Power Allocation for Uniform Illumination with Stochastic LED Arrays

Abstract: In this letter, a simpler heuristic power allocation scheme is proposed for a random LED array to obtain uniform irradiance on the projection surface. This is done by considering a binomial point process (BPP) for modeling the LED location and using the Q-factor as a performance metric. Numerical results are provided to validate the proposed model and demonstrate its simplicity over existing LED geometries.


Introduction
Light has traditionally been used for making objects visible to the naked eye.Lately, there has been tremendous interest in using it for free space communication [1].This has simultaneously been accompanied by significant interest in light emitting diodes (LEDs) that have been replacing conventional light sources in almost all applications [2][3][4].Fair amount of existing literature has focused on achieving uniform irradiance over a planar surface [5][6][7][8], beginning with the problem of finding the optimal LED geometry at the light source to achieve uniform irradiance [9].This was done by using the irradiance distributions at the closest points on the incident surface.The case of LEDs using a freeform lens with a large view angle has been considered in [10].More literature on similar themes is available in [11,12].
The above literature has focused on a regular geometry with equal power allocation to individual source LEDs.While uniform illuminance is desirable, optimal power consumption is an extremely important factor in the design of LED light sources.To address this, recent literature has focused on power allocation, along with flexiblity in the LED source geometry to achieve uniform irradiance.
A trial and error approach for power allocation for uniform irradiance is used in [13] for a combination of circular square geometry in order to illuminate the edges of the incident surface.An evolutionary algorithm based optimization scheme is proposed in [14] to modify the power of LED transmitters to reduce the signal power fluctuation at the receiver.In [15], a genetic algorithm is proposed to optimize the refractive indices of the concentrators on receivers to achieve a uniform distribution of the received power.An optimal LED arrangement to achieve uniform irradiance is investigated as a convex optimization problem in [16].
In all the above, computationally intensive optimization routines were used for power allocation for the source LEDs to realise uniform irradiance on the incident surface.[16] departs from the conventional model by considering arbitary locations for the source LEDs.The most practical scenario would be the case when the LEDs are placed randomly at the source with uniform illumination being achieved through power allocation, keeping the total power constant.This problem is addressed in this paper by considering a BPP based stochastic geometry [17].Further, a simple metaheuristic power allocation scheme is proposed for uniform irradiance on the incident surface.Through numerical results, it is shown that the performance of the BPP model and the associated power allocation is comparable to the model in [13].
Rest of the paper is organized as follows.Section 2 deals with the optical signal and noise models.Section 3 contains details regarding the arrangement of different LED arrays and their performance.Heuristic power allocation for random LED sources is discussed in Section 4. The performance of the proposed model is discussed in Section 5, followed by conclusions in Section 6.

Irradiance
Using the Lambertian radiation pattern to model the LED radiant irradiance, [3,4] where φ is the angle of incident light and is the order of Lambertian emission, with φ 1 2 being the LED semi-angle at half power, provided by the manufacturer.The channel direct current (DC) gain can then be expressed as [3,4] where d is the distance between the LED and the photo-detector, A is the physical area of photodetector,cos (φ) = h d and θ is the inclination of the photodector to the incident surface.Note that all the LEDs are located vertical to the plane where the photo-detector is placed.The received power at the photodetector is given by where P t is the LED power.

SNR
The SNR at the output of the photo detector is where R is the responsivity of the photo-detector and σ 2 is the variance of the Gaussan noise at the photodetector [18].

Problem Definition
Consider the various source geometries for N = 16 LEDs in Fig. 1.Using (5), the respective SNR profiles for the sources are plotted in Fig. 2, when each of the LEDs has equal power.From Fig. 2d, it is obvious that the arrangement in Fig. 1d has a more uniform SNR profile, since the coverage at the edges is better.Circular geometries are limited by their inability to sufficiently illuminate the corners of the incident surface.Thus, with better power allocation, it should be possible for the source in Fig. 1d to achieve more uniformity.

BPP
In a BPP stochastic array, N LEDs are placed randomly within a square of length l at the points (x n , y n ) : The random BPP in Fig. 1c offers more flexiblity in the source geometry, and is useful in applications such as visible light communication [17].However, to achieve a uniform SNR profile using a BPP, optimal power allocation is mandatory.This problem is addressed in the following.

Power Allocation for a BPP array
The received optical power at the photodetector j is H i,j .P ti (7) where H i,j , defined in ( 3) is dependent on the distance d i,j between LED i and photodetector j .The Q-factor, a metric to evaluate the fairness of the system, is defined as [13] where Λ is average SNR and var(Λ) represents its variance.For a BPP, each LED is at a random location, so, heuristically, the power should also depend on the distance of the LED from the center of the array.The proposed power allocation is where P is the total source power, r i is the location of the ith LED from the centre, α is a suitable exponent and P i is the power allocated to the ith LED.An optimal value of α can be obtained using the average mean square error metric defined below. min A simple search routine results in α = 3.1.This value is used in (5) and results in uniform SNR at the receiver plane.

Results
The noise at the photodetector is the sum of the contributions from shot noise and thermal noise, and expressed as [18] where with the parameters defined in Table 1.
A trial and error based power allocation was done for the circle-square geometry in Fig. 1d using the average mean square error (a) Circular geometry (c) BPP (d) Circle-square geometry in [13] Fig. 1: Arrangement of LEDs for different geometries as a metric.This results in uniform illuminance, as shown in Fig. 3a.With the heuristic power allocation scheme in (9), the BPP in Fig. 1d also yields uniform illuminance as shown in Fig. 3b.The Q values defined in (8), for both geometries are listed in Table 2.
It is clear that the Q value for the BPP in Fig. 1c is comparable to the one in Fig. 1d.Also, the mean SNR for random geometry is greater than that for circle-square geometry.However, the variance for the BPP model is greater, which reduces its Q value.

Conclusion
From the above analysis, it is clear that a BPP arrangement of LEDs can achieve uniform illumination.This makes it extremely useful in practical applications like visible light communication where the source geometry is likely to be random.Also, the proposed heuristic for power allocation is much simpler, resulting in reduced computational cost, when compared to existing optimal power allocation schemes.Finding a simple but optimal power allocation scheme for stochastic LED arrays will be the focus of future work.

Table 1 :
Parameters of VLC system 3: Uniform SNR profile with optimal/suboptimal power allocation