Optimized DFT-spread OFDM based visible light ... - OSA Publishing

Vol. 25, No. 22 | 30 Oct 2017 | OPTICS EXPRESS 26468

Optimized DFT-spread OFDM based visible light communications with multiple lighting sources Z I -YANG W U , 1 Y U -L IANG G AO, 1 Z E -K UN WANG , 2 C HUAN YOU , 1 C HUANG YANG , 1 C HONG L UO, 3 AND J IAO WANG 1,* 1 College

of Information Science and Engineering, Northeastern University, Shenyang, China Limited, Shanghai, China 3 Microsoft Research Asia, Beijing, China 2 Broadcom

* [email protected]

Abstract: Discrete Fourier transform spread orthogonal frequency-division multiplexing (DFTS-OFDM) has demonstrated its capability in reducing peak to average ratio (PAPR), while maintaining reliable transmissions. This paper investigates the application of DFT-S-OFDM technology in visible light communications (VLC), and reveals the mechanism on how a multiple lighting distributed layout affects its performance. In addition, an optimization approach of lighting layout is proposed through making a trade-off between the strong interfered areas and the maximum delay spread inside. Eventually, a Gbit/s DFT-S-OFDM based multiple lighting VLC downlink prototype is achieved for the first time in the form of real-time baseband modem and compact size components. © 2017 Optical Society of America OCIS codes: (060.4510) Optical communications; (060.2605) Free-space optical communication.

References and links 1. T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. Consum. Electron. 50(1), 100–107 (2004). 2. H. Li, X. Chen, J. Guo, Z. Gao, and H. Chen, “An analog modulator for 460 MB/S visible light data transmission based on OOK-NRS modulation,” IEEE Wirel. Commun. 22(2), 68–73 (2015). 3. B. Bai, Z. Xu, and Y. Fan, “Joint LED dimming and high capacity visible light communication by overlapping PPM,” in Proc. Annual Wireless and Optical Communications Conference (2010), pp. 1–5. 4. F. M. Wu, C. T. Lin, C. C. Wei, C. W. Chen, H. T. Huang, and C. H. Ho, “1.1-Gb/s White-LED-Based Visible Light Communication Employing Carrier-Less Amplitude and Phase Modulation,” IEEE Photonic. Tech. L., 24(19), 1730–1732 (2012). 5. J. Vucic, C. Kottke, S. Nerreter, K. D. Langer, and J. W. Walewski, “513 Mbit/s visible light communications link based on DMT-modulation of a white LED,” J. Lightwave Technol. 28(24), 3512–3518 (2010). 6. A. H. Azhar, T. A. Tran, and D. O’Brien, “A Gigabit/s Indoor Wireless Transmission Using MIMO-OFDM Visible-Light Communications,” IEEE Photonic. Tech. L. 25(2), 171–174 (2013). 7. J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. 27(3), 189–204 (2009). 8. M. S. Islim, R. X. Ferreira, X. He, E. Xie, S. Videv, S. Viola, S. Watson, N. Bamiedakis, R. Penty, I. White, A. Kelly, E. Gu, H. Haas, and M. Dawson, “Towards 10 Gb/s orthogonal frequency division multiplexing-based visible light communication using a GaN violet micro-LED,” Photonics Research 5(2), A35–A43 (2017). 9. F. Zafar, M. Bakaul, and R. Parthiban, “Laser-Diode-Based Visible Light Communication: Toward Gigabit Class Communication,” IEEE Commun. Mag. 55(2), 144–151 (2017). 10. J. Wang, Y. Xu, X. Ling, R. Zhang, Z. Ding, and C. Zhao, “PAPR analysis for OFDM visible light communication,” Opt. Express 24(24), 27457–27474(2016). 11. A. M. Khalid, G. Cossu, R. Corsini, P. Choudhury, and E. Ciaramella, “1-Gb/s transmission over a phosphorescent white LED by using rate-adaptive discrete multitone modulation,” IEEE Photon. J. 4(5), 1465–1473 (2012). 12. X. Huang, Z. Wang, J. Shi, Y. Wang, and N.Chi, “1.6 Gbit/s phosphorescent white LED based VLC transmission using a cascaded pre-equalization circuit and a differential outputs PIN receiver,” Opt. Express 23(17), 22034–22042 (2015). 13. S. William, Y. Tang, and B. S. Krongold, “DFT-spread OFDM for optical communications,” in Proc. IEEE International Conferenceon Optical Internet (IEEE, 2010), pp.1–3. 14. A. Sahin, R. Yang, E. Bala, M. C. Beluri and R. L. Olesen, “Flexible DFT-S-OFDM: Solutions and Challenges,” IEEE Commun. Mag. 54(11), 106–112 (2016). 15. L. Tao, J. Yu, Q. Yang, M. Luo, Z. He, Y. Shao, and N. Chi, “Spectrally efficient localized carrier distribution scheme for multiple-user DFT-S OFDM ROF-PON wireless access systems,” Opt. Express 20(28), 29665–29672 (2012).

