# Impact of Nonlinear LED Transfer Function on Discrete Multitone Modulation: Analytical Approach

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- 4970 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 22, NOVEMBER 15, 2009 Impact of Nonlinear LED Transfer Function on Discrete Multitone Modulation: Analytical Approach Ioannis Neokosmidis, Thomas Kamalakis, Member, IEEE, Joachim W. Walewski, Beril Inan, and Thomas Sphicopoulos, Member, IEEE Abstract—Light-emitting diodes constitute a low-cost choice for optical transmitters in medium-bit-rate optical links. An example for the latter is local-area networks. However, one of the disadvan- tageous properties of light-emitting diodes is their nonlinear char- acteristic, which may limit the data transmission performance of the system, especially in the case of multiple subcarrier modula- tion, which is starting to attract attention in various applications, such as visible-light communications and data transmission over polymer optical fibers. In this paper, the influence of the nonlinear transfer function of the light-emitting diodes on discrete multitone modulation is studied. The transfer function describes the depen- dence of the emitted optical power on the driving current. Analyt- ical expressions for an idealized link were derived, and these equa- tions allow the estimation of the power of the noise-like, nonlinear crosstalk between the orthogonal subcarriers. The crosstalk com- ponents of the quadrature and in-phase subcarrier components were found to be independent and approximately normally dis- tributed. Using these results, the influence of light-emitting-diode nonlinearity on the performance of the system was investigated. The main finding was that systems using a small number of sub- carriers and/or high QAM level exhibit a large signal-to-noise-ratio penalty due to the nonlinear crosstalk. The model was applied to systems with white and resonant-cavity light-emitting diodes. It is shown that the nonlinearity may severely limit the performance of the system, particularly in the case of resonant-cavity light-emit- ting diodes, which exhibit a strong nonlinear behavior. Index Terms—Discrete multitone (DMT) modulation, light-emit- ting diode (LED), nonlinear distortion, optical communication. I. INTRODUCTION I N recent years, there has occurred a rapid development ofoptical communication systems that can provide Tbit/s con- nectivity in core and metropolitan-area networks [1]. As optical technologies begin to migrate into the access- and home-net- work area, which are characterized by a much higher sensitivity Manuscript received January 08, 2009; revised June 04, 2009 and July 07, 2009. First published July 31, 2009; current version published September 10, 2009. This work was supported in part by the European Community’s Seventh Framework Program FP7/2007-2013 under Grant 213311, also referred to as OMEGA. I. Neokosmidis and T. Sphicopoulos are with the Department of Informatics and Telecommunications, University of Athens, Athens GR-15784, Greece (e-mail: i.neokosmidis@di.uoa.gr; thomas@di.uoa.gr). T. Kamalakis is with the Department of Informatics and Telematics, Harokopio University, Harokopou 89, Athens, GR17671, Greece (e-mail: thkam@hua.gr). J. W. Walewski is with the Siemens AG, Corporate Technology, Information and Communications, Munich, Germany (e-mail: joachim.walewski@siemens. com). B. Inan is with the Technische Universitaet Muenchen (TUM), Munich, Ger- many (e-mail: beril.inan@tum.de). Digital Object Identifier 10.1109/JLT.2009.2028903 to the initial capital expenditure, the cost of optical components and their ease of use becomes a critical factor for their deploy- ment and their future prospects. Light-emitting diodes (LEDs) can be used in low-cost, medium-bit-rate transmitters, and reso- nant-cavity LEDs (RC-LEDs) are capable of providing bit rates up to 1 Gbit/s in local-area networks, where the link distances are limited to less than 5 km [2]. LEDs also provide a cost-effec- tive solution for optical-wireless transmitters in both indoor and outdoor systems [3], [4]. In the infrared range, such systems pro- vide wireless local-area-network connectivity in the order of 50 Mbit/s and above, in both line of sight and the diffuse regime [5], [6]. It is also possible to modulate the light emitted by lighting LEDs, hence providing illumination and wireless connectivity at the same time. These systems are usually referred to as vis- ible-light communication (VLC) systems [7], [8]. There are several properties of LEDs that can affect the per- formance of a communication system. One of them is the non- linearity of their transfer function, i.e., the dependence of the emitted optical power on the driving current. Due to interfer- ence from fluorescent lighting of up to several hundreds of kHz it is beneficial to encrypt the data in VLC on subcarriers. By applying multilevel QAM modulation on discrete multitones (DMT) [9], Grubor et al. demonstrated data transmission rates in excess of 100 Mbit/s [10]. Although existing