Convex Optimization Approach to MultiLevel Modulation for Dimmable Visible Light Communications under LED Efficiency Droop
 Author: Lee Sang Hyun, Park IlKyu, Kwon Jae Kyun
 Publish: Journal of the Optical Society of Korea Volume 20, Issue1, p29~35, 25 Feb 2016

ABSTRACT
This paper deals with a design method and capacity loss of an efficient multilevel modulation scheme for dimmable visible light communications (VLC) systems that use lightemitting diodes (LEDs) with efficiency droop. To this end, the impact of such an impairment on dimmable VLC is addressed with respect to multilevel modulations based on pulseamplitude modulation (PAM) via datarate optimization formulation.

KEYWORD
Visible light communications , Efficiency droop , Dimming , Convex optimization

I. INTRODUCTION
Recent efforts to use the lightemitting diode (LED) in visible light communications (VLC) offer a new alternative for shortrange communications that conveys information via light intensity [1, 2]. The original purpose of the lighting invokes an inherent constraint whereby the average optical intensity matches the dimming requirement imposed by a user. There have been a number of studies to achieve transmission techniques satisfying the constraint based on binary modulations [36], which are normally subject to limitations in the data rate. Therefore, the need for a higher data rate naturally leads to the use of multilevel modulations [7],
1) such as pulseamplitude modulation (PAM), and poses a new challenge for efficient multilevel modulation adapting to the dimming requirement.One of the major challenging issues in LED research is efficiency droop [8]. In particular, nitridebased LEDs suffer from a reduction of internal quantum efficiency with increasing input, which results in saturation of output optical power. This phenomenon becomes one of the main obstacles to widespread deployment of gallium nitride (GaN)based LEDs in lighting applications. Although this brings forth an impairment in terms of data rate for the communications feature, any impact on VLC has not been fully investigated. Droop causes impairment of energy efficiency and cooling, and failure in devices. Moreover, it negatively affects communications in two aspects: on nonlinear electricaltooptical conversion and by reducing optical range. In theory, an LED’s output optical range determines communications capacity, whereas input signal, in practice, is processed in the electrical domain prior to nonlinear conversion. Both aspects are important, and an electrical signal should be carefully designed to maximize capacity with consideration of nonlinear conversion for the optical signal. This paper considers this detrimental effect on intensitymodulationbased optical communications and presents a design approach for an efficient modulation scheme equipped with an adaptation to the arbitrary dimming requirement. To this end, the maximally achievable rate of a VLC channel under efficiency droop is first presented, and optimization with the objective of maximizing the data rate is formulated. In view of the peak intensity restriction for VLC transceivers [7], this optimization naturally ends up as a simple convex problem that can be solved efficiently and exactly. This approach enables us to identify the optimal modulation for VLC channels.
II. LED EFFICIENCY DROOP MODEL
The numerical model of LED droop was first presented by Piprek [8] and is briefly introduced here. LED efficiency indicates a transfer ratio between the electrical and optical energies. However, the LEDs suffer from loss of electrons and photons during this transfer process. Accordingly, the total quantum efficiency,
η _{EQE} is given byη _{EQE} =η _{IQE}η _{EXE} . Here,η _{IQE} is the internal quantum efficiency (IQE) and means the ratio of photons generated by electronhole recombination to the total number of electrons injected into the LED which is related to electron loss. Andη _{EXE} is the optical extraction efficiency (EXE) and means the ratio of photons emitted into free space to the total number of photons generated by the electronhole recombination process. Efficiency droop is mainly caused by a gradual decrease in IQE as the injection current density surpasses typical values ranging between 0.1 and 10 A/cm^{2} [9]. IQE can be further defined as the fraction of the total injected current to the current that feeds radiative recombinationI_{rad} . The total current is split into carriers that recombine by generating photons,I_{rad} , and carriers that are lost to other nonradiative processes, such as ShockleyReadHall (SRH) recombination and Auger recombination inside quantum wells (QWs) and carrier leakage outside QWs. Accordingly, the total current injected into the LED is the sum of these four current components, which is the basis for the principal droop mechanism whereI_{rad} is exceeded by others for increasing injection current. The sum of the first three current components inside QWs, denoted byI_{QW} , can be expressed with an ABC model [8] given byI_{QW} =qV_{QW} (An +Bn ^{2}+Cn ^{3}), whereq is the electron charge,V_{QW} is the active volume of all QWs, andn is the carrier concentration.A ,B , andC represent SRH recombination, radiative recombination, and the Auger coefficient, respectively. As for leakage current sources for a droopcausing mechanism, many have been supposed, including asymmetry in electron and hole transport, electronattracting properties of the electron blocking layer, electron escape from the QW, and lack of electron capture into QWs [9]. Because the leakage current is not simply described in a simple equation, a simple relationship of the leakage currentI_{leak} toI_{QW} is fitted [8] in the formI_{leak} =aI_{QW}^{m} . Therefore, the overall expression for IQE is given bywhere injection efficiency
