On the benefit of DMT modulation in nonlinear VLC systems

Transcription

1 On the benefit of DMT modulation in nonlinear VLC systems Hua Qian, 1 Sunzeng Cai, 1 Saijie Yao, 1 Ting Zhou, 1 Yang Yang, 1 and Xudong Wang 2 1 Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 25, China Shanghai Research Center for Wireless Communications, Shanghai 25, China 2 University of Michigan - Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, Shanghai 224, China hua.qian@shrcwc.org Abstract: In a visible light communication (VLC) system, the nonlinear characteristic of the light emitting diode (LED) in transmitter is a limiting factor of system performance. Modern modulation signals with large peakto-power-ratio (PAPR) suffers uneven distortion. The nonlinear response directly impacts the intensity modulation and direct detection VLC system with pulse-amplitude modulation (PAM). The amplitude of the PAM signal is distorted unevenly and large signal is vulnerable to noise. Orthogonal linear transformations, such as discrete multi-tone (DMT) modulation, can spread the nonlinear effects evenly to each data symbol, thus perform better than PAM signals. In this paper, we provide theoretical analysis on the benefit of DMT modulation in nonlinear VLC system. We show that the DMT modulation is a better choice than the PAM modulation for the VLC system as the DMT modulation is more robust against nonlinearity. We also show that the post-distortion nonlinear elimination method, which is applied at the receiver, can be a reliable solution to the nonlinear VLC system. Simulation results show that the post-distortion greatly improves the system performance for the DMT modulation. 215 Optical Society of America OCIS codes: (6.265) Free-space optical communication; (7.434) Nonlinear optical signal processing. References and links 1. Cisco visual networking index: Forecast and methodology, (Cisco System, 213). 2. Visible light communication(vlc) - A pontential solution to the global wireless spectrum shortage, (GBI Research, Techology Report, 211). 3. H. Elgala, R. Mesleh, and H. Haas, Indoor optical wireless communication: potential and state-of-the-art, IEEE Commun. Mag. 49(9), (211). 4. S. Dimitrov and H. Hass, Information rate of OFDM-based optical wireless communication systems with nonlinear distortion, J. Lightwave Technol. 31(6), (213). 5. L. Grobe, A. Paraskevopoulos, J. Hilt, D. Schulz, F. Lassak, F. Hartlieb, C. Kottke, V. Jungnickel, and K. Langer, High-speed visible light communication systems, IEEE Commun. Mag. 51(12), 6-66 (213). 6. Y. Wang, X. Huang, J. Zhang, Y. Wang, and N. Chi, Enhanced performance of visible light communication employing 512-QAM N-SC-FDE and DD-LMS, Opt. Express 22(13) (214). 7. T. Komine and M. Nakagawa, Integrated system of white LED visible-light communication and power-line communication, IEEE T. Consum. Electr. 49(1), (23). 8. H. Elgala, R. Mesleh, and H. Haas, Indoor broadcasting via white LEDs and OFDM, IEEE T. Consum. Electr. 55(3), (29). (C) 215 OSA 9 Feb 215 Vol. 23, No. 3 DOI:1.1364/OE OPTICS EXPRESS 2618

