Impact of nonlinear LED transfer function on discrete multitone

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REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 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 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 multicarrier 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 deterioration of individual subcarriers. However, even in this case only clipping is considered. In contrast, we investigate the impact of continuousgradient 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 the present 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 performance 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 inter-subcarrier crosstalk are derived. Also, the model is extended in order to include additive white Gaussian noise (AWGN) in the flatfrequency-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 RCLEDs. Conclusions are provided in section V. II. NONLINEAR CHARACTERISTIC OF THE LED Figure 1 shows the experimental setup that was used for measuring the static transfer function of a phosphorescent single-chip LED (NICHIA, NSPW500CS). The DC

2

impedance of the LED was matched to 50 Ω 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 photodiode (Thorlabs, PDA10AEC). Both the applied voltage and the current through the diode are measured with multi-meters (Voltcraft, VC220). The output power, Pout, of the LED is measured with the photo detector. The latter is measured as a function of the DC driving current. The choice of an optimum degree for a polynomial representing the LED transfer function has been discussed in detail by Walewski [22], who showed that albeit polynomial orders of as high as five are needed to realistically model measured transfer functions, a second-order polynomial already provides a fair description, as is demonstrated in Figure 2. The polynomial function in question is Pout = b0 + b1 ( I in − I DC ) + b2 ( I in − I DC )

2

(1)

where IDC = 30.5 mA. The coefficients b0, b1, and b2 are the DC term, the linear gain, and the second-order nonlinearity coefficient, respectively. DC Power Supply (0-5V)

Current Meter

Resistor 22 Ohm

LED

Wireless Blue Channel Filter

Amplified Photodetector

Voltmeter

Figure 1: Experimental setup for measuring the non-linear transfer function of a single-chip white LED.

Figure 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 Figure 1 was used. Markers: measurement data; solid line: secondorder polynomial fit to the data.

The polynomial expansion in Equation (1) is the model used in the following derivation of analytical expressions for the inter-carrier distortion in DMT as a result of transferfunction nonlinearity. It should be noted that the proposed model is only valid for modulation 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 INTER-CARRIER 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 non-linear 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

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < quadrature noise components are approximately 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: u N −1 (2) I in (t ) = I DC + 0 ∑ sk e j 2π fk t + c.c., 0 ≤ t ≤ T 2 k =1 where IDC is the bias current, k is the subcarrier number, Ν-1 is the total number of subcarriers, sk is the sent symbol on subcarrier k, fk is the subcarrier frequency, c.c. is the complex conjugate, and Τ is the duration of a DMT symbol. The subcarrier frequencies are given by fk = k/T. Notice that the DC carrier is not modulated [10]. In Equation (2) u0 is a current amplitude, which is chosen to keep the current Iin(t) inside a given operational range Iin∈[IDC-ΔI IDC+ΔI].The modulation index MI is defined as: ΔI (3) MI = I DC In our calculations, we have chosen ΔΙ=ΙDC=30.5mA (i.e. MI=1) 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: sk = ( akI + jakQ ) (4) where sk is the symbol value and akI and akQ are the in-phase and the quadrature components, respectively. For a Μ=22r QAM modulation (n = 2r bits per symbol, i.e. for quadratic constellations), one can use the following Equation for the inphase and the quadrature components of the symbols: (5) aI, Q = 2u − ( M + 1) where u = 1, 2,..., M Using Equation (5) as well as the fact that |sk| is maximized when both akI and akQ are maximized, that is when u = M , one can easily show that the AC component of the current, i.e. iAC=Iin-IDC, is bounded by: i AC < 2( N − 1)

M − 1 u0

(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 combination of QAM symbols from the corners of the constellation diagram such that sk = sk e and

π

sk = sk e

− j ( k −1)

4

− jk

π 4

for odd values of k

for even values of k. In this case the

maximum amplitude of the current is obtained by replacing the 2 with the factor 1.064 2 + 1 / 2 .