DMT chips with a similar modulation bandwidth make this approach very at- tractive [11], it also comes with a potential drawback, since the above nonlinearity causes subcarrier interaction, and the data stream from other subcarriers increases the noise floor under each subcarrier [9]. The interrogation into the impacts of nonlinearity on OFDM and DSL can be divided into two main areas of activities: clipping of otherwise linear transfer functions and nonlinear transfer functions exhibiting continuous gradients. Clipping in OFDM and DMT is the practice of putting a symmetric upper and lower bound on the AC-portion of the signal. The impetus behind this is to limit the potentially very large crest factor of these signals and, thus, to increase the channel throughput. However, while increasing the power submitted, clipping also entails noise, whose characteristics have been widely studied [12]–[14]. Since, due to the discontinuity of their gradients and their otherwise linear behavior, the considered transfer functions essentially differ from nonlinear transfer functions of LEDs, the findings of these publications are of limited value to us. Concerning studies considering continuous-gradient transfer function, one has to divide them in those addressing OFDM and those addressing DMT. Studies addressing OFDM are of limited value for us, since the OFDM signal by its nature is bipolar, it is inherently point symmetric and the respective transfer function can be described by Taylor series containing 0733-8724/$26.00 © 2009 IEEE
- NEOKOSMIDIS et al.: IMPACT OF NONLINEAR LED TRANSFER FUNCTION ON DISCRETE MULTITONE MODULATION 4971 Fig. 1. Experimental setup for measuring the nonlinear transfer function of a single-chip white LED. only odd-powered elements [15]. Also, due to its bandpass characteristics, many of the interference terms lie out of band [15]. Examples of such studies are those of Tang et al. [16], Chorti et al. [17] and O’Droma et al. [18]. Baseband multicar- rier is also used in analogue video transmission, and they come closest to the issue studied in our work. Example works are those of Frigo et al. [13], [19]. While the former only addresses clipping effects, the latter presents a general approach based on probability-density transfer functions describing the deteriora- tion of individual subcarriers. However, even in this case, only clipping is considered. In contrast, we investigate the impact of continuous-gradient nonlinear LED transfer functions on the overall performance of a DMT transmitter system. Based on a measured static transfer function of a white LED, a quadratic polynomial is used as the parameter-free model of the transfer function. This polynomial approximation has been widely used in the past to model the nonlinearity of LEDs or laser diodes [20]. In this paper, it is used as a starting point to obtain closed-form expressions for the intercarrier crosstalk power and the study of the statistical properties of this crosstalk noise. The analytical formulas can be used to study of the impact of LED nonlinearity on the perfor- mance of DMT without the need for time-consuming numerical simulations. The remainder of this paper is organized as follows. In Section II, the measured nonlinear transfer function of a white single-chip LED is presented and polynomial fitting is used to obtain a second-order parameter-free model of the transfer function. In Section III, the DMT system is described in detail and analytical formula for the power of the intersubcarrier crosstalk are derived. Also, the model is extended in order to include additive white Gaussian noise (AWGN) in the flat-frequency-response channel, as described elsewhere in the literature [6], [21]. In Section IV, our model is used to study various aspects of the DMT system, for instance, the dependence of the nonlinear crosstalk on the total number of subcarriers and the modulation level for both white and RC-LEDs. Conclusions are provided in Section V. II. NONLINEAR CHARACTERISTIC OF THE LED Fig. 1 shows the experimental setup that was used for mea- suring the static transfer function of a phosphorescent single- chip LED (NICHIA, NSPW500CS). The DC impedance of the LED was matched to 50 with a serial resistor, and the DC voltage was supplied by a commercial power source (Agilent, E3620A). The emitted light was directed onto an amplified pho- todiode (Thorlabs, PDA10A-EC). Both the applied voltage and the current through the diode are measured with multimeters (Voltcraft, VC220). The output power, , of the LED is mea- sured with the photo detector. The latter is measured as a func- tion of the DC driving current. The choice of an optimum de- gree for a polynomial representing the LED transfer function has been discussed in detail by Walewski [22], who showed that Fig. 2. Three measurement sets of the LED optical output power as a function of the DC driving current obtained for a single-chip white LED. The setup in Fig. 1 was used. Markers: measurement data; solid line: second-order polyno- mial fit to the data. albeit