2 9. J. Armstrong and A. Lowery, Power efficient optical OFDM, IET Electr. Lett. 42(6), (26). 1. D. J. Barros, S. K. Wilson, and J. M. Kahn, Comparison of orthogonal frequency-division multiplexing and pulse-amplitude modulation in indoor optical wireless links, IEEE T. Commun. 6(1), (212). 11. R. Mesleh, H. Elgala, and H. Haas, On the performance of different OFDM based optical wireless communication systems, IEEE/OSA J. Opt. Commun. Network. 3(8), (211). 12. K. Asatani and T. Kimura, Analyses of LED nonlinear distortions, IEEE J. Solid-st. Circ 13(1), (1978). 13. D. Tsonev, S. Sinanovic, and H. Haas, Complete modeling of nonlinear distortion in OFDM-based optical wireless communication, J. Lightwave Technol. 31(18), (213). 14. I. Neokosmidis, T. Kamalakis, J. W. Walewski, B. Inan, and T. Sphicopoulos, Impact of nonlinear LED transfer function on discrete multitone modulation: analytical approach, J. Lightwave Technol. 27(22), (29). 15. D. Lee, K. Choi, K.-D. Kim, and Y. Park, Visible light wireless communications based on predistorted OFDM, Opt. Commun. 285(7), (212). 16. H. Qian, J. Chen, S. Yao, and Z. Yu, One-bit sigma-delta modulator for nonlinear visible light communication systems, IEEE Photon. Technol. Lett. doc. ID 1.119/LPT (posted 4 December 214, in press). 17. R. Mesleh, H. Elgala and H. Haas, LED nonlinearity mitigation techniques in optical wireless OFDM communication systems, IEEE/OSA J. Opt. Commun. Network. 4(11), (212). 18. H. Qian, S. Yao, S. Cai, and T. Zhou, Adaptive postdistortion for nonlinear LEDs in visible light communications, IEEE Photon. J. 6(4), 1 8 (214). 19. OSRAM GmbH, Datasheet: LE UW S2LN, (Sep. 211). 2. S. Benedetto and E. Biglieri, Principles of Digital Transmission with Wireless Applications ( Kluwer Academic Publishers, July 1999). 21. J. G. Proakis, Digital Communications (McGraw-Hill, 3rd ed., 1995). 22. J. Armstrong and B. Schmidt, Comparison of asymmetrically clipped optical OFDM and DC-biased optical OFDM in AWGN, IEEE Commun. Lett. 12(5), (28). 23. Y. Xiang, M. Zhang, X. Tang, J. Wu and D. Han, A post-processing channel estimation method for DCO- OFDM visible light communication, IEEE 8th International Symposium on Communication Systems, Networks & Digital Signal Processing (CSNDSP) (IEEE, 212), pp P. Schniter, Low-complexity equalization of OFDM in doubly selective channels, IEEE T. Signal Process. 52(4), (24). 25. D. R. Brillinger, Time Series: Data Analysis and Theory (San Francisco: Holden-day Inc., 1981). 26. Å. Björck, Solving linear least squares problems by gram-schmidt orthogonalization, BIT Numer. Math. 7(1), 1-21 (1967). 27. R. Raich and G. T. Zhou, Orthogonal polynomials for complex Gaussian processes, IEEE T. Signal Process. 52, (24). 28. N. Ahmed, T. Natarajan, and K. R. Rao, Discrete cosine transform, IEEE T. Comput. 1(1), 9-93 (1974). 29. J. Tellado, L. M. Hoo, and J. M. Cioffi, Maximum-likelihood detection of nonlinearly distorted multicarrier symbols by iterative decoding, IEEE T. Commun 51(2), (23). 1. Introduction The data traffic of wireless communications has been growing exponentially in the past decade. According to recent market forecast, more than a Zettabytes (1 21 bytes) of data would be transmitted over the air each year after 215 [1]. On the other hand, the wireless spectrum is very limited and is not able to meet the continuously increasing demand for mobile data traffic. Expansion of the wireless spectrum into the visible light spectrum have a huge potential to future wireless communications [2]. The cost of the front end LED and photodiode (PD) is low. Moreover, the VLC system can be easily obtained by upgrading the current illumination infrastructure [3]. Many efforts have been made on the transmission rate of the VLC system [4, 5]. In [6], 4.2Gb/s data rate was achieved using wavelength multiplexing division with redgreen-blue (RGB) LED. In a VLC system, PAM is a natural choice as the LED and the PD applies light intensity modulation and direct detection (IM/DD), which only deal with real and positive signals [7]. In the VLC system, the spectrum efficiency is critical as the frequency response of LED is limited. In radio wireless communication system, orthogonal frequency division multiplexing (OFDM) modulation or discrete multi-tone (DMT) modulation provides high spectrum efficiency and is robust against inter-symbol-interference (ISI) and frequency selective fading channel. Many (C) 215 OSA 9 Feb 215 Vol. 23, No. 3 DOI:1.1364/OE OPTICS EXPRESS 2619

3 modifications are applied to the DMT modulation to meet the requirement of the VLC system for a real and positive signal. By Hermitian symmetry, the real OFDM signal can be obtained by inverse discrete fourier transform (IDFT) of frequency domain data. One example is the polarity OFDM technology which focuses on direct offset current DC-biased OFDM (DCO- OFDM) [8]. Other modifications include asymmetrically clipped optical OFDM (ACO-OFDM) [9], PAM-DMT [1] and nonpolar U-OFDM. These modifications have similar performance in Gaussian channel [11]. Similar to the power amplifier in the transmitter of a wireless communication system, the LED in the transmitter of a VLC system is the major source of nonlinearity [12, 13]. In the presence of nonlinearity, the system performance such as error vector magnitude (EVM) and bit error rate (BER) greatly degrades with high PAPR signals. For the PAM modulation, the nonlinearity directly changes on the signal constellation. Large signals are compressed, while the small signals are little affected. The distortion in the large signals makes them vulnerable to noise. For the DMT modulation, the nonlinear distortion works on the time domain signal, which is linearly transformed from signal domain [14]. At the receiver, after the inverse transformation, the nonlinear effects are spread into each subcarrier. The overall system performance improves with DMT modulation. In this paper, we do not intend to challenge the observation that OFDM has large PAPR and is vulnerable to the LED nonlinearity. The single carrier system may also be subject to large PAPR and may also be sensitive to the PA nonlinearity. We show theoretical results of the nonlinear effects after DMT. In order to compensate for the nonlinear effects of the LEDs, adaptive digital predistortion (DPD) can be applied in the transmitter at the expense of an additional feedback path [15]. In this letter, a one-bit sigma-delta modulator (SDM) is introduced to the transmit signal to mitigation [16]. Iterative signal clipping (ISC) can be used to reduce the signal clipping distortion [17]. In VLC system, the out-of-band spectral regrowth is not an issue as the adjacent channel is not occupied in general. Post-distortion nonlinear elimination, which works at the receiver, can be a reliable solution to the VLC system since no additional hardware blocks are needed [18]. In this paper, we provide an alternative post-distortion algorithm with iteration. We show that with DMT modulation, the post-distortion algorithm works well; while with single carrier system, the benefit of post-distortion is limited. The benefit of DMT modulation is further validated. The rest of paper is organized as follows. Section 2 describes the VLC system and shows the nonlinear effects to the PAM modulation. Section 3 works on the nonlinear effects to the DMT modulation. We show that with the DMT modulation, the nonlinearity in the transformed domain is evenly distributed. System performance can be improved with the DMT modulation. In Section 4, we propose a post-distortion nonlinear elimination algorithm to further mitigate the nonlinear effects in the VLC system. The effectiveness of the proposed algorithm is validated with simulations. Section 5 concludes this paper. 2. Nonlinear VLC system 2.1. System architecture Figure 1 shows basic block diagrams of a VLC system. The LED s intensity is modulated by the baseband input signal x(n) at the transmitter. A DC bias is applied to the input signal to shift the input signal to positive region. Since only the alternating current (AC) components of the LED output carry the data information, the direct current (DC) components are omitted in the LED model representation. The transmitted signal after the nonlinear LED is denoted by y(n). A typical multi-path channel model with additive white Gaussian noise (AWGN) is added in the VLC system. At the receiver, the received signal r(n) is obtained by PD with direct detection. The LED at the transmitter and the PD at the receiver are both nonlinear devices, which (C) 215 OSA 9 Feb 215 Vol. 23, No. 3 DOI:1.1364/OE OPTICS EXPRESS 262