(

)

Although, iAC will never take the value of equation (6), it is close enough to the actual maximum. Furthermore, it helps to increase 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,

3

it underestimates the amplitude of the DMT signal for few QAM symbols and few carriers (iAC, Mestdagh / iAC, actual = 0.92). Therefore, Mestdagh et al.’s approach fails in always providing an unclipped DMT signal and was not further pursued in our investigation. ΔI u0 = (7) 2( N − 1) M − 1

Inserting Equation (2) into (1), one can easily derive the optical output power as a function of the DMT driving current: b b u N −1 Pout = 0 + 1 0 ∑ sk e j 2π fk t 2 2 k =1 (8) 2 N −1 N −1 N −1 N −1 bu ⎧ j 2π f + f t j 2π f − f t ⎫ + 2 0 ⎨∑∑ sk sl e ( k l ) + ∑∑ sk sl* e ( k l ) ⎬ + c.c. 4 ⎩ k =1 l =1 k =1 l =1 ⎭ The third term in the sum of Equation (8) represents intermodulation products at frequencies fk+fl and fk-fl, which give rise to non-linear 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 m is: T 2 ⎪⎫ ⎪⎧ (9) Re ⎨ ∫ e − j 2π fm t Pout dt ⎬ AmI = u 0T ⎩⎪ 0 ⎭⎪ ⎫⎪ ⎧⎪ T 2 (10) Im ⎨ ∫ e − j 2π fm t Pout dt ⎬ u 0T ⎪⎩ 0 ⎪⎭ Inserting Equation (8) into (9) and (10), one obtains the following Equation for the symbol estimates at the receiver: AmI = amI + cmI (11) AmQ =

AmQ = amQ + cmQ

(12)

where ⎧⎪ ⎫⎪ b2 u0 Re ⎨∑ sk sl + 2∑ sk sl* ⎬ 2b1 Λm ⎪⎩ Τm ⎪⎭ bu ⎪⎧ ⎪⎫ cmQ = 2 0 Im ⎨∑ sk sl + 2∑ sk sl* ⎬ 2b1 Λm ⎪⎩ Τm ⎪⎭ Here, the sets Tm and Λm are given by cmI =

{( k , l ) 1 ≤ k , l ≤ N − 1 and k + l = m} = {( k , l ) 1 ≤ k , l ≤ N − 1 and k − l = m}

(13) (14)

Τm =

(15)

Λm

(16)

B. Analytical formulas for the variation of the inter-carrier crosstalk The conditions posed in the sums of Equations (13) and (14) stem from the condition that the frequency of the interference has to coincide with a subcarrier in order to contribute to the detected noise. Using Equations (13) and (14), one can estimate the variances σQ2(m) and σI2(m) 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 n. Fortuitously, for exactly

28 26 24 22 20 18 16 14

SXRI (dB)

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. ==0 for m≠k. After some mathematical manipulation and 2 exploiting that akI2 = akQ = ( M − 1) / 3 , as provided in the literature [24], one obtains for even subcarrier numbers m:

2 2 ⎪⎧ 4M − 8M + 4 ⎛ M − 1 ⎞ ⎪⎫ (18) N m 4 2 4 + − − ( )⎜ ⎨ ⎟ ⎬ 9 ⎝ 3 ⎠ ⎭⎪ ⎩⎪

while for odd subcarrier numbers m the variances are given by: 2

2

⎛ b2u0 ⎞ ⎛ M − 1 ⎞ (19) ⎟ ( 2 N − m − 3) ⎜ ⎟ 2 b ⎝ 3 ⎠ ⎝ 1 ⎠

σ I2 ( m) = σ Q2 ( m) = 4 ⎜

Equations (17) to (19) are very useful since they allow the estimation of the noise variances without the need for numerical simulations. When the number of bits per symbol n 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 Equations (17) to (19) we compared them against results obtained from Monte Carlo (MC) simulations based on Equations (13) and (14). Convenient figures of merit for the comparison of inter-carrier crosstalk are the ratio of the crosstalk variance and the symbol distance, viz. d2 1 SXRI (m) = (20) 4 σ I2 (m)

