Accuracy analysis and improvement of visible light positioning based on VLC system using orthogonal frequency division multiple access

Yitong Xu; Zixiong Wang; Peixi Liu; Jian Chen; Shiying Han; Changyuan Yu; Jinlong Yu

doi:10.1364/OE.25.032618

1. Introduction

Light-emitting diode (LED) is an attractive illumination device for its long lifetime, low energy consumption, and low heat generation [1]. Since LED is a kind of current-driving device, when the high data rate modulation signal is added to the direct current (DC) of LED, both communication and illumination can be achieved simultaneously, without causing any flicker [2]. Besides visible light communications (VLC), indoor positioning is another gift brought by LED. The accuracy of visible light positioning (VLP) is on the order of centimeter, which is much better than global positioning system (GPS) and other radio frequency (RF) counterparts [3,4]. Hence, VLP is a great candidate for indoor positioning, tracking and navigation [4–6].

In VLP system, triangulation is a widely used positioning technology, including time of arrival (TOA), time difference of arrival (TDOA), angle of arrival (AOA), and received signal strength (RSS) [7,8]. In [9], Wang et al analyzed the Cramér-Rao bound (CRB) of positioning accuracy in the TOA technique. However, perfect synchronization between transmitter and receiver is a prerequisite. In the TDOA technique, the LEDs transmit information simultaneously [7]. However, a local oscillator is required to recover the signal, which increases the complexity of VLP. In the AOA technique, both the image sensor and acceleration sensor are used to distinguish the received signals from light sources with different colors [10]. Therefore, it is not suitable for illumination. The positioning accuracy based on RSS technique has also been investigated in the literature. In [11], Wu et al proposed a scheme to improve the accuracy of VLP by excluding the LED with low RSS. However, the theoretical analysis of positioning accuracy is absent. Lin et al demonstrated the feasibility of VLP using orthogonal frequency division multiple access (OFDMA) [12,13]. Nevertheless, all the subcarriers are exclusively assigned between VLP and VLC in these works. Specifically, only three tones are selected for localization, and the rest subcarriers are used for communication. Therefore, the VLC throughput is degraded by using this scheme, when introducing VLP in the same system.

In this paper, we propose a system that realizes VLP and VLC simultaneously in the same band using OFDMA. In such a system, each LED occupies particular subcarrier blocks of the whole spectrum. The receiver position in terms of transmission distances is estimated by measuring the channel gain of each subcarrier block. The advantages of this VLP system are two-fold. Firstly, fewer orthogonal frequency division multiplexing (OFDM) frames are required to obtain the stable estimated position, since all the subcarriers are involved in positioning. Secondly, the positioning is achieved by using arbitrary modulation format, such as M-ary quadrature amplitude modulation (M-QAM) OFDM, which is widely used in VLC system to boost the throughput [14]. We study the positioning error in this VLP system using OFDMA, and find that the critical factors in positioning error are the differences among the errors of square of estimated transmission distances. The positioning error is investigated in both overdetermined and determined VLP systems. In overdetermined VLP system, the positioning error fluctuates when different LEDs are selected as the master LED. The fluctuation of positioning error is cancelled by optimally selecting the master LED. The overdetermined VLP system is changed to determined VLP system by excluding the LED with longest transmission distance. In this case, the positioning error of the determined VLP system is smaller than that of the overdetermined VLP system. The positioning error of determined VLP system is reduced further by using power allocation scheme. The power allocation coefficients are obtained according to the estimated channel gains. This method is fast and effective. The well-matched theoretical and Monte-Carlo (MC) simulation results validate the VLP based on VLC system using OFDMA. Part of this work has been presented in [15].

The rest of the paper is organized as follows. Section 2 describes the VLP based on VLC system using OFDMA. Section 3 analyzes the positioning errors in overdetermined VLP system, determined VLP system and determined VLP system using power allocation scheme. Finally, the concluding remarks are summarized in Section 4.