#306158 Journal © 2017

https://doi.org/10.1364/OE.25.026468 Received 1 Sep 2017; revised 8 Oct 2017; accepted 9 Oct 2017; published 16 Oct 2017


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1.

Introduction

Visible light communication (VLC) is one of the most promising optical wireless communications (OWC) for commercial applications, with a potential to provide a huge unlicensed transmission bandwidth. Meanwhile, the explosive growth of wireless data traffic shows an active demand for VLC [1]. VLC can adopt a variety of intensity modulation/ direct detection (IM/DD) methods including on-off keying (OOK) [2], pulse position modulation (PPM) [3], carrierless amplitude and phase modulation (CAP) [4], discrete multi-tone modulation (DMT) or orthogonal frequencydivision multiplexing (OFDM) [5, 6]. However, these techniques originally designed for radio frequency (RF) systems cannot be directly applied to VLC systems. Certain adaptations need to be made to match the characteristics of VLC. Among the above mentioned IM/DD methods, OFDM appears to be the most promising one. It has been widely used to achieve high-speed communications in RF systems. More recently, its potential has been actively explored in VLC [7, 8]. In RF communications, the high peak to average ratio (PAPR) of OFDM has raised some concerns, especially in high-order modulations. In VLC, this is a more critical issue, due to light-emitting diodes’ (LED) saturated optical power limitation and the potential damage it may cause to human eyes [9]. Systematic investigation on PAPR distributions of various prevalent VLC OFDM schemes has been carried out by Wang et al. [10], providing a comprehensive review and guideline for VLC OFDM designs. One of the mainstream PAPR reduction approaches is clipping. It is usually deployed in DC-biased optical OFDM (DCO-OFDM) or asymmetrically clipped optical OFDM (ACO-OFDM), and many other modified OFDM techniques [5,11,12]. Another choice is discrete Fourier transform spread OFDM (DFT-S-OFDM), an active approach to achieve low PAPR with better bit error rate (BER) performance [13]. DFT-S-OFDM shows a low implementation complexity, and it could offer a flexible physical layer modulation scheme for 5G [14]. For optical


