Multi-User Sum-Rate Optimization for Visible Light ...

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Multi-User Sum-Rate Optimization for Visible Light Communications with Lighting Constraints Rui Jiang, Zhaocheng Wang, Senior Member, IEEE, Qi Wang, and Linglong Dai, Senior Member, IEEE

Abstract—In visible light communication (VLC) systems, white light emitting diodes (LEDs) are used as illumination sources and transmitters simultaneously. Compared to the phosphorconverted LEDs, multi-chip LEDs have higher modulation bandwidth. Consequently, the multi-chip based VLC systems have great potential for high data rate transmission. Since each chip of the multi-chip LEDs can be modulated independently, parallel communication channels are viable for information transmission. In this paper, in order to maximize the multi-user sum-rate for the multi-chip based multi-input single-output VLC systems, an electrical and optical power allocation scheme is proposed in consideration of the luminance, chromaticity, amplitude and bit error rate constraints. From the perspective of human color vision, the chromaticity constraint is defined within a MacAdam ellipse. As a result, the degree of freedom can be achieved by relaxing the chromaticity constraint from a fixed color point to an elliptic region. Numerical results demonstrate that with the increase of the total luminous flux, the maximum sum-rates present an open-down parabolic tendency due to the limited dynamic range of LEDs. Higher data rate can be achieved under higher correlated color temperature (CCT) for the variation of light components. In addition, the simulation results indicate that the shapes of the chromaticity constrained region (either ellipse or quadrangle) have little impact on the multi-user sum-rate at the same CCT. Index Terms—Visible light communication, multi-user sumrate, chromaticity constraint, MacAdam ellipse, conic optimization.

I. I NTRODUCTION Driven by rapid advancement of energy-efficient light emitting diodes (LEDs) for indoor illumination over the past decade, visible light communication (VLC) has become an emerging short-range wireless communication technology [1]– [3]. As one candidate of optical wireless communication (OWC), it exploits vast unregulated visible light spectrum (from 480 nm to 780 nm) to convey information. Based on intensity modulation with direct detection (IM/DD), the amplitude of the electrical signal determines the forward current passing through LEDs and the radiated optical power [4]. The authors are with Tsinghua National Laboratory for Information Science and Technology (TNList), Department of Electronic Engineering, Tsinghua University, Beijing 100084, China (E-mails: [email protected]; [email protected]; [email protected]; [email protected]). This work was supported in part by National Key Basic Research Program of China under Grant 2013CB329203, in part by National Natural Science Foundation of China under Grant 61571267, in part by Shenzhen Peacock Plan (No. 1108170036003286), in part by Guangdong Science and Technology Planning Project (2014B010120001), and in part by Shenzhen Fundamental Research Project (JCYJ20150401112337177). Copyright (c) 2016 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected].

Since visible light does not cause any interference to the radio frequency (RF) signal, VLC can coexist with its RF counterpart and be a complementary technology to provide high data rate transmission. Compared to conventional RF systems, VLC systems are usually free from multipath fading. In addition, due to the small wavelength of visible light, the optical signal is confined to enclosed space and consequently, high security can be guaranteed. In VLC systems, LEDs are used as illumination sources and transmitters simultaneously. There are two common types of white LEDs used for illumination: phosphor-converted LEDs (pc-LEDs) and multi-chip LEDs [5]. In pc-LED packages, one or more visible light-emitting phosphors are coated on a LED chip emitting short-wavelength light. The pc-LEDs employ some of the short-wavelength light to pump the phosphors and further generate long-wavelength light while the rest of the short-wavelength light is leaked out. By mixing these lights of different wavelengths, the white light can be produced. Typical commercial pc-LEDs utilize the cerium doped yttrium aluminum garnet (Ce:YAG) phosphor to generate the yellow light and mix it with the blue light emitted by the gallium nitride based LED chip [6]. With the improvement of modern manufacturing technology, the luminous efficacy of pc-LED has been improved to above 150 lm/W [7]. However, the intrinsic modulation bandwidth of pc-LED is still limited to several MHz due to the slow relaxation time of the phosphor [8]. On the other hand, multi-chip LEDs exploit three or more LED chips to emit different monochromatic lights and mix them together according to the predefined ratio to produce white light, i.e., red-green-blue (RGB) LEDs. Although multichip LEDs are more complex and expensive, their intrinsic modulation bandwidth is several times larger than that of pc-LEDs [9]. Besides that, they can be utilized for multiple parallel data transmission. Since wireless communication is combined with illumination in VLC systems, both electrical domain and optical domain constraints should be taken into consideration. Several prior work have been done in this field. A subcarrier-reuse and power-redistribution algorithm is proposed to improve the throughput for the discrete-multi-tone-based VLC system in the presence of a restricted total electrical power [10]. In [11], quality-of-service requirements such as data rate, fairness and statistical delay are considered in resource allocation for heterogeneous visible-light and radio frequency femtocell. The spectral efficiency is maximized with adaptive modulation under the constraints of the optical power and a target bit error rate (BER) [12]. As brightness and chromaticity are the basic characteristics of the color perceived by human eyes,

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luminance as well as chromaticity requirements should be investigated in the design of VLC systems. Regardless of chromaticity requirement, luminance control is considered in [13]– [15]. Given constant luminance levels, a linear precoder in a coordinated broadcasting system is designed to minimize the sum of minimum mean-squared error [13]. In [14], a power allocation method in optical multiple-input multiple-output (MIMO) VLC systems under sum optical power constraint is proposed. In [15], a precoding and biasing model for multiuser multi-input single-output (MU-MISO) VLC systmes is demonstrated with the constraint of expected luminance level. Recently, some works have been done in consideration of chromaticity constraint. The transmitting rate is maximized in a point-to-point VLC sysetm under the chromaticity constraint as well as transmission power constraint [16]. In [17], by employing different resource of RGB LEDs, constellation design subject to optical power and color constraints is proposed. However, it is worth noting that only fixed color chromaticity of LED is investigated in these papers. In fact, human eyes could not notice the small difference between two different colors in the small area in Commission Internationale de L’Eclairage (CIE) 1931 color space chromaticity diagram called MacAdam ellipse [18]. Thus, chromaticity variation within MacAdam ellipse could be tolerated in practical VLC scenarios. In this paper, we consider a MU-MISO VLC system where a multi-chip LED is used as the transmitter. Multiple independent parallel information bits are modulated onto the monochromatic lights emitted by the LED chips. Indoor users utilize the monochromatic filter and photodiode (PD) to detect the information. An electrical and optical power allocation scheme is proposed to maximize the multi-user sum-rate subject to four constrains in electrical domain and optical domain: luminance constraint, chromaticity constraint, amplitude constraint and BER constraint. Different from the abovementioned prior work, the degree of freedom can be achieved from relaxing the chromaticity constraint from a fixed color point to an elliptic region (i.e., MacAdam ellipse). Given the MacAdam ellipse chromaticity constraint, the sum-rate maximization problem is then formulated to be non-convex. Further, it is transformed into a conic optimization problem and can be solved with classic optimization algorithms. Our results show that large optical power might not contribute to high data rate due to the decrease of the electrical power influenced by the dynamic range of LEDs. Besides that, there is a small variation of multi-user sum-rate over a large luminous flux range. For comparison, a quadrangle chromaticity constraint is also considered. It is concluded that the shapes of the chromaticity constrained region have little impact on the multi-user sum-rate at the same correlated color temperature (CCT). The remainder of this paper is organized as follows: Section II introduces color perception by human eyes and Section III presents the MU-MISO VLC system model based on a multi-chip LED. In Section IV, to maximize multi-user sum-rate, a power allocation scheme for multiple LED chips is proposed under four constraints in electrical and optical domains. Numerical results are presented in Section V and

Fig. 1. CIE 1931 color space chromaticity diagram with wavelengths displayed in nanometers. The blackbody line is shown within the range of correlated color temperature from 1500 K to 10000 K.

the conclusions are drawn in Section VI. II. LED C OLORIMETRY A. Color Characteristics To describe the light brightness perceived by human eyes, spectral luminous efficiency function V (λ) was defined in 1924 by CIE. Based on V (λ) funciton, the perceived light power is measured as luminous flux, which is expressed as [19] ∫ Φ = Km P (λ)V (λ)dλ, (1) λ

where Km is a constant of 683 lm/W to convert irradiance to illuminance and P (λ) is the power spectral distribution. In CIE 1931 XYZ color space, three tristimulus values (X, Y, Z) associated with three color matching functions (¯ x(λ), y¯(λ) and z¯(λ)) are defined to describe any color perception. The tristimulus values (X, Y, Z) are regarded as the amounts of three primaries in a tri-chromatic additive color model, which are expressed as [19] ∫ X = Km Y = Km

P (λ)¯ x(λ)dλ,

(2a)

P (λ)¯ y (λ)dλ,

(2b)

P (λ)¯ z (λ)dλ.

(2c)

∫λ ∫λ

Z = Km λ

Since color matching function y¯(λ) is deliberately defined to be equal to luminous efficiency function V (λ), Y can represent the luminance of the light. The chromaticity of the color can be represented by two coordinate points x and y in the CIE 1931 color space chromaticity diagram as shown in Fig. 1, which are defined as [19] X , X +Y +Z Y y= . X +Y +Z

x=

(3a) (3b)

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is because human eyes have different sensitivities to different colors. The ξ-step MacAdam ellipse can be expressed as g11 dx2 + 2g12 dxdy + g22 dy 2 = ξ 2 ,

Fig. 2. MacAdam ellipses in CIE 1931 color space chromaticity diagram with 10 times the actual size in MacAdam’ experiment.

