Threefold layered optical wireless network: illumination, communication, and positioning

Anna Maria Vegni; Mauro Biagi

doi:10.1364/OE.482935

1. Introduction

Visible light communication (VLC) is expected as a prominent technology for future Beyond 5G (B5G) wireless networks due to its spectral, energy, and cost efficiencies [1,2]. Also known as a green technology, it is largely adopted in indoor environments, working as a complementary solution to radio frequency (RF) networks, especially at places where RF signals fail to support end-users requirements.

The main benefits of VLC rely on its achievable high data rates, while guaranteeing illumination and security requirements. The remarkable research advances in VLC motivated researchers to employ light-based access points (APs) also for localization services [3], thus resulting in visible light positioning (VLP) technology. This is considered as a key technology for B5G wireless networks [4] and is particularly important as the widely-used RF-based positioning technology, i.e., the global positioning system (GPS), fails at indoor environments. However, the system requirements for data transmission and positioning applications are quite different. This is simply as the former applications aim at maximizing the spectral efficiency, while the latter ones target on minimizing the localization resolution, which measures the ability to separate two users in the spatial domain. In this context, from radionavigation theory, at least three sources at known positions (i.e., anchor nodes) are required to estimate the user position with high accuracy.

Leveraging on the previous considerations, in practical realizations, the constraint of having at least three APs in each position of a given environment is not always feasible. As for instance, in [5] it was demonstrated that in an indoor area of $9 \times 10$ m$^2$, only two APs can be sufficient to satisfy certain illumination and data rate requirements in a multi-user scenario, but they do not meet localization requirements. It follows that adding more access points is needed for localization purpose, but at the same time it is necessary to find a proper number of APs for allowing the achievement of desired localization accuracy and also avoid increasing installation costs, and inter-LED interference [5].

In this paper, we focus on the three pillars of VLC networks, aiming to provide (i) illumination, (ii) positioning, and (iii) communication in a unified solution. Furthermore, it must be specified that for what concerns illumination we focus on a twofold optimization since we pay attention both to minimum illumination to grant and also its distribution by balancing maximum and minimum levels. For each feature, we consider different Quality of Service (QoS) requirements and aim to optimize the LED deployment that can meet such requirements. Specifically, we distinguish three different optimization problems, each of them considers different kinds of white LEDs based on the tasks they are designed to provide, i.e., communication or localization. Problem $1$ seeks for the minimum number of white LEDs that can provide at the same time both illumination, communication, and localization constraints, in each position of a given environment. Since in this case the LEDs can act both for communication and localization, they are defined as “peer” LEDs. On the other side, Problem $2$ seeks for the minimum number of white LEDs aiming to provide only localization and illumination constraints, neglecting data communication purpose. This case suits in all those environments where localization service is required and communication has been already setup (e.g., industry, hospitals, etc.). In this second approach, LEDs are defined as localization-only devices. Finally, Problem $3$ considers two different types of white LEDs, i.e., those designed for communication and those for localization purpose. In this case, the optimization problem aims to distinguish the minimum number of communication-LEDs and localization-LEDs, meeting communication and localization accuracy requirements, respectively. This problem reflects a VLC system enriched with the ability to support positioning service, by means of some (extra) non-data, i.e., localization-only APs, to be installed at different positions in order to improve the localization accuracy within the whole environment. By the term “localization-only”, we mean that such APs do not transmit data but only send some predefined signals with certain power to facilitate the positioning process. Therefore, these APs do not cause interference to the existing data-APs.

In this paper, we propose the network architecture design and, more, three LEDs deployment optimization approaches. Solutions are shown for different environments (i.e., indoor room sizes) and network requirements. Specifically, for each LED deployment approach, we focus on a minimization process that aims to guarantee specific QoS requirements, such as the illumination and localization accuracy below a given threshold, and the achievable data rate higher than a given threshold, while providing a minimum number of APs. Extended simulation results highlight a minimum of $4$ ($8$) white LEDs are needed to provide at least $30$ Mbps, with at least $250$ lux, in each position in a $4\times 5 \times 3$ m$^3$ ($9\times 10 \times 3$ m$^3$) room, respectively. However, adding also localization requirements, the minimum number of LEDs increases to at least $7$ and $41$ in a $4\times 5 \times 3$ m$^3$ and $9\times 10 \times 3$ m$^3$ room, respectively.