SXRQ (m) =

d2 1 4 σ Q2 (m)

(21)

where SXRI(m) and SXRQ(m) are the in-phase and quadrature crosstalk ratios for subcarrier m, respectively, and d is the minimum distance between QAM symbols. For the QAM constellation defined in Equation (5) one finds that d = 2 [24]. It is also useful to define the average signal-to-crosstalk ratio for subcarrier m, given by: 1 SXR ( m ) = ( SXRQ ( m ) + SXRI ( m ) ) (22) 2 In Figure 3(a) and 3(b), SXRI and SXRQ obtained from MC simulations are compared to the analytical formulas and for various subcarrier numbers m in a DMT system with 7 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 m = 0 is not included in the figures, since the DC component does not carry any signal.

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(a)

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4 QAM equation 64 QAM equation 4 QAM simulation 64 QAM simulation

1

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4 m

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(b)

Figure 3: Crosstalk ratios SXR [see Equations (20) and (21)] as a function of subcarrier number m 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 subcarriers is 7.

It is also interesting to note that one could include a cubic term in the polynomial expansion in (1). A series of Monte Carlo simulations 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 N=127 and M=4 the obtained SXR value is only -0.0014dB lower when the cubic term is included. C. Statistical Nature of the Non-linear Crosstalk Noise Next, we consider the distribution function of the intercarrier crosstalk. In Figure 4(a) and 4(b), the probability density functions (PDFs) of cmI and cmQ, respectively [see Equations (13) and (14)], are illustrated for the first channel (m=1), in the case of a DMT system with 7 subcarriers and M = 16 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 inter-carrier crosstalk can be roughly approximated by a Gaussian PDF. The approximation becomes better as the number of carriers is increased. This is illustrated in Figure 5, where similar PDFs are plotted for 255 subcarriers and M=16 QAM levels. Similar results are obtained for other subcarrier indices. 0

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Monte Carlo Gaussian Approximation

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10 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 cmI (a)

Monte Carlo Gaussian Approximation

10 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 cmQ (b)

Figure 4: PDFs of the of non-linear inter-carrier 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 modulation with 7 subcarriers and M=16 QAM states per subcarrier. The PDF of a Gaussian random variable with the same variance is also shown (solid line). 10

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pdf

⎛ b2u0 ⎞ ⎟ ⎝ 2b1 ⎠

σ Q2 (m) = ⎜

2 ⎧⎪ 8M 2 − 40M + 32 ⎛ M − 1 ⎞ ⎫⎪ (17) + 4 ( 2N − m − 4) ⎜ ⎨ ⎟ ⎬ 45 ⎝ 3 ⎠ ⎪⎭ ⎪⎩

pdf

2

pdf

⎛b u ⎞ σ ( m) = ⎜ 2 0 ⎟ ⎝ 2b1 ⎠ 2 I

SXRQ (dB)

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REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < PDF of a Gaussian random variable with the same variance is also shown (solid line).

Next, it is of interest whether the distortion terms cmI and cmQ are uncorrelated. To this end we investigated into the correlation coefficient of these terms, i.e.

