Hadamard-Coded Modulation for Visible Light Communications Mohammad Noshad, Member, IEEE, and Maïté Brandt-Pearce, Senior Member, IEEE

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1 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH Hadamard-Coded Moduation for Visibe Light Communications Mohammad Noshad, Member, IEEE, and Maïté Brandt-Pearce, Senior Member, IEEE Abstract Visibe ight communication (VLC systems using the indoor ighting system to aso provide downink communications require high-average optica powers to satisfy the iumination needs. This can cause high-ampitude signas common in higherorder moduation schemes to be cipped by the peak power constraint of the ight emitting diode (LED and ead to high-signa distortion. In this paper, we introduce Hadamard coded moduation (HCM to achieve ow error probabiities in LED-based VLC systems needing high-average optica powers. This technique uses a fast Wash Hadamard transform (FWHT to moduate the data as an aternative moduation technique to orthogona frequency division mutipexing (OFDM. HCM achieves a better performance for high-iumination eves because of its sma peak to average power ratio (PAPR. The power efficiency of HCM can be improved by reducing the DC part of the transmitted signas. The resuting so-caed DC-reduced HCM (DCR-HCM is we suited to environments requiring dimmer ighting. Index Terms Visibe ight communications (VLC, Hadamard matrix, Wash-Hadamard transform, orthogona frequency division mutipexing (OFDM, peak-to-average power ratio (PAPR, LED noninearity. I. INTRODUCTION V ISIBLE ight communications (VLC is an emerging technoogy for indoor wireess networking that can offer energy efficient Gbps streaming through the ighting system. The idea is to transmit downink data by moduating white ight emitting diodes (LED that are aready being used by energy efficient and cost effective ighting systems. The unreguated bandwidth avaiabe in VLC technoogy can reieve the traffic on radio-frequency (RF communications. However, using LEDs as sources adds restrictions on the moduation schemes and codes that can be used. Main imitations of these LEDs are their imited peak optica power, noninear transfer function, and imited moduation bandwidth [1]. Therefore, moduation and coding schemes with high spectra efficiencies are required to provide a high data-rate connection. In this work we propose a new moduation technique to achieve reiabe and high-speed data transmission in noninear-led-based VLC systems. Manuscript received June 18, 2015; revised October 29, 2015; accepted December 30, Date of pubication January 22, 2016; date of current version March 15, This work was supported by the U.S. Department of Energy (DOE through the SBIR program under Grant DE-SC The associate editor coordinating the review of this paper and approving it for pubication was H. Haas. M. Noshad is with VLNComm LLC, Charottesvie, VA USA (e-mai: noshad@vncomm.com. M. Brandt-Pearce is with the Chares L. Brown Department of Eectrica and Computer Engineering, University of Virginia, Charottesvie, VA USA (e-mai: mb-p@virginia.edu. Coor versions of one or more of the figures in this paper are avaiabe onine at Digita Object Identifier /TCOMM Orthogona frequency-division mutipexing (OFDM is an efficient moduation technique for high speed data communication through bandimited channes, and is being widey used in modern systems because of its high spectra efficiency and robustness against narrow-band interference [2]. OFDM signas are generated by appying an inverse fast Fourier transform (IFFT on the data stream at the transmitter and decoded using a fast Fourier transform (FFT at the receiver. OFDM has been adapted to work in energy efficient optica communications because of its better average-power efficiency compared to other schemes [3]. Asymmetricay cipped optica OFDM (ACO-OFDM, DC-biased optica OFDM (DCO- OFDM and asymmetricay cipped DC biased optica OFDM (ADO-OFDM are modified forms of OFDM proposed for intensity-moduation direct-detection (IM/DD optica communication systems [4] [6]. These OFDM techniques use Hermitian symmetry to generate rea signas from the data sequence, trading-off oss in the encoding rate. Due to the ow rate of these techniques, high order quadratic ampitude moduation (QAM has to be empoyed to achieve high spectra efficiencies, which degrades the energy efficiency. As in the origina OFDM, these optica communications techniques generate signas with arge peaks, which can end up cipped by the peak optica power constraint of the optica sources. This cipping causes a distortion of the OFDM signas that becomes