2. VLP based on VLC system using OFDMA

Fig. 1 shows the diagram of the system that realizes VLP and VLC simultaneously in the same band using OFDMA. In this system, the whole bandwidth is divided to subcarrier blocks. The particular subcarrier blocks are exclusively assigned to the corresponding LEDs. The LEDs transmit different data sequences to the photo-detector (PD) for VLC. The VLC’s receiver measures the power of particular data sequences to estimate the channel gains and consequently the transmission distances. Based on the estimated transmission distances and the LEDs’ locations, the VLP is realized without degrading the VLC’s throughput. In this section, firstly we introduce the VLC system model using M-QAM OFDM. After that, we describe the system that realizes VLP and VLC simultaneously in the same band in detail.

Fig. 1 Simultaneous realization of VLP and VLC using OFDMA.

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2.1. VLC model

Both VLC and VLP use LED as the light source. The LED is a current-driving device. It is driven by the DC I₀ and alternating current (AC) i(t) components for illumination and communication, respectively. We use M-QAM OFDM signal as the AC driving component i(t). The magnitude of i(t) is confined by the modulation index (MI), which is defined as the ratio of maximum absolute value of i(t) to I₀, i.e., $MI = \frac{max | i (t) |}{I_{0}}$ . It is assumed that the LED works in the linear region, i.e., the LED’s driving signal is proportional to their output power. Let K denote the ratio of LED instantaneous power to the driving signal. The LED’s output power is

P_{T} (t) = K [I_{0} + i (t)] = P_{0} + K i (t),

where P₀ = KI₀ is the LED’s DC output power and Ki(t) is the LED’s AC output power.

The modulated LED output light propagates through indoor VLC channel. If we do not consider the light reflections, the VLC channel gain is defined as [1]

G = \frac{(m + 1) A}{2 π D^{2}} {cos}^{m} (ϕ) cos (ψ),

where A is the PD’s area, D is the transmission distance between LED and PD, ϕ is the angle of radiation, and ψ is the angle of incidence. The order of Lambertian radiation pattern is m =− ln 2/ ln (cos Φ_1/2), where Φ_1/2 is the transmitter’s semiangle at half power [16]. Let h denote the vertical distance between the LED and PD. Hence,

cos (ψ) = cos (ϕ) = \frac{h}{D}

. The channel gain in Eq. (2) becomes

G = \frac{(m + 1) A h^{m + 1}}{2 π} \frac{1}{D^{m + 3}} .

The received optical signal is P_R (t)= RGP_T (t)= RGP₀ + RGKi (t), where R is the responsivity of PD. The DC component RGP₀ is removed by the DC block after photo-detection. Therefore, the converted electrical signal is given by

y (t) = s (t) + n (t) = R G δ (t - τ) * K i (t) + n (t) .

s(t)= RGδ (t − τ) * Ki (t) is the recovered information signal, where * represents convolution and τ is the propagation delay. n (t) is the additive white Gaussian noise (AWGN). After passing through the fast Fourier transform (FFT) module, the recovered data is given by

Y (f) = S (f) + N (f) = R G K I (f) exp (- j 2 π f τ) + N (f) .

The communication performance in terms of bit error rate (BER) is evaluated by comparing the detected data and original data. The VLC system using OFDM is shown in Fig. 2. Note that cyclic prefix (CP) module is not shown Fig. 2, since the light reflections are not considered.

Fig. 2 The system that realizes VLC and VLP simultaneously in the same band.

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2.2. VLP and VLC system using OFDMA

In this subsection, we realize VLP in the existing VLC system as described in Section 2.1. The system is shown in Figs. 1 and 2, where VLC and VLP are realized simultaneously in the same band using OFDMA. Suppose that there are N LEDs mounted in the ceiling. As shown in Fig. 3, the whole bandwidth is divided into 2N subcarrier blocks. Each LED occupies two subcarrier blocks. The data allocated to DC term and Nyquist term are zeros [17], since Hermitian symmetry is used to guarantee the real output of OFDM signal. The controller in Fig. 1 assigns the data to different subcarrier blocks, and lets the OFDM signals at the output of inverse fast Fourier transform (IFFT) modules to drive the corresponding LEDs. The LEDs emit the modulated light, which is collected by the receiver. The receiver converts the light into one electrical signal, and analyzes the signal’s power in each subcarrier block to estimate the channel gain. The channel gain is estimated by comparing the power of received signal with the power of transmitted signal, which is given by