4 yn ( ) x( n) rn ( ) Fig. 1. Block diagrams of a VLC transceiver. Relative Luminous Flux φ v /φ vin (3.2v) V TOV =2.7, I max =1A, φ max Flux =2.2 DC Bias.5 Calculated from data sheet Curve fitted with a 7th order polynomial model Forward voltage V (V) in Probablity density Forward voltage V (V) in PAM DMT (a) The input voltage to the output light intensity transfer function of a commercial white LED. (b) Probability density functions of the input PAM signal and DMT signal. Fig. 2. Nonlinear characteristics of LED and corresponding input signal distribution. may greatly degrade the system performance, such as EVM and BER. In practice, the PD at the receiver is nonlinear when the luminous intensity at the receiver is large. In this case, the received signal is large and the signal to noise ratio (SNR) at the receiver is satisfactory. In this paper, we only consider the nonlinearity of the LED. The received signal r(n) can be written as r(n)=h ( y(n) ) + v(n), (1) where h( ) is the lumped the frequency response of the LED and the channel response with unity gain, and v(n) is additive white Gaussian noise. Figure 2(a) shows the nonlinear response between the input voltage and the output light intensity of a commercial white LED (part number: OSRAM LE UW S2LN) [19], which is obtained by combining the P-I and I-V curves from the data sheet. As shown in Fig. 2(a), the LED s turn on voltage (TOV) is 2.7 V. In order to guarantee that the LED works in the operation region, a DC bias is superimposed to the input signal. In addition, the maximum input voltage of 3.7 V, which is limited by a maximum permissible current of the LED. For reference, the probability density functions of input signals used in our simulation are shown in Fig. 2(b). The PAPRs of the two input signals are both around 9 db. The LED s nonlinearity can be modeled as a memoryless polynomial model [2]: y(n)= f LED ( x(n) ) = P p=1a p x p (n), (2) where f LED ( ) is LED s nonlinear transfer function, P is highest order of the polynomial terms, and a p is the model coefficient of the pth-order monomial term. To simplify the discussion, the DC term (3.2 V in Fig. 2(a)) is drop out and the input signal amplitude is normalized to [ 1,1]. In this example, the normalized model coefficients with P = 7 are shown in Table 1. The dashed (C) 215 OSA 9 Feb 215 Vol. 23, No. 3 DOI:1.1364/OE OPTICS EXPRESS 2621