arger by increasing the transmitted average power. Consequenty, in VLC systems where high average optica powers are required for iumination, some forms of OFDM can suffer from signa cipping. This probem can be aeviated by using peak to average power ratio (PAPR reduction techniques that trade-off compexity and energy inefficiency [7]. For exampe, Hadamard matrices can be used as precoders in OFDM systems to decrease the PAPR [8], [9], reduce the BER [10] and increase the resistance of the signas against frequency seective fading [11]. The chaenge of supporting a wide range of dimming eves is another big drawback in the appication of these modified forms of OFDM to VLC systems. There have been significant efforts to address this probem [12]. Reverse poarity optica-ofdm (RPO-OFDM combines puse width moduation (PWM with OFDM to change the dimming eve of the transmitted signas [13]. Pused moduation techniques are another soution to achieve reiabe high-speed communication at high optica average power eves in VLC systems [14]. Among these techniques, those that use the optica sources in their on/off mode are preferred since they avoid the noninear effects of the LEDs [15]. Athough using sources ony in their on/off mode imits the spectra efficiency of the system in singe-led systems IEEE. 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2 1168 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH 2016 [16], systems with mutipe LED sources have the potentia to use mutieve signaing, which can be empoyed to use the avaiabe bandwidth more efficienty [17] [19]. In [20] we introduce a mutieve moduation technique named HCM that uses the Hadamard matrices as a moduation technique (rather than a precoder. In this technique, the data is moduated using a fast Wash-Hadamard transform (FWHT and the receiver uses an inverse fast Wash-Hadamard transform (IFWHT to decode the received signas. Therefore, it has a ow compexity and can expoit the bandwidth effectivey. We aso propose a modification referred to as the DC-reduced HCM (DCR-HCM technique. Because of their ow PAPR, HCM and DCR-HCM can provide high iumination eves in VLC systems without being too affected by LED-induced distortion. The technique is reminiscent of the approach in [21] to use the bipoar Hadamard transform for channe orthogonaization in code division mutipe access (CDMA systems because of their ow PAPR. In this paper we anayze these moduations and study their performance in LED-based VLC systems. We propose using sinc puses for efficient use of the avaiabe LED bandwidth and for fair comparison with OFDM. We show that using DCR- HCM the energy efficiency of HCM can be improved. This improvement becomes more significant by increasing the size of the FWHT. DCR-HCM is aso abe to achieve ower BER eves compared to HCM and OFDM due to its reduced DC eve, which decreases the ampitudes of the transmitted signas and makes them ess ikey to be cipped by the peak-power imit of the LEDs. As in OFDM systems, a cycic prefix is used to avoid interference between adjacent symbos in bandimited channes, and then symbo-ength intereaving is appied on the HCM signas, as was proposed in [22], to decrease the effect of the intra-symbo interference. This approach is shown to ower the error probabiity of high data-rate transmissions through VLC channes. The rest of the paper is organized as foows. Section II describes the VLC system mode incuding the LED noninear transfer function and VLC channe characteristics. Section III introduces the principes of HCM and presents modified forms of HCM to increase its energy efficiency and make it renitent against inter-symbo interference (ISI in dispersive VLC channes. Numerica resuts are presented in Section IV that compare the performance of the HCM and its modified forms to OFDM in VLC systems. Finay, concusions are drawn in Section V. II. PROBLEM DESCRIPTION This section describes the principes of a VLC system. Modes for the optica sources and VLC channe are discussed. In this work we represent vectors with bodfaced ower-case etters, and bodfaced upper-case etters are reserved for matrices. The identity matrix is represented by I. The notation A T denotes the transpose of the matrix A, and x indicates the conjugate of the compex number x. The notations A x and a x are respectivey used to show the matrix and vector that are obtained by subtracting a scaar x from a eements of the matrix A and vector a. We define the compement of the vector a as a := (1 a, and that of the matrix A as A := (1 A. In this paper, a ( and A ( represent the th right cycic-shifts of vector a and matrix