Therefore, for the ith chip of the multi-chip LED, assuming the luminous flux emitted is Φi and the chromaticity coordinate of the emitted monochromatic light is (xi , yi ), based on Grassmann’s laws of additive color mixture, the chromaticity coordinate of the mixed white light can be calculated by [19] ∑Nt xi aT Φ i=1 y Φi x ˆ = T = ∑Nt 1i , b Φ i=1 yi Φi ∑Nt T Φi 1 Φ , yˆ = T = ∑Ni=1 t 1 b Φ Φi i=1

(4a) (4b)

yi

where Nt is the number of the LED chips, Φ = [Φ1 , Φ2 , · · · , ΦNt ]T is the luminous flux vector, a = xNt T [ xy11 , xy22 , · · · , yN ] and b = [ y11 , y12 , · · · , yN1 ]T are the cot t efficient vectors. B. MacAdam Ellipse As human eyes have limitation on color discrimination, quantifying chromaticity difference is subjective. MacAdam ellipse is used as a statistical measurement tool to describe the small chromaticity difference between two colors in the chromaticity diagram at the same luminance [18]. When two colors are located inside the same MacAdam ellipse, average human observer could not discern the color difference. The original ellipse in MacAdam’ experiment is called as 1step MacAdam ellipse. In practice, larger ξ-step (ξ > 1) MacAdam ellipses are usually used, which implies that the lengths of its major and minor axes are ξ times the lengths of the original ellipse’s major and minor axes in MacAdam’s experiment. As shown in Fig. 2, there are twenty-five 10step MacAdam ellipses with different center points in the chromaticity diagram, which are 10 times the size of the 1step MacAdam ellipses to make them observed clearly. Each of them varies in size and orientation. It can be observed that the blue region is much smaller than the green region. This

(5)

where g11 , g12 and g22 are constant coefficients to describe the orientation and size of each ellipse; dx and dy are the differences of x and y coordinates between the color points on the ellipse and the center point of the ellipse. MacAdam ellipse has been extensively used to describe the chromaticity specification for fluorescent lamps and solid state lighting products such as LEDs. American National Standards Institute (ANSI) suggests chromaticity tolerance for fluorescent lamps is defined by a 4-step MacAdam ellipse while a 7step MacAdam ellipse can be used for solid state lighting [20], [21]. Although quadrangles instead of ellipses are applied for LEDs in ANSI specification, the quadrangles are almost overlapping with 7-step MacAdam ellipses. On the other hand, some manufacturers such as OSRAM require chromaticity tolerance for LEDs should be within 3-step MacAdam ellipse area. III. S YSTEM M ODEL We consider an indoor MU-MISO VLC system to support Nr indoor users, which is shown in Fig. 3. A multi-chip LED with Nt chips for illumination is used as the transmitter array. Each LED chip emits one monochromatic light and serves a single user. Every indoor user utilizes the optical filter and PD to concentrate corresponding monochromatic light. Besides that, crosstalk among these independent parallel channels is not considered in this paper. For each channel, pulse amplitude modulation (PAM) is employed to modulate the information bits as it has higher spectrum efficiency over on-off Keying (OOK) modulation and avoid high peak to average power ratio (PAPR) in optical orthogonal frequency-division multiplexing (OFDM) modulation. The modulated signal di for every transmitting chain i (i = 1, 2, · · · , Nt ) is normalized to be in [−1, 1] with zero mean. As LED has a nonlinear transfer function, a timedomain predistortion is utilized such that the emitted optical power is linear to the loaded forward current [22]. To fully exploit the dynamic range of LEDs, amplifiers are used to increase the signal electrical power. This indicates that the modulated signal di would be amplified with a coefficient denoted as γi . Since IM/DD is adopted in VLC systems, a DCbias dDC,i is added to the amplified electrical signal to make it non-negative with the bias-tee circuits. Then, the electrical signal can be expressed as si = γi di + dDC,i ,

i = 1, 2, · · · , Nt .

(6)

It should be noted that in VLC systems, the optical power is the mean of the input signals while the electrical power is the mean square of the input signals. Due to the persistence of vision, human eyes would not be sensitive to the rapid fluctuation of the emitted optical power. Given that the mean of the modulated signal is zero, the perceived brightness is only related to the DC-bias. In this paper, a line-of-sight (LOS) scenario for VLC systems is investigated as in [13], [23], where the diffuse link

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Fig. 3.

Block diagram of the proposed MU-MISO VLC system based on a multi-chip LED.

caused by the reflection of the light is neglected since it is much weaker than LOS link between the transmitter and the receiver. As the channel frequency response is relatively flat near DC, the channel gain hij can be estimated by the DC gain between the ith transmitter and j th receiver, which is expressed as [4] { (m+1)Aj δj cos(φij )m cos(ϕij ) if 0 6 ϕij 6 Ψc , 2 2πτij hij = 0 if ϕij > Ψc , (7) where m is the order of Lambertian emission; τij is the distance between the ith transmitter and the j th receiver; φij is the emission angle at the ith transmitter; ϕij is the angle of incidence at the j th receiver; Ψc is the field of view (FOV) of the receiver; δj is the receiver responsivity and Aj is the collection area of the j th receiver, which is given by [23] Aj =

χ2j AP D,j , sin2 (Ψc )

(8)

where χj denotes the concentrator refractive index and AP D,j denotes the physical detection area of the j th PD. In visible light communication systems, shot noise is a major noise source in the wireless optical link. There are various sources of shot noise such as dark current noise, quantum noise (or photon fluctuation noise) and background radiation (ambient light) noise [24]. Dark current noise is caused by random generation of electrons and holes within the depletion region without photon-induced excitation, which is signal-independent. Quantum noise is caused by the random arrival rate of photons from the data carrying optical source. On the other hand, background radiation noise is caused by the reception of the photons generated by the environment. Localized point source (e.g., the Sun) and extended sources (e.g., the sky) are two types of sources of this noise. In many practical systems, the received signal to noise ratio (SNR) is limited by the background shot noise that is much stronger than the quantum noise from the the data carrying optical source and other noise sources even with the use of the optical filters [24], [25]. Thus, we model the noise as additive white