5 line in Fig. 2(a) shows the estimated P-V response. This polynomial model is valid beyond the working range of the LED, which also models the clipping distortion. Table 1. Normalized model coefficients of the LED. Coefficients a 1 a 2 a 3 a 4 a 5 a 6 a 7 Values From Fig. 2(a), we observe that the nonlinearity of LED in the transmitter changes the signal amplitude. In Fig. 3, we show the offset added to the signal amplitude with the nonlinear LED. The offset introduced by the nonlinear distortion is defined as the difference between the ideal linear output and the actual nonlinear output normalized by the ideal linear output, or ε d = f (x(n)) x(n) / x(n). In Fig. 3, the x-axis shows the normalized amplitude of the input signal, and y-axis shows the nonlinear offset in decibel scale. When the signal amplitude is within [.125,.135], the nonlinear distortion is negligible and is less than 3 db, which is not shown in the figure. When the input signal amplitude is large, which generated by the nonlinear distortion becomes significant. The offset can be as large as 7.5 db for the maximum input amplitude, such distortion is not acceptable when decision is made on received symbols. From Fig. 3, we conclude that the EVM/BER performance is dominated by the outer constellation points. In other words, the single carrier modulation is sensitive to the LED s nonlinearity in a VLC system. Normalized Distortion Offset (db) Normalized Range Fig. 3. Normalized nonlinear offset for single carrier modulation. The solid line shows the offset generated by the nonlinearity. When the signal amplitude is within [.125,.135], the nonlinear distortion is less than 3 db and is not shown in the figure Single carrier modulation Since the LED and the PD are using light IM/DD, PAM is a natural choice. To make a fair comparison, we choose 64 quadrature amplitude modulation (QAM) with a raised cosine pulse shaping filter of.25 roll-off factor to compare the single carrier system and multi carrier system. In this situation, the PAPRs of single carrier and multi carrier signals are similar. Since the LED only transmits real signals, the real and imaginary part of the original complex symbols are transmitted serially. In this case, the modulation scheme can be regarded as 8-PAM. We can now discuss the error performance of M-PAM modulation with the LED s nonlinearity. The distinct inputs are denoted by A 1,A 2,...,A M, and the corresponding nonlinear outputs (C) 215 OSA 9 Feb 215 Vol. 23, No. 3 DOI:1.1364/OE OPTICS EXPRESS 2622

6 are denoted by f LED (A 1 ), f LED (A 2 ),, f LED (A M ). For a memoryless nonlinearity, the error probability P for optimal decision in AWGN channel can be defined as [21] P = M i=1 P i = M i=1 {p Ai } f Ai (y)dy γ i, (3) where P i is the optimal decision error probability of A i, p Ai is the transmit probability of A i, γ i is the error decision region where the decision on the received signal is made in error, and f Ai (y) is the probability density function (PDF) of received signal given by f Ai (y)= 1 2πσv e (y f LED(A i )) 2 /2σ 2 v, (4) where σ 2 v is the noise power. The error probability P in Eq. (3) can be further simplified as P = M i=2 where Q-function is defined as Q(a)= ( ) a 1/ 2π e y 2 /2 dy. 2 ( M Q fled (A i ) f LED (A i 1 ) ), (5) 2σ v Fig. 4. Normalized constellation diagram of 64-QAM for single carrier modulation. The blue dots are the original 64-QAM constellation and the red dots are the distorted nonlinear signal with AWGN noise. The LED nonlinearity is a 7th-order polynomial model whose coefficients are given by Table 1. The SNR at the receiver is 25 db. A total of 64 symbols are shown. With a strong nonlinearity f LED ( ), the signal may be greatly distorted and the error probability can be large. Figure 4 shows the normalized constellation diagram of 64-QAM for single carrier modulation. In this simulation, the input signal is modulated by 64-QAM, the LED nonlinearity is a 7th-order polynomial model whose coefficients are given by Table 1. The SNR at the receiver is 25 db. A total of 64 symbols are shown. In Fig. 4, the blue dots are the original 64-QAM constellation and the red dots are the distorted signal with AWGN noise. In this simulation, we observe that the signal with small input is not affected by the nonlinearity. The distorted signal from different constellations is scattered in clusters and can be clearly distinguished. While the signal with large input is compressed. The distorted signal from different signal constellations is blurred and erroneous decision may be reached. (C) 215 OSA 9 Feb 215 Vol. 23, No. 3 DOI:1.1364/OE OPTICS EXPRESS 2623

7 3. Nonlinear distortion with orthogonal linear transformation In Fig. 2(b), we show that in time domain, the PAM signal and the DMT signal have the same average power and exercise similar peak power. The amount of distortion of the two signals is about the same. For the single carrier system, the impact of nonlinear distortion is not the same as that for input signals with different amplitudes. Large signals suffers more distortion than small signals. System performance is limited by the distortion to large signals. This observation motivates us to think: If the distortion can be redistributed evenly to all input signals with different amplitudes, the system performance may be improved. Orthogonal linear transformation, such as DMT or OFDM, which does not change the variance of the nonlinear distortion, can be applied to change the distribution of distortions in the signal domain Nonlinear distortion with DMT X k x n yn rn X k Rk Fig. 5. Block diagrams of a multi-carrier VLC system with post-distortion nonlinear elimination algorithm. Figure 5 shows a multi-carrier VLC system with an orthogonal linear transformation, where X k is the modulated input signal, x(n) is the time domain signal orthogonally transformed from X k. In a VLC system, the conventional OFDM signals are modified to other specific OFDM formats, such as DCO-OFDM or ACO-OFDM as the LED only works with real and positive input signal [22]. In this paper, without loss of generality, we assume a DCO-OFDM signal, which is given by N 1 x(n)= k= X k e j2πnk/n, (6) where N is the total number of data subcarriers and N is even, X k is the information symbol on the k-th subcarrier and X k satisfies X k k = 1,,N/2 1, X k = X N k k = N/2 + 1,,N 1, (7) k =,N/2, where denotes the complex conjugate. To meet the real signal requirement, the data symbols on the negative subcarriers are required to be the complex conjugate of the data symbols on the positive subcarriers. The data symbols X k for < k < N/2 can be assumed to be independent, identically distributed (i.i.d.) random variables whose values are taken from a given signal constellations, such as the QAM. From Eq. (7), we also know that X N k e j2πn(n k)/n = X k e j2πnk/n. Then the time domain transmitted signal x(n) is a linear combination of i.i.d. random variables X k e j2πnk/n + X k e j2πnk/n ), for < k < N/2. If the number of subcarriers N is sufficiently large, by the central limit theorem, the signal x(n) can be approximated by a Gaussian process. (C) 215 OSA 9 Feb 215 Vol. 23, No. 3 DOI:1.1364/OE OPTICS EXPRESS 2624