A, respectivey. A. Iumination Requirements Effect According to the Iuminating Engineering Society of North America (IESNA, the standard iumination eve for most indoor environments (cassroom, conference-room, ecture ha, offices, etc. is between 300 and 500 ux at 0.8 m height from the foor [23, Tabe 32.I]. In the daytime, a portion of the indoor iumination needs coud be provided by dayight, and the ights can then be dimmed to reduce the energy consumed. The dimming eve has a noninear reation to the average optica power [24], and affects the performance of the VLC system. In order to evauate the performance of a VLC ink, it is essentia to determine the optica power eve that corresponds to a desired iumination eve. This can be done using uminance efficiency of radiation (LER, which is defined as the uminous fux per unit optica power. Athough the theoretica imit of LER for white LEDs is m/w [25], commerciay avaiabe white LEDs have an LER of m/w. One of the most efficient currenty avaiabe white LEDs is Cree s XLamp XT-E white LED with an LER of 148 m/w. Since 1ux= 1m/m 2, an iumination eve of 500 ux from a ight source with an LER of 148 m/w corresponds to 33.8 μwaverage optica power on a photo-detector with an effective area of 0.1 cm 2. B. Noninearity in Optica Sources The optica source is a key component of any optica communication system, as it generates optica power as a function of the moduated input eectrica signa and converts the information into an optica beam. Because of the structure of LEDs, the output optica power and the forward current are reated by a noninear function. The maximum optica power in these sources is imited to a peak-power, and this can resut in the cipping of arge peaks in the moduated signa. In optica communication systems using mutieve or continuous vaued signaing, the noninearity of the optica sources introduces a distortion on the transmitted optica signa. Predistortion is a soution to inearize the reation between the output optica power and the forward current over a range. This technique requires an accurate mode for the design of the predistortion and inearization over the dynamic range of the optica source. A poynomia mode is presented in [26] to describe the noninear transfer function of the optica sources, through which a predistortion function can be designed to inearize the output optica power in terms of the input current. A probem with the predistortion technique is that the noninear transfer function of optica sources can change due to many factors, one of which is the temperature of the transmitter. LEDs and asers tend to dissipate a portion of the input energy as heat, which increases the temperature of the device over time and changes the noninear reation between the output power and forward current. This means that one predistortion

3 NOSHAD AND BRANDT-PEARCE: HADAMARD CODED MODULATION FOR VISIBLE LIGHT COMMUNICATIONS 1169 Fig. 2. VLC system front end. Fig. 1. (a Noninear transfer function of an LED (soid and its inearization after pre-distortion (dashed, and (b transfer function of and idea LED with a imited peak-power. function is not abe to keep the device inear over time, and dynamic feedback is needed to modify the mode of the instantaneous noninear transfer function of the optica source [27] and activey match the predistortion to that mode in order to keep the reation between the optica power and input current inear. This makes the design of the transmitter more compex. In this paper we assume no predistortion is empoyed. As discussed in [22], in VLC systems that empoy arrays of LEDs as sources, pused moduation techniques can sove the probem by using the LEDs in an on/off mode. In these moduation techniques, mutieve signas can be generated by independenty turning on and off each eement of the LEDarray. In this way mutieve signaing can be used without concern for the effects of the LED noninearity, and the optica signa eve remains proportiona to the intended moduating signa 1. Based on a simiar idea, quantized OFDM is proposed to utiize the fu dynamic range of LEDs by using LED arrays and empoying discrete power eve stepping [28]. Beow we show that HCM symbos can aso be generated using an LED-array without being affected by the LED noninearity. In this paper we mode the LED as an idea peak-power imited source, i.e., a hard imiter, which generates a power ranging from 0 to the LED peak power, P max, proportiona to the forward input current (Fig. 1-(b. Ignoring the bandwidth imit of the LED, the ony distortion on the transmitted signas is assumed to be caused by cipping the transmitted signas at 0 and the peak-power