This paper is organized as follows. Section 2. provides a state-of-the-art of the most relevant studies about indoor VLP. Section 3. deals with the proposed indoor VLC/VLP system and describes the main elements of network architecture. Different design constraints related to the threefold pillar of illumination, communication and positioning are introduced in Section 4, that are fully integrated in our proposed approach. Then, the APs deployment optimization problem is introduced in Section 5, where we distinguish three different approaches in case of (i) “peer”, (ii) localization-only, and (iii) joint communication and localization LEDs. Extended simulation results are described in Section 6, in case of variable geometric features and room setup. Results have been achieved also in case of blockage and variable tilting angle, in order to consider a most realistic scenario. Finally, conclusions are drawn at the end of this paper.

2. Related works

Traditionally, VLC has been investigated from the point of view of data communication. At the same time, the topic of VLP has been largely investigated and still represents an open issue, especially in the context of B5G networks that are expected to require high accuracy localization. The use of Optical Wireless Communications (OWC), and in particular Line of Sight (LoS) VLC connectivity links, is envisioned to provide under-centimeter accuracies, mainly due to its high bandwidth and link directivity.

VLC and VLP look as separated issues that should be investigated each in a dedicated manner. Commonly, research on VLP is split by VLC component, while both VLP and VLC should be fully integrated according to the vision of next generation services. For instance, in [6], Tran and Ha propose an innovative and easily implemented solution for indoor positioning, based on dual-function machine learning (ML) algorithms. As a benefit, increased positioning accuracy is observed under the negative effect of multipath reflections. An interesting approach is presented by Chaudhary et al. [7], where the received signal strengthand the information of tilt angles of transmitters and receivers are exploited, as well as multipath reflections. The tilting effect has been shown to be beneficial in VLP systems considering both LoS and Non-Line of Sight (NLoS) transmission modes.

Contributions dealing with both VLP and VLC as a joint system are limited. For instance, Li et al. [8] investigate the optimal configuration of LEDs in indoor environments from the sole point of view of communications, expressed through the constraints quality of experience (QoE), and also illumination. Singh et al. [9] present an optimal LED power allocation framework that maximizes the average data rate across the room, while satisfying the bit error rate and illumination constraint. This solution is mainly focused on LED energy-saving, and does not consider a fully integrated VLC/VLP indoor system, but still splits the communication pillar from the localization one. Also, Dang et al. [10] propose a new metric to evaluate the positioning accuracy based on information theory. The optimum LED deployment for communication and positioning has been investigated, and it was found that LEDs concentrating in the middle of the room are efficient for communication-optimal scenario, while for positioning-optimal scenario, LEDs should be placed at the corners of the room. However, the proposed approach still separates the localization and communication goals, since no optimal solution for joint VLC and VLP has been identified. Yazar and Gezic [11] address the problem of localization performance maximization for pulse design, with consideration of practical limitations related to power consumption, illumination levels, and effective bandwidths. In [12], Xu et al. propose a system that realizes simultaneously both positioning and communications in the same band by using orthogonal frequency division multiple access (OFDMA). Position estimation is carried out through received signal strength measured in each subcarrier block, and therefore, VLP does not deteriorate the performance of VLC. However, in this work, the authors do not consider communication requirements, but only focus on localization accuracy.