polynomial orders of as high as five are needed to realis- tically model measured transfer functions, a second-order poly- nomial already provides a fair description, as is demonstrated in Fig. 2. The polynomial function in question is (1) where mA. The coefficients , , and are the DC term, the linear gain, and the second-order nonlinearity co- efficient, respectively. The polynomial expansion in (1) is the model used in the fol- lowing derivation of analytical expressions for the intercarrier distortion in DMT as a result of transfer-function nonlinearity. It should be noted that the proposed model is only valid for mod- ulation frequencies well below the LED 3-dB bandwidth, since (1) is actually the static transfer function of the LED. A more complete description would require the use of a dynamic model based on the solution of the active region carrier density rate equation [23]. Deriving closed form formulas for the intercarrier crosstalk power is much more involved in this case, however. III. ANALYTICAL EXPRESSIONS FOR INTERCARRIER DISTORTION In this section, the DMT waveform distortion due to the LED nonlinearity is investigated. It is shown that the nonlinearity of the LED transfer function adds a nonlinear crosstalk component to each subcarrier, and closed-form formulas are obtained for the power of this distortion. It will also be numerically shown that this crosstalk noise is approximately normally distributed, and that the in-phase and quadrature noise components are ap- proximately independent. The model is also extended to include the contribution of AWGN stemming from thermal noise at the receiver and/or ambient light noise. A. DMT Waveform Distortion In order to simplify the analysis, cyclic prefixes are ignored. In this case, the current signal driving the LED can be written as (2) where is the bias current, is the subcarrier number, is the total number of subcarriers, is the sent symbol on sub- carrier , is the subcarrier frequency, is the complex con-
- 4972 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 22, NOVEMBER 15, 2009 jugate, and is the duration of a DMT symbol. The subcarrier frequencies are given by . Notice that the DC carrier is not modulated [10]. In (2), is a current amplitude, which is chosen to keep the current inside a given operational range .The modulation index MI is defined as (3) In our calculations, we have chosen mA (i.e., ) and the operational range is, thus, [0 61mA]. We do not consider clipping of the driving current. The DMT symbols are obtained from a QAM symbol constellation (4) where is the symbol value and and are the in-phase and the quadrature components, respectively. For a QAM modulation ( bits per symbol, i.e., for quadratic constellations), one can use the following equation for the in-phase and the quadrature components of the symbols: (5) Using (5) as well as the fact that is maximized when both and are maximized, that is when , one can easily show that the AC component of the current, i.e., , is bounded by (6) Another approach to calculate the maximum value of the AC current has been proposed by Mestdagh [14]. In this model, the current is maximized if one chooses a suitable combina- tion of QAM symbols from the corners of the constellation di- agram such that for odd values of and for even values of . In this case the maximum amplitude of the current is obtained by replacing the with the factor . Although, will never take the value of (6), it is close enough to the actual maximum. Furthermore, it helps to in- crease the channel throughput avoiding the clipping noise. On the other hand, Mestdagh’s maximum is only valid for high QAM levels and number of sub-carriers. On the contrary, it un- derestimates the amplitude of the DMT signal for few QAM symbols and few carriers . Therefore, Mestdagh et al.’s approach fails in always providing an unclipped DMT signal and was not further pursued in our investigation (7) Inserting (2) into (1), one can easily derive the optical output power as a function of the DMT driving current (8) The third term in the sum of (8) represents intermodulation products at frequencies and , which give rise to nonlinear crosstalk noise, since for randomly distributed input data, the terms in the third term are uncorrelated to the data stream purveyed by term two. To estimate the impact of this noise on the performance of the system, one needs to calculate the decoded symbols. In a matched receiver, the decoded symbol of carrier is (9) (10) Inserting (8) into (9) and (10), one obtains the following equa- tion for the symbol estimates at the receiver: (11) (12) where (13) (14) Here, the sets and are given by and (15) and (16) B. Analytical Formulas for the Variation of the Intercarrier Crosstalk The conditions posed in the sums of (13) and (14) stem from the condition that the frequency of the interference has to coin- cide with a subcarrier in order to contribute to the detected noise. Using (13) and (14), one can estimate the variances and of the quadrature and in-phase component of the inter- carrier crosstalk. As stated above, our approach is only valid for an even number of bits per symbol . Fortuitously, for exactly this case both variances can be written in a closed form. For this, one invokes the assumption that symbols of different subcarriers are statistically independent, viz. for . After some mathematical manipulation and exploiting that , as provided in the literature [24], one obtains for even subcarrier numbers (17) (18)