\tilde{G} = {\frac{𝔼 [{| Y (f) |}^{2}]}{R^{2} K^{2} 𝔼 [{| I (f) |}^{2}]}}^{\frac{1}{2}} = {\frac{R^{2} G^{2} K^{2} 𝔼 [{| I (f) |}^{2}] + 𝔼 [{| N (f) |}^{2}]}{R^{2} K^{2} 𝔼 [{| I (f) |}^{2}]}}^{\frac{1}{2}},

where the intermediate item 2𝔼 [ℜ{S(f) N^∗ (f)}] in the numerator vanishes. This method to estimate channel gain is suitable for arbitrary modulation format in VLC system.

Fig. 3 Spectrum allocation of VLP system using OFDMA. †: Hermitian symmetry.

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According to Eqs. (3) and (6), the transmission distance is estimated. The estimated transmission distances and the LEDs’ locations are necessary to estimate the receiver’s position. Therefore, the critical issue in VLP using OFDMA is how to obtain the LED’s location from the power of each subcarrier block. We use pulse position modulation (PPM) code to distinguish LEDs. The frame structure is shown in Fig. 4. The first frame uses PPM code, while the rest frames do not. More specifically, each LED is assigned an identification (ID) number in the first frame, where the ID number’s length is not longer than half of the subcarrier block’s length. The ID represents the LED’s location. When the LED’s ID is ‘0’, the PPM code is ‘01’, i.e., the data is allocated to the second subcarrier. Otherwise, the PPM code is ‘10’, i.e., the data is allocated to the first subcarrier. By using PPM, the data is allocated to only one of the two adjacent subcarriers. The LED’s ID is decoded by comparing the power of adjacent subcarriers in each subcarrier block. Therefore, the LED’s location is obtained.

Fig. 4 Frame structure for VLP system using OFDMA. †: Hermitian symmetry. Dark: subcarriers allocated by data.

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In the rest frames, the data is allocated to all the available subcarriers for both VLP and VLC, if the assignment of LEDs for VLP is not changed. The transmission distances in terms of channel gains are obtained by evaluating the average received power of the data in each subcarrier and comparing it with the transmitted power. Consequently, the receiver’s location is estimated by using the LEDs’ locations and estimated transmission distances.

3. Accuracy analysis of VLP using OFDMA

In this section, we analyze the positioning accuracy of VLP system using OFDMA, based on the LEDs’ locations and estimated transmission distances. According to Fig. 2, there are N LEDs mounted in the ceiling. Their locations are denoted as [X_i, Y_i, Z_i]^T, i ∈ 𝕃 ={1, 2,..., N}. Let [X̃, Ỹ, Z̃]^T denote the receiver’s estimated location. Using the geometry property, we have

{‖ [\tilde{X}, \tilde{Y}, \tilde{Z}] - [X_{i}, Y_{i}, Z_{i}] ‖}_{2}^{2} = {\tilde{D}}_{i}^{2}, i \in 𝕃,

where D̃_i are the estimated transmission distances. Suppose that the height of receiver is known. The two-dimensional positioning is obtained by letting one equation subtract other equations in Eq. (7), which has the following form

A \tilde{p} = \tilde{b} .

The items in Eq. (8) are

A = 2 [\begin{matrix} X_{S_{1}} - X_{M} & Y_{S_{1}} - Y_{M} \\ X_{S_{2}} - X_{M} & Y_{S_{2}} - Y_{M} \\ ⋮ & ⋮ \\ X_{S_{N - 1}} - X_{M} & Y_{S_{N - 1}} - Y_{M} \end{matrix}],

\tilde{p} = [\begin{matrix} \tilde{X} \\ \tilde{Y} \end{matrix}] = [\begin{matrix} X \\ Y \end{matrix}] + [\begin{matrix} Δ X \\ Δ Y \end{matrix}],

and

\tilde{b} = [\begin{matrix} X_{S_{1}}^{2} - Y_{M}^{2} + Y_{S_{1}}^{2} - Y_{M}^{2} + {\tilde{D}}_{M}^{2} - {\tilde{D}}_{S_{1}}^{2} \\ X_{S_{2}}^{2} - X_{M}^{2} + Y_{S_{2}}^{2} - Y_{M}^{2} + {\tilde{D}}_{M}^{2} - {\tilde{D}}_{S_{2}}^{2} \\ ⋮ \\ X_{S_{N - 1}}^{2} - X_{M}^{2} + Y_{S_{N - 1}}^{2} - Y_{M}^{2} + {\tilde{D}}_{M}^{2} - {\tilde{D}}_{S_{N - 1}}^{2} \end{matrix}] .

