IN INDOOR environments, light emitting diode (LED) based

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1 5496 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 34, NO. 23, DECEMBER 1, 2016 Improved Lower Bounds for Ranging in Synhronous Visible Light Positioning Systems Musa Furkan Keskin, Erdal Gonendik, and Sinan Gezii, Senior Member, IEEE Abstrat In this study, the Ziv Zakai bound ZZB is derived for synhronous visible light positioning VLP systems. The proposed ZZB extrats ranging information from the prior information, the time delay parameter, and the hannel attenuation fator based on the Lambertian pattern. In addition to the ZZB, the Bayesian Cramér Rao bound CRB and the weighted CRB WCRB are alulated for synhronous VLP systems. Furthermore, a losed-form expression is obtained for the expetation of the onditional CRB ECRB. Numerial examples are presented to ompare the bounds against eah other and against the maximum a posteriori probability MAP estimator. It is observed that the ZZB an provide a reasonable lower limit on the performane of MAP estimators. On the other hand, the WCRB and the ECRB onverge to the ZZB in regions of low and high soure optial powers, respetively; however, they are not tight in other regions. Index Terms Estimation, Lambertian pattern, positioning, visible light, Ziv-Zakai bound. I. INTRODUCTION IN INDOOR environments, light emitting diode LED based visible light systems an be used for aurate positioning, high speed data transmission, and illumination simultaneously [1] [9]. The topi of this manusript is related to positioning via visible light systems. Visible light positioning VLP systems with high loalization auray an be employed in numerous appliations inluding robot navigation and asset traking [4], [10]. Reently, various studies have been performed on VLP systems, and high positioning auraies have been reported based on experiments and simulations; e.g., [11] [13]. The aim in this manusript is to obtain theoretial limits on distane range estimation in a synhronous VLP system, whih provide performane benhmarks for pratial estimators. Depending on the presene/absene of synhronization among LED transmitters and visible light ommuniation VLC reeivers, VLP systems an be ategorized as synhronous and asynhronous. In an asynhronous VLP system, LED transmitters are not synhronized with VLC reeivers, and the main parameter utilized for positioning is the reeived signal strength RSS or, power of the inoming signal based on the Lambertian formula [7], [8], [11], [13] [15]. On the other hand, Manusript reeived January 25, 2016; revised July 19, 2016 and September 20, 2016; aepted Otober 19, Date of publiation Otober 23, 2016; date of urrent version November 17, This work was supported in part by the Distinguished Young Sientist Award of Turkish Aademy of Sienes TUBA-GEBIP The authors are with the Department of Eletrial and Eletronis Engineering, Bilkent University, Ankara, Turkey keskin@ee.bilkent. edu.tr; erdal@ee.bilkent.edu.tr; gezii@ee.bilkent.edu.tr. Color versions of one or more of the figures in this paper are available online at Digital Objet Identifier /JLT in a synhronous VLP system, where LED transmitters and VLC reeivers are synhronized, both the time-of-arrival TOA and the RSS parameters an be utilized for distane hene, position estimation [5], [16]. Also, in the presene of multiple photo-detetors at the VLC reeiver, the angle-of-arrival AOA parameter an also be employed for positioning [9], [17], [18]. The AOA an be alulated from the TOA differenes or the RSS differenes among the photo-detetors depending on the presene/absene of synhronization. Theoretial auray limits are important for VLP systems to present benhmarks for numerous studies in the literature. In [5], [6], [16], [19], the Cramér-Rao bound CRB is onsidered for providing theoretial limits for various VLP systems. In [6], the CRB on range distane estimation is derived for an asynhronous VLP system based on RSS measurements, and the effets of system parameters, suh as the signal bandwidth, LED onfiguration and transmitter height, are investigated. The study in [5] fouses on a synhronous VLP system and presents the CRB on TOA based range estimation. It also analyzes the impat of various system parameters, suh as the area of the photo detetor, soure optial power, and enter frequeny, on ranging auray. In [16], the CRBs and maximum likelihood estimators MLEs are investigated for both synhronous and asynhronous VLP systems. In the synhronous ase, both the hannel attenuation fator RSS and the TOA parameters are utilized while only the RSS parameter is used in the asynhronous ase. Comparisons are performed among the synhronous and asynhronous senarios based on the analytial CRB expressions [16]. In [19], a hybrid AOA and RSS based three-dimensional loalization is investigated for an asynhronous VLP system, where AOA based and RSS based loalization algorithms employ, respetively, a least-squares estimator and an analytial learning rule based on the Newton- Raphson method. Also, the CRB is derived for RSS based three dimensional loalization for a generi deployment senario. Unlike the theoretial limits in [5], [6], [16], [19], the aim in this study is to provide theoretial limits for a synhronous VLP system by onsidering the effets of prior information, as well. Although the CRB an provide tight limits on mean-squared errors MSEs of unbiased estimators in high signal-to-noise ratio SNR onditions, it an be quite loose for low SNRs [20]. In addition, the CRB derivations do not onsider any prior statistial information about the range or, position parameter, whih an in fat be available in indoor environments; e.g., based on physial dimensions and known system parameters suh as the field of view of the photo detetor. To address these issues, the Ziv-Zakai bound ZZB an be used as a benhmark IEEE. Personal use is permitted, but republiation/redistribution requires IEEE permission. See standards/publiations/rights/index.html for more information.