In all previous works, VLP has been addressed from the point of view of achievable accuracy, ignoring the network design and the LED deployment related issues, specially in a joint VLC/VLP approach. Indeed, VLP should not neglect neither communication nor illumination constraints, as they are the essential pillars of VLC systems and VLP should be built upon them. Leveraging on such research gap, in this paper we aim to build a fully integrated VLC and VLP system, aiming to satisfy both communication, localization, and illumination requirements. This paper represents an evolution of a previous work [5], dealing with the optimal deployment of an indoor VLC, according to different data rate requirements. Moreover, in [5] we aimed at considering the outage in a probabilistic sense, while here localization is also taken into account as well as illumination distribution and the constraints are punctually considered. Moreover, differently from [5], we present a framework to communicate, that is, the whole system architecture of the threefold layered optical wireless network. Hence, that work presented the optimal number of white LEDs achieving different data rates in a given environment, while neglecting localization and illumination constraints. Now, we investigate an evolved approach dealing with the three main pillars of VLC technology, resulting in an indoor illumination-plus-communication-plus-localization system.

This paper presents an optimization technique related to three different criteria, such as the illumination, the achievable data rate, and the localization estimation error, applied to indoor VLC systems. The optimal deployment solutions will be presented as expressed in terms of minimum number of LEDs achieving at the same time different localization, illumination, and communication requirements. The objective is then unique since this contribution presents a threefold optimization approach. From an overview of the state-of-the-art, the network optimization of VLC systems has been focused on communication-only or positioning-only aspects. A joint optimization of communication and positioning, in addition to illumination requirements to fulfill, has not been investigated yet in visible light systems. In RF systems as well, the issue of cell coverage optimization is related to data rate only, and localization is not investigated as a joint metric to optimize. For instance, in [13] the authors investigate the cell coverage optimization problem for a massive multiple-input multiple-output (MIMO) uplink, and their goal is the maximization of cell throughput, which is shown to be mainly determined by the user distribution within several key geometrical regions. Similarly, in [14] Wei and Hwang present a multiple-objective optimization model in a multi-user 5G networks where users are grouped based on their propagation conditions. However, also in this case the optimal cell size is estimated to maximize the total network capacity and minimize the deployment cost, and do not consider localization service.

3. Proposed system and network architecture

Let us consider an indoor VLC network where $N$ APs are assumed able to provide both illumination, localization, and communication services. In such a network, the APs might be insufficient to support all the services with the required quality levels. For example, the value of $N$ may be sufficient to provide illumination and/or communication requests and insufficient to grant the desired level of localization accuracy. This implicitly suggests that we aim at determining a sufficient number of APs so as to guarantee all the systems requests/constraints.

3.1 System signaling

The APs are simple LED devices each one with an identification code that is represented by a direct-sequence optical code division multiple access (DS-OCDMA), [15]. Since the communication is bi-directional we assume the nodes to be equipped with a photodiode (PD) for receiving signals from APs and also IR-LEDs to transmit towards the ceiling of the room. For this reason we consider that the uplink requires the ceiling to have IR-PDs according to the framework in [16].

Herein, we consider a two-phase localization process, such that each phase corresponds to a different localization level, as follows:

3.1.1 Phase I

In this phase, the user detects the received signals. Due to orthogonality of the codes, it is able to measure the signal strength of each AP since under the realistic approach of orthogonality (it is reasonable in downlink) by considering all the possible sequences of a given spreading factor, the user collects the power levels from each AP.

As anticipated, due to the orthogonality of optical codes and to clarify the decoding process at the user side, in this paper we assume that the $N$ APs do not cause interference to each other. This does not mean that we neglect signal overlapping, however we consider the opportunity, by the receiver, to resolve each code. Specifically, the APs transmit orthogonal sequences by employing Walsh-Hadamard codes so as to grant orthogonality. In this context, we should regard the special requirements for VLC transmission, particularly the condition that the transmitted symbols should be real and positive. Such requirements should be carefully considered before applying normal CDMA scheme, i.e., that used for the RF systems. Herein, we consider the simple solution of employing the VLC-adapted version of the DS-CDMA, in which a fixed DC offset is added to convert the bipolar signal into a unipolar signal before modulating the light source, [17,18].

