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Journals >Photonics Research >Volume 3 >Issue 6 >Page 296 > Article

Photonics Research
Vol. 3, Issue 6, 296 (2015)

Impact of multipath effects on theoretical accuracy of TOA-based indoor VLC positioning system

Xuqing Sun¹, Jingyuan Duan^1、*, Yonggang Zou², and Ancun Shi¹

Author Affiliations

¹Optoelectronic System Laboratory, Institute of Semiconductors, Chinese Academy of Sciences, Beijing, China

²National Key Laboratory on High Power Semiconductor Lasers, Changchun University of Science and Technology,Changchun, Jilin 130022, China

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DOI: 10.1364/PRJ.3.000296 Cite this Article Set citation alerts

Xuqing Sun, Jingyuan Duan, Yonggang Zou, Ancun Shi. Impact of multipath effects on theoretical accuracy of TOA-based indoor VLC positioning system[J]. Photonics Research, 2015, 3(6): 296 Copy Citation Text

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Abstract

This paper discusses the time-of-arrival (TOA) based indoor visible light communication (VLC) positioning system in a non-line-of-sight environment. The propagation delay is assumed to be gamma distributed. The generalized Cramer–Rao lower bound for multipath propagation is derived as the theoretical accuracy limitation. The performance of the positioning system is affected by the shape parameter and the scale parameter of gamma distribution. The influences on positioning accuracy of multipath effects are analyzed through discussing the physical meaning of the gamma distribution parameters. It is concluded that the lower bound of positioning accuracy is attained when variance of the non-line-of-sight propagation-induced path lengths is zero. The simulation result proves that the theoretical positioning accuracy is in the order of centimeters with the given scenario.

Keywords

Free-space optical communication Optical communications

r (t) = α \cdot R \cdot x (t - τ) + n (t),

(1)

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α = \frac{m + 1}{2 π} \cdot \frac{h^{m + 1}}{d^{m + 3}} \cdot S,

(2)

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x (t) = A \cdot (1 + \cos (2 π f t)) \cdot (1 + \cos (2 π t / T_{TOA})) 0 \leq t \leq T_{TOA},

(3)

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θ = {(l_{1}, l_{2}, \dots, l_{M})}^{T} .

(4)

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\hat{τ_{z}} = τ_{z} + ξ, for z \in Z .

(5)

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τ_{z} = \frac{1}{c} \cdot (\sqrt{{(x_{z} - x)}^{2} + {(y_{z} - y)}^{2}} + l_{z}), for z \in Z,

(6)

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f (\hat{τ_{z}} | θ) \propto \prod_{z = 1}^{Z} \exp {- \frac{1}{2 σ_{z}^{2}} {(\hat{τ_{z}} - τ_{z})}^{2}} .

(7)

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E [(\hat{θ} - θ) \cdot {(\hat{θ} - θ)}^{T}] \geq J_{1}^{- 1},

(8)

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J_{1} = E [(\frac{\partial}{\partial θ} \ln f (\hat{τ_{z}} | θ)) \cdot {(\frac{\partial}{\partial θ} \ln f (\hat{τ_{z}} | θ))}^{T}] = E [\frac{\partial}{\partial θ} τ_{z} \cdot (\frac{\partial}{\partial τ_{z}} \ln f (\hat{τ_{z}} | θ)) \cdot {(\frac{\partial}{\partial τ_{z}} \ln f (\hat{τ_{z}} | θ))}^{T} \cdot {(\frac{\partial}{\partial θ} τ_{z})}^{T}] = H \cdot J_{τ_{z}} \cdot H^{T} .

(9)

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H = \frac{1}{c} \underset{M \times Z}{\underset{︸}{(\begin{matrix} 1 & \dots & \begin{matrix} \dots & \dots & 0 \end{matrix} \\ ⋮ & ⋱ & \begin{matrix} ⋮ \end{matrix} \\ 0 & \dots & \begin{matrix} 1 & \dots & 0 \end{matrix} \end{matrix})}} .

(10)

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J_{τ_{z}} = 4 π^{2} β^{2} α^{2} R^{2} E \underset{Z \times Z}{\underset{︸}{(\begin{matrix} \frac{1}{σ_{1}^{2}} & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & \dots & \frac{1}{σ_{Z}^{2}} \end{matrix})}} .