2 KESKIN et al.: IMPROVED LOWER BOUNDS FOR RANGING IN SYNCHRONOUS VISIBLE LIGHT POSITIONING SYSTEMS 5497 for ranging in VLP systems. The ZZB an provide tight limits on MSEs of estimators in all SNR onditions, and it also utilizes the available prior information [20], [21]. The study in [22] derives the ZZB on range estimation in an asynhronous VLP system based on RSS measurements and provides omparisons with the maximum a-posteriori probability MAP and the minimum mean-squared error MMSE estimators. In this paper, the ZZB on ranging is derived for a synhronous VLP system by utilizing the prior information and the ranging information from both the time delay TOA parameter and the hannel attenuation fator RSS via the Lambertian pattern. Based on the ZZB, effets of various system parameters, suh as the Lambertian order, the area of the photo detetor, and the soure optial power, are analyzed in terms of ranging auray, and design guidelines are provided for pratial VLP systems. In addition, the expetation of the CRB ECRB is alulated and a losed-form expression is obtained for uniform prior information. The ECRB expression both illustrates the effets of prior information and provides a low-omplexity alternative to the ZZB in high SNR onditions. Moreover, the Bayesian CRB BCRB and the weighted CRB WCRB are derived in order to present theoretial limits that effetively utilize the prior information, and they are ompared against the ZZB. The main ontributions of this study an be summarized as follows: 1 The ZZB on ranging is derived for a synhronous VLC system by utilizing prior information together with the ranging information extrated from the time delay parameter and the hannel attenuation fator. The provided ZZB expression is different from those for synhronous RF systems [20], [23], [24] due to the fats that i synhronous VLP systems utilize both time delay and reeived signal power information whereas synhronous RF systems use time delay information only, and ii the Lambertian formula is available for VLP systems to speify the reeived signal power, whih is not valid for RF systems. 2 A losed-form ECRB expression is derived for ranging in synhronous VLC systems, whih onverges to the ZZB in the high SNR regime. 3 The BCRB and the WCRB expressions are provided for a synhronous VLC system, whih have not been available in the literature. 4 Performane of the MAP estimator is ompared against the theoretial limits. It is demonstrated that the theoretial limits on the performane of the MAP estimators an be haraterized by the ZZB, whih provides important guidelines for designers of pratial VLP systems. In addition, the ECRB and the WCRB are observed to onverge to the ZZB in the high and low SNR regimes, respetively. The rest of the paper is organized as follows: The system model is introdued in Setion II. The ZZB for synhronous VLP systems is derived in Setion III, and a losed-form ECRB expression is provided in Setion IV. The BCRB and the WCRB expressions are obtained in Setion V. Numerial results investigating the performane of the MAP estimator and the bounds are presented in Setion VI, followed by some onluding remarks in Setion VII. II. SYSTEM MODEL Consider an LED transmitter and a VLC reeiver that are loated at a distane of x from eah other. A line-of-sight LOS senario is assumed, whih is ommonly the ase for visible light systems [4], [5]. Then, the reeived signal at the VLC reeiver is stated as [5] rt =αr p st τ+nt 1 for t [,T 2 ], where and T 2 determine the observation interval, α is the attenuation fator of the optial hannel α > 0, R p is the responsivity of the photo detetor, st is the transmitted signal whih is nonzero over an interval of [0,T s ], τ is the time-of-arrival TOA, and nt is zero-mean additive white Gaussian noise with spetral density level σ 2. Considering a synhronous system as in [5], the TOA parameter is modeled as τ = x 2 where x is the distane range between the LED transmitter and the VLC reeiver, and is the speed of light. It is assumed that the signal omponent in 1 is ontained ompletely in the observation interval [,T 2 ]; that is, τ [,T 2 T s ].In1, the hannel attenuation fator α is modeled as α = m +1 2π osm φosθ S x 2 3 where m is the Lambertian order, φ is the irradiation angle, θ is the inidene angle, and S is the area of the photo detetor at the VLC reeiver [5]. For larity of theoretial expressions, it is assumed, as in [5], [6], [11], [22], that the LED transmitter is pointing downwards whih is ommonly the ase sine LEDs are employed also for illumination and the photo detetor at the VLC reeiver is pointing upwards. Then, φ = θ and osφ =osθ =h/x, where h denotes the height of the LED transmitter relative to the VLC reeiver. The theoretial expressions in this study an also be extended to the ases with arbitrary transmitter and reeiver orientations, whih however leads to lengthy and inonvenient expressions. In addition, as in [5], [6], [8], [11], [22], it is assumed that the LED transmitter is at a known height with respet to the VLC reeiver; i.e., possible loations of the VLC reeiver are onfined to a two-dimensional plane. This assumption is valid in various pratial appliations; e.g., when the VLC reeiver is attahed to a art or a robot that is traked via a VLP system sine VLC reeivers have fixed and known heights in those appliations e.g., [4, Fig. 3]. Under these assumptions, 3 redues to α = m +1 m +1 h S 2π x x 2 γx m 3 4 where γ is a known onstant defined as γ m +1hm +1 S 5 2π III. ZIV-ZAKAI BOUND ZZB The ZZB provides a lower limit on MSEs of estimators based on a relation in terms of the probability of error in a binary

3 5498 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 34, NO. 23, DECEMBER 1, 2016 hypothesis-testing problem. It is expressed as [21] ξ 1 wϑ+p min ϑ, ϑ + δdϑ δ dδ where ξ = E{ ˆx x 2 } is the MSE of an estimator ˆx, w represents the prior probability density funtion PDF of parameter x, and P min ϑ, ϑ + δ denotes the probability of error orresponding to the optimal deision rule for the following hypothesis-testing problem: H 0 : prt x = ϑ 7 : prt x = ϑ + δ In pratial indoor senarios, lower and upper limits on the range parameter x are available based on physial dimensions of the environment and the field of view of the photo detetor. Hene, it is reasonable to assume that the prior PDF of x is zero outside the interval [,D 2 ], where and D 2 denote the minimum and maximum possible distanes, respetively. For the signal model in 1, the observation interval [,T 2 ] an be related to and D 2 as = / and T 2 = D 2 / + T s. In this ase, the ZZB in 6 is expressed as ξ 1 2 δ 0 wϑ+ P min ϑ, ϑ + δdϑ δ dδ. 8 For example, if the prior PDF of x orresponds to uniform distribution over [,D 2 ] that is, wx =1/D 2 if x [,D 2 ] and wx =0otherwise, the ZZB in 8 redues to 1 D 2 δ ξ P min ϑ, ϑ + δdϑδdδ. 9 D 2 0 To obtain an expliit expression for the ZZB, P min ϑ, ϑ + δ in 8 should be speified. Based on the PDF wx of x, the prior probabilities of hypotheses H 0 and in 7 are equal to wϑ/wϑ+ and /wϑ+, respetively. Then, the optimal deision rule for the problem in 7 is the MAP rule [25], whih is expressed as prt x = ϑ + δ wϑ+ H 0 wϑ prt x = ϑ. 10 wϑ+ After taking the natural logarithm of both sides, 10 beomes log prt x = ϑ + log prt x = ϑ + δ H 0 wϑ log. 11 From 1, 2, and 4, the log-likelihood funtion is expressed as [26] log prt x =k 1 T 2 2σ 2 rt R pγ x m +3 s t x 2 dt 12 where k is a onstant that does not depend on x. From 12, the deision rule in 11 be stated as T 2 rt γϑ m 3 R p s t ϑ 2 dt T 2 H 0 rt γ ϑ + δ m 3 R p s t ϑ + δ 2σ 2 log wϑ whih redues, via some rearrangement, to where and C rs ϑ + δ ϑ + δ m +3 C H 0 rsϑ ϑ m σ2 wϑ ϑ2m +6 R p γ log C rs x E s T 2 2 dt R p γe s 1 2 ϑ + δ 2m rts t x dt 15 st 2 dt 16 is the energy of signal st. The probability of error for the deision rule in 14 is alulated as P min ϑ, ϑ + δ = wϑ wϑ+ P Ĥ 1 H 0 + wϑ+ P Ĥ 0 17 where P Ĥ i H j denotes the probability of deiding for hypothesis H i when H j is true. Under H 0, it an be shown from 1, 4, 7, and 15 that C rs ϑ and C rs ϑ + δ are jointly Gaussian distributed as [ ] Crs ϑ N C rs ϑ + δ [ γrp E s ϑ m +3 γr p E s ρ δ ϑ m +3 ] [ σ 2 E s σs 2E, σs 2E ρ δ ρ δ σ 2E s ] 18 where N μ, Σ represents Gaussian distribution with mean μ and ovariane matrix Σ, and ρ represents the normalized auto-orrelation funtion of st, whih is defined as ρτ 1 stst τdt. 19 E s From 14 and 18, P Ĥ 1 H 0 an be alulated as P Ĥ 1 H 0 = Rp γe s gϑ, ϑ + δ+ σ 2 γr Q p log wϑ/ σ2 E s gϑ, ϑ + δ