Hence, Phase I identifies the APs covering a certain location, while Phase II calculates the distances to those APs. Analytically speaking, let us describe the channel LoS propagation as a single coefficient. In fact, dealing with VLC channel model, this is typically composed of both LoS and diffuse (multi-path) components. However, it was observed in [19] that, diffuse components can be ignored in typical indoor scenarios as the majority of the collected energy at the photodetector (more than $95\%$) comes from the LoS component. Therefore, in this paper we mainly assume that the VLC link has a dominant LoS component [20]. More in detail, we assume that the LED-based AP follows the Lambertian radiation pattern, and that the VLC AP is directed downwards and the user PD is directed upwards. It follows that, the channel gain at a given distance $d_{j,k}$ [m] between the $j$-th LED and the $k$-th PD is expressed as [21]:

(1)$${h}_{j,k} = \frac{(m+1)A_k}{2\pi d^2_{j,k}}T_s(\psi_k) G(\psi_k) \cos^n(\phi_j)\cos(\psi_k),$$

where $m$ is the Lambertian emission order, given as $-\log _2 [\cos (\phi _{\frac {1}{2}}) ]$, with $\phi _{\frac {1}{2}}$ as the LED’s semi-angle at half-power. In Eq. (1), $T_s(\psi _k)$ and $G(\psi _k)$ are the gains of the optical filter and of the optical non-imaging concentrator of the $k$-th PD, respectively. We assume $T_s(\psi _k)=1$ and $G(\psi _C)=n^2/\sin ^2(\psi _C)$, with $n$ as the refractive index, $\psi _k$ and $\psi _C$ as the angle of incidence to the $k$-th PD and the Field of View (FOV) of the $k$-th PD, respectively. Furthermore, $A_k$ [cm$^2$] is the physical area of the $k$-th PD and $\phi _j$ is the angle of irradiance of the $j$-th LED.

Formally, let us first express the received signal of the $k$-th PD at the $l$-th chip as follows:

(2)$$r_{l,k} = \sum_{j=1}^{N}(c_{l,j}+P_{bias}) h_{j,k} + w_{l,k} \quad \text{for} \quad l = 1,2,\dots, {SF},$$

where $c_{l,j}$ is the $l$-th chip of the $j$-th AP sequence, while $w_{l,k}$ is the additive white Gaussian noise sampled at chip level, and $P_{bias}$ is the light bias. It is important to note at this stage that using a light bias $P_{bias}$ allows the transmitter to obtain non-negative chip values. Moreover, it can be considered as a dummy spreading sequence where all the chips are set to $1$ so excluding the sequence of all ones from those available by the APs. It it important to highlight here that there is a relationship between the spreading factor ($SF$) to be used and the number of APs. Since one sequence is used for the light bias, the SF should be properly set according to $SF=2^{\lceil \log _2 (N+1) \rceil }$. Moving to a sequence-base description, we have for the $k$-th PD the following vector/matrix description:

(3)$$\mathbf{r}_{k} = \mathbf{h}_k^T\mathbf{C} + \mathbf{w}_k = [h_{1,k} \, h_{2,k} \, \dots \, h_{j,k} \dots h_{N,k}] \left[ \begin{array}{c} c_{1,1} \, c_{2,1}\, \dots \, c_{SF, 1} \\ c_{1,2} \, c_{2,2} \, \dots \, c_{SF,2} \\ \dots \; \\ c_{1,j} \, c_{2,j}\, \dots \, c_{SF, j} \\ \dots \; \\ c_{1,N} \, c_{2,N}\, \dots \, c_{SF,N} \\ \end{array} \right] + [w_{1,k} \, w_{2,k} \, \dots \, w_{l,k} \, \dots w_{SF,k}],$$

where $\mathbf {h}_k$ is the $N$-elements column vector reporting the DC gains, while the $(N \times SF)$ matrix $\mathbf {C}$ reports the spreading sequences of the APs in the $N$ rows. Moreover the ($1 \times SF$) vector $\mathbf {w}_k$ gathers the noise samples on a sequence basis, while $\mathbf {r}_{k}$ is the sum of all the sequences weighted by the channel coefficients. From Eq. (3) it is possible to infer that all the chips related to the same transmitting node (e.g., $c_{1,1} \, c_{2,1} \, \dots c_{SF,1}$) are multiplied by the same channel gain (e.g., $h_{1,k}$). However each received chip is the linear combination of all chips in the same position within the spreading sequences, weighted by the channel coefficients.