In Eq. (9), we name the LED as the master LED, whose coordinate is [X_M,Y_M]^T. The master LED corresponds to the equation that subtracts other equations in Eq. (7). The remainder LEDs are named as the slave LEDs, whose coordinates are [X_{S_j},Y_{S_j}]^T, j = 1, 2,..., N − 1. Thus, 𝕃, M ≠ S_j. Let p =[X,Y]^T and Δp =[ΔX, ΔY]^T denote the receiver’s actual position and the positioning error, respectively. Eq. (8) are recast as

A (p + Δ p) = b + Δ b,

where

b = [\begin{matrix} X_{S_{1}}^{2} - Y_{M}^{2} + Y_{S_{1}}^{2} - Y_{M}^{2} + D_{M}^{2} - D_{S_{1}}^{2} \\ X_{S_{2}}^{2} - X_{M}^{2} + Y_{S_{2}}^{2} - Y_{M}^{2} + D_{M}^{2} - D_{S_{2}}^{2} \\ ⋮ \\ X_{S_{N - 1}}^{2} - X_{M}^{2} + Y_{S_{N - 1}}^{2} - Y_{M}^{2} + D_{M}^{2} - D_{S_{N - 1}}^{2} \end{matrix}], Δ b = [\begin{matrix} Δ D_{M}^{2} - Δ D_{S_{1}}^{2} \\ Δ D_{M}^{2} - Δ D_{S_{2}}^{2} \\ ⋮ \\ Δ D_{M}^{2} - Δ D_{S_{N - 1}}^{2} \end{matrix}] .

By using least squares estimation (LSE) method [18] to solve Eq. (12), we can obtain the positioning error vector Δp as

Δ p = [\begin{matrix} Δ X \\ Δ Y \end{matrix}] = {(A^{T} A)}^{- 1} A^{T} Δ b .

The positioning error ‖Δp‖₂ is the 2-norm of the positioning error vector Δp. Eq. (13) indicates that the differences between

Δ D_{M}^{2}

and

Δ D_{S j}^{2}

are critical to the positioning error. Choosing different LEDs as the master LED leads to different positioning errors.

According to Eqs. (3) and (6), the square of estimated transmission distance for the ith LED, ∀i ∈ 𝕃 is

{\tilde{D}}_{i}^{2} = {\frac{1}{D_{i}^{2 (m + 3)}} + \frac{σ_{N}^{2}}{{[\frac{R K (m + 1) A h^{m + 1}}{2 π}]}^{2} 𝔼 [{| I (f) |}^{2}]}}^{- \frac{1}{m + 3}},

where

σ_{N}^{2} = 𝔼 [{| N (f) |}^{2}]

is the noise power. The error of square of estimated transmission distance

Δ D_{i}^{2}

is defined as,

Δ D_{i}^{2} = {\tilde{D}}_{i}^{2} - D_{i}^{2} = {\frac{1}{D_{i}^{2 (m + 3)}} + \frac{σ_{N}^{2}}{{[\frac{R K (m + 1) A h^{m + 1}}{2 π}]}^{2} 𝔼 [| I (f) |]}}^{- \frac{1}{m + 3}} - D_{i}^{2} .

In the rest of this paper, we will study the positioning error ‖Δp‖₂ in terms of

Δ D_{i}^{2}

by both theoretical analysis and MC simulation. Note that the MC simulation results are averaged, according to Eq. (6).