Gaussian noise (AWGN) in VLC systems as in [8], [12], [14], [17]. Then, the received signal rj can be expressed as rj = hij si + wj ,

(9)

where wj is the Gaussian noise (i.e., wj ∼ N (0, σ 2 )). IV. P OWER A LLOCATION A. Constraint Analysis As LED is used for both illumination and communication at the same time, several constraints are discussed in this paper. For illumination, luminance and chromaticity constraints are imposed. While for communication, signal amplitude and BER constraints are investigated. Besides that, we consider a typical transmission scheme, where the transmitter employs all its Nt colors (i.e., Nt parallel channels) to send the signals to the Nt receivers owning individual optical filters (i.e., Nr = Nt ). Then each user would only receive the information transmitted by one LED chip with corresponding optical filter. 1) Luminance Constraint When multi-chip LEDs utilize Nt chips to transmit information, indoor users should not perceive the luminance variation of the white light and we have 1T ϕ =

Nt ∑

ϕi = Pt ,

(10)

i=1

where Pt is total luminous flux of the multi-chip LED. It should be pointed out that the average luminous flux instead of instantaneous luminous flux is optimized in this paper because the light intensity perceived by human eye is determined by the average luminous flux. Besides, observed from (6), optimization the instantaneous luminous flux would increase the complexity of the signal detection and is impractical due to its additional cost for the prior information of the DC bias for each signal. 2) Chromaticity Constraint Since human eyes could not distinguish small color difference, small chromaticity change of the white light can be tolerated. If the chromaticity coordinate of the white light

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moves within the MacAdam ellipse, human eyes almost could not notice the light color variation. Based on (4) and (5), the chromaticity constraint can be expressed as g11 (

aT ϕ 1T ϕ aT ϕ 2 − x ) + 2g ( − x )( − y0 ) 0 12 0 bT ϕ bT ϕ bT ϕ 1T ϕ +g22 ( T − y0 )2 6 ξ 2 , b ϕ

(11)

where x0 and y0 are the chromaticity coordinate points of the center point inside the MacAdam ellipse. 3) Amplitude Constraint In VLC systems, the electrical signal should be nonnegative (i.e., si > 0). Thus we have γi 6 dDC,i = ci ϕi and ci is the constant coefficient to convert the luminous flux to the forward current. Moreover, since LEDs have a maximum forward current, the electrical signal exceeding the maximum permissible amplitude Ai would suffer clipping distortion. Therefore, the maximum amplitude constraint need to be considered as well (i.e., si 6 Ai ), and we have γi 6 Ai − dDC,i = Ai − ci ϕi . As a result, the amplitude constraint can be given by 0 6 γi 6 min(Ai − ci ϕi , ci ϕi ),

i = 1, · · · , Nt .

C. Optimization Solution Since the compound function of the first inequality constraint in the optimization problem P1 is non-convex, P1 should be cast into a convex problem in order to find a global optimization solution. First, we utilize polar coordinate to transform the rotated MacAdam ellipse expressed by (5) to the ellipse with a standard form, which is given by p2 q2 + = ξ2, α2 β2

where α is the semi-major axis and β is the semi-minor axis. From (5), α and β can be given by [26] √ α= √ β=

In this subsection, we propose an electrical and optical power allocation scheme to maximize the multi-user sum-rate under the abovementioned constrains. In consideration of BER constraint (i.e., BERi 6 BERt ), the achievable data rate for the ith user can be given by [14], [15] i = 1, · · · , Nt ,

(13)

√ where ρ = (− log(5BERt ))−1 . Considering the amplification coefficient γi and luminous flux ϕi , the sum-rate maximization problem can be formulated as

P1 : max ϕ,γ

Nt ∑

log(1 +

i=1

ρhi γi ) σi

aT ϕ aT ϕ 1T ϕ − x0 )2 + 2g12 ( T − x0 )( T − y0 ) T b ϕ b ϕ b ϕ 1T ϕ + g22 ( T − y0 )2 6 ξ 2 , b ϕ 0 ≼ γ ≼ min(A − c ◦ ϕ, c ◦ ϕ), ϕ ≽ 0,

s.t. g11 (

1T ϕ = Pt ,

(16a)

2 √ . 2 (g11 + g22 ) + (g11 − g22 )2 + 4g12

(16b)

The chromaticity point (p, q) on the standard ellipse can be written as

B. Problem Formulation

ρhi γi ), σi

2 √ , 2 (g11 + g22 ) − (g11 − g22 )2 + 4g12

(12)

4) BER Constraint A BER threshold is defined to ensure the quality of service. For the ith user, we have BERi 6 BERt .