In order to proceed to detect the $h_{j,k}$ value, the point is to remove the light bias as for Eq. (2). The light bias is a constant signal. This can be considered as a dummy user whose spreading sequence is characterized by constant chips (that is a unit vector 1 with $SF$ elements). Since it is assigned to a –dummy– user, the spreading sequence is orthogonal with respect to the other $(SF-1)$ sequences. Hence, by operating the despreading operation related to the $j$-th sequence, we have that all the spreading sequences are orthogonal to the $j$-th one, including the dummy sequence characterized by all ones. It follows that the estimated $\tilde {h}_{j,k}$ can be inferred as

(4)$$\tilde{h}_{j,k} = \frac{1}{SF}\mathbf{r}_k \mathbf{C}[k,:]^T= \frac{1}{SF}\left(\textbf{h}_k^T \mathbf{C}\mathbf{C}[k,:]^T+[w_{1,k} w_{2,k} \dots w_{l,k} \dots w_{SF,k}]\mathbf{C}[k,:]^T\right) ,$$

where $\mathbf {C}[k,:]$ is the $k$-th row of the matrix, that is, the signature (spreading sequence) related to the $k$-th AP. It is important to highlight that the product of $\mathbf {C}$ by $\mathbf {C}[k,:]^T$ leads to obtain a column vector since the dimensions of $\mathbf {C}$ are $(N\times SF)$, while those of $\mathbf {C}[k,:]^T$ are $(SF \times 1)$. Moreover, since the product of a row of $\mathbf {C}$ by the vector $\mathbf {C}[k,:]^T$ is a scalar product, we have that due to the orthogonality of the sequences only one element of the vector given by $\mathbf {C}\mathbf {C}[k,:]^T$ will be different from $0$ and it equates $SF$, exactly in the $k$-th position of the vector. In fact, due to the orthogonality property of the spreading sequences we have that $\mathbf {C}[l,:]\mathbf {C}[k,:]^T=0$ when $k\neq l$. As a consequence, the product of the vector collecting the channel gains $\textbf {h}_k$ by the above mentioned quasi-null vector given by ($\mathbf {C}\mathbf {C}[k,:]^T$) is characterized only by the term $h_{j,k}$ with the exception of an additional filtered noise $[w_{1,k} w_{2,k} \dots w_{l,k} \dots w_{SF,k}]\mathbf {C}[k,:]^T$.

3.1.2 Phase II

Having the power levels detected, each node can send the measures by using an uplink connection (that is mandatory to have, also for the communication process and it can be infra-red [16]) hence the network is responsible for process the information and apply a simple triangulation algorithm by selecting the three highest signal levels so as to provide accurate localization. It should be remarked here, that the exact user location might not be found without phase I and by depending only on this phase. This is because most of the algorithms applied in this phase mainly depend on calculating the distances between the user and the APs forming the overlapping area. Thus, these algorithms identify the user location relative to the positions of the associated APs, and not to a reference (global) point within the entire environment, unless the positions of these APs are known to that reference point. This latter information is provided by Phase I.

The system design proposed in this paper is, principally, independent of the algorithm used in this phase. However, the applied algorithm will have an indirect impact on the system design through the estimation accuracy, which is a key performance criteria to be regarded by system designers.