communications, Tao et al. [15] demonstrates a modified localized carrier distribution scheme of DFT-S-OFDM for multiple-user RoF-PON system over 40 km fiber link and wireless link, and analyzed its noise spread principle [16]. Shi et al. [17] verifies the feasibility of a Gbit/s DFT-S-OFDM for a VLC system for the first time. Meanwhile, Zina et al. [18] proposes an optimized circular deployment scheme resulting in an enhancement of its performance, giving a new aspect of lighting layout design. All these works promote the theories and development of lighting sources’ layout and DFT-S-OFDM based optical communications. However, some details in practical DFT-S-OFDM based VLC still have not been understood clearly. A practical VLC downlink usually adopts multiple lighting equipment for expanding illumination coverage. This indicates stronger interferences and multiple path propagations among the distributed lighting sources. In addition, optics receivers also get interfered by low-frequency noises, such as bias boost convertors for avalanche photodiodes (APD), or other artificial light sources and switching ripples from LED drivers [19]. So it requires a subcarrier shifting method to skip low-frequency noises and to avoid occupying other users’ band. To the best of our knowledge, there is no research on how these practical factors will impact DFT-S-OFDM based VLC system design. This paper conducts a series of theoretical analysis and experiments, and proposes a notable mechanism on how the multipath effect and spectrum selection will affect the performance of DFT-S-OFDM based VLC. In addition, this paper propose an optimization method for baseband modems and distributed lighting sources layout. It minimizes the influence of multipath effect and relaxes the limitation on baseband modems, achieving an optimal channel capacity. These arguments are verified in a practical VLC downlink platform. This platform employs high-efficient switching power in LED DC supplying at the transmitting side. At the receiving side, we use a boost convertor to generate 100V bias for APD from a battery voltage, achieving a high density mobile VLC receiver with compact size components. Multi-user’s bands can be assigned flexibly. We utilize subcarrier shifting directly without any performance deterioration from multipath effect. Finally, we implement a real-time DFT-S-OFDM based multiple-user VLC downlink system in a practical room. The peak real-time data transmission rate reaches about 1Gbit/s under the pre-forward error correction (FEC) limit (below a bit-error-rate (BER) of 3.8 × 10−3 ). The potential of DFT-S-OFDM based VLC is further expanded in massive indoor data transmissions. 2.

Overview of low-frequency interferences in a practical VLC scenario

LEDs in VLC require high power to transmit further and keep standard illumination. So a high efficient switching power for supplying is necessary. However, such power supply circuits will induce a large amount of switching noises from their output ripples. Most of commercial switching convertors are designed with full consideration of efficiency, while the ripples are seldom considered. In order to enhance the power density, ∼MHz high frequency control is used in drivers [20].These ripples have a fundamental frequency at several MHz, and harmonics up to tens of MHz. Passive filtering networks may mitigate these ripples to a certain extent, but they cannot be completely eliminated. In addition, some conventional lighting lamps use rectifier supplies with an output waveform as half-wave or full wave rectifier bridge circuits. Furthermore, high-efficiency electronic ballasts start up the fluorescent lamps at frequencies of tens to hundreds of kHz [21]. These ripples make the receiving photoelectric sensors more easily to be saturated and interfere the low-frequency subcarriers. High sensitive APD based receiver in a compact size or a mobile VLC device form requires a DC-DC boost converter to generate a low-noise, high-voltage reverse bias from a battery voltage. But switching circuit radiation of APDs’ bias supply shows an electromagnetic interferences (EMI) problem as shown in Fig. 1. Its ripple’s spectrum has a wider harmonic distribution from about 300 kHz to 7.3 MHz. Comparing to a -100dBm receiving power, these interferences are


Fig. 1. Low-frequency interferences from APD bias boosters.

Fig. 2. DFT-Spread OFDM scheme with spectrum shifting.

strong enough to threaten the modulator. Commercial APD bias boost converters could achieve low-ripple with LC filters in the output stage, but the EMI especially from the inductor inside is always a beset. With higher density of the receiver, the radiation will be responded much more strongly. For instance, when a high gain transimpedance amplifier (TIA) is assigned near the boost converter to achieve a denser layout, the low-frequency noises will get tremendously augmented. In addition, for white phosphorescent LEDs, the low frequency response curve is very steep, because of the yellow component from stimulated phosphorescence. Even red-green-blue (RGB) LEDs shows a poor character from DC to ∼MHz [22]. Their bandwidths are accounted from a higher reference frequency in most applications [2, 12, 22]. Although there is about several hundred of MHz bandwidth reported to be deployed as subcarriers of commercial LEDs, the low frequency is not suitable for transmissions. To sum up, it demands a spectrum shifting procedure to keep the subcarriers from these unsuitable frequencies, so that the channel can be accurately and robustly estimated. Conventional OFDM schemes usually adopt quadrature modulation in order to transmit real and complex-valued signals onto one main carrier simultaneously, which moves the spectrum to a higher frequency. However, due to the limitation of LEDs, it is preferable to make the frequency-domain vectors Hermitian Symmetric to obtain a real-valued time-domain signal instead (also known as DMT) [5]. Thus, we need to select the subcarrier mapping frequency carefully before IFFT, moving the spectrum to a safe range and skipping the subcarriers that are severely disturbed or dropped sharply in response curve, as illustrated in Fig. 2. 3. 3.1.