of the LED. Based on the centra imit theorem, we mode the cipping induced distortion by an attenuation and an additive Gaussian noise with variance [29] 0 σ 2 cip = x 2 f (xdx + (x P max 2 f (xdx, (1 P max where f ( is the probabiity density function (pdf of the ampitude of the signa sent to the LED. C. VLC Channe Mode The impuse-response of a VLC channe consists of ineof-sight (LOS and non-ine-of-sight (NLOS parts. In VLC systems, the NLOS part of the impuse response is due to refections of the ight from the was and other objects and usuay causes inter-symbo interference at symbo-rates higher than 1 Note that the LED noninearity coud sti affect the puse-shape, an effect that is ignored in this paper. 50 Msps. The resuts of [30] and [31] can be used to find the impuse-response of a VLC channe with a given room geometry. Given the samping period, which is assumed to be the same as the ength of the time-sots in this work, an equivaent discrete impuse-response of the VLC channe, α ={α }, can be cacuated from the continuous impuse-response. For mathematica simpicity, in this work we assume the normaized discrete impuse-response of a VLC channe for which we have α = 1, and the channe oss is ignored for notationa = convenience. We mode the noise in the system as an additive white Gaussian noise (AWGN source, which is a good approximation for high background ight scenarios. The front-end of our VLC system mode is depicted in Fig. 2. III. HADAMARD CODED MODULATION Hadamard coded moduation (HCM, which uses a binary Hadamard matrix to moduate the input data in bocks of ength N, is introduced in [20] as an aternative to OFDM. As described in [20], the HCM signa x = [x 0, x 1,, x N 1 ] T is generated from the data sequence u = [0, u 1,, u N 1 ] T as ( x = H N u + H N u, (2 where H N is the binary Hadamard matrix of order N [32], H N is the compement of H N, and the matrix (H N H N is the bipoar Hadamard matrix. The components of u areassumed{ tobem-ary puse ampitude moduated (PAM, where u n 0, } 1 M 1, 2 M 1,...,1 for n = 0, 1,..., N 1. The compexity of HCM is the same as OFDM since an N-size FWHT aso has a computationa compexity of order N og 2 N. Simiar to [17], two structures can be used for the HCM transmitter. In the first structure, shown in Fig. 3-(a, the HCM symbos generated are sent to an ampitude moduator that then moduates the optica source. This structure, which we ca the singe-source structure, can be used with power-ine communication (PLC integrated VLC networks, where the data is send to the LED bubs via the power ines and each component of the LED array cannot be moduated separatey. In the singe-source structure, as mentioned earier in Section II-B, the noninear transfer function of the optica source causes unequa spacing between the transmitted power eves, which makes the symbos more susceptibe to noise, and therefore, a predistorter is required to make the power eves equa. A contro circuit is aso needed to compensate for the drift due to the therma changes, which eads to an increased compexity of the transmitter. In the second structure, the noninearity probem of the optica sources is soved by using each LED in an array in its on

4 1170 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH 2016 Fig. 3. Bock diagram of the HCM transmitter using FWHT: (a singe-source structure and (b LED array structure. Fig. 5. (a An HCM signa, and (b the transmitted puses using sinc puseshaping. power. Assuming a photo-detector with responsivity equa to 1, the decoded data can be rewritten as v = P u + ñ, (4 N Fig. 4. Bock diagram of the HCM receiver using IFWHT. or off mode. This structure, which is referred to as the LEDarray structure, directy moduates a set of LEDs with one Hadamard code as shown in Fig. 3-(b. Given that the u n s are M-PAM moduated, the LED-array structure uses a tota of N (M 1 LEDs to moduate the data, i.e., (M 1 LEDs for each Hadamard code. This structure can be used in VLC systems in which each LED can be moduated independenty. The LED-array structure guarantees equa spacing between the output optica power eves, and avoids the effect of the LED noninearity on the transmitted optica signa. Using HCM, this structure is abe to transmit ony average powers ess than P max /2, where P max is the peak optica power of the LED array. In either case, the decoded vector v is obtained from the received vector y as v = 1 (H T N N y HT N y + P 2 [1 N, 1, 1...,1]T, (3 which can be reaized by an inverse FWHT (IFWHT as shown in Fig. 4. The noise due to the channe, n, is assumed to be an AWGN vector with auto-covariance matrix σn 2 I, and hence, in the