4 KESKIN et al.: IMPROVED LOWER BOUNDS FOR RANGING IN SYNCHRONOUS VISIBLE LIGHT POSITIONING SYSTEMS 5499 where Qy = 1 2π y e t2 /2 dt denotes the Q-funtion, and 1 1 2ρδ/ gϑ, ϑ + δ + ϑ + δ 2m +6 ϑ2m +6 ϑϑ + δ m Also, via similar derivations, P Ĥ 0 an be obtained as follows: P Ĥ 0 = Rp γe s gϑ, ϑ + δ σ 2 γr Q p log wϑ/. σ2 E s gϑ, ϑ + δ Then, the probability of error for the deision rule in 14 an be evaluated via 17, 20, and 22, whih an be expressed in a ompat form as follows: 1 i=0 P min ϑ, ϑ + δ = wϑ + iδp Ĥ 1 i H i. 23 wϑ+ Based on the obtained minimum probability of error expression in 23, the ZZB in 8 an be alulated. As a speial ase, when the prior PDF of x is uniform over [,D 2 ], the logarithm terms in 20 and 22 beome zero, and P min ϑ, ϑ + δ in 23 an be simplified as follows: P min ϑ, ϑ + δ =0.5P Ĥ 1 H P Ĥ0 R p γ E s gϑ, ϑ + δ = Q. 24 2σ For the uniform prior ase, the ZZB an be alulated based on 9 and 24. Sine the integral limits in 8 and 9 are finite, the ZZB an aurately be evaluated via numerial approahes. From 8, 20, 22, and 23, it is observed that the ZZB redues as E s inreases; that is, improved ranging auray is ahieved with higher transmitted signal energy, as expeted. It is also noted that the ZZB expression in 8 and 23 is different from both the ZZB expression in asynhronous VLP systems [22] sine the range related information from both the time delay parameter and the hannel attenuation fator is employed in the synhronous ase. Remark 1: It is important to emphasize that the ZZB expression in 8 and 23 has important distintions ompared to the ZZB expressions for synhronous RF based ranging systems e.g., [20] due to the fats that i the synhronous VLP system utilizes both time delay and reeived signal power hannel attenuation fator information whereas synhronous RF systems use time delay information only sine the reeived power parameter arries negligible information ompared to the time delay parameter in most pratial RF loalization systems, and ii the Lambertian equation in 4 is available for VLP systems to relate the hannel attenuation fator the reeived signal power to distane x in LOS visible light hannels, whih is not valid for RF systems. Overall, the Lambertian formula is utilized, together with the time delay information and the prior information, for the purpose of range estimation in this study. IV. ECRB DERIVATIONS In this setion, the Cramér-Rao bound CRB expressions for range estimation in VLP systems are investigated to provide omparisons against the ZZB. For a given value of the unknown parameter, the onditional CRB presents a lower limit on the MSEs of unbiased estimators, whih is expressed as [21] { } 2 1 log prt x E{ ˆx x 2 } E x J F x 1 =CRBx 25 where ˆx is an unbiased estimate of x and the expetation operators are onditioned on x. For the estimation of the range parameter x, the onditional CRB in the synhronous ase an be obtained from 25 as [16] CRBx =J F x 1 = σx m +4 /γr p 2 m +3 2 E s + Ẽsx/ 2 26 where Ẽs T s 0 s t 2 dt, with s t denoting the first-order derivative of st. 1 The onditional CRB expression in 26 is a funtion of the unknown parameter x, and no prior information is onsidered in the derivation of this bound. The expetation of the onditional CRB ECRB: is obtained by alulating the average of the onditional CRB over the prior distribution of the unknown parameter [21], whih results in the following expression for the onsidered senario: ECRB = E {CRBx} = wxcrbx dx 27 where CRBx denotes the onditional CRB in 26, and wx is the prior PDF of x, whih is zero outside [,D 2 ].Forthe uniform prior PDF, the ECRB is speified as in the following lemma: Lemma 1: Suppose that the prior PDF of x is speified by a uniform distribution over [,D 2 ]. Then, the ECRB in the synhronous ase is given by ECRB = σ/γr p 2 x 2m +8 D 2 Ẽs/ 2 x 2 dx 28 + a with a m E s /Ẽs, whih an be stated as in the following expression when 2m is an integer: ECRB = σ/γr p 2 a m H m,d 2,a D 2 Ẽs/ 2 m +3 + i=0 a i 2m +3 i+1 D2 D1 2m +3 i+1 2m +3 i For the expression in 26, it is assumed that s0 = st s, whih is ommonly the ase [16].