Optimization of baseband modem and lighting layout Order errors in demodulated sequences

In practical VLC downlink transmissions, signals to be sent are separated into several paths of LEDs driver circuits. Each of the LEDs illuminates and transmits the same signal simultaneously, extending the coverage of radiation. But they induce several strong delayed line-of-sight (LOS) paths. These signals interfere with each other, and deteriorate selective frequency fading. For simplicity of analysis, we adopt a common practice to assume a cubic environment Ω that


occupies a certain volume. For the receiver R and the i − th transmitter Si ∈ Ω, the completed impulse response hΩ can be defined as hΩ (t; S, R) =

I −1 K−1 Õ Õ

hi(k) (t; Si, R)

(1)

i=0 k=0

where hi is the impulse response of the transmitter Si , k is the index of the multipath among K paths, and I is the amount of transmitters. As Uysal et al. [23] gives, some practical scenarios exhibit a delay spread of over 15ns, limiting the coherent bandwidth as about 67MHz, and the interfered frequency inside the first Nyquist zone is about 33.5MHz. Komine et al. [24] gives an effective adaptive equalization to solve it. But all these equalizers remain a capability limit and outage areas in a room. After this multipath propagation in hΩ , the received signal after synchronization is rs rs (m) =

K−1 Õ

{pk [x 0 k ((m − τk )) M RM (m)] ⊗ hl (m)} + nr (m)

(2)

k=0

where τk is the k − th path’s timing error points, pk is the relative received optical power scale factor of the k − th path, M is the total number of subcarriers and also the points of IFFT in the modulators, x 0 k is the demodulated quadrature amplitude modulation (QAM) sequence component from the k path, hl is the impulse response of LED with low-pass characteristic which will be solved by a pre-equalizer, nr donates the channel and system noises including timing jitter, and RM is a rectangular sequence. Since timing jitter’s effect is equivalent to an added noise [25], and its effects on DFT-S OFDM in synchronization and ICI are similar to those on the conventional OFDM, we could use oversampling to weaken its influence. Meanwhile, effects from timing jitter are relatively smaller than those from multipath interferences, so this paper will focus on the problem from multipath rather than other noises. Then rs gets OFDM demodulation, the M-point FFT, and we obtain its signal components Rs from the demodulation result. K−1 M−1 Õ 2π 1 Õ 2π pk Hl (g)Xk 0(g) exp(− j g · τk ) Rs (g) = √ rs (m) exp(− j m · g) = M M M m=0 k=0

(3)

where X k0 (g) is the frequency domain signal for each of the multipath, Hl is the frequency response of LED, and we only focus on the signal components themselves. Next, we extract and separate the N data vectors of DFT-spread block and the pilots in each user block above from the M subcarriers. Then, a post-equalization is deployed to acquire the estimation of data signal on each subcarrier, assuming pre-equalization eliminates the impact of LED as much as possible, then we get b f) = R(

K−1 Õ k=0

pk Xk ( f ) exp[− j

K−1 Õ 2π pk Xk ( f ) exp[− jϕ( f ) − j ϕ b( f )] f · τk − j ϕ b( f )] = M k=0

(4)

where ϕ b( f ) is the phase error estimation from the pilots. The next procedure is the opposite operation of PAPR suppression, IDFT in N-point. Finally, the output QAM sequences is under a QAM demapping procedure by hard decision or soft decision method to obtain the original data stream. Conventionally, the timing errors from multipath can be corrected by pilots’ estimation, because the timing errors turn into phasic errors with the help of cyclic prefix (CP). It also makes the subcarriers immune to the temporal circular shift, which only causes phasic errors but none of order errors. However, in DFT-S-OFDM, the original QAM signals in time domain are spread