υi = log(1 +

(15)

(14)

where γ = [γ1 , γ2 , · · · , γNt ] is the amplication coefficient vector; A = [A1 , A2 , · · · , ANt ] is the maximum amplitude vector; c = [c1 , c2 , · · · , cNt ] is the luminous flux–forward current conversion coefficient vector; the notation ≼ denotes the generalized ( inequality and the notation ◦ denotes Hadamard ) product i.e., for two matrices E and F , (E◦F )ij = Eij ∗Fij .

p=x ¯ cos θ + y¯ sin θ,

(17a)

q = −¯ x sin θ + y¯ cos θ,

(17b)

T

T

ϕ ϕ where we have x ¯ = abT ϕ − x0 , y¯ = 1bT ϕ − y0 , and the rotated angle is donated as θ, which is given by

θ=

 0,    π,  2

for g12 = 0 and g11 < g22 ; for g12 = 0 and g11 > g22 ;

( ) 1 −1 g11 −g22 cot , for g12 = ̸ 0 and g11 < g22 ;  2 2g  (12 )    π + 1 cot−1 g11 −g22 , for g = 0 and g11 > g22 . 12 ̸ 2 2 2g12 (18)

Thus, the first constraint in P1 is equivalent to the expressions as follows  2 q2 p    + 2 6 ξ2,  2  α β    aT ϕ 1T ϕ p = ( T − x0 ) cos θ + ( T − y0 ) sin θ,  b ϕ b ϕ    T T  1 ϕ a ϕ    q = ( T − y0 ) cos θ − ( T − x0 ) sin θ. (19) b ϕ b ϕ ρh

Nt 1 ρh2 Moreover, we define κ = [ ρh σ1 , σ2 , · · · , σNt ]. Due to the monotonicity of logarithmic function, the objective function is equivalent to

det(I + diag(κ ◦ γ)) =

Nt ∏

(1 +

i=1

ρhi γi ), σi

(20)

where det(·) denotes the determinant of a matrix and diag(·) denotes a diagonal matrix.

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Therefore, the optimization problem P1 can be reformulated as P2 : max det(I + diag(κ ◦ γ)) ϕ,γ

s.t.

p2 q2 + 6 ξ2, α2 β2 0 ≼ γ ≼ min(A − c ◦ ϕ, c ◦ ϕ), ϕ ≽ 0, aT ϕ 1T ϕ − x0 ) cos θ + ( T − y0 ) sin θ, T b ϕ b ϕ 1T ϕ aT ϕ q = ( T − y0 ) cos θ − ( T − x0 ) sin θ, b ϕ b ϕ 1T ϕ = Pt . (21) p=(

Since the fourth and fifth constraints (i.e., the two equality constraints involving p, q and ϕ) in P2 are not affine or linear, we define two intermediate variables m = αp bT ϕ and n = q T β b ϕ. After that, the problem can be transformed into P3 : max det(I + diag(κ ◦ γ)) ϕ,γ

s.t. m2 + n2 6 (ξbT ϕ)2 , 0 ≼ γ ≼ min(A − c ◦ ϕ, c ◦ ϕ), ϕ ≽ 0, 1 m = [(aT − x0 bT ) cos θ + (1T − y0 bT ) sin θ]ϕ, α 1 T n = [(1 − y0 bT ) cos θ − (aT − x0 bT ) sin θ]ϕ, β T 1 ϕ = Pt . (22) Using a real-valued slack variable t, the first inequality constraint in P3 is equivalent to { 2 m + n2 6 t2 , 0 6 t 6 ξbT ϕ. (23) It can be observed that the first(inequality in (23) defines a { second order or Lorentz cone i.e., (z, w) ∈ R2 × R | }) z T z 6 w2 , w > 0 [27]. Thus, the problem becomes a conic optimization problem, which is given by P4 : max det(I + diag(κ ◦ γ)) ϕ,γ

D. Quadrangle Problem Since quadrangles are used for chromaticity binning of LEDs and can be specified by CCT and Duv (denoted as the closest distance from the Planckian locus on the (u′, 2/3v′) chromaticity diagram), quadrangles instead of ellipses are used by ANSI to describe the chromaticity tolerance for LED products [21]. Thus, the chromaticity constraint based on quadrangles and the corresponding optimization problem are investigated. The chromaticity of the mixed white light within a specific quadrangle on the chromaticity diagram can be given by yˆ 6 k11 x ˆ + k12 ,

(25a)

yˆ > k21 x ˆ + k22 , yˆ 6 k31 x ˆ + k32 , yˆ > k41 x ˆ + k42 ,

(25b) (25c) (25d)

where kij is a constant for ∀i = 1, 2, 3, 4 and ∀j = 1, 2. After that, the problem of multi-user sum-rate maximization subject to the chromaticity/luminance/amplitude/BER constraints can be formulated as

P5 : max

Nt ∑

ϕ,γ

log(1 +

i=1

ρhi γi ) σi

1T ϕ aT ϕ s.t. 6 k + k12 , 11 bT ϕ bT ϕ aT ϕ 1T ϕ > k + k22 , 21 bT ϕ bT ϕ 1T ϕ aT ϕ 6 k31 T + k32 , T b ϕ b ϕ T aT ϕ 1 ϕ > k41 T + k42 , T b ϕ b ϕ 0 ≼ γ ≼ min(A − c ◦ ϕ, c ◦ ϕ), ϕ ≽ 0, 1T ϕ = P t . As bT ϕ = Φy11 + be rewritten as

Φ2 y2

+ ··· +

(26) Φ Nt yNt

is non-negative, P5 can

s.t. m2 + n2 6 t2 , 0 6 t 6 ξbT ϕ, 0 ≼ γ ≼ min(A − c ◦ ϕ, c ◦ ϕ), ϕ ≽ 0, 1 m = [(aT − x0 bT ) cos θ + (1T − y0 bT ) sin θ]ϕ, α 1 T n = [(1 − y0 bT ) cos θ − (aT − x0 bT ) sin θ]ϕ, β T 1 ϕ = Pt . (24) Conic optimization is a typical class of convex optimization. Several optimization algorithms such as infeasible pathfollowing algorithms have been proposed to solve this problem [28]. In this paper, we use the CVX, a Matlab optimization software package to obtain the optimal solutions [29].