absence of noninearity, the output signa of a non-dispersive channe, i.e., α = 0for = 0, is given by y = (P/N x + n, where P is the uncipped peak transmitted where ñ = N 1 independent components. ( H T N HT N n is a N 1 noise vector with A. Puse Shaping to Increase Spectra Efficiency In practice, transmitting rectanguar puses requires a arge bandwidth and is not spectray efficient. In order to overcome this probem, we use sinc puses instead of rectanguar ones to transmit data. But since negative signas cannot be sent over the optica ink, we add a DC bias to the signas to make them positive. Fig. 5-(b iustrates the transmitted puses for the three rectanguar puses shown in Fig. 5-(a. Repacing rectanguar puses with sinc puses reduces the SNR by 0.83 db. B. BER Cacuation Through (4, the BER of M-PAM HCM for non-dispersive AWGN channes can be cacuated from [33] as (M 1 BER HCM M og 2 M Q 3 P 2 /N γ(m 2 1 σ 2 N + σ 2, (5 cip where γ represents the penaty in SNR due to the puse-shaping, which is 1.21 in this work, σn 2 is the variance of the additive

5 NOSHAD AND BRANDT-PEARCE: HADAMARD CODED MODULATION FOR VISIBLE LIGHT COMMUNICATIONS 1171 Gaussian noise at the receiver and σcip 2 is the variance of the cipping noise. For the LED-array transmitter structure in Fig. 3-(b, σcip 2 = 0 and no further anaysis is needed. Since each of the N 1 coumns of H N that are used to moduate the data has an equa number of zeros and ones, HCM signas have a PAPR of 2, and therefore, for the singe-source transmitter structure in Fig. 3-(a, its signas are not cipped by LEDs for average power eves ess than P max /2 and the cipping noise is zero, i.e., σcip 2 = 0. For average powers arger than P max/2, we use (1 to find σcip 2. In order to find the pdf of x, we first consider u to be a binary vector, and then we generaize the resuts to the case when the components of u are M-PAM. For the binary case, x n {0, 1, 2,...,N 1, N} for n = 0, 1,...,N 1, and the probabiity that x n = k is equa to Fig. 6. (a An HCM signa, and (b its corresponding DC reduced signa. Pr (x n = k = ( N k ( 1 2 N, k = 0, 1,...,N. (6 Through (1, the cipping noise for binary HCM is σ 2 cip = ( 1 2 N N k= N Pmax P ( k P N P max 2 ( N k, (7 where x is the smaest integer arger than x. For M-PAM HCM, the components of x take vaues 1 from a arger set as x n {0, M 1, 2 M 1,...,N} for n = 0, 1,...,N 1, and the probabiity of x n = M 1 k is ( Pr x n = k ( 1 N = C(m, N, k, k = 0, 1,...,N, m 1 m (8 where C(m, N, k s are the extended binomia coefficients defined as the coefficients of x k in the expansion of (1 + x + x ,x (m 1 N [34]. C. Increasing Energy Efficiency Using DC-Reduced HCM As shown in [20], the DC part of HCM signas (before the puse-shaping can be reduced without osing any information, making HCM more average power efficient. This is important when for iumination reasons the ight shoud be dimmed, i.e., not operated at its brightest eve. Let the first component of u be set to zero and ony N 1 codewords of the Hadamard matrix be moduated, as proposed in [20]. In this scheme, which is caed DC-reduced HCM (DCR-HCM, the average transmitted power is reduced by sending (x min x instead of x. The reduced DC eve is per HCM symbo and its vaue can be different for each symbo. The same receiver structure as in Fig. 4 can be used to decode the received signas. Fig. 6 shows an exampe of DC reduction in an HCM symbo of size N = 8, where the transmitted energy of the HCM symbo in Fig. 6-(a is reduced by a factor of 3/7 in its corresponding DCR-HCM symbo in Fig. 6-(b. This technique can ony be easiy impemented for the singe-source transmitter structure, and an intermediate circuit is required to appy this technique to the LED-array structure. Fig. 7. Probabiity mass function for (a binary HCM, and (b binary DCR-HCM. The DC reduction technique decreases the probabiity of arge ampitudes of x, which makes the signas ess ikey to be cipped by the optica source, and therefore, DCR-HCM can achieve ower BERs at ower average power eves compared to HCM in peak-power imited systems. The probabiity mass function of the transmitted signa, Pr(x = k, is potted using Monte-Caro simuation for binary HCM and DCR-HCM respectivey in Fig. 7-(a and Fig. 7-(b for N = 128. According to these resuts, the peak of Pr (x = k is shifted to ower x s for DCR-HCM and the high ampitudes in DCR-HCM signas have ower probabiities compared to that of HCM 2. 2 Note that the DC vaue coud instead be increased to its maximum vaue for scenarios that require even brighter iumination. This idea is entirey anaogous to DCR-HCM and is therefore not discussed further here.