σ I ( m)σ Q ( m)

(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 coefficient was estimated by aid of Equation (23) and the procedure repeated for a total of 20 000 times. The resulting histograms of the correlation coefficient were found to be centered on zero, which supports our hypothesis, that the distortion terms indeed are uncorrelated. If one instead of the correlation coefficient plots the distribution of t = RIQ

k −2 RIQ

(26)

The noise components nmI and nmQ are independent and identically distributed, with zero mean and the variance is: N σ G2 (m) = 0 (27) 2T with N0 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 SNRb=Eb/N0, where Eb is the average energy per bit. Assuming a DMT waveform [see Equation (2)], ignoring the inter-carrier crosstalk noise, relying on the fact that 2 2 , and that inside a DMT symbol akI = akQ = ( M − 1) / 3 period there are (N-1) QAM symbols or (N-1)log2M bits, it is straightforward to show that: E u 2 ( M − 1) SNRb = b = 0 (28) N 0 3N 0 log 2 M Sub: 3, QAM: 4

(24)

one expects a student t distribution in case both distortions are normally distributed [25]. In Figure 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 RIQ is indeed symmetrical but is close distributed around zero than the theoretical distribution. This deviation can be attributed to a non-normal distribution of the interferences themselves. While the histogram supports the assumption that the correlation coefficient is zero we cannot test this hypothesis, since the distribution function of RIQ is unknown. As discussed before, for an increasing number of total subcarriers the distribution of the interferences becomes more normal, and, as can be seen in Figure 6, the distribution of t [see Equation(24)] is much closer to the theoretical one. For these cases we can apply hypothesis testing based on the student t distribution, and for all the subcarrier and QAM-level combinations addressed in this work the hypothesis that the correlation coefficient is not distributed according to the student t 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 hypothesis of the correlation coefficient being unequal zero can also be rejected. Since two uncorrelated normally distributed entities are statistically independent this is hence also the case for the in- and quadrature inter-carrier interferences cmI and cmQ. 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 and/or ambient light noise. This can be done in a straight forward manner by adding an AWGN term n(t) in Equation (8). The symbol estimates in Equation (11) will thus contain additional Gaussian noise components nmI and nmQ, i.e. AmI = amI + cmI + nmI (25)

Sub: 3, QAM: 64

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cmI cmQ

AmQ = amQ + cmQ + nmQ

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Sub: 63, QAM: 64

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Frequency

RIQ ( m ) =

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100 0 t

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0 -4

-2

0 t

2

4

Figure 6: Histograms of t for the first subcarrier. The quantity t is the transformed value of the correlation coefficient between cmI and cmQ (see Equation (24)), as compared with the student t distribution (solid line) for four sets of QAM level and total number of subcarriers (‘sub’).

IV. SIMULATION RESULTS AND DISCUSSION A. System Bitrate The analytical model described in section III is used to further analyze the impact of the non-linear LED transfer function on the performance of a point-to-point DMT system. Since the DC subcarrier is not modulated, the symbol rate Rs of the system is Rs=(N-1)Δf=fN-1. Hence, Rs does not depend on N and M. The bit rate Rb of the signal is Rb=Rslog2M and varies with M. 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 M used in this paper. TABLE I BIT RATES FOR A SYMBOL RATE OF 24.2 MHZ AND VARIOUS QAM MODULATION LEVELS.

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 4 0.2

16 0.4

64 0.6

256 0.8

1024 1

16 QAM 256 QAM

1

2 m

3

(a)

31 carriers 4 QAM 64 QAM 1024 QAM

40 35 30 25 20 15 10 5

16 QAM 256 QAM

1 4 7 10 13 16 19 22 25 28 31 (c) m

50 45 40 35 30 25 20 15

15 carriers 4 QAM 64 QAM 1024 QAM

16 QAM 256 QAM

1

3

5

7

m

9

(b)

255 carriers 4 QAM 64 QAM 1024 QAM

0

16 QAM 256 QAM

128 256 384 Subcarrier total

512 (a)

SXR (dB)

SXR (dB)

4 QAM 64 QAM 1024 QAM

45 40 35 30 25 20 15 10 5 0

-2

16 QAM 256 QAM

4 QAM Linear 4 QAM Nonlinear 16 QAM Linear 16 QAM Nonlinear 64 QAM Linear 64 QAM Nonlinear 256 QAM Linear 256 QAM Nonlinear