5 5500 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 34, NO. 23, DECEMBER 1, 2016 where H m,d 2,a { tan 1 D 2 / a tan 1 / a / a if m Z lnd a lnd a, if 2m Z + m/ Z + 30 Proof: The generi expression in 28 diretly follows from 26 and 27. To derive the speifi expressions in 29 and 30, onsider the division of x 2m +8 by x 2 + a, whih results in the following relation: x 2m +8 =x 2 + a j a i x 2m +3 i 1 j a j+1 x i=0 2m +3 j 31 where j m +3 is an integer. Then, the integral in 28 beomes x 2m +8 x 2 + a dx = j a i x 2m +3 i dx i=0 1 j a j+1 x 2m +3 j x 2 dx a If m is a positive integer, j = m +3an be employed to obtain the result speified by 29 and the first part of 30. Note that the last integral term in 32 beomes x 2 + a 1 dx in this ase, whih leads to the tan 1 terms in 30. Similarly, if m is not an integer but 2m is a positive integer, then j = m +3 an be used to derive the expression speified by 29 and the seond part of 30. Note that the last integral term in 32 beomes x x 2 + a 1 dx in this ase, whih results in the logarithm terms in 30. The ECRB may not provide a lower bound on the performane of MAP estimators sine the onditional CRBs, whih are the basis for the ECRB as desribed above, do not take the prior information into aount [21]. However, at high SNRs, the ECRB an onverge to the ZZB, whih is expeted sine the prior information beomes negligible ompared to the information gathered from the measurements in high SNR onditions. Overall, the ECRB provides useful benhmarks for omparisons against the ZZB and helps quantify the range related information gathered from prior information, as investigated in Setion VI. In addition, the ECRB expressions provide a low-omplexity approah ompared to the ZZB expressions for alulating the theoretial limits on range estimation in high SNR senarios. V. BAYESIAN CRB BCRB AND WEIGHTED CRB WCRB In order to inorporate the prior information into the lower bound effetively, the Bayesian CRB BCRB an be onsidered [21]. The BCRB is expressed as { } { 2 } 2 1 log prt x log wx ξ E + E x x 33 where ξ = E{ ˆx x 2 } denotes the MSE of an estimator ˆx [21]. In 33, the first expetation operator is with respet to both rt and x while the seond expetation is over parameter x only. From 25 and 26, the first term in 33 an be alulated as follows: { } 2 log prt x E = E {J F x} x = wx m +32 E s + Ẽsx/ 2 σx m +4 /γr p 2 dx. 34 For a given prior PDF, the BCRB an be obtained based on 33 and 34. One of the limitations of the BCRB is due to the existene and absolute integrability requirement for the partial derivative of the joint PDF of the observation and the parameter [21]. Therefore, it may not be appliable in some senarios. For example, when the range parameter is uniformly distributed over [,D 2 ], the BCRB does not exist. The weighted CRB WCRB provides an alternative to the BCRB and handles the existene problem. It is defined as [21] E{qx} 2 ξ { E{q 2 xj F x} + E q 2 x d logw xqx dx 2 } 35 where ξ is the MSE of any estimator, J F x is as in 25, qx is a weighting funtion, and the expetations are with respet to x. As in [21], the following weighting funtion an be employed: ν x D1 qx = 1 x D ν 1 36 D 2 D 2 for x [,D 2 ] and qx =0otherwise, where ν is a parameter used to enhane the bound. Namely, the value of ν that maximizes the bound in 35 is employed to obtain the tightest bound. For the uniform prior PDF, E{qx} in 35 is alulated from 36 as follows: 1 E{qx} = qxdx = βν +1,ν+1 37 D 2 where βa, b 1 0 xa 1 1 x b 1 dx denotes the beta funtion. In addition, the seond term in the denominator of 35 an be expressed for the uniform prior PDF as [22] { } 2 d logwxqx E q 2 νβ2ν +1, 2ν 1 x = dx D Also, the first term in the denominator of 35 an be alulated based on 26 and 36 as E { q 2 xj F x } = m +3 2 E s + Ẽs 2 γ 2 R 2 p/σ 2 D 2 4ν +1 x 2ν D 2 x 2ν dx x2ν +8 x 2ν D 2 x 2ν dx x2m

6 KESKIN et al.: IMPROVED LOWER BOUNDS FOR RANGING IN SYNCHRONOUS VISIBLE LIGHT POSITIONING SYSTEMS 5501 Fig. 1. ZZB versus soure optial power for various values of the Lambertian order, where S =1m 2. Fig. 2. ZZB versus soure optial power for various values of the area of the photo detetor, where m =10. Then, the WCRB in 35 an be evaluated via In order to obtain the tightest bound, the value of ν that yields the maximum lower bound is obtained. Remark 2: The theoretial limits obtained in this study do not onsider the effets of multipath see 1. In the presene of multipath propagation, higher MSEs would be observed in general; hene, the lower bounds for the LOS senario provided in this manusript present lower limits for the multipath senario, as well. The tightness of the bounds depends on the severity of multipath effets. VI. NUMERICAL RESULTS In this setion, numerial examples are provided to investigate the theoretial limits. In the examples, h in 4 is set to 5 m and the prior PDF of the