8 After the nonlinear LED, the transmitted signal is given by y(n) = f LED ( x(n) ). The corresponding received time domain signal is given by Eq. (1). After the inverse transformation and conventional MMSE channel estimation and equalization [23,24], the received signal R k is given by, N 1 R k = k= where V k is the noise on the k-th sub-carrier. Substituting Eq. (2) into Eq. (8), we have r(n)e j2πkn/n N 1 ( ) = f LED x(n) e j2πkn/n +V k, (8) n= R k = N 1 P n= p=1 N 1 = a 1 X k + a p x p (n)e j2πkn/n +V k P n= p=2 a p x p (n)e j2πkn/n +V k. (9) The received signal R k, as shown in Eq. (9), is composed by the linear term a 1 X k, the distortion term N 1 n= P p=2 a px p (n)e j2πkn/n, and the noise term V k. The linear term corresponds to the original input X k. The noise term V k is additive white Gaussian noise, which is true if the frequency response of the LED and the channel effect to the inband signal are not significant. The distortion term, which is a superposition multiple identically distributed random variables, can also be approximated by Gaussian noise when N is large by cental limit theorem. To further understand the property of the nonlinear distortion in the transformed domain, we first work on the covariance function of the distorted signal. The auto-covariance function of the distorted signal y(n) is given by c 2y (l) = E{y(n),y(n + l)} E{y(n)}E{y(n + l)} = cum{y(n),y(n + l)}, (1) where cum{ } denotes the cumulant operation defined in [25](page 19). By definition, the pthorder cumulant is given by: c px (l 1,l 2,,l p 1 )=cum{x(n),x(n + l 1 ),,x(n + l p 1 )}. In our case, x(n) is a real Gaussian distributed random variable, the 2nd-order cumulant c 2x (l) is exactly the auto-covariance function of x(n). For p > 2, all the pth-order cumulant terms c px (l 1,l 2,,l p 1 ) disappear. We have c px (l 1,l 2,,l p 1 )=, p > 2. (11) Substituting the polynomial nonlinearity Eq. (2) into the auto-covariance function Eq. (1), we have { P c 2y (l) = cum p=1 = P P p=1 q=1 a p x p (n), P q=1 } a q x q (n + l) a p a q cum{x p (n),x q (n + l)}. (12) In Eq. (12), the cumulant of the product term t pq = cum{x p (n),x q (n+l)} can be obtained by the Leonov-Shiryaev formula [25], which can be constructed by a two-way table shown below (C) 215 OSA 9 Feb 215 Vol. 23, No. 3 DOI:1.1364/OE OPTICS EXPRESS 2625

9 and finding its indecomposable partitions, x(n) x(n) }{{} p x(n + l) x(n + l). }{{} q The following examples show the computation process obtaining the cumulant of the product terms. For p = 1,q = 1, the two-way table has only 1 indecomposable partition of {x(n),x(n + l)}. Wehave t 11 (l)=cum{x(n),x(n + l)} = c 2x (l). (13) For p = 1,q = 3, the two-way table can be divided into indecomposable partitions of {x(n),x(n + l)}, {x(n + l),x(n + l)},or{x(n),x(n + l),x(n + l),x(n + l)}.wehave t 13 (l) = cum{x(n),x 3 (n + l)} = 3cum{x(n),x(n + l)}cum{x(n + l),x(n + l)} +cum{x(n),x(n + l),x(n + l),x(n + l)} = 3c 2x ()c 2x (l)+c 4x (,l,l,l) = 3σ 2 x c 2x (l), (14) where σ 2 x = c 2x () is the variance of the input signal x, and c 4x (,l,l,l)= according to Eq. (11). Since there are 3 distinct indecomposable partitions in the form of {x(n),x(n+l)}, {x(n+ l),x(n + l)}, coefficient 3 shows up in Eq. (14). For p = 3,q = 3, the indecomposable partitions can be {x(n),x(n + l)}, {x(n),x(n + l)}, {x(n),x(n+l)},or{x(n),x(n+l)}, {x(n),x(n)}, {x(n+l),x(n+l)},or{x(n),x(n),x(n),x(n+ l)}, {x(n + l),x(n + l)}, etc. We have t 33 (l)=cum{x 3 (n),x 3 (n + l)} = 6c 3 2x (l)+9σ 4 x c 2x (l). (15) 3.2. Example and simulation High order cumulant can be obtained in a similar way. For the nonlinear LED that we discussed earlier, a memoryless polynomial model can be applied with polynomial order P = 7 and the normalized model coefficients are given by Table 1. The auto-covariance function of the distorted signal y(n) is given by c 2y (l) = cum{y(n),y(n + l)} = = 288a 2 7c 7 2x(l) 7 7 p=1 q=1 a p a q t pq + ( 12a σ 2 x a σ 4 x a 2 7) c 5 2x (l) + ( 6a σ 2 x a 3 a 5 + 6σ 4 x a σ 4 x a 3 a σ 6 x a 5 a σ 8 x a 2 7) c 3 2x (l) + ( a σ 2 x a 1 a 3 + 9σ 4 x a σ 4 x a 1 a σ 6 x a 3 a σ 6 x a 1 a σ 8 x a σ 8 x a 3 a σ 1 x a 5 a σ 1 x 2a 2 7) c2x (l). (16) Since x(n) is i.i.d. Gaussian distributed (for large N), the auto-covariance function c 2x (l) has an impulse response, { σ 2 c 2x (l)= x, l =, (17), l. (C) 215 OSA 9 Feb 215 Vol. 23, No. 3 DOI:1.1364/OE OPTICS EXPRESS 2626