10

11 13 15

-3

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-4

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-5

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-6

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1

43

85 127 169 211 253 m (d)

Figure 7: Average signal-to-crosstalk ratio (SXR) as a function of the subcarrier number m for various QAM modulation levels and total number of subcarriers. 45 40 35 30 25 20 15 10 5 0

3 carriers

-1

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BER

40 35 30 25 20 15 10 5

3 carriers 4 QAM 64 QAM 1024 QAM

SXR (dB)

35 30 25 20 15 10 5 0

SXR (dB)

SXR (dB)

SXR (dB)

B. Non-linear degradation Figure 7 illustrates the values of the average signal-tocrosstalk ratio SXR obtained for various DMT parameter settings. The results are based on Equations (17) to (21). There are several interesting features that can be drawn from this Figure. First, the lowest subcarrier (m=1) is always the subcarrier with the worst distortion. This is consistent with Equations (17) to (19), which show that the power of the crosstalk noise decreases as m increases. It is also interesting to note that the lowest subcarrier has about 3dB lower SXR than the highest subcarrier. Another point that can be drawn from Figure 7 is the fact that the SXR generally tends to improve as the number of QAM levels is reduced. As shown in the figure, there is about 20 dB degradation of SXR when M = 1024 QAM instead of M = 4 QAM is used. In most cases, there is a 4 to 5dB decrease in SXR increasing the modulation level from M=22r to M=22(r+1) QAM. The degradation with increasing M is not surprising. Although the distance between neighboring symbols in the original QAM constellation does not depend on M, 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 statistical properties vary little upon a change of M.

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 Figure 8, where the SXR is plotted as a function of the total number of subcarriers for (a) the first and (b) the central subcarrier, respectively. The explanation of this behavior can be found in the fact that according to Equation (7), the parameter u02 in Equations (17) to (19) decreases as 1/N 2. This decrease is much faster than the increase of the possible combinations satisfying the conditions k+l=m and k-l=m, the number of which is equal to N-2. From another point of view, the use of more subcarriers will result in the original DMT waveform Iin(t) having large spikes while still remaining inside a given operational range Iin∈[IDC-ΔI IDC+ΔI]. As the number of subcarriers is increased, these spikes will become sharper and sharper, followed by longer periods of time, where Iin(t) will possess a small amplitude. Since the influence of nonlinearity grows with an increase in Iin(t), only high-current spikes will be affected by the LED non-linear characteristic, while the distortion for the rest of the signal will be small. Thus, the occurrence of nonlinear 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 SNRb as defined in (28) decreases with an increase in the number of subcarriers N since u0 is inversely proportional to N. This means that, as N increases, obtaining the same SNRb requires a lower N0. Since N0 is due to both ambient light noise and thermal noise, in practice, it is not possible to reduce N0 beyond a certain minimum value leading to an upper bound for the achievable SNRb.

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Figure 8: Estimation of SXR for (a) the first subcarrier and (b) the central subcarrier (m= ⎡(N-1)/2⎤) as a function of the number of subcarriers

It is also interesting to mention that, as the number of subcarriers is increased, SXR is improved and hence an

0

5

10 15 20 25 (b) SNRb (dB) Figure 9: Bit-error ratio (BER) of the first subcarrier as a function of the signal-to-noise ratio per bit (SNRb) for various QAM modulation levels and for (a) 3 subcarriers and (b) 255 subcarriers.