- NEOKOSMIDIS et al.: IMPACT OF NONLINEAR LED TRANSFER FUNCTION ON DISCRETE MULTITONE MODULATION 4973 Fig. 3. Crosstalk ratios ��� [see equations (20) and (21)] as a function of subcarrier number� for 4 QAM and 64 QAM modulation for (a) the real part and (b) the imaginary part of the received symbols. The total number of subcar- riers is 7. while for odd subcarrier numbers , the variances are given by (19) Equations (17)–(19) are very useful since they allow the esti- mation of the noise variances without the need for numerical simulations. When the number of bits per symbol is odd, it is more difficult to derive a closed form expression for the crosstalk variances and one can resort to numerical approaches like Monte Carlo (MC) simulation in order to calculate them. To assess the validity of (17)to (19) we compared them against results obtained from Monte Carlo (MC) simulations based on (13) and (14). Convenient figures of merit for the comparison of intercarrier crosstalk are the ratio of the crosstalk variance and the symbol distance, viz. (20) (21) where and are the in-phase and quadra- ture crosstalk ratios for subcarrier , respectively, and is the minimum distance between QAM symbols. For the QAM con- stellation defined in (5), one finds that [24]. It is also useful to define the average signal-to-crosstalk ratio for subcar- rier , given by (22) In Fig. 3(a) and (b), and obtained from MC simulations are compared to the analytical formulas and for var- ious subcarrier numbers in a DMT system with seven total subcarriers and 4-QAM and 64-QAM modulation, respectively. We conducted similar comparisons for various total numbers of subcarriers and yielded a similar excellent agreement. Note that the value is not included in the figures, since the DC component does not carry any signal. It is also interesting to note that one could include a cubic term in the polynomial expansion in (1). A series of Monte Carlo sim- ulations were performed in order to estimate the influence of the cubic term and it was concluded that its influence is negligible. For example in the case of and the obtained value is only 0.0014 dB lower when the cubic term is included. Fig. 4. PDFs of the of nonlinear intercarrier crosstalk obtained by Monte Carlo simulation (dots) of (a) the in-phase component [equation (13)] and (b) the out-of-phase component [equation (14)] of the received symbol for a DMT mod- ulation with seven subcarriers and � � �� QAM states per subcarrier. The PDF of a Gaussian random variable with the same variance is also shown (solid line). Fig. 5. PDFs of the of nonlinear intercarrier crosstalk obtained by Monte Carlo simulation (dots) of for (a) the in-phase component [equation (13)] and (b) the out-phase component [equation (14)] of the received symbol for a DMT modu- lation with 255 subcarriers and� � �� QAM states per subcarrier. The PDF of a Gaussian random variable with the same variance is also shown (solid line). C. Statistical Nature of the Nonlinear Crosstalk Noise Next, we consider the distribution function of the intercar- rier crosstalk. In Fig. 4(a) and (b), the probability density func- tions (PDFs) of and , respectively [see (13) and (14)], are illustrated for the first channel , in the case of a DMT system with seven subcarriers and QAM levels. Also plotted is the PDF of a zero-mean Gaussian random variable with the same variance. The results indicate that the PDFs of the intercarrier crosstalk can be roughly approximated by a Gaussian PDF. The approximation becomes better as the number of carriers is increased. This is illustrated in Fig. 5, where similar PDFs are plotted for 255 subcarriers and QAM levels. Similar results are obtained for other subcarrier indices. Next, it is of interest whether the distortion terms and are uncorrelated. To this end we investigated into the cor- relation coefficient of these terms, i.e., (23) for which we assume that the expectation of both distortion terms is zero, which has already been shown to be supported by simulation. The correlation coefficient was estimated from repeated Monte Carlo runs. After 50 runs the correlation coef- ficient was estimated by aid of (23) and the procedure repeated for a total of 20 000 times. The resulting histograms of the cor- relation coefficient were found to be centered on zero, which