Fig. 3. One of the pilot’s waveform after synchronization in the receiver.

into their relative frequency domain, and then converted back into their relative time domain. So there is an order in DFT-S-OFDM signals, and the demodulated QAM sequences would be circular shifted. If the timing errors cannot be estimated correctly, it will cause a critical BER derogation when different latency paths’ signals converge, as shown in Fig. 3. These timing errors should have been compensated by post-equalization through pilot’s phasic error estimation. However, there is a limitation to their phase detection, each pilot’s estimation has a period of 2π. The estimation of phase errors in a DFT-S-OFDM system works only when the latency of timing or the CP components remained after the synchronization do not exceed the limitation of its periodicity. In fact, the changes in phase have no periodicity but monotonically decrease. In view of the multipath propagation model, impulse response of the diffused path is spread, so that there is a low-pass characteristic in the channel model [26]. So the phase should be monotonically decreasing approximately as the frequency increases. Apparently, the extra periods from the cyclic shifted CP component can’t be estimated. The over-range path components will be demodulated in wrong orders and get superposition in the final result. This is an important factor which induces error vector magnitude (EVM) in QAM vectors. Next, this paper will elaborate on this view. The relevant latency estimation for the pilot with index fp is b τp = ϕ b( fp )M 2π fp , with a range field as [−0.5Tp, 0.5Tp ], where Tp is its periodic sampling points. Thus, the maximum latency estimation range belongs to the longest period pilot (mostly the first pilot in each user block). When the latency or timing error exceeds this limitation, the QAM constellation will not rotate too much because of the periodicity, but the residue induces a cyclic shift to its order of the time domain vector sequence. Since the phasic estimation is constrained by −π < ϕ b(min{ fp }) < π, the error of the estimation for the delay path components over the codomain will be enlarged to a function of frequency and the timing error ϕ(min{ fp }) ≤ π σk ( f ), ϕ( f ) − ϕ b( f ) = (5) f f ϕ(min{ fp }) > π σk ( f ) + 2π τf ) ≈ 2π M (τk − b M ηk max{TP }, where σk ( f ) donates the basic residual error between the actual subcarrier phase and the estimation result, and ηk is a proportionality coefficient. Then we focus on those paths over the periodic restriction. When the derogation of SNR is not big enough to damage pilots, σk ( f ) approaches zero comparing to 2π f /Mη max{TP }. Therefore, assuming the amplitude is equalized properly, the post-equalization result can be presented by b f )) = R(

K−1 Õ

pk · Xk ( f ) exp[− j

k=0

2π N f · (ηk · max{TP })], ηk ∈ N+ N M

(6)

and the QAM vector sequences after M point IDFT are b r (n) =

K−1 Õ k=0

pk x[[n −

N ηk · max{TP }]] RN (n), ηk ∈ N+ M N

(7)


Here, we can analytically present its order and predict the cyclic shift error precisely. According to b r , every single delayed path component over the limitation can be demodulated in a cyclic shift form as rbk (n) = pk · x((n − δk )) N RN (n) (8) where there is a cyclic shift point number δk for each excessive path in this equation δk =

N N (τk − b τf ) = η · max{TP }, η ∈ N+, k > 0 M M

(9)

Theoretically, once over the limitation, each rbk component performs as a noise signal superimposed on the strongest synchronized LOS signal, no matter how many points are delayed. The SNR of accumulated sequence b r is therefore reduced rapidly and converge towards Õ SN R = p0 /(σnr + pi ) (10) i ∈∆

where p0 donates the strongest synchronized LOS signal power scale factor , σnr presents the relative power scale factor of noise components from channel and system including the added noise caused by jitter, and pi is the received power scale factor from those excessive paths defined by set ∆ = {i |i ∈ Ω, τi ≥ max{Tp } }. The BER after MQ − Q AM demodulation raises simultaneously [27] s ! ! 4 3 1 (11) BE R ≈ Q 1− p SN R log2 MQ MQ − 1 MQ where Q[·] is the Gaussian co-error function. As for the individual damage to the vectors of QAM, the EVM method is more pertinent [27] v v u u 2 t t NR N R Õ Õ 1 1 Õ 2 (12) EV M = pi x[[n − δi ]] N RN (n) r (n) − x(n)| = |b NR P M NR P M n=1 i ∈∆

