1、Optimal Transmission Power in a Nonlinear VLC SystemIn a visible light communication （VLC） system， the light emitting diode （LED） is nonlinear for large signals， which limits the transmission power or equivalently the coverage of the VLC system. When the input signal amplitude is large， the nonlinea
2、r distortion creates harmonic and intermodulation distortion， which degrades the transmission error vector magnitude （EVM）. To evaluate the impact of nonlinearity on system performance， the signal to noise and distortion ratio （SNDR） is applied， defined as the linear signal power over the thermal no
3、ise plus the front end nonlinear distortion. At a given noise level， the optimal system performance can be achieved by maximizing the SNDR， which results in high transmission rate or long transmission range for the VLC system. In this paper， we provide theoretical analysis on the optimization of SND
4、R with a nonlinear Hammerstein model of LED. Simulation results and lab experiments validate the theoretical analysis. nonlinearity； light emitting diode （LED） ； SNDR 1 Introduction isible light communication （ VLC） systems become attractive as they utilize the unlicensed visible light spectrum that
5、 has 1000 times larger bandwidth than the conventional radio frequency （RF） spectrum 1. Moreover， existing illumination devices and illumination infrastructure can be easily upgraded to accommodate wireless data transmission 2. In a VLC system， spectrum efficiency is very critical because the freque
6、ncy response of the light emission diode （LED） is limited. Orthogonal frequency division modulation （OFDM） ， which is widely used in wireless communications because of its high spectral efficiency， can be applied to the VLC system with modifications. In the VLC system， the signal is modulated on lig
7、ht intensity （IM） of LED 3. At the receiver side， the photon detector （PD） is applied with direct detection （DD） of the light intensity. Since the LED at the transmitter and the PD at the receiver only deal with real and positive signals， conventional OFDM signals need to be modified to meet such th
8、e requirement. The real?valued OFDM can be obtained by introducing Hermitian symmetry， such as direct current biased optical OFDM （DCO?OFDM） 4， asymmetrically clipped optical OFDM （ACO?OFDM） 5， pulse?amplitude?modulated discrete multitone modulation （PAM?DMT） 6， and unipolar OFDM （U?OFDM） 7. Many ef
9、forts have been made to improve the transmission rate of the VLC system. The authors in 8 achieved 40 Mb/s data rate on 25 MHz bandwidth at a distance of 2 m with the normal room illumination level. In 9， 1.1 Gb/s data rate was achieved at a distance of 23 cm by employing carrier?less amplitude and
10、phase modulation （CAP）. Besides， 4.2 Gb/s data rate was achieved using wavelength multiplexing division with red?green?blue （RGB） LED at a distance of 10 cm 10. It can be found that there is an inverse relationship between the transmission range and data rate. In conventional wireless communication
11、systems， the data rate and transmission range can be improved simultaneously if the transmission signal power increases. However， nonlinear effects in the VLC system are more severe with large input signals than those with small signals. The nonlinear effects significantly degrade system performance
12、 and limit the application of spectral efficient modulation schemes. Therefore， the research on nonlinear effects in the VLC system is necessary. The authors in 11 studied the VLC system performance with clipping effects. In order to compensate for the nonlinear effects of LED， digital predistortion
13、 （DPD） was applied in the transmitter at the expense of an additional feedback path 12， 13. Adaptive post?distortion algorithm for nonlinear LEDs delivered similar performance as the DPD at no additional hardware cost 14. The Volterra equalization used to compensate for the nonlinearity of LEDwith m
14、emory effects was discussed in 15. The authors in 16 optimized the SNDR in the family of dynamic?constrained memoryless nonlinearities and found out the optimal nonlinear mapping. In order to mitigate the nonlinearity of LED， we have also proposed two methods. We used a one?bit sigma?delta modulator
15、 to convert the multi?level input signal into the binary input signal with signal LED， thus avoiding LED nonlinearity 17. Moreover， a new system architecture was proposed with micro?LED arrays， which provides digital controls to each element. The multi?level signal is realized with multiple elements
16、 in the micro?LED array， and a linear transmission is achieved for signals with large peak?to?average power ratio （PAPR） 18. In this paper， we study the LED nonlinearity and provide theoretical analysis on the optimization of SNDR with a general nonlinear Hammerstein model of LED. The rest of paper
17、is organized as follows. Section 2 provides a setup of VLC system. The performance metric SNDR is introduced for the nonlinear LED model. The optimal transmission power is obtained with theoretical derivation. Section 3 shows simulation results as well as experiment measurements of the SNDR optimiza
18、tion. These results validate the theoretical analysis. Section 4 concludes this paper. 2 Optimization of SNDR In a typical VLC system （Fig. 1a） ， the information bits are coded and modulated first. The transmit signal is generated by the inverse discrete Fourier transform （IDFT） ， which is realized
19、by inverse Fast Fourier transform （IFFT） algorithm. A direct current （DC） bias is applied to ensure that the LED works properly as an illumination device. The input signal x（n） directly modulates the lighting intensity of the LED and generates the lighting signal y（n）. A typical power delay profile