- 4974 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 22, NOVEMBER 15, 2009 Fig. 6. Histograms of � for the first subcarrier. The quantity � is the transformed value of the correlation coefficient between � and � [see equation (24)], as compared with the student � distribution (solid line) for four sets of QAM level and total number of subcarriers (‘sub’). supports our hypothesis, that the distortion terms indeed are un- correlated. If one instead of the correlation coefficient plots the distribution of (24) one expects a student distribution in case both distortions are normally distributed [25]. In Fig. 6, we show the histogram of the correlation coefficients for four cases and compare them with the theoretical distribution. For low sub-carrier numbers the historgram of is indeed symmetrical but is close dis- tributed around zero than the theoretical distribution. This devi- ation can be attributed to a non-normal distribution of the inter- ferences themselves. While the histogram supports the assump- tion that the correlation coefficient is zero we cannot test this hypothesis, since the distribution function of is unknown. As discussed before, for an increasing number of total subcar- riers the distribution of the interferences becomes more normal, and, as can be seen in Fig. 6, the distribution of [see (24)] is much closer to the theoretical one. For these cases, we can apply hypothesis testing based on the student distribution, and for all the subcarrier and QAM-level combinations addressed in this work the hypothesis that the correlation coefficient is not dis- tributed according to the student distribution could be rejected with an alpha of 0.05 for all simulations with QAM level higher than 4 and more than seven subcarriers. Therefore, the hypoth- esis of the correlation coefficient being unequal zero can also be rejected. Since two uncorrelated normally distributed enti- ties are statistically independent this is, hence, also the case for the in- and quadrature intercarrier interferences and . D. Inclusion of Additive White Gaussian Noise The model presented above can be generalized by including AWGN, which may originate from thermal noise at the receiver TABLE I BIT RATES FOR A SYMBOL RATE OF 24.2 MHz AND VARIOUS QAM MODULATION LEVELS and/or ambient light noise. This can be done in a straight for- ward manner by adding an AWGN term in (8). The symbol estimates in (11) will, thus, contain additional Gaussian noise components and , i.e., (25) (26) The noise components and are independent and identically distributed, with zero mean and the variance is (27) with the double-sided power spectral density of the noise. To characterize the influence of the AWGN one figure of merit is the signal-to-noise ratio per bit, defined as , where is the average energy per bit. Assuming a DMT waveform [see (2)], ignoring the intercarrier crosstalk noise, relying on the fact that , and that inside a DMT symbol period there are QAM symbols or bits, it is straightforward to show that (28) IV. SIMULATION RESULTS AND DISCUSSION A. System Bitrate The analytical model described in Section III is used to further analyze the impact of the nonlinear LED transfer function on the performance of a point-to-point DMT system. Since the DC subcarrier is not modulated, the symbol rate of the system is . Hence, does not depend on and . The bit rate of the signal is and varies with . It is already noted that the proposed model is only valid for modulation frequencies well below the LED 3 dB bandwidth. Since the frequency of the last subcarrier, and, hence, the maximum allowable signal bandwidth cannot be determined, the data rates are normalized assuming that the signal at the maximum data rate (e.g., for 1024 QAM levels) utilizes the whole allowable bandwidth. Table I summarizes the values of the normalized bit rates for several values of used in this paper. B. Nonlinear Degradation Fig. 7 illustrates the values of the average signal-to-crosstalk ratio obtained for various DMT parameter settings. The results are based on (17)to (21). There are several interesting features that can be drawn from this figure. First, the lowest subcarrier is always the subcarrier with the worst distortion. This is consistent with (17)to (19), which show that
- NEOKOSMIDIS et al.: IMPACT OF NONLINEAR LED TRANSFER FUNCTION ON DISCRETE MULTITONE MODULATION 4975 Fig. 7. Average signal-to-crosstalk ratio ����� as a function of the subcarrier number� for various QAM modulation levels and total number of subcarriers. the power of the crosstalk noise decreases as increases. It is also interesting to note that the lowest subcarrier has about 3 dB lower SXR than the highest subcarrier. Another point that can be drawn from Fig. 7 is the fact that the generally tends to improve as the number of QAM levels is reduced. As shown in the figure, there is about 20-dB degradation of when QAM instead of QAM is used. In most cases, there is a 4 to 5 dB decrease in increasing the modu- lation level from to QAM. The degrada- tion with increasing is not surprising. Although the distance between neighboring symbols in the original QAM constella- tion does not depend on , the average power at the transmitter must be kept constant, and, hence, the actual distance between the symbols in the scaled waveform (2) is decreased, rendering the signal more susceptible to