n=1

2 − 2/LQ BE R ≈ Q log2 LQ

s

3log2 LQ

2 2 2 LQ − 1 EV M log2 MQ

! (13)

where LQ represents the levels of the MQ − Q AM in each dimension, and NR donates the amount of received QAM vectors, PM is the maximum normalized power for each vector. Note the cyclic shift point number δk is relevant with N/M. So as the ratio between DFT points and subcarriers’ amount (FFT/IFFT points) decays, the point of cyclic shift for the corresponding latency path will be reduced. But it contributes nothing to mitigate BER, literally speaking, once the cyclic shift exists, the BER is therefore determined. Apparently, there is a cyclic shift error free limit for DFT-S-OFDM system in a practical channel, designating a specific limit for the feasible interval of the relevant parameters. The feasible interval without any cyclic shift error only exists when 2π (14) ϕ(min{ fp }) = − τk · min{ fp } < π M then we obtain the most conservative limit for cyclic shift error free |max{τk }| · min{ fp } < M/2

(15)

Actually not all the latency paths carries weight, their impacts depend on the relative optical magnitude of each component. Generally, in a multiple lighting scenario, the diffused paths


Fig. 4. (a) The feasible interval of DFT-S-OFDM parameters. (b) Strong coherent region (in bule) in three patterns.

exhibit much lower power than the LOS paths. The maximum delay spread can be approximated as the latency of the longest LOS path. The received optical powers of the other delay LOS paths are diminished because of the Lambertian pattern of LED sources and the field of view (FOV) limit of receiver. Since the omnidirectional receiver is promising in mobile VLC [28], the FOV is enlarged so that powers of received delay paths will be enhanced subsequently. In addition, we can define a threshold power for receivers, so that its corresponding delay interval becomes the maximum delay spread. Therefore, we can simplify the limit equation as |τmax | · fp1 < M/2

(16)

where τmax is the maximum delay point number, and fp1 is the first pilot’s carrier index in each user block. Note the selection of fp1 is based on avoiding the noises from low-frequency ripples mentioned in the lase section. Further, when the sample rate Fs of FFT and IFFT in OFDM is certain, a more generalized limit is given by |Tm | · Fp1 < 1/2

(17)

where the real maximum delay spread is Tm =τmax /Fs , and the real frequency of the first pilot in each block is Fp1 = Fs · fp1 M. Thus, a cyclic shift error free limit is that the product of the maximum delay spread and the minimum pilot’s frequency must not exceeds 1/2. Of course, M should not be too small, because a small M leads to a large Fp1 even if there is only one user or no spectrum shifting. Otherwise, the tolerance of Tm will be very small. So we can plan a feasible interval for DFT-Spread OFDM in VLC, ensuring an optimal interval below the FEC limit of 3.8 × 10−3 , as shown in Figs. 4(a). 3.2.

Optimization of distributed lighting layout

Mostly in an indoor scenario, distributed lighting infrastructures enlarge the signal coverage, where the received power ratios among different two LOS paths are high in most areas. So the coherent frequency fading can be mitigated by an adaptive equalizer. However, on some coordinate points, the maximum and the second strong received optical power among the four LOS paths are very close numerically [24]. We define such outage area as strong coherent region in this paper, where interferences in the corresponding frequencies are strong enough to invalidate the equalizers. The power ratio between two LOS paths will affect the BER performance. The maximum ratio which keeps a certain BER we set is the threshold for deciding whether a point is in strong