3.1. Simulation setup

The simulation is carried out in a spacious room (16 m × 16 m × 5 m). The positioning error performance is studied in the space of 8 m × 8m × 5 m, which is the inner frame as shown in Fig. 5. Since the receiver is quite far from the walls, we do not consider the reflection effect on positioning error. We assume that there are four identical LEDs mounted in the ceiling, where the LEDs are distributed evenly. The four LEDs’ locations are [2m, 2m, 5m]^T, [−2m, 2m, 5m]^T, [−2m, −2m, 5m]^T and [2m, −2m, 5m]^T, respectively. The receiver is parallel to all the LEDs. The LED’s semiangle at half power is 60^○, which means that the order of Lambertian radiation pattern m = 1. The MI is no larger than 0.6. The responsivity of PD is R = 0.6 A/W. The LED launching power is 3 W and the total bandwidth B_d is 10 MHz, if not explicitly stated.

Fig. 5 Locations of LEDs and receivers. Red: LEDs, blue: receivers, inner frame: receiver region.

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3.2. Positioning error in overdetermined VLP system

The VLP system is overdetermined when all the four LEDs (N = 4) are used for positioning, as described in Eq. (13). The positioning error is mainly determined by the items in Δb, i.e., the differences between $Δ D_{M}^{2}$ and $Δ D_{S_{j}}^{2}$ , j = 1, 2, 3. In this subsection, we study the performances of $Δ D_{M}^{2}$ and $Δ D_{M}^{2} - Δ D_{S_{j}}^{2}$ , ∀M, S_j ∈ 𝕃, M ≠ S_j. The receiver locates at [−4m, −3m, 1m]^T, as shown in Fig. 5. The transmission distances between the four LEDs and the receiver are 8.78 m, 6.71 m, 4.58 m and 7.28 m, respectively. The performance of $Δ D_{i}^{2}$ , i ∈ 𝕃, under these transmission distances are shown in Fig. 6(a). All $Δ D_{i}^{2}$ the are negative. $Δ D_{i}^{2}$ becomes smaller as the transmission distance increases. The smallest $Δ D_{i}^{2}$ is $Δ D_{1}^{2}$ (−1.357 m²), where LED 1 is with the longest transmission distance. The largest $Δ D_{i}^{2}$ is $Δ D_{3}^{2}$ (−0.002 m²), where LED 3 is with the shortest transmission distance. The theoretical and MC simulations match very well, which validates Eq. (15).

Fig. 6 (a) $Δ D_{i}^{2}$ , i = 1, 2, 3, 4, (b) $Δ D_{M}^{2} - Δ D_{S_{j}}^{2}$ and ‖Δp‖₂, blue: slave LED 1, green: slave LED 2, pink: slave LED 3, black: slave LED 4, red: ‖Δp‖₂.

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The performance of $Δ D_{M}^{2} - Δ D_{S_{j}}^{2}$ and ‖Δp‖₂ are shown in Fig. 6(b). When LED 1 is selected as the master LED (M = 1), $Δ D_{M}^{2} - Δ D_{2}^{2}$ , $Δ D_{M}^{2} - Δ D_{3}^{2}$ and $Δ D_{M}^{2} - Δ D_{4}^{2}$ are most distinct. In this case, the norm of Δb is 2.09 m² and the corresponding positioning error ‖Δp‖₂ is 0.145 m. When LEDs 2, 3, and 4 are selected as the master LED (M = 2, 3, 4), their positioning errors ‖Δp‖₂ are 0.121 m, 0.085 m, and 0.117 m, respectively. Therefore, the fluctuation of positioning error occurs in the overdetermined VLP system, and the positioning error is minimum when LED 3 is the master LED. We should select the LED with the shortest transmission distance as the master LED to avoid the fluctuation of positioning error.

This strategy is verified by investigating the positioning error at the room’s diagonal, where the receiver locates from [−4m, −4m, 1m]^T to [4m, 4m, 1m]^T, as shown in Fig. 5. The transmission distances follow D₃ < D₂ = D₄ < D₁, when the receiver’s horizontal location is less than zero. The positioning error follows ‖Δp₃‖₂ <‖Δp₂‖₂ =‖Δp₄‖₂ <‖Δp₁‖₂, when the corresponding LED is selected as the master LED, as shown in Fig. 7. The trend of positioning error is inverse, when the receiver locates at the other half of the diagonal. We can see from Figs. 6 and 7 that, the longer transmission distance leads to the more distinct $Δ D_{M}^{2} - Δ D_{S_{1}}^{2}$ , $Δ D_{M}^{2} - Δ D_{S_{2}}^{2}$ and $Δ D_{M}^{2} - Δ D_{S_{3}}^{2}$ , and thereafter leads to larger positioning error. Hence, we should select the LED with the shortest transmission distance as the master LED to avoid the positioning error fluctuation and to minimize the positioning error in overdetermined VLP system.