the crosstalk noise, whose statis- tical properties vary little upon a change of . It is also interesting to mention that, as the number of subcar- riers is increased, is improved, and, hence, an LED-based optical wireless multiple subcarrier system exhibits different behavior than its wired, Four-Wave-Mixing limted counterpart [26]. The improvement becomes even more obvious in Fig. 8, where the is plotted as a function of the total number of subcarriers for (a) the first and (b) the central subcarrier, re- spectively. The explanation of this behavior can be found in the fact that according to (7), the parameter in (17)to (19) decreases as . This decrease is much faster than the in- crease of the possible combinations satisfying the conditions and , the number of which is equal to . From another point of view, the use of more subcarriers will result in the original DMT waveform having large spikes while still remaining inside a given operational range . As the number of subcarriers is increased, these spikes will become sharper and sharper, fol- lowed by longer periods of time, where will possess a small amplitude. Since the influence of nonlinearity grows with an increase in , only high-current spikes will be affected by the LED nonlinear characteristic, while the distortion for the rest of the signal will be small. Thus, the occurrence of nonlinear Fig. 8. Estimation of SXR for (a) the first subcarrier and (b) the central subcar- rier �� � ��� � ������ as a function of the number of subcarriers. Fig. 9. Bit-error ratio ����� of the first subcarrier as a function of the signal-to-noise ratio per bit ���� � for various QAM modulation levels and for (a) three subcarriers and (b) 255 subcarriers. distortion, and, hence, its impact on the overall signal integrity, decreases with an increase in the total number of subcarriers. There is one point to note, however: The linear signal-to-noise ration as defined in (28) decreases with an increase in the number of subcarriers since is inversely proportional to . This means that, as increases, obtaining the same re- quires a lower . Since is due to both ambient light noise and thermal noise, in practice, it is not possible to reduce beyond a certain minimum value leading to an upper bound for the achievable . C. Power Penalty The influence of the LED nonlinearity on the performance of a system degraded by AWGN is illustrated in Fig. 9, where the bit-error ratio (BER) is plotted as a function of the [see (28)]. Note that takes only the AWGN into ac- count, not the intercarrier crosstalk. The values of the BER are then calculated with and without the nonlinear crosstalk (dotted and solid lines, respectively). Fig. 9(a) and (b) corresponds to the case with three and 255 subcarriers, respectively. The BER
- 4976 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 22, NOVEMBER 15, 2009 Fig. 10. Bit-error ratio (BER) as a function of the signal-to-noise ratio per bit for various QAM modulation levels and for (a) three subcarriers and (b) 255 subcarriers (RC-LED). was calculated for the first subcarrier (worst case scenario) and the intercarrier crosstalk is assumed to follow Gaussian statis- tics (see Section III-C). In a system with three subcarriers, the performance is severely degraded compared to the linear case for a QAM constellation with 64 symbols and above. For 64 QAM symbols and dB, the BER is increased from to 0.0013, almost three orders of magnitude. On the other hand, the performance of a system with fewer QAM symbols or with larger number of subcarriers is less severely af- fected. It is, therefore, understood that depending on the system parameters the intercarrier cross talk due to LED nonlinearity may drastically influence the performance of a system with a small number of subcarriers. So far, we have only discussed the impact of transfer function nonlinearity for a white LED. As it turns out, there are LEDs exhibiting an even stronger nonlinearity. One example is a red RC-LED from AVAGO (HFBR-1521Z). We recorded the static transfer function of this LED and the parameters of the polyno- mial fitting were found to be , mA and mA . In our simulations, the DC current was set to 20 mA, while . Fig. 10 depicts the with and without the intercarrier crosstalk as a function of the signal-to-noise ratio per bit for the RC-LED. A system using the RC-LED exhibits the same behavior as the one with the white LED. For example, the performance is improved when less QAM states per subcarrier are used and/or more subcarriers are included in the system. It is obvious that the performance of the system incorporating the RC-LED is worse compared to the one with the white LED, which is caused by the stronger non- linearity of its transfer function. It is interesting to note that for three subcarriers, a floor higher than is observed for more than 16 QAM symbols. Fig. 11. Signal-to-noise-ratio penalty of the first subcarrier as a function of the QAM level for achieving ��� � �� (white LED). Fig. 12. Signal-to-noise ratio penalty of the first subcarrier as a function