distane, x, is taken to be uniform over the interval [,D 2 ], where =5mand D 2 =10m f. 8 and 9. As in [5], the responsivity of the photo detetor is taken as R p =0.4mA/mW, and the spetral density level of the noise is set to σ 2 = W/Hz. Also, signal st in 1 is modeled as [5] st =A 1 os 2πt/T s 1 + os2πf t 40 for t [0,T s ], where f denotes the enter frequeny, and A orresponds to the average emitted optial power; that is, soure optial power. In the first example, T s =0.1 ms, f =1MHz, and the area S of the photo detetor at the VLC reeiver is set to 1 m 2.InFig.1, the ZZBs in Setion III are plotted versus the soure optial power A in 40 for various values of the Lambertian order m. It is observed that the ranging auray degrades as m inreases for pratial values of the soure optial power. Although the exat relation between the ZZB and m an be dedued from 5, 9, and 24, an intuitive explanation an also be provided as follows: Parameter m determines the diretionality of the LED transmitter, and a large value of m orresponds to a fast power deay as the irradiation angle inreases from zero see 3. Hene, lower SNRs are expeted at higher distanes for larger values of m, whih an lead to higher ZZBs, as observed in Fig. 1. In Fig. 2, the ZZBs in Setion III are presented versus the soure optial power for various values of S, the area of the photo detetor at the VLC reeiver, where T s =0.1 ms, f =1MHz, and m =10are employed. From the figure, it is observed that the ZZB inreases i.e., the estimation auray degrades as S dereases. This observation an be explained based on the ZZB expression in 9 and 24 as follows: From 5, γ is proportional to S, and from [5, eqn. 18], σ is proportional to S. Hene, the γ/σ term in 24 hanges in proportion to S, whih leads to lower ZZBs as S inreases due to the monotone dereasing nature of the Q-funtion. In other words, as the area of the photo detetor inreases, higher SNRs are obtained at the VLC reeiver and lower ZZBs are ahieved. Next, the ZZB in Setion III, the ECRB in Setion IV, and the WCRB in Setion V are investigated in Fig. 3, together with the performane of the MAP estimator, where T s =0.1 ms, f =1MHz, S =1m 2, and m =1. The MAP estimator an be obtained based on the ML estimator in [16, eqn. 18] by onfining the searh spae for the distane parameter x to the interval [,D 2 ] sine the prior distribution of x is uniform over [,D 2 ] with =5m and D 2 =10mFig.3shows that the ECRB onverges to the ZZB at high soure optial powers; i.e., at high SNRs, sine the prior information beomes less important as the SNR inreases. However, for lower optial powers, the ECRB gets signifiantly higher than the ZZB sine the ECRB alulations do not effetively utilize the prior information, whih beomes signifiant in the low SNR regime. 2 On 2 In the ECRB alulations in 27, the prior information is used to alulate the average of the onditional CRBs; however, eah onditional CRB expression is obtained without utilizing the prior information. Hene, the ECRBs do not effetively utilize the prior information.

7 5502 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 34, NO. 23, DECEMBER 1, 2016 Fig. 3. RMSE versus soure optial power for the MAP estimator, the ZZB, the ECRB, and the WCRB, where T s =0.1ms, f =1MHz, S =1m 2,and m =1. the other hand, the WCRB is lose to the ZZB at low SNRs but it beomes looser as the SNR inreases. The main reason for this behavior is that the WCRB and the BCRB may not provide a tight bound at high SNRs when the onditional Fisher information depends on the unknown parameter [21, p. 7], whih is the ase for the onsidered VLP system that is, the onditional Fisher information in 26 depends on the unknown parameter x, the distane between the LED transmitter and the VLC reeiver. In addition, it is observed from Fig. 3 that the ZZB provides a reasonably tight bound for the performane of the MAP estimator in all SNR regions. Furthermore, sine the MAP estimator utilizes the prior information, its performane annot be lower bounded by the ECRB in the low SNR regime, whih does not effetively utilize the prior information. Therefore, the ECRB expression an provide useful lower bounds only in the high SNR regime, where the prior information is not ruial in the estimation proess ompared to the information obtained from the reeived signal. In the final example, the same parameters as in the previous senario are employed exept that a larger value of f is used, namely, f =50MHz. The results presented in Fig. 4 illustrate that the RMSEs are redued i.e., the ranging performane is improved in the medium and high SNR regimes ompared to the previous senario, whih an be explained as follows: In a synhronous VLP system, in addition to the prior information, information from both the time delay parameter and the hannel attenuation fator an be utilized for range estimation. Sine the