Fig. 7 Positioning error in overdetermined VLP system using OFDMA, when the receiver locates at the diagonal of the room.

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3.3. Positioning error in determined VLP system

In Section 3.2, we analyzed the positioning error performance in overdetermined VLP system using OFDMA, because usually N (>3) LEDs are mounted in the ceiling for the convenience of illumination. However, three out of N LEDs are enough for two-dimensional VLP system. The overdetermined VLP system is consequently changed to determined VLP system. In this subsection, we study the positioning error in the determined VLP system using OFDMA.

As shown in Fig. 5, there are four LEDs mounted in the ceiling. We use only three of them for positioning, i.e., excluding one LED in the VLP system using OFDMA. In this case, the whole band is occupied by three LEDs, i.e., N = 3 in Fig. 4. The expression of positioning error vector Δp in Eq. (13) becomes

Δ p = [\begin{matrix} Δ X \\ Δ Y \end{matrix}] = (A^{' T} A^{'}) A^{' T} Δ b^{'},

where

A^{'} = 2 [\begin{matrix} X_{S_{1}} - X_{M} & Y_{S_{1}} - Y_{M} \\ X_{S_{2}} - X_{M} & Y_{S_{2}} - Y_{M} \end{matrix}]

and

Δ b^{'} = [\begin{matrix} Δ D_{M}^{2} - Δ D_{S_{1}}^{2} \\ Δ D_{M}^{2} - Δ D_{S_{2}}^{2} \end{matrix}]

. Since the positioning system is determined, the selection of master LED does not affect the positioning error.

The positioning error and estimated position are shown in Fig. 8, when the receiver locates at [−4m, −3m, 1m]^T. The positioning error is highly affected by the excluded LED. The positioning errors are 0.029 m, 0.145 m, 0.212 m, and 0.158 m, by excluding LEDs 1 to 4, respectively. The positioning error is 0.085 m in overdetermined VLP system, as described in Section 3.2. The positioning error is reduced, only by excluding LED 1 in the overdetermined VLP system. Since D₁ > D₄ > D₂ > D₃, we should excluding the LED with longest transmission distance to achieve the reduction of positioning error in determined VLP system.

Fig. 8 Estimated position for the receiver at [−4m, −3m, 1m]^T in overdetermined (filled) and determined (hollow) VLP systems using OFDMA.

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The performance of positioning error in the determined VLP system using OFDMA is also studied in Fig. 9, where the receiver locates at the room’s diagonal from [−4m, −4m, 1m]^T to [4m, 4m, 1m]^T. The transmission distances follow D₃ < D₂ = D₄ < D₁, when the receiver’s horizontal location is less than zero. The positioning error follows ‖Δp₁‖₂ <‖Δp₂‖₂ =‖Δp₄‖₂ < ‖Δp₃‖₂, when the corresponding LED is excluded in the determined VLP system. Comparing Fig. 9 with Fig. 7, the positioning error is reduced only when the LED with largest transmission distance (LED 1) is excluded. For example, when the receiver locates at [−4m, −4m, 1m]^T, the positioning errors in the overdetermined and determined VLP system using OFDMA are 0.164 m and 0.049 m, respectively. This determined VLP system brings 0.115 m improvement of positioning accuracy. However, the positioning errors are increased, when the rest LEDs are excluded in the determined VLP system. This trend of positioning error is inverse, when the receiver locates as the other half of the diagonal.

Fig. 9 Positioning error in determined VLP system using OFDMA for the receivers at the diagonal of the room.