of the QAM level for achieving ��� � �� �� (red RC-LED). To further illustrate the effect of nonlinear intercarrier crosstalk, the penalty was evaluated for various QAM levels and numbers of subcarriers. The penalty is defined as the difference between the required for achieving a specific with and without the nonlinear crosstalk. In Fig. 11, the penalty of the first subcarrier (worst case) for a BER equal to is plotted versus the QAM level. Once again it is apparent that the noise stemming from in- tercarrier crosstalk is more pronounced for few subcarriers or higher QAM level. For three subcarriers, the penalty is 0.9 dB and 3.8 dB for and QAM symbols, respectively. For seven subcarriers and 64 QAM symbols, the difference in the required approaches 1.7 dB. In the case of the RC-LED the SNR penalty values are higher, as illustrated in Fig. 12. This is again due to the stronger non- linearity of the transfer function. Notice that the SNR penalty is calculated for , which corresponds to the BER floor of the system with more than 16 QAM symbols. Notice also difference in maximum QAM level in Figs. 11 and 12. It seems, therefore, that in both cases (white and RC-LED), the intersubcarrier crosstalk noise noticeably affects the perfor- mance of the system. Also, the nonlinear crosstalk becomes one of the limiting factors in an optical communication system es- pecially in the case of few subcarriers and high number of QAM levels. V. CONCLUSION In this paper, the impact of the nonlinear transfer function of LEDs on the performance of a QAM-DMT data transmis- sion system was analyzed. Analytical formulas were derived,
- NEOKOSMIDIS et al.: IMPACT OF NONLINEAR LED TRANSFER FUNCTION ON DISCRETE MULTITONE MODULATION 4977 which allow the estimation of the nonlinear crosstalk on each subcarrier when the total number of subcarrier channels and the number of quadrature-amplitude levels is known. It was shown that the crosstalk components of the quadrature and the in-phase components are independent and approximately nor- mally distributed. The model was extended to incorporate the in- fluence of an AWGN component, which may originate from the thermal noise at the receiver and/or ambient light noise. Using this model, the influence of the nonlinear LED transfer func- tion was investigated, and it was shown that for an unclipped signal, the system performance is degraded as the number of subcarriers is reduced or higher QAM modulation is used. The model was applied to evaluate the performance degradation in the case of a white LED (previously used in a VLC system) and a RC-LED transmitter (commonly used in polymeric optical fiber transmission systems). Stricter performance limits are posed for the RC-LED due to its stronger nonlinearity. Increasing the number of subcarriers or reducing the number of QAM symbols may alleviate the effect of nonlinearity, thus improving the per- formance of the system. ACKNOWLEDGMENT The authors would like to thank their colleagues for their con- tributions. This information reflects the consortiums view; the Community is not liable for any use that may be made of any of the information contained therein. REFERENCES [1] B. Mukherjee, “WDM optical communication networks: Progress and challenges,” IEEE J. Sel. Areas Commun., vol. 18, no. 10, pp. 1810–1824, Oct. 2000. [2] E. F. Schubert, Light-Emitting Diodes, 2nd ed. Cambridge, U.K.: Cambridge Univ. Press, 2006. 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Spruyt, and B. Biran, “Analysis of clipping effect in DMT-based ADSL systems,” in Proc. IEEE Int. Conf. Communica- tions, 1994, pp. 293–300. [15] C.-L. Liu, “The effect on nonlinearity on a QPSK-OFDM-QAM signal,” IEEE Trans. Consum. Electron., pp. 443–447, 1997. [16] Y. Tang, W. Shieh, X. Yi, and R. Evans, “Optimum design for RF-to-optical up-converter in coherent optical OFDM systems,” IEEE Photon. Technol. Lett., vol. 19, no. 7, Apr. 1, 2007. [17] A. Chorti and M. Brookes, “On the effects of memoryless nonlinearities on M-QAM and DQPSK OFDM signals,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 8, Aug. 2006. [18] M. O’Droma, N. Mgebrishvili, and A. Gloacher, “Simulation-based analysis of nonlinearites in multi-carrier OFDM signals,” in Proc. 5th Workshop on Singal Processing Advances in Wireless Communica- tions, 2004, pp. 611–615. [19] N. J. Frigo, “A model of intermodulation distiortion in non-linear multicarrier systems,” IEEE Trans. Commun., vol. 42, no. 2/3/4, pp. 1216–1222, 1994. [20] J. C. Daly, “Fiber optic intermodulation distortion,” IEEE Trans. Commun., vol. Com-30, no. 8, Aug. 1982. [21] A. R. Hayes, Z. Ghassemlooy, N. L. Seed, and R. McLaughlin, “Base- line wander effects on systems employing digital pulse interval modu- lation,” IEE Proc. Optoelect., vol. 147, no. 4, pp. 295–300, 2000. [22] J. W. Walewski, Inference of Calibration Curves Non-Linear Transfer Function by Use of Orthogonal Polynomials, 2008. [23] R. Windisch, A. Knobloch, M. Kuijk, C. Rooman, B. Dutta, P. Kiesel, G. Borghs, G. H. Dohler, and P. Heremans, “Large signal modulation of