η _{inj} is given byη _{inj} =I_{QW} / (I_{QW} +aI_{QW}^{m} ), thereby accounting for the leakage. For the mathematical formulation, the electrical input and the optical output are denoted byU andX , respectively. We consider Eq. (1) asη _{IQE} =X (n ) /U (n ). Thus, Eq. (1) corresponds to a parametric form (or an implicit function) for the relationship betweenU andX . To obtain the relationship of an explicit functionX ≡g (U ), which is independent of parametern , we fixn to evaluate the numerator and denominator of Eq. (1) and to set as the output and input, respectively. This can be repeated for increasingn until a smooth curve is obtained. The results do not depend onn and are used to define explicit functionX ≡g (U ).III. EFFICIENT MULTILEVEL MODULATION
An efficient multilevel modulation is designed based on optimization with respect to symbol levels. Let
X be the optically modulated signal communicated with VLC. From the practical consideration of intensity modulation [10], the signal levels ofX are assumed to be upper and lowerbounded, i.e., all symbols are contained in a nonnegative finite range of signal levels. The optical signal undergoes a communications channel that can be modeled [11] with an additive white Gaussian noise (AWGN) channel. Then, the signal received at the photodetector (denoted byY ) is expressed as a corrupted version ofX by additive noiseZ with zero mean and standard deviation σ, i.e.,Y =X +Z . The best achievable data rate for this AWGN channel is known to equal the mutual information [12] between channel inputX and outputY , numerically evaluated usingwhere the first term is differential entropy
h (Y ) of channel outputY . Therefore, the design objective is to find the best set of symbol levels forX , such that the data rate in Eq. (2) is maximized under several constraints resulting from signal detection, dimming support, efficiency droop, etc. Note that channel outputY has a continuous probability distribution expressed as a mixture of multiple Gaussian distributions, and thus, the expression for data rate in Eq. (2) involves an integral. Therefore, determining the best configuration ofX in an integral objective is very challenging. However, it is known that the best input distribution for an AWGN channel with bounded input levels has nonzero probabilities only at a finite set of input values [13, 14]. In other words, distribution ofX has nonzero values only at a few separated levels. Therefore, the resulting multilevel modulation naturally resembles PAM, except that spacings between symbol levels are not uniform.Multilevel modulation is assumed to have, at most,
L different levels. However, the exact number of symbol levels (denoted byM ) is unknown. Letx_{i} be thei th smallest level ofX . Thus,x _{1} andx_{L} correspond to the zero level and the highest level that the optical transmitter emits, respectively. The dimming requirement whereby the mean intensity of the optical signal is equal toD ∈[0, 1], set externally by a user, is expressed aswhere
p_{X} (x_{i} ) is the probability thatX takes the value ofx_{i} . In addition, differential entropyh (Y ) in Eq. (2) becomesLet
U be an electrical signal of a communications message. Because of efficiency droop in the optical intensity modulation using LED, electrical signalU undergoes distortion during the modulation. The resulting optical signalX is assumed to have a relationship with electrical signalU such thatX =g (U ). Therefore, the main goal readily reduces to the determination of the best distribution ofU , such that the resulting electrical signal, or optical one, can convey the maximum amount of information via VLC. It is naturally speculated that the best distribution ofU has uneven probabilities at discrete levels of a symbol, according to the preceding argument onX . Thus,U is assumed to takeL different symbol levels based on the relationshipx_{i} =g (u_{i} ). In addition, the probability ofU taking thei th levelu_{i} is denoted byp_{i} . Then, it holds thatp_{X} (x_{i} ) =p_{X} (g (u_{i} )) =p_{i} .Next, for ease of numerical evaluation, the integral in Eq. (4) is discretized into a numerical integration. A finite interval of width 2
σ z _{max}, symmetric about the peak, is considered instead of [∞, ∞]. Then, the finite interval is quantized with uniformly spacedK levels, and its two ends are defined as Δ_{1} =x _{1} σ z _{max}, and Δ_{K} =x_{L} +σ z _{max}. The configuration of parametersz _{max} = 5, andK =L = 100 suffices to yield solutions of reasonable speed and accuracy, althoughK should be in fact chosen sufficiently larger thanL . The discrete version of Eq. (4) is simply given bywith coefficient
c_{ij} given byNote that function