10 With Eq. (17), it is straightforward to show that for p 2, c p 2x (l) is also an impulse response with c p 2p 2x (l) =σx (DFT) on both sides of Eq. (16), we obtain when l =, and c p 2x (l) = when l. Taking discrete Fourier Transform S 2y (k) = 1368a 2 7σx a 5 a 7 σx 12 +(945a a 3a 7 )σx 1 +(21a 3 a a 1 a 7 )σx 8 +(15a a 1 a 5 )σx 6 + 6a 1 a 3 σx 4 + a 2 1σx 2, (18) where S 2y (k) is the power spectrum density function of the nonlinear output y(n). Given a set of nonlinear coefficients {a p } P p=1, S 2y(k) is a function of the input signal power σ 2 x. This result suggests that the distorted signal has equal power in data domain. The impact of distortion terms to each data symbol is the same, whereas the impact of distortion terms to the data symbol with the PAM modulation directly relates to the input signal power. We further validate this observation with simulation. In this simulation, a memoryless polynomial model with coefficients listed in Table 1 is applied. DCO-OFDM systems with 64 and 124 data subcarriers are assumed. The normalized average power of input original signal, distorted output signal and distortion signal is obtained and shown in Fig. 6. Figures 6(a) - 6(b) show the normalized power for 64 and 124 data subcarriers, respectively. The blue solid line shows the signal power of original signal; the black dash line shows the signal power of distorted output signal; and the red line shows the power of distortion term. The results are obtained by averaging 1 independent realizations. From this simulation, we observe that in all cases, the input signal power, the output signal power, and the distortion signal power are almost flat. The nonlinear distortion is indeed spread to different subcarriers evenly subcarriers subcarriers Normalized Power Normalized Power Snbcarrier Index Subcarrier Index (a) (b) Fig. 6. The normalized power for DCO-OFDM system with 64, 124 data subcarriers, respectively. The blue solid line shows the signal power of input signal; the black dash line shows the signal power of output distorted signal; and the red solid line shows the power of distortion term. The results are obtained by averaging 1 independent realizations. Figure 7 shows the constellation diagram of 64-QAM with DCO-OFDM modulation. Simulation setup of this case is the same as that in Fig. 4. A total of 1 independent DCO-OFDM symbols with 64 subcarriers are shown. In Fig.7, the blue dots are the original 64-QAM constellation and the red dots are the distorted nonlinear signal with the nonlinearity and AWGN noise. In this simulation, we observe that signal with small input suffers similar distortion and noise as the signal with large input. While in Fig. 4, signal with small amplitude is almost not affected by the nonlinearity. On the other hand, in Fig. 7, the distorted nonlinear signal is still close to its original constellation. The distortion does not provide an offset to the original signal (C) 215 OSA 9 Feb 215 Vol. 23, No. 3 DOI:1.1364/OE OPTICS EXPRESS 2627