P6 : max det(I + diag(κ ◦ γ)) ϕ,γ

s.t. k11 aT ϕ + k12 bT ϕ − 1T ϕ > 0, k21 aT ϕ + k22 bT ϕ − 1T ϕ 6 0, k31 aT ϕ + k32 bT ϕ − 1T ϕ > 0, k41 aT ϕ + k42 bT ϕ − 1T ϕ 6 0, 0 ≼ γ ≼ min(A − c ◦ ϕ, c ◦ ϕ), ϕ ≽ 0, 1T ϕ = P t .

(27)

Since P6 is a problem of determinant maximization of a matrix subject to linear matrix inequalities, interior point algorithms can be applied to solve this problem [30], [31].

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TABLE I S IMULATION PARAMETERS FOR MULTI - CHIP LED

9 Red-Green-Blue Chips Red Chip Green Chip Blue Chip

Achievable Sum-rate (bit/s/hz)

8

Parameters Type of white LED

7 6 5 4 3 2 1 0 0

20

40

60

80

100

120

140

160

180

200

Luminous Flux (lm)

Fig. 4. Maximum sum-rate for different luminous flux when CCT= 5000 K.

V. N UMERICAL R ESULTS In this section, some numerical results about the multi-user sum-rate in multi-chip LED based VLC system are presented, whereby RGB LED (Cree Xlamp MC-E) is chosen as the transmitter. Three LED chips employ red/green/blue lights to transmit information individually. We assume that three indoor users are at the same distance from the transmitter and use perfect optical filters to receive their own information. The parameters used in the simulation are listed in Table I. For each user, the target BER is defined as 10−3 . Besides that, the total luminous flux ranges from 10 lm to 200 lm. The numerical results of maximum sum-rate under different luminous flux are shown in Fig. 4. The step of MacAdam ellipse is set to be 7 and the CCT value is defined as 5000 K. It can be observed that the curves of the achievable data rate for red chip and green chip present the open-down parabolic shapes and has a peak point at 100 lm. Unlike conventional RF system where more electrical power allocated for the transmitter is in favor of higher achievable data rate, higher optical power for LED might even decrease the data rate. The reason is that optical power is determined by the DC bias and narrower dynamic range would be utilized to amplify the modulated signal in case of clipping distortion when DC bias deviates from the middle position of the dynamic range of LED chips, or vice versa. For this reason, the same results can be obtained for complex modulation formats such as DC-biased optical orthogonal frequency-division multiplexing (DCO-OFDM), which have been verified in [32] without considering the chromaticity constraint. Moreover, the variation of the data rate for red and green chips further causes the change of the sum-rate according to the same tendency. Due to the small portion of the blue component in the white light, the DC bias for the blue chip is always below the middle position of the dynamic range of blue LED chip in the luminous flux range. Consequently, the data rate for the blue chip increases slowly when DC bias approaches to the middle position. In addition, it is notable that there is a small change of sum-rate over a long luminous flux range. From 70 lm to 150 lm, the sum-rate variation is about 1 bit/s/Hz.

Number of LED chips Nt Target BER for each user Maximum forward current for each chip Distance between LED and users τi Detect area A Lambertian emission order m FoV of the receiver Angle of the irradiance Angle of the incidence Concentrator refractive index Receiver responsivity Noise Variance Red Peak wavelength of each chip Green Blue Red Chromaticity coordinate of each chip Green Blue Luminance flux - Forward current Red conversion coefficient for each chip Green Blue

BASED

VLC

SYSTEM

Values RGB LED (Cree Xlamp MC-E) 3 10−3 700 mA 2m 1 cm2 1 60◦ 30◦ 40◦ 1.5 0.5 A/W 0.013 mA2 625 nm 530 nm 460 nm (0.7006, 0.2993) (0.1547, 0.8059) (0.1440, 0.0297) 0.0114 A/lm 0.0052 A/lm 0.0427 A/lm

