technique can only beeasily implemented for the single-source transmitter structure,and an intermediate circuit is required to apply this techniqueto the LED-array structure.
Fig. 6. (a) An HCM signal, and (b) its corresponding DC reduced signal.
Fig. 7. Probability mass function for (a) binary HCM, and (b) binaryDCR-HCM.
The DC reduction technique decreases the probability oflarge amplitudes of x, which makes the signals less likely tobe clipped by the optical source, and therefore, DCR-HCMcan achieve lower BERs at lower average power levels com-pared to HCM in peak-power limited systems. The probabilitymass function of the transmitted signal, Pr(x = k), is plottedusing Monte-Carlo simulation for binary HCM and DCR-HCMrespectively in Fig. 7-(a) and Fig. 7-(b) for N = 128. Accordingto these results, the peak of Pr (x = k) is shifted to lower x’s forDCR-HCM and the high amplitudes in DCR-HCM signals havelower probabilities compared to that of HCM2.
2Note that the DC value could instead be increased to its maximum value forscenarios that require even brighter illumination. This idea is entirely analogousto DCR-HCM and is therefore not discussed further here.
1172 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH 2016
Fig. 8. Energy efficiency of DCR-HCM versus the order of the Hadamardtransform, log2 N , for M = 2, 4, 16 and 64.
The energy efficiency of DCR-HCM, denoted η, is defined as
η = E {xn}E {xn} − E{min x} . (9)
In this definition we have used the fact that all compo-nents of x have the same mean, i.e., E{xn} is fixed for alln = 0, 1, . . . , N − 1. Fig. 8 shows η as a function of theorder of the Hadamard transform. In this figure, the datasequence is assumed to be M-ary PAM modulated. Accordingto these results, the energy efficiency of DCR-HCM increasesalmost linearly with log2 N , and also increases slightly withincreasing M .
D. Dispersive Channels
VLC experience dispersive channels that create inter-symbolinterference (ISI) on the transmitted signals, and therefore, anypractical modulation technique must be resistant against ISI. InOFDM, a cyclic prefix is used as a guard interval in order toeliminate the intersymbol interference from adjacent symbols.It also allows the linear convolution of a frequency-selectivemultipath channel to be modeled as a circular convolution byrepeating the end of the symbol, which simplifies channel esti-mation and equalization at the receiver. Likewise, a cyclic prefixis used for HCM symbols to avoid interference from other sym-bols, and therefore, the interference on a symbol is intra-symbolinterference that is caused by its own pulses. This also allows usto use cyclic shifts of transmitted vectors instead of their rightshifts in our analysis. Under these assumptions, given x is sent,and ignoring background light and other noise, the received sig-nal is proportional to the nonlinearly distorted (clipped) versionof∑�
α�x(�), and therefore,
y = F
(∑�
α�x(�)
)+ n, (10)
where F(·) describes the LED nonlinearity. In this section, twotechniques are proposed to handle the dispersive nature of thechannel: interleaving and equalization.
Hadamard matrices consist of rows that are cyclic shifts,which increases the similarity between Hadamard codewords
Fig. 9. Schematic view of an interleaved HCM system using an for dispersivechannels.
in dispersive channels and makes HCM vulnerable to ISI.Interleaving is an effective solution that reduces the ISI bydecreasing the cross-correlation of the codewords with theircyclic shifts [22]. In this technique, as shown in Fig. 9, asymbol-length interleaver and a deinterleaver are used at thetransmitter and receiver, respectively, to reduce the effects ofintra-symbol ISI due to a dispersive channel. The interleaver isa permutation matrix, π , and the deinterleaver is its inverse,π−1. Hence, xπ is sent instead of x. For a non-equalizingreceiver, the best interleaver matrix is the one that evenly dis-tributes the interference over all symbols, and can be foundusing binary linear programming [22]. In non-dispersive chan-nels, the performance of interleaved HCM is the same as HCMsince ππ−1 = I.
Assuming the transmitted signal x is not affected by the LEDnonlinearity, the noiseless output of the channel is proportionalto∑�
α� (πx)(�). Then the decoded signal can be written as
v = P
N
∑�
α�
(HN − HN
)π−1I(�)π
(uHN + uHN
)
+ P
2[1 − N , 1, 1 . . . , 1]T + n. (11)
Defining G := ∑�
α�I(�), (11) can be written as
v = P(
HN − HN
)πTGπ
(HN − HN
) (2u − 1)
2N+ P
2+ n.