coherent region. It is also decided by capability limit of the equalization algorithm. In this subsection, we embark on investigating three patterns of interferences and the way to make a layout optimization. We use a simplified point light source model instead of a light array [29]. As shown in Figs. 4(b), the four lights are distributed on S0(−a, a), S1(−a, −a), S2(a, a) and S3(a, −a), respectively, on the ceiling plane with height of h. The size of this room is 2w × 2w × h. As will be readily seen, the room can be divided into 8 equivalent triangle regions, and we can easily analyze the triangle (−w, w), (−w, 0), (0, 0) to cover the whole. The first interference pattern is path S0 to R1 gets interfered with path S1 to R1. Path S0 to R1 is the strongest and the headmost, with path S1 to R1 following closely, and they dominate this pattern. The second pattern is that the strongest and the earliest paths S0 and S1 to R2, which is close enough to or right on the x axis, get interfered with the second strong paths S2, S3 to R2. The third pattern is that the earliest path S0 to R3, which is close enough to or right on the line y = −x, gets interfered with paths S1, S2 to R3, and the intensity contrast between them is not unique. These three patterns divide each triangle into 3 areas. However, two of the areas are lines strictly, and turn into strips only when the time and spatial resolution of the numerical analysis is relatively low. So we focus on the first pattern to track down the principal contradiction. The predicate function of strong coherence is ∫ h(0) (t; S0, R)dt P[S0, S1, R] = ∫∞ − rth 2 ≤ 0 (18) (0) (t; S , R)dt h 1 ∞ ∫ where rth is the square root of strong coherent threshold, and ∞ δ(t − dc )dt = 1, so we have P[S0, S1, R] = P[R(x, y)]=

4 cosm θ 0 cos ϕ0 d2 (S1, R) d(S1, R) 2 − rth 2 ≤ 0 − r = th d(S0, R) cosm θ 1 cos ϕ1 d 2 (S0, R)

(19)

where d(S, R) is the distance between S and R. Thus, we take the parameters into the equation, and get 1 + rth 2 4rth a2 + h2 = 0, R(x, y) ∈ ∆ (20) C0 : (x + a)2 + (y + a ) − 1 − rth (1 − r)2 as the borderline of the strong coherent region, where ∆ is the triangle confined by (−w, w), (−w, 0), (0, 0). Further, we calculate its area to find an optimized position of light source with sustainable area and latency. By integrating these equations, we conclude the strong coherent region area expression as a function of a Asc (a) = w 2 /2 − s(z + a) − s(−w + a)

(21)

where 2ax x 2 x √ 2 E2 x ar s(x) = − + E − x2 + arcsin , z = + 1−r 2 2 2 2 1−r

s

E2 a2 4ra2 − , E = − h2 2 (1 − r)2 (1 − r)2 (22) We give a strong coherent region proportion threshold Ath , for example 20%, and let 2Asc (a)/w 2 ≤ Ath

(23)

After solving the inequality, we therefore get the feasible interval of a, which is the key of lighting layout. So once w, h, Ath as 20% and capability upper bound of equalization are given, interval of a is readily solved.


Then we considerate the latency among the LOS paths. According to the interference principle, the interference frequencies are (1 + n)/2τc, n ∈ N+ , where τc, is the latency point under Fs . In the first Nyquist sampling area, the interfered subcarrier index is M/2τc . Therefore, we need to confine τc in the strong coherent region to ease the burden of equalization. As the largest latency Tm donates the lowest coherent bandwidth, we only take the latency between path S3 to R against path S0 to R (in the strong coherent region) into account L[R(x, y)] = [d(S3, R) − d(S0, R)]/c (24) √ as d(S3, R) > d(S0, R), and function f (x) = x/ x 2 + k 2 is monotonically increasing, we obtain ∂L 1 x − a ∂L 1 y + a y−a x+a > 0, 0 = q −q ® ∂a c 2 2 2 2 2(a − w) + h ¬ « 2(a + w) + h

(27)

so we conclude that as a decreases, the maximum Tm monotonically decreases. Tm inside the strong coherent region is always equal or less than that on the vertex (−w, w), hence, the minimum a in the feasible interval we conducted above is the single optimal layout solution . ao = min {a Asc (a) ≤ w 2 Ath 2 } (28) 0