C. Power Penalty The influence of the LED nonlinearity on the performance of a system degraded by AWGN is illustrated in Figure 9,

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT)
REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 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 transmission system was analyzed. Analytical formulas were derived, which allow the estimation of the non-linear 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 normally distributed. The model was extended to incorporate the influence 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 function 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 RCLED 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 performance of the system. ACKNOWLEDGMENT The research leading to these results received partial funding from the European Community's Seventh Framework Program FP7/2007-2013 under grant agreement n° 213311, also referred to as OMEGA. The authors acknowledge the contributions of their colleagues. 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] [2] [3] [4] [5] [6]

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B. Mukherjee, “WDM Optical Communication Networks: Progress and Challenges”, IEEE J. Select Areas in Communications Vol. 18, No 10, October 2000, pp. 1810-1824 E. F. Schubert, Light-Emitting Diodes, 2nd ed., Cambridge University Press, 2006 See for example the IrDA specifications for LED based optical wireless devices (http://www.irda.org/) An example of the use of LED in an output point to point 10 Mbps data link developed by Ronja project (http://ronja.twibright.com/) For example the wireless Local Area Network (LAN) by JVC http://www.jvc.com/press/index.jsp?item=420 G. W. Marsh and J. M. Kahn, "Performance Evaluation of Experimental 50-Mb/s Diffuse Infrared Wireless Link using On-Off Keying with Decision-Feedback Equalization", IEEE Trans. on Commun., vol. 44, no. 11, pp. 1496-1504, November 1996 E. T. Won, "IEEE 802.15 IG-VLC Closing Report", 1st Meeting as a Interesting Group, Taipei, Taiwan, January 2008, https://mentor.ieee.org/802.15/file/08/15-08-0084-01-0vlc-closingreport-january-meeting.pdf T. Komine, M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Transactions on Consumer Electronics, Vol. 50, No. 1, pp. 100- 107, 2004

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J. A. C. Bingham, “Multicarrier modulation for data transmission: An idea whose time has come,” IEEE Communications Magazine, vol. 28, pp. 5-14, May 1990. J. Grubor, S. Ch. J. Lee, K-D. Langer, T. Koonen, and J. W. Walewski, “Wireless High-Speed Data Transmission with Phosphorescent WhiteLight LEDs,” Proceedings 33rd European Conference and Exhibition on Optical Communication, Vol. 6, Post-Deadline Papers, PD3.6, 2007 See for example the survey of DSL applications (http://www.amazon.com/Implementation-Applications-TechnologyPhilip-Golden/dp/0849334233/ref=pd_bxgy_b_img_b) J. H. van Vleck and D. Middleton, The Spectrum of Clipped Noise, Proceedings of the IEEE, Vol 54., No. 1, pp. 2-19, 1966 N. J. Frigo, M. R. Phillips, and G. E. Bodeep, Clipping Distortion in Lightwave CATV Systems: Models, Simulations, and Measurements, JLT, Vol. 11, No. 1, pp. 138-146, 1993 D. J. G. Mestdagh, P. Spruyt, and B. Biran, Analysis of Clipping Effect in DMT-based ADSL Systems, IEEE International Conference on Communications, pp. 293-300, 1994 C.-L. Liu, “The effect on nonlinearity on a QPSK-OFDM-QAM Signal,” IEEE Transactions on Consumer Electronics, pp. 443-447, 1997 Y. Tang, W. Shieh, X. Yi, and R. Evans, “Optimum Design for RF-toOptical Up-Converter in Coherent Optical OFDM Systems,” IEEE Photonics Technology Letters, Vol. 19, NO. 7, APRIL 1, 2007 A. Chorti and M. Brookes, “On the Effects of Memoryless Nonlinearities on M-QAM and DQPSK OFDM Signals,” IEEE Trans. on Microwave Theory and Techniques, Vol. 54, No 8, August 2006 M. O’ Droma, N. Mgebrishvili, and A. Gloacher, “Simulation-Based Analysis of Nonlinearites in Multi-Carrier OFDM Signals,” 5th Workshop on Singal Processing Advances in Wireless Communications, pp. 611-615, 2004 N. J. Frigo, “A Model of Intermodulation Distiortion in Non-Linear Multicarrier Systems,” IEEE Tansactions on Communications, Vol. 42, No. 2/3/4, pp. 1216-1222, 1994 J. C. Daly, “Fiber optic intermodulation distortion,” IEEE Trans. on Communication, vol. Com-30, No. 8, Aug. 1982. A. R. Hayes, Z. Ghassemlooy, N. L. Seed, and R. McLaughlin, “Baseline Wander Effects on Systems Employing Digital Pulse Interval Modulation,” IEE Proc. Optoelect., Vol 147, No.4, pp.295-300, 2000 J. W. Walewski, “Inference of Calibration Curves Non-Linear Transfer Function by use of Orthogonal Polynomials,” In preperation, 2008. 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. Quantum Electronics, Vol. 36, No. 12, pp. 1445-1453 (2000). J. G. Proakis, Digital Communication, 4thed., McGraw-Hill, N. York, 95 Pugh, E. M. and Winslow, G. H. The Analysis of Physical Measurements. Reading, MA: Addison-Wesley, 1966. R. Hui, B. Zhu, R. Huang, C. T. Allen, K. R. Demarest and D. Richards “Subcarrier multiplexing for high-speed optical transmission,” IEEE J. of Lightwave Technology, Vol. 20, No. 3, Mar. 2002.