(12)
In addition, equalization is an effective technique to estimatethe data at the output of a noisy dispersive VLC channel [35],[36]. Here we use minimum mean square error (MMSE) equal-ization, in which the goal is to find the matrix W that minimizesthe trace of E
{(u − u)(u − u)T
}, where u = W(v − P/2) +
1/2 is an estimate of the data sent, u, based on the observationof the decoded vector v. According to [37], the optimum W isgiven by
W = CuvC−1v , (13)
where Cuv is cross-covariance matrix between u and v, and Cvis auto-covariance matrix of v. For (13), the error is
LMMSE = tr{
Cu − CuvC−1v Cvu
}. (14)
For HCM, using (2) and (3) we get
Cuv = P
4N
(HN − HN
)πTGπ
(HN − HN
), (15)
and
Cv = σN 2
NI + P2
4N
(HN − HN
)πTGGTπ
(HN − HN
),
(16)
where we have used the symmetry property of Hadamardmatrices, i.e., HN = HT
N . In dispersive channels, HCM andDCR-HCM requires more complex equalizers compared toOFDM techniques.
NOSHAD AND BRANDT-PEARCE: HADAMARD CODED MODULATION FOR VISIBLE LIGHT COMMUNICATIONS 1173
Fig. 10. Analytical (dashed lines) and simulated (solid lines) BER of HCM,DRC-HCM, ACO-OFDM, DCO-OFDM, and RPO-OFDM versus the averagereceived power in an ideal channel, i.e., α0 = 1.
IV. NUMERICAL RESULTS
In this section, numerical results using simulation and anal-ysis are presented to compare the performance of HCM andDCR-HCM to ACO-OFDM, DCO-OFDM, and RPO-OFDM.In the simulations, the sources are assumed to be ideal peak-power limited sources as shown in Fig. 1-(b) and the peakreceived power is assumed to be 0.1 mW. The optical sourceis modulated as in Fig. 3-(a). The transmitter and receiverare assumed to be perfectly synchronized. These results aregenerated using sinc pulse-shaping, shown in Fig. 5-(b).
The BERs of HCM, DCR-HCM, ACO-OFDM, RPO-OFDM, and DCO-OFDM are plotted versus the averagereceived optical power in Fig. 10 for N = 128. The parame-ters of these techniques are chosen such that all have spectralefficiency of 1, i.e., the same data-rate. The analytical BERsare in good agreement with Monte-Carlo simulation results.For HCM and DCR-HCM, (5) and (1) are used to plot theanalytical results, while those of ACO-OFDM, DCO-OFDMand RPO-OFDM are plotted using the results of [3], [13],[38]. ACO-OFDM and RPO-OFDM both use 16-QAM andDCO-OFDM uses QPSK to modulate the data, while on-offkeying (OOK) is used to modulate the data in HCM and DCR-HCM. The results are plotted assuming an AWGN channel for anoise level of σ 2
N = 2 μW. The BER of ACO-OFDM decreasesby increasing the average optical power until it reaches anoptimum point, and increases afterwards due to the clippingimposed distortion. DCO-OFDM has the same behavior butfor higher average powers. The average optical power of DCO-OFDM is proportional to its DC level, and hence, it reaches itslowest BER for an average power of half of the peak power.In contrast, RPO-OFDM is able to achieve low BER for awide range of average received optical powers and is more suit-able to dimmer VLC systems. Although ACO-OFDM is morepower efficient, DCO-OFDM, HCM and DCR-HCM are bet-ter choices for VLC systems since they achieve lower BER athigh average powers. As one can see, HCM and DCR-HCM canachieve lower BERs compared to OFDM techniques. Betweenthese two, DCR-HCM uses signals with lower amplitudes and
Fig. 11. The minimum theoretically achievable BER versus the spectral effi-ciency for HCM, DCR-HCM, ACO-OFDM, DCO-OFDM, and RPO-OFDM.
is therefore more resistant to clipping. Note that by changingthe DC level of the HCM signals, lower BERs can be achievedfor all average power levels between −15.6 dB and −11.8 dB.