6 1172 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH 2016 Fig. 9. Schematic view of an intereaved HCM system using an for dispersive channes. Fig. 8. Energy efficiency of DCR-HCM versus the order of the Hadamard transform, og 2 N,forM = 2, 4, 16 and 64. The energy efficiency of DCR-HCM, denoted η, is defined as E {x n } η = E {x n } E{min x}. (9 In this definition we have used the fact that a components of x have the same mean, i.e., E{x n } is fixed for a n = 0, 1,..., N 1. Fig. 8 shows η as a function of the order of the Hadamard transform. In this figure, the data sequence is assumed to be M-ary PAM moduated. According to these resuts, the energy efficiency of DCR-HCM increases amost ineary with og 2 N, and aso increases sighty with increasing M. D. Dispersive Channes VLC experience dispersive channes that create inter-symbo interference (ISI on the transmitted signas, and therefore, any practica moduation technique must be resistant against ISI. In OFDM, a cycic prefix is used as a guard interva in order to eiminate the intersymbo interference from adjacent symbos. It aso aows the inear convoution of a frequency-seective mutipath channe to be modeed as a circuar convoution by repeating the end of the symbo, which simpifies channe estimation and equaization at the receiver. Likewise, a cycic prefix is used for HCM symbos to avoid interference from other symbos, and therefore, the interference on a symbo is intra-symbo interference that is caused by its own puses. This aso aows us to use cycic shifts of transmitted vectors instead of their right shifts in our anaysis. Under these assumptions, given x is sent, and ignoring background ight and other noise, the received signa is proportiona to the nonineary distorted (cipped version of α x (, and therefore, ( y = F α x ( + n, (10 where F( describes the LED noninearity. In this section, two techniques are proposed to hande the dispersive nature of the channe: intereaving and equaization. Hadamard matrices consist of rows that are cycic shifts, which increases the simiarity between Hadamard codewords in dispersive channes and makes HCM vunerabe to ISI. Intereaving is an effective soution that reduces the ISI by decreasing the cross-correation of the codewords with their cycic shifts [22]. In this technique, as shown in Fig. 9, a symbo-ength intereaver and a deintereaver are used at the transmitter and receiver, respectivey, to reduce the effects of intra-symbo ISI due to a dispersive channe. The intereaver is a permutation matrix, π, and the deintereaver is its inverse, π 1. Hence, xπ is sent instead of x. For a non-equaizing receiver, the best intereaver matrix is the one that eveny distributes the interference over a symbos, and can be found using binary inear programming [22]. In non-dispersive channes, the performance of intereaved HCM is the same as HCM since ππ 1 = I. Assuming the transmitted signa x is not affected by the LED noninearity, the noiseess output of the channe is proportiona to α (πx (. Then the decoded signa can be written as v = P ( α H N H N π 1 I ( π (uh N + uh N N + P 2 [1 N, 1, 1...,1]T + ñ. (11 Defining G := α I (, (11 can be written as ( (2u 1 v = P (H N H N π T Gπ H N H N + P 2N 2 + ñ. (12 In addition, equaization is an effective technique to estimate the data at the output of a noisy dispersive VLC channe [35], [36]. Here we use minimum mean square error (MMSE equaization, in which the goa is to find the matrix W that minimizes the trace of E { (u û(u û T}, where û = W(v P/2 + 1/2 is an estimate of the data sent, u, based on the observation of the decoded vector v. According to [37], the optimum W is given by W = C uv C 1 v, (13 where C uv is cross-covariance matrix between u and v, and C v is auto-covariance matrix of v. For (13, the error is { } LMMSE = tr C u C uv C 1 v C vu. (14 For HCM, using (2 and (3 we get C uv = P (H N H N π T Gπ (H N H N, (15 4N and C v = σ N 2 N I + P2 4N (H N H N π T GG T π (H N H N, (16 where we have used the symmetry property of Hadamard matrices, i.e., H N = H T N. In dispersive channes, HCM and DCR-HCM requires more compex equaizers compared to OFDM techniques.