TABLE II C HROMATICITY CCT 2700 K 3000 K 3500 K 4000 K 5000 K 6500 K

COORDINATES FOR THE

Center point (0.459,0.412) (0.440,0.403) (0.411,0.393) (0.380,0.380) (0.346,0.359) (0.313,0.337)

g11 40 × 104 39 × 104 38 × 104 39.5 × 104 56 × 104 86 × 104

M AC A DAM 2g12 −39 × 104 −39 × 104 −40 × 104 −43 × 104 −50 × 104 −80 × 104

ELLIPSE

[20]

g22 28 × 104 27.5 × 104 25 × 104 26 × 104 28 × 104 45 × 104

In Fig. 5, the percentages of three light components to achieve the maximum sum rate are compared when the CCT is defined as 5000 K and chromaticity tolerance is within a 7-step MacAdam ellipse. It can be seen that the percentages of all components almost remain unchanged, which are 0.3, 0.675 and 0.025 respectively. The fluctuation of each component is very small. The main reason is that the chromaticity variation is still restricted in a small range in order to satisfy the white light constraint. As is shown in Fig. 6, different steps of the MacAdam ellipse for 5000 K CCT value are compared. It can be observed that different steps have an influence on the multi-user sumrate. The maximum gap between the 1-step and 10-step is more than 0.5 bit/s/Hz. Next, the maximum sum-rates for different CCT values under the chromaticity constraint based on MacAdam ellipse are compared. Six CCT values are investigated according to ANSI C78.377–2008. They are 2700 K, 3000 K (warm white), 3500 K(White), 4000 K (Cool White), 5000 K, 6500 K (Daylight) respectively. A 7-step MacAdam ellipse is chosen for every CCT value in the simulation. The parameters of the corresponding MacAdam ellipse are listed in Table II. It should be pointed out that for some CCT value and luminous flux, no feasible solution of the optimization problem would be found. As is illustrated in Fig. 7, the curves of the sumrate for each CCT value all have an open-down parabolic

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Achievable Sum-rate (bit/s/hz)

Red Chip Green Chip Blue Chip

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Fig. 5. Percentages of three light components to achieve the maximum sum-rate for different luminous flux when CCT= 5000 K.

Fig. 7. Maximum achievable sum-rate comparison for different CCT values under chromaticity constraint based on MacAdam ellipse.

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Fig. 6. Comparison of different steps of MacAdam ellispe under different luminous flux when CCT= 5000 K. Fig. 8. Maximum achievable sum-rate comparison for different CCT values under chromaticity constraint based on quadrangle.

tendency. With the increase of the CCT value, higher sumrate can be achieved. This is because when a larger CCT value is considered, the proportion of the blue component becomes higher. The increase of the data rate in the blue light link has the greatest impacts on the sum-rate compared to the other two links for a larger CCT. In addition, for a lower CCT value, the decrease of the electrical power as well as the data rate in the red light link is much severer under larger total luminous flux (above 100 lm). Thus, the difference of the sum-rate among all CCT values becomes more obvious. Since quadrangles are used by ANSI to describe the chromaticity tolerance for LEDs, the sum-rate under chromaticity constraint based on quadrangle for seven standard CCT values

(i.e., 2700 K, 3000 K, 3500 K, 4000 K, 4500 K, 5000 K, 6500 K) are discussed. The chromaticity coordinates of the vertexes of the quadrangles are listed in Table III. As shown in Fig. 8, the curves for different CCT values have the same variation tendency. Likewise, the higher data rate can be achieved with larger CCT values. In Fig. 9, the sum-rates for the same CCT values (i.e., 2700K, 4000K, 5000 K and 6500 K) under elliptical and quadrangular chromaticity constrained region are compared. It can be seen that for the same CCT value, the shapes of the chromaticity constrained region barely have any influence on the sum-rate.

TABLE III C HROMATICITY CCT Vertex 1 Vertex 2 Vertex 3 Vertex 4

2700 K (0.4813, 0.4319) (0.4562, 0.4260) (0.4373, 0.3893) (0.4593, 0.3944)

3000 K (0.4562, 0.4260) (0.4299, 0.4165) (0.4147, 0.3814) (0.4373, 0.3893)

COORDINATES FOR THE QUADRANGLE

3500 K (0.4299, 0.4165) (0.3996, 0.4015) (0.3899, 0.3690) (0.4147, 0.3814)

4000 K (0.4006, 0.4044) (0.3736, 0.3874) (0.3670, 0.3578) (0.3898, 0.3716)

[21] 5000 K (0.3551, 0.3760) (0.3376, 0.3616) (0.3366, 0.3369) (0.3515, 0.3487)

6500 K (0.3205, 0.3481) (0.3208, 0.3304) (0.3068, 0.3113) (0.3221, 0.3261)

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4 Ellipse (CCT=2700 K) Ellipse (CCT=4000 K) Ellipse (CCT=5000 K) Ellipse (CCT=6500 K) Quadrangle (CCT=2700 K) Quadrangle (CCT=4000 K) Quadrangle (CCT=5000 K) Quadrangle (CCT=6500 K)

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Fig. 9. Comparison of maximum achievable sum-rate under chromaticity constraint based on MacAdam ellipse or quadrangle.