The minimum theoretical achievable BER, defined as thelowest BER possible over all power levels, is plotted versusthe spectral efficiency in Fig. 11 for HCM, DCR-HCM, ACO-OFDM, DCO-OFDM and RPO-OFDM for a noise level ofσ 2
N = 0.5 μW. Analytical BER expressions taken from (5) forHCM and DCR-HCM and from [3], [13], [38] for ACO-OFDM,DCO-OFDM and RPO-OFDM are used to plot these results.For each technique, the optimum average power that minimizesthe BER is found analytically and then the BER is calculatedfor that average power. For DCO-OFDM, the DC level is set toPmax/2 since it minimizes the clipping noise, and hence, cor-responds to the minimum BER among all possible DC levels.Monte-Carlo simulation is used to find the probability massfunction of DCR-HCM, and then it is used in (1) to find theclipping noise power. According to [13], RPO-OFDM achievesits best performance when the duty cycle is zero, for whichRPO-OFDM becomes the same as ACO-OFDM, and there-fore, its lowest achievable BER is the same as ACO-OFDM.Note that ACO-OFDM is only better than DCO-OFDM atlow spectral efficiencies, and it requires high-order QAM toget spectral efficiencies larger than 2, which is impractical.Hence, ACO-OFDM and RPO-OFDM are not suitable for spec-trally efficient VLC systems. HCM and DCR-HCM are ableto provide a lower achievable BER for all spectral efficienciestested.
The simulated BER of 16-QAM ACO-OFDM is comparedto that of OOK DCR-HCM in Fig. 12 for a highly dispersivechannel with a discrete-time equivalent impulse response ofα0 = 0.4, α1 = 0.3 and α2 = 0.3 (using N = 128 samples persymbol) using an MMSE equalizer for DCR-HCM and one-tapequalizer for ACO-OFDM. Both techniques use a cyclic prefixof length 4. The noise is assumed to be σ 2
N = 6 μW. Accordingto these results, DCR-HCM can achieve lower BERs comparedto ACO-OFDM. The BER of the system using MMSE equal-ization on the interleaved DCR-HCM is also plotted versusthe average optical power. The interleaver is found using thebinary linear program described in [22]. As shown in Fig. 12,interleaving improves the performance of the MMSE equalizer
1174 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH 2016
Fig. 12. Simulated BER of ACO-OFDM, DCR-HCM, and interleaved DCR-HCM in a dispersive channel with impulse response α0 = 0.4, α1 = 0.3 andα2 = 0.3 using an MMSE equalizer for DCR-HCM and one-tap equalizer forACO-OFDM.
by almost an order of magnitude by spreading the interferenceequivalently over all Hadamard codewords.
V. CONCLUSION
In this paper, HCM and its modified form, DCR-HCM, areproposed as alternative techniques to OFDM for LED-basedVLC systems that require high illumination levels. HCM andDCR-HCM achieve lower BERs compared to ACO-OFDM forhigh average power since they transmit signals with lower peakamplitudes. The energy efficiency of HCM can be improvedby reducing the DC part of the transmitted signals without los-ing any information. This efficiency is shown to increase withthe size of the Hadamard transform and the size of the PAMconstellation used. The performance of HCM and DCR-HCMis shown to surpass that of ACO-OFDM and DCO-OFDM,and they are able to achieve 2 to 3 orders of magnitude lowerBERs. Interleaving along with MMSE equalization can effec-tively decrease the BER of HCM by an order of magnitude indispersive VLC channels.
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Mohammad Noshad (M’07) received the Ph.D.degree in electrical engineering from the Universityof Virginia, Charlottesville, VA, USA. From May2010 to December 2010, he was a Researcher withi2cat foundation in Barcelona, Spain. His researchinterests include visible light communications, free-space optical communications, information and cod-ing theory, and statistical machine learning. He wasthe recipient of the Best Paper Award at the IEEEGLOBECOM 2012.
Maïté Brandt-Pearce (M’86–SM’99) is a Professorof Electrical and Computer Engineering and theExecutive Associate Dean for Academic Affairs ofthe School of Engineering and Applied Sciences,University of Virginia, Charlottesville, VA, USA.She received the Ph.D. degree in electrical engineer-ing from Rice University, Houston, TX, USA, in1993. Her research interests include nonlinear effectsin fiber-optics, free-space optical communications,cross-layer design of optical networks subject tophysical layer degradations, body area networks, and
radar signal processing. She serves on the Editorial Board of the IEEE/OSAJOURNAL OF OPTICAL COMMUNICATIONS AND NETWORKS and PhotonicNetwork Communications (Springer). She was a Co-Editor of a book entitledCross-Layer Design in Optical Networks (Springer Optical Networks Series,2013). She was the General Chair of the Asilomar Conference on Signals,Systems, and Computers, in 2009 and the Co-Chair of the Optical NetworkSymposium at GLOBECOM 2015. She was the recipient of the NSF CAREERAward and the NSF RIA. She was the corecipient of Best Paper Awards at ICC2006 and GLOBECOM 2012.