4. Design constraints

As previously anticipated, we focus on optimizing the proposed network architecture, according to three constraints. The first one is the localization that can be useful especially for identifying the LEDs that should provide the communication service so as to allow the transmission by multiple LEDs. In this regard the communication rate with a target error probability constitutes the second constraint. Finally, the third one twofold since it is related to the minimum illumination level and the ratio between minimum and maximum.

4.1 Localization error

Let us consider the task of estimating the position of the $k$-th user, which is covered by a number of APs that are assumed to be greater or equal to $3$, otherwise it is not possible to grant the absence of ambiguity. Although this is a very well known result from the literature, it must be underlined that using less than $3$ APs leads to have huge estimation error and/or location ambiguity.

More in detail, we consider the $k$-th user is in a 3D position, where the $z$-component is assumed as constant. So, the localization algorithm provides the estimation error considering all the three components along $x$, $y$, and $z$, then representing a 3D localization. We assume the localization estimation relies on the received signal strength indicator (RSSI) metric [22], and we refer to the Cramer-Rao bound (CRB), that is the estimation error variance of an efficient estimator [23]. According to [24], we exploit the best three signals the $k$-th user receives so as to obtain the following expression for the CRB, mathematically indicated as $\Theta$ [m$^2$], i.e.,

(5)$$\Theta =\sum_{i\in \mathcal{I}_3}\frac{c^2d_{j,k}^2}{2h_{j,k}^2\frac{P_L}{N_0 B_L }\left(4c^2+\frac{d_{j,k}^2(2\pi B_L)^3}{3}+2cd_{j,k}\right)},$$

where $c$ [m/s] is the speed of light in free space, $P_L$ [W] is the electrical transmission power used for localization purposes that already accounts for optical-to-electrical conversion, thus including the PD responsivity and the load resistance. Furthermore, $N_0$ [W/Hz] is the noise spectral density, and $B_L$ [Hz] is the bandwidth of the signal to be utilized for localization purposes. It is important to highlight that the set $\mathcal {I}_3$ collects the labels of the three best signal components at the $k$-th PD. As it is possible to notice, when the ratio between the received power (or transmitted since they are direct proportional) and the noise increases, i.e., $P_L/N_0$, we have that the CRB quickly decreases. From this point of view, we expect to have the mean square error $\sigma _\epsilon ^2$ that perfectly fits the value of CRB. This can be performed with RSS methods, [22].

4.2 Communication/access power constraint

Although each LED utilizes its own DS-OCDMA sequence, the access within a cell is assumed occurring according to a TDMA fashion. In order to grant a minimum rate we focus on some geometrical, transmission as well as, access features. In this regard, the achievable rate per single user for the VLC link is expressed according to the following well-known formula [25]:

(6)$$R = {B} \log_2\bigg(1+ \frac{\zeta}{\Gamma}\bigg), \quad \text{bps}$$

where $B$ [Hz] is the communication bandwidth, that is, the bandwidth related to the signal utilized for the communication in downlink between LED and PD, $\zeta$ [dB] describes the signal-to-noise-ratio (SNR) and $\Gamma$ is a factor that considers the SNR increase allowing to achieve required performance in terms of symbol error rate (SER). As an example we can consider a unity SNR value that allows to achieve a spectral efficiency of $1$ bit/s/Hz. However at $0$ dB of SNR the performance in terms of error rate are really poor. The $\Gamma$ term allows to consider the error rate, and it is defined as

(7)$$\Gamma={1\over 3}\left[Q^{{-}1}\left({SER\over 4}\right)\right]^{2},$$

where $Q$ function is expressed as

(8)$$Q(x)=\int_{x}^{\infty}{{e^{{-}u^{3}/2}}\over{\sqrt{2\pi}}}du.$$

This allows to grant not only a rate but also an error rate.