Ioannis Neokosmidis was born in Athens in 1977. He obtained the BSc degree in Physics, the MSc degree in Telecommunications and the Ph.D. degree in transmission limitations due to nonlinear phenomena from the University of Athens, Greece in 1999, 2002 and 2007 respectively. He is currently a Research Associate for the Optical Communications Laboratory, University of Athens. His research interests include nonlinearities, optical networks and components, photonic crystals and wireless optical systems. Thomas Kamalakis was born in Athens, Greece, in 1975. He received the B.Sc. degree in informatics, the M.Sc. degree (with distinction) in telecommunications and the Ph.D. degree in the design and modelling of Arrayed Waveguide Grating devices from the University of Athens, Athens, Greece, in 1997, 1999, and 2004, respectively. He is a lecturer at the Department of Informatics and Telematics of the Harokopio University of Athens and a research associate in the Optical Communications Laboratory of the University of Athens. His research interests include photonic crystal devices and nonlinear effects in optical fibers. Joachim W. Walewski graduated from the Christian Albrechts University, Germany, with a diploma degree in physics (Dipl.-Phys.) in 1995 and he earned a PhD from the Lund Institute of Technology, Sweden, in 2002 for his research on applied laser spectroscopy. From 1996 to 1997 he was a visiting

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < scientist at the Tampere University of Technology, Finland, engaging in laserassisted diagnostics of CVD diamond. From 2001 to 2003 he served as junior lecturer at the Lund Institute of Technology, and from 2003 to 2006 he held positions as research associate and finally as assistant scientist at the Engine Research Center of the University of Wisconsin – Madison, U.S.A. At the latter two institutions, his research was focused on the development and application of laser-spectroscopic techniques for combustion research. In May 2006 he joined Siemens Corporate Technology, Information & Communications, in Munich, Germany and has focused his efforts on R&D in the field of wireless optical communications and Green ICT. 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. From 2005 to 2006, she worked at ASELSAN Electronic Industries Inc. Ankara, Turkey as a system engineer. She obtained her M.Sc. degree from Eindhoven University of Technology, Eindhoven, The Netherlands, in 2008 at electrical engineering. She carried out her master thesis project at Siemens Corporate Technology, Information & Communications, in Munich, Germany in the field of nonlinearity impact on optical communication. She is currently pursuing her Ph.D. at Siemens Corporate Technology, Information & Communications in cooperation with the Technical University of Munich, Germany . Thomas Sphicopoulos received the degree in physics from Athens University, Athens, Greece, in 1976, the D.E.A. degree and the Ph.D. degree in electronics, 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 in Thomson CSF Central Research Laboratories 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, Athens, Greece, engaged in research on broadband 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 technoeconomics. 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 organizations in the fields of fiber optics networks, spectrum management techniques, and technology convergence.

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