11 constellation. While in Fig. 4, there is an obvious large offset added to the signal with large amplitude, which can be a disaster for the detector design Fig. 7. Normalized constellation diagram of 64-QAM for DCO-OFDM modulation. The blue dots are the original 64-QAM constellation and the red dots are the distorted signal with AWGN noise. The SNR at the receiver is 25 db. A total of 1 independent DCO- OFDM symbols are shown. The benefit of DMT modulation can be further validated by BER performance simulations in the presence of nonlinearity. The nonlinearity setup is inherited from previous examples. For single carrier system, the 64-QAM signal is obtained by modulating I/Q signal into separate 8 PAM signals. For the multi-carrier system, a 1/4-symbol length cyclic prefix is applied to the DCO-OFDM system with 64 subcarriers. To simulate the BER performance, a total number of information bits are simulated. The frequency response of the LED is shown in Fig. 8(a). We assume that the channel response can be lumped in the frequency response of the LED. The overall frequency response is estimated and equalized with MMSE algorithm. Figure 8(b) shows the BER performance of different modulations. The black line shows the ideal BER without nonlinearity; the red line shows the BER with PAM modulation; and the blue line shows the BER with DMT modulation. We observe that the PAM modulation is sensitive to the LED s nonlinearity, which could not achieve BER = 1 2 when SNR increases. For comparison, the performance of the DMT modulation provides about 5 db performance improvement when SNR is large. In addition, we also notice that the nonlinearity introduces an error floor to both PAM modulation as well as DMT modulation. The error floor for the DMT modulation is lower than that for the PAM modulation Extension to general polynomial nonlinearities In previous section, we work on a specific LED example with 7th-order nonlinear polynomial model. The distortion in the frequency domain has a white spectrum. For each data subcarrier, the impact of the distortion is the same. The result can be extended to general polynomial nonlinearities. Let us define a set of orthonormal polynomial basis functions ψ p (x(n)) that satisfy E[ψ p (x(n)),ψ q (x(n))] = { 1 p = q, p q, (19) where E[ψ p (x(n)),ψ q (x(n))] = ψ p (x(n))ψ q (x(n)) f p (x)dx, (C) 215 OSA 9 Feb 215 Vol. 23, No. 3 DOI:1.1364/OE OPTICS EXPRESS 2628

12 1 LED Frequency Response (db) Frequency x 1 7 (a) Measured frequency response of VLC channel including the frequency response of the LED and the multi-path channel effects. BER PAM W Nonlinearity DMT W Nonlinearity DCT W Nonlinearity Linear system SNR (db) (b) Bit error rate performance of the proposed schemes. Fig. 8. Channel response and the BER performance of the proposed schemes. 1 3 and f p (x) is the probability density function (PDF) of the input signal x(n). Assuming the pth order orthonormal nonlinearity has the form ψ p (x(n)) = p q=1 U qpx q (n), the orthonormal polynomial basis functions can be obtained iteratively by Gram-Schmidt process [26] or by matrix inversion as suggested in [27]. With the orthogonal polynomials, Eq. (2) can be rewritten as y(n) = P p=1 α p ψ p (x(n)), (2) where α p is the model coefficient for the orthonormal polynomial basis functions. Similar to Eq. (12), the auto-covariance function of the output signal is given by c 2y (l) = cum{y(n),y(n + l)} = { P cum α p ψ p (x(n)), p=1 = P P p=1 q=1 P q=1 } α q ψ q (x(n)) α p α q cum{ψ p (x(n)),ψ q (x(n + l))}. (21) With the definition of the orthonormality, after derivation, the product term of the cumulant can be obtained, { c p cum{ψ p (x(n)),ψ q (x(n + l))} = 2x (l) p = q, (22) p q. Combining Eqs. (22) and (21), we obtain c 2y (l)= P p=1 αpc 2 p 2x (l). (23) Substituting Eq. (23) into Eq. (16), and taking DFT on both sides, we obtain S 2y (k)= P p= α 2 pσ 2p x. (24) (C) 215 OSA 9 Feb 215 Vol. 23, No. 3 DOI:1.1364/OE OPTICS EXPRESS 2629

13 For general polynomial nonlinearities with arbitrarily order, the distortion signal in the frequency domain has equal power in different subcarriers. The result is consistent with previous examples that work on low order nonlinearities Extension to other orthogonal linear transformations In previous sections, we learn that with DMT modulation, the nonlinear distortion in the data path is converted to additive distortion term in the data domain that applies to the signal constellation evenly. Following the same principle, we may apply other orthogonal linear transformations to obtain the time-domain signal. When the nonlinear distortion in the time domain is converted to the transformed domain, the central limit theorem still applies. The nonlinear distortion in the transformed domain is whitened and affects each data symbol in the same way. Similar benefit can be achieved with other orthogonal linear transforms. In this section, we show the nonlinear performance with an example orthogonal linear transformation, inverse discrete cosine transform (IDCT) [28]. The IDCT to the input signal is given by N 1 [ ] (k +.5)π x(n)=c(n) X k cos n, (25) N n= where 1 c(n)= N n =, 2 N n. In Fig. 8(b), the magenta line shows the BER performance of IDCT modulation in the presence of nonlinearity. The simulation setup is the same as the example used in DMT modulation. The simulation result with 64-point DCO-OFDM is replaced with 64-point IDCT. Since the output at IDCT modulation is complex, we transmit the in-phase part and the quadrature part in serial. The IDCT modulation achieves similar performance as the DCO-OFDM modulation and is advantageous over the PAM modulation. 4. Post-Distortion algorithm for DMT in nonlinear systems The nonlinearity in the VLC system can be different from that in the wireless radio communications system. The VLC system can operate at high nonlinear region without considering spectral emission to adjacent channels. The system performance, on the other hand, can be affected severely by the nonlinearity in the LED. As shown in Fig. 8(b), although the BER performance of the DMT modulation is greatly improved comparing to that of the PAM, it still suffers significant loss in the presence of nonlinearity. In order to further improve the BER performance, the nonlinearity needs to be estimated and compensated. Post-distortion nonlinear elimination technique is of particular interest since this technique can be applied to existing VLC systems without changing the hardware. For a single carrier system, when the inverse nonlinearity applies to the received signal, the noise added to large signal is amplified. The overall performance improvement may be limited. For a multi-carrier system, the distortion is applied to the time domain signal. In frequency domain, the impact of the nonlinearity is added to each subcarrier evenly. The post-distortion nonlinear elimination algorithm provides the same performance improvement to every subcarrier. In Fig. 5, a post-distortion block is applied before the conventional demodulation block to mitigate the LED s nonlinearity. An iterative post-distortion algorithm can be applied to eliminate the nonlinear distortion item iteratively [29]. The adaptive post-distortion algorithm sug- (C) 215 OSA 9 Feb 215 Vol. 23, No. 3 DOI:1.1364/OE OPTICS EXPRESS 263