(11)

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J_{1} = \frac{4 π^{2} \cdot f^{2} \cdot R^{2} \cdot E \cdot α^{2}}{3 c^{2}} \underset{M \times M}{\underset{︸}{(\begin{matrix} \frac{1}{N_{01}} & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & \dots & \frac{1}{N_{0 M}} \end{matrix})}} = \frac{f^{2} \cdot R^{2} \cdot E \cdot {(m + 1)}^{2} \cdot h^{2 (m + 1)} \cdot S^{2}}{3 \cdot N_{0 i} \cdot c^{2} \cdot d_{i}^{2 (m + 3)}} i \in {1, \dots, M},

(12)

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J_{1}^{- 1} \geq \frac{\sqrt{3 N_{0 i}} \cdot c \cdot d_{i}^{(m + 3)}}{f \cdot R \cdot \sqrt{E} \cdot (m + 1) \cdot h^{(m + 1)} \cdot S},

(13)

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β^{2} \approx \frac{1}{3} f^{2} .

(14)

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G (l | a, b) = \frac{a^{b}}{Γ (b)} \exp (- a \cdot l) \cdot l^{b - 1}, l \in θ,

(15)

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E [(\hat{θ} - θ) \cdot {(\hat{θ} - θ)}^{T}] \geq J_{2}^{- 1},

(16)

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J_{2} = J_{D} + J_{P} .

(17)

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J_{D} = J_{1}, J_{P} = E [(\frac{\partial}{\partial θ} \ln p_{θ} (θ)) \cdot {(\frac{\partial}{\partial θ} \ln p_{θ} (θ))}^{T}] = E {[\frac{\partial}{\partial l} \log G (l | a, b)]}^{2} = a^{2} + \frac{a^{b} \cdot {(b - 1)}^{2}}{Γ (b)} \int_{0}^{\infty} \exp (- a l) \cdot l^{b - 3} d l + \frac{2 a^{b + 1} \cdot (b - 1)}{Γ (b)} \int_{0}^{\infty} \exp (- a l) \cdot l^{b - 2} d l .

(18)

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J_{P} = E {[\frac{\partial}{\partial l} \log G (l | a, b)]}^{2} = a^{2} .

(19)

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J_{P} = E {[\frac{\partial}{\partial l} \log G (l | a, b)]}^{2} \to \infty .

(20)

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J_{P} = E {[\frac{\partial}{\partial l} \log G (l | a, b)]}^{2} = \frac{a^{2}}{b - 2} (4 b - 7) .

(21)

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J_{P} = (\begin{matrix} \frac{a_{1}^{2}}{b_{1} - 2} (4 b_{1} - 7) & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & \dots & \frac{a_{M}^{2}}{b_{M} - 2} (4 b_{M} - 7) \end{matrix}) .

(22)

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J_{2} = J_{1} + J_{P} = \frac{f^{2} \cdot R^{2} \cdot E \cdot {(m + 1)}^{2} \cdot h^{2 (m + 1)} \cdot S^{2}}{3 \cdot N_{0 i} \cdot c^{2} \cdot d_{i}^{2 (m + 3)}} + \frac{a_{i}^{2}}{b_{i} - 2} (4 b_{i} - 7) i \in {1, \dots, M} .

(23)

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J_{2}^{- 1} \geq \frac{1}{\sqrt{\frac{f^{2} \cdot R^{2} \cdot E \cdot {(m + 1)}^{2} \cdot h^{2 (m + 1)} \cdot S^{2}}{3 \cdot N_{0 i} \cdot c^{2} \cdot d_{i}^{2 (m + 3)}} + \frac{a_{i}^{2}}{b_{i} - 2} (4 b_{i} - 7)}} i \in {1, \dots, M} .

(24)

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\frac{1}{\sqrt{\frac{f^{2} \cdot R^{2} \cdot E \cdot {(m + 1)}^{2} \cdot h^{2 (m + 1)} \cdot S^{2}}{3 \cdot N_{0 i} \cdot c^{2} \cdot d_{i}^{2 (m + 3)}} + \frac{a_{i}^{2}}{b_{i} - 2} (4 b_{i} - 7)}} \leq J^{- 1} \leq \frac{\sqrt{3 N_{0 i}} \cdot c \cdot d_{i}^{(m + 3)}}{f \cdot R \cdot \sqrt{E} \cdot (m + 1) \cdot h^{(m + 1)} \cdot S} .

(25)

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N_{0} = 2 \cdot q \cdot R \cdot p \cdot S \cdot Δ λ,

(26)

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Xuqing Sun, Jingyuan Duan, Yonggang Zou, Ancun Shi. Impact of multipath effects on theoretical accuracy of TOA-based indoor VLC positioning system[J]. Photonics Research, 2015, 3(6): 296

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