7 NOSHAD AND BRANDT-PEARCE: HADAMARD CODED MODULATION FOR VISIBLE LIGHT COMMUNICATIONS 1173 Fig. 10. Anaytica (dashed ines and simuated (soid ines BER of HCM, DRC-HCM, ACO-OFDM, DCO-OFDM, and RPO-OFDM versus the average received power in an idea channe, i.e., α 0 = 1. IV. NUMERICAL RESULTS In this section, numerica resuts using simuation and anaysis are presented to compare the performance of HCM and DCR-HCM to ACO-OFDM, DCO-OFDM, and RPO-OFDM. In the simuations, the sources are assumed to be idea peakpower imited sources as shown in Fig. 1-(b and the peak received power is assumed to be 0.1 mw. The optica source is moduated as in Fig. 3-(a. The transmitter and receiver are assumed to be perfecty synchronized. These resuts are generated using sinc puse-shaping, shown in Fig. 5-(b. The BERs of HCM, DCR-HCM, ACO-OFDM, RPO- OFDM, and DCO-OFDM are potted versus the average received optica power in Fig. 10 for N = 128. The parameters of these techniques are chosen such that a have spectra efficiency of 1, i.e., the same data-rate. The anaytica BERs are in good agreement with Monte-Caro simuation resuts. For HCM and DCR-HCM, (5 and (1 are used to pot the anaytica resuts, whie those of ACO-OFDM, DCO-OFDM and RPO-OFDM are potted using the resuts of [3], [13], [38]. ACO-OFDM and RPO-OFDM both use 16-QAM and DCO-OFDM uses QPSK to moduate the data, whie on-off keying (OOK is used to moduate the data in HCM and DCR- HCM. The resuts are potted assuming an AWGN channe for a noise eve of σn 2 = 2 μw. The BER of ACO-OFDM decreases by increasing the average optica power unti it reaches an optimum point, and increases afterwards due to the cipping imposed distortion. DCO-OFDM has the same behavior but for higher average powers. The average optica power of DCO- OFDM is proportiona to its DC eve, and hence, it reaches its owest BER for an average power of haf of the peak power. In contrast, RPO-OFDM is abe to achieve ow BER for a wide range of average received optica powers and is more suitabe to dimmer VLC systems. Athough ACO-OFDM is more power efficient, DCO-OFDM, HCM and DCR-HCM are better choices for VLC systems since they achieve ower BER at high average powers. As one can see, HCM and DCR-HCM can achieve ower BERs compared to OFDM techniques. Between these two, DCR-HCM uses signas with ower ampitudes and Fig. 11. The minimum theoreticay achievabe BER versus the spectra efficiency for HCM, DCR-HCM, ACO-OFDM, DCO-OFDM, and RPO-OFDM. is therefore more resistant to cipping. Note that by changing the DC eve of the HCM signas, ower BERs can be achieved for a average power eves between 15.6 db and 11.8 db. The minimum theoretica achievabe BER, defined as the owest BER possibe over a power eves, is potted versus the spectra efficiency in Fig. 11 for HCM, DCR-HCM, ACO- OFDM, DCO-OFDM and RPO-OFDM for a noise eve of σn 2 = 0.5 μw. Anaytica BER expressions taken from (5 for HCM and DCR-HCM and from [3], [13], [38] for ACO-OFDM, DCO-OFDM and RPO-OFDM are used to pot these resuts. For each technique, the optimum average power that minimizes the BER is found anayticay and then the BER is cacuated for that average power. For DCO-OFDM, the DC eve is set to P max /2 since it minimizes the cipping noise, and hence, corresponds to the minimum BER among a possibe DC eves. Monte-Caro simuation is used to find the probabiity mass function of DCR-HCM, and then it is used in (1 to find the cipping noise power. According to [13], RPO-OFDM achieves its best performance when the duty cyce is zero, for which RPO-OFDM becomes the same as ACO-OFDM, and therefore, its owest achievabe BER is the same as ACO-OFDM. Note that ACO-OFDM is ony better than DCO-OFDM at ow spectra efficiencies, and it requires high-order QAM to get spectra efficiencies arger than 2, which is impractica. Hence, ACO-OFDM and RPO-OFDM are not suitabe for spectray efficient VLC systems. HCM and DCR-HCM are abe to provide a ower achievabe BER for a spectra efficiencies tested. The simuated BER of 16-QAM ACO-OFDM is compared to that of OOK DCR-HCM in Fig. 12 for a highy dispersive channe with a discrete-time equivaent impuse response of α 0 = 0.4, α 1 = 0.3 and α 2 = 0.3 (usingn = 128 sampes per symbo using an MMSE equaizer for DCR-HCM and one-tap equaizer for ACO-OFDM. Both techniques use a cycic prefix of ength 4. The noise is assumed to be σn 2 = 6 μw. According to these resuts, DCR-HCM can achieve ower BERs compared to ACO-OFDM. The BER of the system using MMSE equaization on the intereaved DCR-HCM is aso potted versus the average optica power. The intereaver is found using the binary inear program described in [22]. As shown in Fig. 12, intereaving improves the performance of the MMSE equaizer