information gathered from the time delay parameter inreases with f [16], improved estimation performane an be observed at suffiiently high SNRs, where the prior information beomes less signifiant than the information gathered from the time delay parameter and the hannel attenuation fator. However, in the low SNR regime, the prior information beomes the most signifiant soure of information, whih leads to similar performane for the MAP estimators in Figs. 3 and 4. In addition, it Fig. 4. RMSE versus soure optial power for the MAP estimator, the ZZB, the ECRB, and the WCRB, where T s =0.1ms, f =50MHz, S =1m 2, and m =1. is noted from Fig. 4 that the MAP estimator annot get very lose to the theoretial limits at high SNRs, whih is due to the finite sampling rate namely, s employed in the simulations. In partiular, the finite resolution of the searh for the distane parameter an introdue additional errors in the high SNR regime where the theoretial auray limits are quite low see [16, eqn. 18]. Please refer to [16, Se. IV] for a detailed disussion. VII. CONCLUDING REMARKS AND EXTENSIONS In this study, the ZZB has been derived for range estimation in synhronous VLP systems. The proposed ZZB exploits ranging information from the prior information, the time delay parameter, and the hannel attenuation fator. In addition, a losed-form ECRB expression has been obtained, and the BCRB and the WCRB derivations have been presented for synhronous VLP systems. Via the numerial examples, the bounds have been ompared against eah other and against the MAP estimator. The ZZB has been shown to provide a reasonable lower bound for the MAP estimator. Hene, it an provide important guidelines for design of pratial VLP systems. For example, based on the ZZB expression, effets of various system parameters, suh as the Lambertian order, the area of the photo detetor, and the soure optial power, on ranging auray an be analyzed. On the other hand, the ECRB and the WCRB BCRB an provide useful bounds in the high and low SNR regimes, respetively. For the theoretial limits in Setions III V, the generi expressions have been presented first, and then the partiular expressions have been obtained for the speial ase of uniform prior distribution for the distane parameter x. As another speial ase with pratial importane, the senario in whih the VLC reeiver is uniformly distributed on the floor ground an be onsidered. In that ase, a two dimensional uniform distribution an be employed over the area where the VLC reeiver an

8 KESKIN et al.: IMPROVED LOWER BOUNDS FOR RANGING IN SYNCHRONOUS VISIBLE LIGHT POSITIONING SYSTEMS 5503 Based on 42, the deision rule in 11 an be expressed, after some manipulation, as follows: log p ε ɛe R p γ C rs ϑ + δ,ɛ σ 2 ϑ + δ m +3 dɛ log p ε ɛe R p γ C rs ϑ,ɛ σ 2 ϑ m +3 dɛ H 0 R2 pγ 2 E s 2σ ϑ + δ 2m +6 ϑ 2m +6 wϑ + log 43 Fig. 5. The senario in whih the VLC reeiver is loated in the gray irular area aording to a uniform distribution. ommuniate with the LED transmitter. Let A v denote this area. Based on the minimum and the maximum possible distanes, whih are denoted by and D 2, respetively see Setion III and the fat that the LED transmitter and the VLC reeiver are pointing in vertial diretions, area A v an be represented by a irle with a radius of D2 2 D2 1, the enter of whih is loated at the projetion of the LED transmitter to the floor please see Fig. 5 for an illustration. 3 The use of suh a irular area an be justified by the field of view of the VLC reeiver, whih imposes an upper limit on the inidene angle θ see 3 for ommuniation between the LED transmitter and the VLC reeiver [22]. When the position of the VLC reeiver is uniformly distributed over the irular area A v, it an be shown, via some manipulation of random variables, that the distane x between the LED transmitter and the VLC reeiver is haraterized by the following prior PDF: { 2x/D2 2 D1 2, if x D 2 wx = 41 0, otherwise. The ZZB bound an easily be evaluated for the prior PDF in 41 by inserting it into 8, 17, 20, and 22. Similarly, the ECRB expressions an be speified based on 27 and 41, whih leads to similar expressions to those in In fat, the use of the prior PDF in 41 instead of the uniform PDF mainly inreases the degree of x in the numerator of the integral in 28; hene, the derivations stay almost the same. In a similar fashion, the BCRB and the WCRB in Setion V an also be evaluated for 8. Hene, speifi expressions for the bounds an be obtained for the prior PDF in 41, as well. In pratial systems, due to synhronization errors and finite resolution of time delay estimates, the relation in 2 may not hold exatly. In order to derive the ZZB in the presene of suh effets, 2 an be updated as τ = x/ + ε, where ε has a PDF denoted by p ε. Then, from 1, the likelihood funtion an be obtained as f. 12 prt x =e k p ε ɛe 1 T σ 2 T rt R p γ 1 x