14 gested in [18] is used for PAM signal. Table 2 summarizes the details of the post-distortion nonlinear elimination algorithm. Table 2. Iterative post-distortion nonlinear elimination algorithm. Input: received frequency domain data R =[R 1,,R N ], nonlinear model coefficients a =[a 1,,a P ], and the iteration number M. Output: estimated frequency domain data ˆX (M) =[ˆX (M) 1,, ˆX (M) N ]. Initialization: Provide soft estimation on the received data, ˆX () = R/a 1. Iteration: For iteration m = 1:M 1) calculate time domain data r (m) = IDFT( ˆX (m 1) ). 2) estimate time domain nonlinear error e (m) = h LED (r (m) ) a 1 r (m). 3) calculate frequency domain nonlinearity error E (m) = DFT (e (m) ). 4) eliminate nonlinearity error, obtain received data ˆX (m) =(R E (m) )/a 1. end In Fig. 9, we show the simulation results of a nonlinear system with and without postdistortion technique. The simulation setup is the same as that for Fig. 8(b). A total of 5 iterations are applied for the post-distortion nonlinear elimination algorithm. Figure 9(a) shows the BER performance of the receiver and Fig. 9(b) shows the EVM performance of the receiver. In both cases, from bottom to up, the black solid line shows the performance of a linear system; the blue dashed line shows the performance of the DMT system with post-distortion; the blue solid line shows the performance of the DMT modulation without post-distortion; the red dashed line shows the performance of the PAM system with post-distortion; the red solid line shows the performance of the PAM system without post-distortion. From Fig. 9(a), we observe that the post-distortion technique improves the BER performance for both the PAM and the DMT modulation with optimal decision. In addition, the BER performance for DMT with post-distortion almost overlaps with the BER performance without nonlinear impairment. The iterative post-distortion technique is ideal for the DMT modulation. On the other hand, although the proposed iterative post-distortion technique also provides improvement to the PAM modulation, the overall system degradation is still not noticeable. Nevertheless, we would like to emphasize that the optimal demodulation decision needs a complete knowledge of nonlinearity of the LED, which may not be realistic at the receiver. The EVM performance in Fig. 9(b) shows the same trend as the BER performance in Fig. 9(a). 5. Conclusion In a VLC system, the nonlinear characteristics of the LED in the transmitter is the limiting factor of the system performance. Conventional single carrier modulation such as PAM, which is a (C) 215 OSA 9 Feb 215 Vol. 23, No. 3 DOI:1.1364/OE OPTICS EXPRESS 2631

15 BER 1 2 EVM (db) PAM W/O PD PAM W PD DMT W/O PD DMT W PD Linear system SNR (db) 25 PAM W/O PD PAM W PD 3 DMT W/O PD DMT W PD Linear system SNR (db) (a) BER performance of the VLC system. (b) EVM performance of the VLC system. Fig. 9. VLC system performance with nonlinear LED. In both cases, from bottom to up, the black solid line shows the performance of a linear system; the blue dashed line shows the performance of the DMT system with post-distortion; the blue solid line shows the performance of the DMT system without post-distortion; the red dashed line shows the performance of the PAM system with post-distortion; the red solid line shows the performance of the PAM system without post-distortion. natural choice in the IM/DD VLC system, is vulnerable to the nonlinear effects. The multicarrier system, such as the DMT modulation, the distortion terms in time domain is spread evenly to each subcarrier in the transformed domain, whereas the distortion terms distort more on the large signal than on the small signal for single carrier modulation. The system performance is improved with the DMT modulation. Furthermore, the iterative post-distortion nonlinear elimination algorithm, which is applied at the receiver side, greatly improves the performance of the DMT modulation in the nonlinear VLC system. Acknowledgments This work was supported in part by the 1 Talents Program of Chinese Academy of Sciences, the National Key Science and Technology Project (No. 214ZX3124), and the Science and Technology Commission Foundation of Shanghai (No ). (C) 215 OSA 9 Feb 215 Vol. 23, No. 3 DOI:1.1364/OE OPTICS EXPRESS 2632