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3.4. Positioning error in determined VLP system with power allocation

In this subsection, we propose a power allocation scheme to reduce the positioning error further in determined VLP system using OFDMA. According to Eq. (16), the positioning error is reduced if the items of Δb′ is are decreased. The power allocation scheme allocates different power to the electrical modulation signal for each LED, which is shown in Fig. 10. The receiver calculates the power allocation coefficient $α_{PA, i, k}^{2}$ based on the estimated channel gain G̃_PA,i,k for the modulation signal of ith LED in kth positioning trial. The $α_{PA, i, k}^{2}$ are sent back to the controller to allocate the power for (k + 1)th positioning trial via the uplink, as shown in Fig. 1. The uplink can be realized by either RF or infrared [19, 20]. The $α_{PA, i, k}^{2}$ is identical to the subcarriers in each subcarrier block. The overall power of electrical modulation signals using the power allocation scheme is the same as that without using power allocation scheme. What’s more, the power of all the recovered electrical signals should be identical based on the estimation of channel gain G̃_PA,i,k.

Fig. 10 VLP system using OFDMA and power allocation.

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Take the receiver at [−4m, −3m, 1m]^T for example. As shown in Section 3.3, excluding LED 1 changes the overdetermined VLP system to determined VLP system. Hence, for each subcarrier block we have

\sum_{i = 2}^{4} α_{PA, i, k}^{2} = 3

α_{PA, 2, k}^{2} {\tilde{G}}_{PA, 2, k}^{2} = α_{PA, 3, k}^{2} {\tilde{G}}_{PA, 3, k}^{2} = α_{PA, 4, k}^{2} {\tilde{G}}_{PA, 4, k}^{2} .

By solving Eqs. (17) and (18), we have

α_{PA, i, k}^{2} = \frac{\frac{3}{{\tilde{G}}_{PA, i, k}^{2}}}{\sum_{i = 2}^{4} \frac{1}{{\tilde{G}}_{PA, i, k}^{2}}}, i = 2, 3, 4 .

Using power allocation scheme, the frequency-domain received signal of the ith LED in Eq. (5) becomes

Y_{PA, i, k} (f) = S_{PA, i, k} (f) + N (f) = R G_{i} K α_{PA, i, k - 1} I (f) exp (- j 2 π f τ) + N (f) .

The estimated channel gain for the ith LED in the kth positioning trial is given by

\begin{array}{l} {\tilde{G}}_{PA, i, k} & = {\frac{𝔼 [{| Y_{PA, i, k} (f) |}^{2}]}{R^{2} K^{2} α_{PA, i, k - 1}^{2} 𝔼 [{| I (f) |}^{2}]}}^{\frac{1}{2}} = {\frac{R^{2} G_{i}^{2} K^{2} α_{PA, i, k - 1}^{2} 𝔼 [{| I (f) |}^{2}] + 𝔼 [{| N (f) |}^{2}]}{R^{2} K^{2} α_{PA, i, k - 1}^{2} 𝔼 [{| I (f) |}^{2}]}}^{\frac{1}{2}} \\ = {G_{i}^{2} + \frac{σ_{N}^{2}}{R^{2} K^{2} 𝔼 [{| I (f) |}^{2}]} \frac{{\tilde{G}}_{PA, i, k - 1}^{2}}{3} \sum_{i = 2}^{4} \frac{1}{{\tilde{G}}_{PA, i, k - 1}^{2}}}^{\frac{1}{2}} . \end{array}

The error of square of estimated transmission distance

Δ D_{PA, i, k}^{2}

using power allocation scheme for the ith LED in the kth positioning trial is given by,

\begin{array}{l} Δ D_{PA, i, k}^{2} & = {\tilde{D}}_{PA, i, k}^{2} - D_{i}^{2} \\ = {\frac{1}{D_{i}^{2 (m + 3)}} + \frac{σ_{N}^{2}}{{[\frac{R K (m + 1) A h^{m + 1}}{2 π}]}^{2} 𝔼 [{| I (f) |}^{2}]} \frac{\sum_{i = 2}^{4} {\tilde{D}}_{PA, i, k - 1}^{2 (m + 3)}}{3 {\tilde{D}}_{PA, i, k - 1}^{2 (m + 3)}}}^{- \frac{1}{m + 3}} - D_{i}^{2} . \end{array}

Note that

{\tilde{D}}_{PA, i, 0}^{2}

is the square of estimated distance (

{\tilde{D}}_{i}^{2}

) without using power allocation scheme, i.e.,

{\tilde{D}}_{PA, i, 0}^{2} = {\tilde{D}}_{i}^{2}

. The power allocation stops, when the

Δ D_{PA, i, k}^{2}

is stable.