VI. C ONCLUSION In this paper, a multi-chip LED based MU-MISO VLC system is investigated, where multi-chip LED employs monochromatic lights to transmit independent parallel data streams. As LED is used as illumination source and transmitter simultaneously, four constraints are considered in this paper including luminance, chromaticity, amplitude and BER. Under these constraints, an electrical and optical power allocation scheme is designed to maximize the multi-user sum-rate. When the chromaticity constrained region is defined within a MacAdam ellipse, the formulated problem is non-convex. With polar coordinate transformation and slack variables, the original problem can be reformulated as a conic optimization problem whereby optimal solutions could be found. Simulation results show that higher optical power might not improve the data rate due to the corresponding lower electrical power. It is also concluded that over a wide luminous flux range (from 70 lm to 150 lm), the variation of the multi-user sum-rate under these constraints is very small. Moreover, higher data rate can be achieved with larger CCT value due to the increase of the blue light component in the mixed light. In addition, for the same CCT value, the shapes of the chromaticity constrained region (ellipse or quadrangle) barely have any influence on the sum-rate. R EFERENCES [1] R. Zhang, J. Wang, Z. Wang, Z. Xu, C. Zhao, and L. Hanzo, “Visible light communications in heterogeneous networks: paving the way for user-centric design,” IEEE Wireless Commun., vol. 22, no. 2, pp. 8–16, Apr. 2015. [2] A. Jovicic, J. Li, and T. Richardson, “Visible light communication: opportunities, challenges and the path to market,” IEEE Commun. Mag., vol. 51, no. 12, pp. 26–32, Dec. 2013. [3] A. Tsiatmas, C. P. M. J. Baggen, F. M. J. Willems, J.-P. M. G. Linnartz, and J. W. M. Bergmans, “An illumination perspective on visible light communications,” IEEE Commun. Mag., vol. 52, no. 7, pp. 64–71, Jul. 2014. [4] J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE, vol. 85, no. 2, pp. 265–298, Feb. 1997. [5] D. A. Steigerwald, J. C. Bhat, D. Collins, R. M. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, and S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Topics Quantum Electron., vol. 8, no. 2, pp. 310–320, Mar. 2002.

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Rui Jiang received his B.E. and M.E. degrees from Beijing Jiaotong University, Beijing, China, in 2011 and 2014, respectively. He is currently working toward his Ph.D. degree at Tsinghua National Laboratory for Information Science and Technology, Department of Electronic Engineering, Tsinghua University. His research interests include visible light communications and optimization theory.

Zhaocheng Wang (M’09–SM’11) received the B.S., M.S., and Ph.D. degrees from Tsinghua University, Beijing, China, in 1991, 1993, and 1996, respectively. From 1996 to 1997, he was a Postdoctoral Fellow with Nanyang Technological University, Singapore. From 1997 to 1999, he was with OKI Techno Centre (Singapore) Pte. Ltd., Singapore, where he was first a Research Engineer and later became a Senior Engineer. From 1999 to 2009, he was with Sony Deutschland GmbH, where he was first a Senior Engineer and later became a Principal Engineer. He iscurrently a Professor of Electronic Engineering with Tsinghua University and serves as the Director of Broadband Communication Key Laboratory, Tsinghua National Laboratory for Information Science and Technology (TNlist). He has authored or coauthored over 100 journal papers (SCI indexed). He is the holder of 34 granted U.S./EU patents. He coauthored two books, one of which, Millimeter Wave Communication Systems, was selected by IEEE Series on Digital and Mobile Communication (Wiley–IEEE Press). His research interests include wireless communications, visible light communications, millimeterwave communications, and digital broadcasting. He is a Fellow of the Institution of Engineering and Technology. Currently, he serves as an Associate Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and the IEEE COMMUNICATIONS LETTERS, and has also served as Technical Program Committee Co-Chair of various international conferences.

Qi Wang (S’15) received his B.E. degree from Tsinghua University, Beijing, China, in 2011. He is currently working toward his Ph.D. degree at Tsinghua National Laboratory for Information Science and Technology, Department of Electronic Engineering, Tsinghua University. From October 2014 to March 2015, he was a visiting scholar at the Centre for Photonic Systems, Electrical Engineering Division, Department of Engineering, University of Cambridge, Cambridge, U.K. He has published over 20 journal papers and several conference papers. His research interests include modulation and signal processing for wireless and visible light communications. He was the recipient of the National Scholarship and the Academic Star of Electronic Engineering Department in Tsinghua University.

Linglong Dai (M’11–SM’14) received the B.S. degree from Zhejiang University, Hangzhou, China, the M.S. degree (highest Hons.) from the China Academy of Telecommunications Technology (CATT), Beijing, China, and the Ph.D. degree (highest Hons.) from Tsinghua University, Beijing, China, in 2003, 2006, and 2011, respectively. From 2011 to 2013, he was a Postdoctoral Fellow with the Department of Electronic Engineering, Tsinghua University, where he has been an Assistant Professor since July 2013. He has authored over 50 IEEE journal papers and over 30 IEEE conference papers. His research interests include wireless communications, with a focus on multicarrier techniques, multiantenna techniques, and multiuser techniques. He currently serves as a Co–Chair of the IEEE Special Interest Group (SIG) on Signal Processing Techniques in 5G Communication Systems. He was the recipient of the Outstanding Ph.D. Graduate of Tsinghua University Award in 2011, the Excellent Doctoral Dissertation of Beijing Award in 2012, the IEEE ICC Best Paper Award in 2013, the National Excellent Doctoral Dissertation Nomination Award in 2013, the IEEE ICC Best Paper Award in 2014, the URSI Young Scientists Award in 2014, the IEEE Transactions on Broadcasting Best Paper Award in 2015, and the IEEE RADIO Young Scientists Award in 2015.