high efficiency light emitting diodes for optical communication,” IEEE J. Quant. Electron., vol. 36, no. 12, pp. 1445–1453, 2000. [24] J. G. Proakis, Digital Communication, 4th ed. New York: McGraw- Hill, 95. [25] E. M. Pugh and G. H. Winslow, The Analysis of Physical Measure- ments. Reading, MA: Addison-Wesley, 1966. [26] R. Hui, B. Zhu, R. Huang, C. T. Allen, K. R. Demarest, and D. Richards, “Subcarrier multiplexing for high-speed optical transmis- sion,” J. Lightw. Technol., vol. 20, no. 3, Mar. 2002. Ioannis Neokosmidis was born in Athens, Greece, in 1977. He received the B.Sc. degree in physics, the M.Sc. degree in telecommunications, and the Ph.D. degree in transmission limitations due to nonlinear phenomena from the Uni- versity of Athens in 1999 and 2002, respectively. He is currently a Research Associate for the Optical Communications Labo- ratory, University of Athens. His research interests include nonlinearities, WDM optical networks and optical components, photonic crystals, and wireless optical systems. Thomas Kamalakis (M’09) was born in Athens, Greece, in 1975. He received the B.Sc. degree in informatics, the M.Sc. degree (with distinction) in telecom- munications, and the Ph.D. degree in the design and modeling of arrayed wave- guide grating devices from the University of Athens in 1997, 1999, and 2004, respectively. He is a Lecturer at the Department of Informatics and Telematics, Harokopio University of Athens, and a research associate in the Optical Communications Laboratory, University of Athens. His research interests include photonic crystal devices, coupled resonator optical waveguides, optical wireless, and nonlinear effects in optical fibers. Dr . Kamalakis is a member of the Optical Society of America and IEEE. Joachim W. Walewski graduated from the Christian Albrechts University, Ger- many, with a diploma degree in physics (Dipl.-Phys.) in 1995 and the Ph.D. de- gree from the Lund Institute of Technology, Sweden, in 2002 for his research on applied laser spectroscopy. From 1996 to 1997, he was a visiting scientist at the Tampere University of Technology, Finland, engaging in laser-assisted diagnostics of CVD diamond. From 2001 to 2003, he served as junior lecturer at the Lund Institute of Tech- nology, and from 2003 to 2006, he held positions as research associate and fi- nally as assistant scientist at the Engine Research Center of the University of Wisconsin-Madison. At the latter two institutions, his research was focused on the development and application of laser-spectroscopic techniques for combus- tion research. In May 2006, he joined Siemens Corporate Technology, Infor- mation and Communications, Munich, Germany and has focused his efforts on R&D in the field of wireless optical communications and Green ICT.
- 4978 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 22, NOVEMBER 15, 2009 Beril Inan was born in Ankara, Turkey, in 1983. She received the B.Sc. degree in electrical and electronics engineering at Middle East Technical University, Ankara, Turkey, in 2005, and the M.Sc. degree from the Eindhoven University of Technology, Eindhoven, The Netherlands, in 2008, in electrical engineering. She carried out her M.S. thesis project at Siemens Corporate Technology, Infor- mation and Communications, Munich, Germany, in the field of nonlinearity im- pact on optical communication. She is currently pursuing her Ph.D. at Siemens Corporate Technology, Information and Communications. From 2005 to 2006, she worked at ASELSAN Electronic Industries Inc., Ankara, Turkey, as a system engineer. Thomas Sphicopoulos (M’87) received the degree in physics from Athens Uni- versity, Athens, Greece, in 1976, the D.E.A. degree and the Ph.D. degree in elec- tronics, both from the University of Paris VI, Paris, France, in 1977 and 1980, respectively, and the D.Sc. degree from the Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland, in 1986. From 1976 to 1977, he worked at Thomson CSF Central Research Labo- ratories on microwave oscillators. From 1977 to 1980, he was an Associate Researcher in Thomson CSF Aeronautics Infrastructure Division. In 1980, he joined the Electromagnetism Laboratory of the Ecole Polytechnique Federal de Lausanne, where he carried out research on applied electromagnetism. Since 1987, he has been with the University of Athens, engaged in research on broad- band communications systems. In 1990, he was elected as an Assistant Professor of communications in the Department of Informatics and Telecommunications, in 1993 as Associate Professor, and since 1998, he has been a Professor. His main scientific interests are optical communication systems and networks and techno-economics. He has lead about 40 National and European research and development projects. He has more than 150 publications in scientific journals and conference proceedings. Since 1999, he has been an advisor in several or- ganizations in the fields of fiber optics networks, spectrum management tech- niques, and technology convergence.

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