m +3 st x ɛ dt dɛ In this ase, orresponds to the height of the LED transmitter relative to the VLC reeiver, whih is also denoted by h see 4. where C rs x, ɛ T 2 rtst x/ ɛdt. Sine it is diffiult to speify the PDF of the deision statistis in 43, a losed-form expression for P min in 23 may not be obtained. However, a Monte-Carlo approah an be adopted to evaluate P min based on the deision rule in 43. Then, the ZZB an be alulated numerially via 8. As future work, theoretial limits for synhronous VLP systems an be onsidered for three-dimensional senarios i.e., when the height of the VLC reeiver is unknown. In that ase, the extended ZZB for vetor parameter estimation [27] should be employed. REFERENCES [1] H. Burhardt, N. Serafimovski, D. Tsonev, S. Videv, and H. Haas, VLC: Beyond point-to-point ommuniation, IEEE Commun. Mag., vol. 52, no. 7, pp , Jul [2] P. Pathak, X. Feng, P. Hu, and P. 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9 5504 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 34, NO. 23, DECEMBER 1, 2016 [15] H.-S. Kim, D.-R. Kim, S.-H. Yang, Y.-H. Son, and S.-K. Han, An indoor visible light ommuniation positioning system using a RF arrier alloation tehnique, J. Lightw. Tehnol., vol. 31, no. 1, pp , Jan [16] M. F. Keskin and S. Gezii, Comparative theoretial analysis of distane estimation in visible light positioning systems, J. Lightw. Tehnol., vol. 34, no. 3, pp , Feb [17] M. Bilgi, A. Seviner, M. Yuksel, and N. Pala, Optial wireless loalization, Wireless Netw., vol. 18, no. 2, pp , Feb [18] S.-H. Yang, H.-S. Kim, Y.-H. Son, and S.-K. Han, Three-dimensional visible light indoor loalization using AOA and RSS with multiple optial reeivers, J. Lightw. Tehnol., vol. 32, no. 14, pp , Jul [19] A. Sahin, Y. S. Eroglu, I. Guven, N. Pala, and M. Yuksel, Hybrid 3D loalization for visible light ommuniation systems, J. Lightw. Tehnol., vol. 33, no. 22, pp , Nov [20] D. Dardari, C.-C. Chong, and M. Win, Improved lower bounds on timeof-arrival estimation error in realisti UWB hannels, in IEEE Int. Conf. Ultra-Wideband, Sep. 2006, pp [21] H. L. Van Trees, and K. L. Bell, Eds., Bayesian Bounds for Parameter Estimation and Nonlinear Filtering/Traking. Hoboken, NJ, USA: Wiley, [22] E. Gonendik and S. Gezii, Fundamental limits on RSS based range estimation in visible light positioning systems, IEEE Commun. Lett., vol. 19, no. 12, pp , De [23] Z. Liu, J. Shen, Z. Yao, M. Lu, and Y. Zhao, The Ziv-Zakai bound for time-of-arrival estimation of new generation GNSS signals, in Pro. Int. Teh. Meeting Inst. Navig., Monterey, CA, USA, Jan. 2016, pp [24] D. Dardari and M. Win, Ziv-Zakai bound on time-of-arrival estimation with statistial hannel knowledge at the reeiver, in Pro. IEEE Int. Conf. Ultra-Wideband, Sep. 2009, pp [25] H. V. Poor, An Introdution to Signal Detetion and Estimation. New York, NY, USA: Springer, [26] Y. Qi and H. Kobayashi, Cramér-Rao lower bound for geoloation in non-line-of-sight environment, in Pro. IEEE Int. Conf. Aoust., Speeh, Signal Proess., May 2002, vol. 3, pp. III-2473 III [27] K. L. Bell, Y. Steinberg, Y. Ephraim, and H. L. V. Trees, Extended Ziv- Zakai lower bound for vetor parameter estimation, IEEE Trans. Inf. Theory, vol. 43, no. 2, pp , Mar Musa Furkan Keskin reeived the B.S. and M.S. degrees, in 2010 and 2012, respetively, from the Department of Eletrial and Eletronis Engineering, Bilkent University, Ankara, Turkey, where he is urrently working toward the Ph.D. degree. His main researh interests inlude statistial signal proessing, wireless ommuniations, and visible light positioning. Erdal Gonendik reeived the B.S. degree from the Department of Eletrial and Eletronis Engineering, Middle East Tehnial University, Ankara, Turkey, and the M.S. degree from Bilkent University, Ankara, where he is urrently working toward the Ph.D. degree at the Department of Eletrial and Eletronis Engineering. His main researh interests inlude statistial signal proessing, wireless ommuniations, and visible light positioning. Sinan Gezii reeived the B.S. degree from Bilkent University, Ankara, Turkey in 2001, and the Ph.D. degree in eletrial engineering from Prineton University, Prineton, NJ, USA, in From 2006 to 2007, he worked at Mitsubishi Eletri Researh Laboratories, Cambridge, MA, USA. Sine 2007, he has been with the Department of Eletrial and Eletronis Engineering, Bilkent University, where he is urrently an Assoiate Professor. His researh interests inlude detetion and estimation theory, wireless ommuniations, and loalization systems. Among his publiations in these areas is the book entitled Ultra-Wideband Positioning Systems: Theoretial Limits, Ranging Algorithms, and Protools Cambridge, U.K.: Cambridge Univ. Press, He is an Assoiate Editor for the IEEE TRANSACTIONS ON COMMUNICA- TIONS, IEEE WIRELESS COMMUNICATIONS LETTERS,andJournal of Communiations and Networks.