In case of a multi-user scenario, we introduce, as an innovative contribution, a correction factor expressed as the product of the lighting cell coverage and the user density per area, in order to measure the average rate per user. In fact, usually when TDMA is considered as enabling access procedure, the correct way of scaling the available bandwidth is to divide the single user rate expression in Eq. (6) by the number of users, since the channel is shared. This is correct when we consider only one LED in the network. When we consider multiple LEDs in the network it is not true that we need to scale the rate by all the users but simply by a fraction of them, corresponding to the (average) number of users covered by a single LED. In this regard, we consider the average number of users covered one –out of $N$– LED by referring the the user density in the room. Following the above considerations, Eq. (6) becomes

(9)$$R = \frac{B}{\lceil{\mathcal{A}\Delta}\rceil} \log_2\bigg(1+ \frac{\zeta}{\Gamma}\bigg), \quad \text{bps}$$

where the term $\mathcal {A}$ [m$^2$] is the area of the footprint of the lighting cell at the considered height, while the term $\Delta$ [user/m$^2$] takes into account for the user density in the room expressed as the number of users per squared meter. This approach is highly realistic since it takes into account how many users can populate the network, and in a spatial sense, the room. This allows to avoid to allocate resources (i.e., time slots) for all the user by each LED. Hence, we focus on this approach that considers how many users are covered by a single LED. More regarding the parameter $\Delta$, this number usually is smaller than $1$ since considering more than $1$ user per squared meter is a bit unrealistic. As a simple example, in a cell with $\mathcal {A}=2$ [m$^2$] and $\Delta =1$ [m$^{-2}$], it is expected that the minimum rate per user will be $B/2 \log _2(1+ \zeta /\Gamma )$. Besides, since it is largely possible that the footprint overlaps, in those positions we assume the user to be able to combine signals since the LEDs, in that case, transmit the same information, and hence the rate becomes

(10)$$R = \frac{B}{\lceil{ \mathcal{A}\Delta}\rceil} \log_2\bigg(1+ \sum_{i \in \mathcal{I}_3}\frac{\zeta_i}{\Gamma}\bigg), \quad \text{bps}$$

where we consider the sum of the three best RSS signals belonging to $\mathcal {I}_3$, measured at a given PD position. It is important to underline that the above modeling for what concerns rate is able to jointly account for both number of users and BER. In fact the SNR gap is able to grant the target BER while the denominator, that is, $\lceil {\mathcal {A}\Delta }\rceil$ takes into account the number of users since we multiply the spatial user density $\Delta$ by the area $\mathcal {A}$, and by considering the ceiling function we resort to the worst case for the rate since we take the upper integer number.

Till now, it appears that the number of LEDs in the network that influences the chip-rate, is unlinked to the communication bandwidth. However, the communication bandwidth is essentially $B=B_t/SF$, where $B_t$ [Hz] is the bandwidth of the signal in air (after spreading). At a first glance this suggests that if we consider a high number of LEDs for a fixed B we have a large increase of $B_t$, or converse, for a fixed $B_t$, a large number of LED reduces $B$. However, since not all the light footprints overlap (for example, light footprint of the LEDs posed in corners of room) we can think to apply the policy of spreading sequence re-use as for carrier frequencies in cellular networks, hence utilizing the same spreading sequence by different LEDs that are far one from each other, thus meaning not interfering in any position of the floor.

4.3 Illumination constraint

Since the main role of a LED is to provide the illumination service, we must take care of providing a minimum level of illumination everywhere in the room, thus meaning that a target light intensity must be granted. From this point of view, the constraint on illumination does not go in contrast with the above two considerations about localization accuracy and achieved rate. Higher is the number of LEDs, higher is the illumination, even though, for sake of energy saving as well as eye safety, uncomfortable situations should be avoided. Indeed, in principle also a maximum illumination level should be set as a constraint since we must pay attention to eye safety issues/regulations.