Fig. 11 illustrates the performance of $Δ D_{PA, i, k}^{2}$ . The values of $Δ D_{PA, i, k}^{2}$ (i = 2, 3, 4) converge after several power allocation trials. Hence, the power allocation is a fast algorithm. The converged values are −0.098 m², −0.043 m² and −0.11 m² for LED 2 to LED 4, respectively. Without using power allocation scheme, $Δ D_{i}^{2}$ are −0.096 m², −0.0022 m² and −0.218 m² for LED 2 to LED 4, respectively, as shown in Fig. 12. Although the values of $Δ D_{PA, i, k}^{2}$ are smaller than the minimum $Δ D_{i}^{2}$ (i = 3), the differences among $Δ D_{PA, i, k}^{2}$ are smaller than those of $Δ D_{i}^{2}$ . The corresponding positioning errors are 0.011 m and 0.029 m when using and without using power allocation scheme, respectively, as shown in Fig. 13. Hence, the power allocation scheme is effective to reduce the positioning error further in determined VLP system using OFDMA.

Fig. 11 $Δ D_{PA, i, k}^{2}$ (k ≥ 1) using power allocation scheme for the receiver at [−4m, −3m, 1m]^T. k = 0: without using power allocation scheme. Blue: theory, green: LED 2, pink: LED 3, black: LED 4.

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Fig. 12 $Δ D_{PA, i, k}^{2}$ (a) without using and (b) using power allocation scheme.

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Fig. 13 Positioning error of determined VLP system using OFDMA in the whole room: (a) without using power allocation scheme, (b) using power allocation scheme.

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Fig. 13 shows the positioning error in the whole room of determined VLP system using and without using power allocation schemes. The power allocation schemes reduces the positioning error, especially when the transmission distances are long. Without using power allocation scheme, the positioning errors are 0.049 m and 0.036 m, when the receiver locates at [−4m, −4m, 1m]^T and [−4m, 0m, 1m]^T, respectively. Using the power allocation scheme, the corresponding positioning errors are reduced to 0.020 m and 0.0072 m for the two locations, respectively. The positioning accuracy are improved by 0.029 m (59%) and 0.029 m (81%), respectively. Therefore, the fast and effective power allocation scheme improves the positioning accuracy further in determined VLP system using OFDMA.

4. Conclusion

In this paper, we proposed a system that realized VLP and VLC simultaneously in the same band using OFDMA. We used the power of received data sequence in each subcarrier block as the metric for the estimation of transmission distance in terms of channel gain. Therefore, VLP did not deteriorate the performance of VLC. In this system, the differences among errors of square of estimated transmission distances are the critical factors in positioning error. We have investigated the following three method to reduce such differences and consequently to reduce the positioning error. In overdertermined VLP system, the positioning error fluctuates due to the redundant LED. To avoid such fluctuation, we selected the LED with the shortest transmission distance as the master LED. Then, we changed the overdetermined VLP system to determined VLP system by excluding one LED for positioning. This method reduced the positioning error, only when we excluded the LED with the longest transmission distance. The positioning error was reduced further in determined VLP system by using power allocation. The power was allocated to modulation signals according to the estimated channel gains on each subcarrier block. This method was fast and effective to improve the positioning performance. The matched theoretical and MC simulation results validated the proposed VLP based on VLC system using OFDMA.

Funding

National Natural Science Foundation of China (NSFC) (61601321, 61601247, 61271239).

References and links

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Accuracy analysis and improvement of visible light positioning based on VLC system using orthogonal frequency division multiple access

Abstract

1. Introduction

2. VLP based on VLC system using OFDMA

2.1. VLC model

2.2. VLP and VLC system using OFDMA

3. Accuracy analysis of VLP using OFDMA

3.1. Simulation setup

3.2. Positioning error in overdetermined VLP system

3.3. Positioning error in determined VLP system

3.4. Positioning error in determined VLP system with power allocation

4. Conclusion

Funding

References and links

Cited By

Figures (13)

Equations (23)

Optics Express