20、of the VLC channel with additive white Gaussian noise （AWGN） is considered 19. The received signal r（n） is obtained by PD at the receiver. The DC component is ignored since it carries no information. The received information bits can be obtained by the baseband processing including synchronization，
21、discrete Fourier transform （DFT） ， channel estimation， equalization， and demodulation. Similar with IDFT， the DFT is realized by the Fast Fourier transform （FFT） algorithm. Fig. 1b shows an experimental VLC system. The vector signal generator Agilent E4438 is used to generate the baseband signal. Wi
22、th a bias power amplifier， the signal is applied to drive the LED. At the receiver side， the avalanche photodiode （APD） is used for reception. The digital signal analyzer Agilent DSA 90804 is used to capture the received signal. The LED and PD are both nonlinear devices. At a reasonable radiant flux
23、 range， the nonlinearity of the PD is not significant and is ignored during the analysis of this paper. Besides， the PD works in linear region in our simulation and experiment. For the intensity modulated LED， the output signal is a nondecreasing function of the input signal， and the output becomes
24、saturate when the input signal is large. Furthermore， most LEDs have limited bandwidths （from kHz to MHz） and the frequency response or the memory effect shows up. The nonlinearity with memory effects between input voltage and output luminous flux can be described by the Hammerstein model that consi
25、sts of a memoryless polynomial nonlinear block and a linear time?invariant （LTI） system block 20. The memoryless nonlinearity can be described by a polynomial model f（?）： fx（n）=p=1Papx（n）p （1） The LTI system can be modeled by an FIR filter g（?） as yn=gfxn=l=0L-1blf（x（n-l） ） （2） In （1） and （2 ） ， y（n
26、） is the output luminous flux of LED； x（n） is the input voltage signal； ap is the pth order coefficient of polynomial model， where the model coefficients can be estimated with LS/RLS solution adaptively， and the computational complexity of RLS algorithm is on the order of O（K*（D+1） ）2） 14； l is the
27、maximum delay tap； and bl is the coefficient of filter. Decomposing the output of polynomial nonlinearity fxn into the linear signal part and the distortion part， the nonlinear mapping （1） can be rewritten as 21 fxn=xn+d（n） （3） where d（n） is the distortion term that is orthogonal to x（n）， i.e.， Ex（n
28、）d（n）=0； is a constant given by=Exfx/ Ex2=Exfx/2x， where 2x is the variance of x（n）. By definition， we have Ef2xn=22x+2d. For a typical VLC channel， the received signal r（n） is rn=hyn+ vn=h（g（f（x（n） ） ） ） + v（n） （4） where h（?） is channel model for VLC 19 and v（n） is the total noise including ambient
29、 lighting noise and thermal noise. Without loss of generality， we assume that the frequency response of the LED and that of the channel are perfectly equalized with conventional equalization algorithms. We have ?（rn）=xn+dn+Gain?v（n） （5） where ?（?） is the inverse function of the cascaded frequency re
30、sponse of the LED and the channel response. ? satisfies ?（?） ? h（g（?） ） = 1 （6） where ? denotes the time domain convolutional. Normalizing the channel gain of the VLC system ?（?）， the variance of the noise remains the same as 2v. The optics SNDR is defined as the linear signal power over the noise p
31、ower and the distortion power， or SNDR=22x2x+2v= Exfx（n）2/2x， Ef2xn-Exnfxn22x+2v （7） From （ 7） ， we observe that the SNDR is determined by the input baseband signal power， nonlinearity of the LED and the noise power. Intuitively， when the input signal is small， the SNDR is small， and vice versa. How
32、ever， when the input signal becomes very large， the nonlinear distortion dominates and the SNDR degrades. There exists an optimal transmission power for a given noise level. To simplify the discussion， we assume that the input signal x（n） follows a Gaussian distribution. This assumption is quite acc
33、urate if modified OFDM signal， which significantly improves the spectral efficiency， is used for VLC systems 5. For a Gaussian random variable， the expectation on the polynomial term Exp（n） is given by 22：Expn=p-1?pxp even0p odd （8） where （?）！ denotes the double factorial operation and p-1?=p-1p-3?3
34、?1 when p is even. Substituting （1） and （8） into （7） ， we have （9）. As an example， for a 5th order nonlinearity， P = 5. The SNDR in （9） reduces to （10）. The nonlinear function of LED is nondecreasing and convex for the input signal range， which implies the first?order derivative ?f（x） /?x 0 and the
35、second?order derivative ?2f（x）/?x20. After some mathematical derivations， the numerator of （9） is concave and the denominator of （9） is convex. From 23， we conclude that the SNDR expression is pseudo?concave， which guarantees that a global maximum can be achieved within the input signal range. Optim
36、ization tools can be used to find the optimal value numerically 24. As an example， for a 3th order nonlinearity， P = 3. The SNDR in （9） reduces to （11） SNDR=a212x+6a1a34x+9a236x3a224x+6a236x+2v （11） SNDR=1.95572x-15.71794x+31.57996x0.00224x+21.05416x+2v （12） We define （13） and （14） and have （15） and
37、 （16） P2x=a212x+6a1a34x+9a236x， （13） Q2x=3a224x+6a236x+2v （14） ?P2x?2x=a21+12a1a32x+27a234x= 1.9558 - 31.43602x+ 94.73974x， （15） ?Q2x?2x=6a222x+18a234x=0.00442x+ 63.16234x （16） When the signal?to?noise ratio （SNR） is estimated， 2v can be express as2x/SNR， the optimal signal power 2x is calculated by ?SNDR?2x=?P2x?2x?Q2x-P2x?Q2x/?2xQ2x2 （17） The valid input range of signal for the nonlinear