h (p ) can be a new objective function, and can easily prove convex with respect to eachpj . Therefore, the final optimization is given bywhere two constraints result from the probability condition on
U and the dimming constraint. Note that functionx logx and all affine (or linear) functions are convex and that their compositions, i.e., functions obtained by applying the output of one function as an input to the other function, and a linear combination of those compositions are also convex. Therefore, the objective function is concave [14]. Since all constraint functions are linear, i.e., convex, and the objective function is simply converted to a convex form by minimizing its negative, Eq. (6) corresponds to a convex optimization problem [15]. Therefore, any convex optimization solver package [16] can exactly obtain the optimal solution of Eq. (6). After the solution is found, the set of such levels that have nontrivial probability (p_{i} > 0) out ofL levels is collected, and modulation orderM is automatically determined by its cardinality. Normally, values ofp_{ij} ,j =1,…,M are not uniform, whereas communications messages are chosen uniformly. Therefore, preprocessing such as inverse source coding [6] is applied to convert messages so they have the obtained optimal distribution before optical modulation.IV. NUMERICAL RESULTS
Numerical results are presented for various ranges of dimming target and data rate. First, the inputoutput relationships for three different droop models are depicted in Fig. 1. Model parameters are given as follows [8]:
A =1ⅹ10^{7} ,B =2ⅹ10^{11},C =1.5ⅹ10^{30},a =0.05 andm =1.58. The first model considers all effects caused by all parameters, while the second model does not consider the Auger effect, i.e., no cubic effect. The third model corresponds to the linear relationship between input and output. Here, the maximum output level is set at 10. The inputoutput relationship varies with respect to the range of input as the efficiency changes gradually. The slope in a lowerintensity regime is steeper than in a higherintensity regime.The maximum data rates for VLC transmission over a nodroop channel and a cubiceffect droop channel are illustrated with respect to dimming target
d and A_{m}/σ in Figs. 2 and 3, where A_{m} is the maximum value ofX . A_{m}/σ represents the channel or signal quality, where A_{m} is the signal range and σ is the noise standard deviation. The square of A_{m}/σ follows the dimension of the conventional signaltonoise ratio. The CVX package [16] is used to solve Eq. (6). Note that the result is globally optimal because of the convexity of the optimization formulation. The resulting data rates are almost symmetric with respect to the range of dimming targets. Spectral efficiency keeps increasing, despite the efficiency droop, and reaches 2.7 bps/Hz at A_{m}/σ = 30 dB. Figure 4 depicts degradation in the data rate of VLC systems that is caused by droop. Since high dimming targets cannot be achieved in situations with efficiency droop, the loss in data rate is 100% for such dimming target regions. Furthermore, in dimming target regions that can be achieved even with efficiency droop, the data rate becomes very low, and the resulting loss is large, compared with situations without efficiency droop. It can also be observed that the loss is larger in a higher noisepower regime. This can be explained based on the shape of the probability distribution of signalY , which determines differential entropyh (Y ) in Eq. (2). If noise power is high,Y is dispersed in a wide region. Efficiency droop causes shrinkage of the spacing between consecutive data symbol levels. Thus, the tails of the resulting distribution ofY on the outside of the data signal region forX , i.e., [0, 10] for situations without droop in this example, remain nearly the same, whereas the probability distribution ofY within the data signal region forX changes significantly. Since the integral region ofY shrinks within the data signal region, i.e., [0, 7.2] for situations with droop in this example, the resulting differential entropy ofY decreases. In other words, the portion ofY that contributes to the integral evaluated in calculating the differential entropy is shortened by efficiency droop. The probability distribution ofY is merged within the data signal region by efficiency droop, and the resulting differential entropy decreases. In highnoise situations withY dispersed widely, the distribution ofY has heavy tails, and the contribution of the probability distribution of the data signal region to the integral for the differential entropy is relatively high. Therefore, degradation of the differential entropy becomes high. By contrast, in lownoise situations where the distribution ofY is concentrated about data symbol levels, the values of probability distribution within the data signal region are mostly close to zero, and shrinkage of the integral region hardly changes the probability distribution ofY for this region. Thus, the change in the value of the integral is relatively small, and differential entropy is degraded little. Therefore, the loss of the data rate is larger in highernoise situations.Figures 5 and 6 show the number of symbol levels for the optimal modulations of various dimming and noise configurations. We can see that the number of symbols for the optimal scheme is finite [13, 14], and the corresponding signal distribution is discrete. The number of symbols increases in low noisepower regimes. This can be understood as follows. In order to obtain maximal entropy