8 1174 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH 2016 Fig. 12. Simuated BER of ACO-OFDM, DCR-HCM, and intereaved DCR- HCM in a dispersive channe with impuse response α 0 = 0.4, α 1 = 0.3 and α 2 = 0.3 using an MMSE equaizer for DCR-HCM and one-tap equaizer for ACO-OFDM. by amost an order of magnitude by spreading the interference equivaenty over a Hadamard codewords. V. CONCLUSION In this paper, HCM and its modified form, DCR-HCM, are proposed as aternative techniques to OFDM for LED-based VLC systems that require high iumination eves. HCM and DCR-HCM achieve ower BERs compared to ACO-OFDM for high average power since they transmit signas with ower peak ampitudes. The energy efficiency of HCM can be improved by reducing the DC part of the transmitted signas without osing any information. This efficiency is shown to increase with the size of the Hadamard transform and the size of the PAM consteation used. 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9 NOSHAD AND BRANDT-PEARCE: HADAMARD CODED MODULATION FOR VISIBLE LIGHT COMMUNICATIONS 1175 [31] K. Lee, H. Park, and J. Barry, Indoor channe characteristics for visibe ight communications, IEEE Commun. Lett., vo.15,no.2,pp , Feb [32] K. J. Horadam, Hadamard Matrices and Their Appications. Princeton, NJ, USA: Princeton Univ. Press, [33] K. Cho and D. Yoon, On the genera BER expression of one- and twodimensiona ampitude moduations, IEEE Trans. Commun., vo. 50, no. 7, pp , Ju [34] S. Eger, Restricted weighted integer compositions and extended binomia coefficients, J. Integer Sequences, vo. 16, [35] H. Le Minh et a., 100-Mb/s NRZ visibe ight communications using a postequaized white LED, IEEE Photon. Techno. Lett., vo. 21, no. 15, pp , Aug [36] M. Biagi, T. Borogovac, and T. D. Litte, Adaptive receiver for indoor visibe ight communications, J. Lightw. Techno., vo. 31, no. 23, pp , [37] J. Proakis and M. Saehi, Digita Communications, 5th ed. New York, NY, USA: McGraw-Hi, [38] R. Meseh, H. Egaa, and H. Haas, On the performance of different OFDM based optica wireess communication systems, IEEE J. Opt. Commun. Netw., vo. 3, no. 8, pp , Aug Maïté Brandt-Pearce (M 86 SM 99 is a Professor of Eectrica and Computer Engineering and the Executive Associate Dean for Academic Affairs of the Schoo of Engineering and Appied Sciences, University of Virginia, Charottesvie, VA, USA. She received the Ph.D. degree in eectrica engineering from Rice University, Houston, TX, USA, in Her research interests incude noninear effects in fiber-optics, free-space optica communications, cross-ayer design of optica networks subject to physica ayer degradations, body area networks, and radar signa processing. She serves on the Editoria Board of the IEEE/OSA JOURNAL OF OPTICAL COMMUNICATIONS AND NETWORKS and Photonic Network Communications (Springer. She was a Co-Editor of a book entited Cross-Layer Design in Optica Networks (Springer Optica Networks Series, She was the Genera Chair of the Asiomar Conference on Signas, Systems, and Computers, in 2009 and the Co-Chair of the Optica Network Symposium at GLOBECOM She was the recipient of the NSF CAREER Award and the NSF RIA. She was the corecipient of Best Paper Awards at ICC 2006 and GLOBECOM Mohammad Noshad (M 07 received the Ph.D. degree in eectrica engineering from the University of Virginia, Charottesvie, VA, USA. From May 2010 to December 2010, he was a Researcher with i2cat foundation in Barceona, Spain. His research interests incude visibe ight communications, freespace optica communications, information and coding theory, and statistica machine earning. He was the recipient of the Best Paper Award at the IEEE GLOBECOM 2012.