h (Y ) in Eq. (2), received signalY should be spread as uniformly as possible. Since the communications channel is an additive noise, the tails in the distribution of received signalY depend on the positions of symbols for transmitted signalX , and thus, received signalY needs to be spread beyond the available region where the symbol of transmitted signalX can be placed. To this end, it is desirable for the two outmost symbols of transmitted signalX to be placed at two ends of the available region. In lownoise situations where received signalY is not spread widely, many symbols for transmitted signalX need to be uniformly spread in the available signal region so the resulting distribution assimilates with the uniform distribution having the largest differential entropy. On the other hand, in highnoise situations where the distribution of received signalY has long heavy tails, a small number of symbols for transmitted signalX is sufficient to ensure that the resulting distribution is spread widely. Thus, symbols of transmitted signalX are placed far from the center of the available symbol region. Therefore, highnoise situations lead to binary symbol mapping with two symbols placed at the two ends and, as noise power decreases, signal distributions with more symbols placed in between are created. This observation evidences the results of discrete distribution. As the noise power decreases, the number of modulation symbols increases. In most cases of low A_{m}/σ , binary onoff keying modulation is still a good transmission scheme for both droop and nodroop models. However, for a lownoise regime with good signal quality, almost all levels have nonzero probability values, i.e., the distribution becomes similar to uniform distribution. In addition, the number of symbols increases drastically for a good signalquality regime, while other regions have only a few levels. This implies that analoglike modulation is very efficient for very goodquality regions, while the efficiency droop decreases the density of the information that can be conveyed with a given power. Figure 7 shows the decrease in symbol numbers according to efficiency droop. In a high dimming target region close to 1, the dimming target cannot be achieved, and the resulting number of symbols is zero. Therefore, degradation becomes 100%. In a region of low dimming targets and high noise power, the number of data symbols is two, regardless of the existence of efficiency droop, and degradation is 0%. In a low dimming target region, the number of data symbols increases from 2 to 3 and 4 as noise power decreases. Since the rate of the increase is higher in situations without efficiency droop, there are some regions where the resulting number of symbols is 3 without droop, but 2 with droop, thereby causing a 33% decrease in the number of symbols. Subsequently, if the number of symbols for droop situations becomes 3, the resulting degradation becomes 0%. Likewise, the degradation becomes 25% if the resulting number of symbols is 4 in situations without efficiency droop. Therefore, the overall degradation in the number of data symbols fluctuates, as shown in Fig. 7.Figures 8 and 9 show the symbollevel distributions for electrical and optical signals, respectively, for several configurations of target dimming rate. The horizontal axis represents power values shown in Fig. 1. In conversion from an electrical signal to an optical signal, droop decreases the width of the signal span, i.e., the symbol region. Also, the symbol spacing between consecutive symbol levels of the optical signal becomes nonuniform. In addition, we can see that the number of symbols decreases in situations with efficiency droop. Figure 10 compares the data rates among different models for several target dimming values. We see that some regions of high dimming targets cannot be achieved with efficiency droop, and performance degradation is remarkable with high dimming targets.
V. CONCLUSION
This paper presents a multilevel transmission scheme for VLC systems implemented using an LED with efficiency droop. To find the best transmission scheme, convex optimization is formulated and solved using a simple convex solver package. Numerical results show that, although the droop incurs various impairments in devices, a substantial decrease in communications capacity can be minimized if the signal is carefully designed.

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[FIG. 1.] Inputoutput relationship of the LED.

[FIG. 2.] Spectral efficiency with LED efficiency droop.

[FIG. 3.] Spectral efficiency without LED efficiency droop.

[FIG. 4.] Loss in spectral efficiency.

[FIG. 5.] Modulation orders for optimal modulations with LED efficiency droop.

[FIG. 6.] Modulation orders for optimal modulations without LED efficiency droop.

[FIG. 7.] Loss in modulation order.

[FIG. 8.] Optimal distributions of an electrical signal for various dimming targets.

[FIG. 9.] Optimal distributions of an optical signal for various dimming targets.

[FIG. 10.] Comparison among different efficiency droop models for various dimming targets.