There is also another important aspect in the illumination service that deals with uniformity. Generally speaking, obtaining a flat illumination level all over the room/environment is often an unfeasible target. Hence, the key point to solve the minimum requirements, a certain degree of uniformity and preserve eye-safety is to set two constraints. One is on the minimum illumination level achievable in the room, while the other one is a fairness ratio in the illumination that can be defined as

(11)$$\mathcal{F}= I_{\min}/I_{\max},$$

which tends to zero when the minimum illumination measured in the room $I_{\min }$ approaches very low value with respect to the maximum one $I_{\max }$, while it approaches $1$ when the minimum is very close to the maximum, i.e., $I_{\min }\approx I_{\max }$. Besides requiring a certain degree of fairness is equivalent, due to the request of a minimum illumination level, also to requiring the maximum illumination level usually set according to eye-safety regulations and/or to use of room. It is known that different room use leads to different illumination requirements.

In principle, the illumination level at a given position depends mainly on the transmission profile of the light source, in addition to the distance to that light source. Further, notice that the illumination is an additive feature, i.e., if a certain location is covered by multiple light sources then the total illumination at that location is the sum of the illumination levels due to the individual light sources. Analytically speaking, let the $k$-th user be covered by multiple APs, whose indexes define the set $\mathcal {I}$. Then, the total illumination perceived by the $k$-th user is given by

(12)$$I_k = \sum_{i \in \mathcal{I}} P_{i} h_{i,k},$$

where $P_{i}$ [W] is the optical transmission power of the $i$-th AP.

4.4 Regarding blockage

Since the propagation is for the most part in LoS, the event of blockage, that occurs when an obstacle impedes the signal to propagate from LED to PD, is more than relevant and affects system performance. About this aspect two elements must be properly highlighted. First, blockage cannot be inserted directly in the optimization since it can occur in an unpredictable way and this deals both with time (when) and position (where), and also in which way (how). If fact, let us suppose that a user hand is posed so as to cover the PD. This can happen at different distances from the PD itself, and this is not in the possibility of being predicted or assumed. Furthermore, even though blockage can be linked to movements of the user in the network, our optimization is performed on the whole room, thus meaning that also in case of mobility all the constraints are met.

The second element is that we show in the numerical performance the effect of blocking, by posing an obstruction at different distances from the PD. The blockage can affect the results in terms of constraints met or not met. As last comment, about the network planning, we consider two typologies of room the dimensions of which are different. However, no furniture (layout) has been considered, differently from [5] where user spatial distribution has been assumed. This is because we require in each point of the room to meet the constraints. If we consider the spatial distribution of objects in the room (i.e., tables, closet, and so forth) a change in the distribution may lead to fail to meet the constraints if the optimization is performed for a specific room configuration.

5. LEDs-constrained optimal placement

In this section, we aim at solving one key problem, and two additional ones that represent two particular cases. The main problem consists in building a network from the scratch by minimizing the number of access points needed to guarantee (i) a given location estimation accuracy, (ii) a minimum data rate level, (iii) a minimum illumination level, all measured in a given position, and also (iv) a minimum illumination fairness, measured all over the room. Analytically speaking, we have the following constrained problem to be met for each position of coordinates represented by the couple $(x,y)$, i.e.,

(13a)$$\begin{aligned}\textbf{ Problem 1 :} \;\textrm{min} \, & N \end{aligned}$$

(13b)$$\begin{aligned} \text{s.t.} \, & \Theta(x,y) \leq \Theta^\ast \end{aligned}$$

(13c)$$\begin{aligned} \, & R(x,y) > R^\ast \end{aligned}$$

(13d)$$\begin{aligned} \, & I(x,y) \geq I^\ast \end{aligned}$$

(13e)$$\begin{aligned} \, & \mathcal{F} \geq \mathcal{F}^\ast \end{aligned}$$

where $\Theta (x,y)$, $R(x,y)$, and $I(x,y)$ are the CRB value, the achieved rate, and the illumination level, at each location, respectively, while $\mathcal {F}$ is the illumination fairness all over the room. Since $I(x,y)\geq I^{\ast }$ and $\mathcal {F}\geq \mathcal {F}^{\ast }$, it follows that $I^{\ast }\geq I_{\min }$ and then the limitation on the maximum illumination level is provided by the following relationship, i.e.,

(14)$$I_{\max} \leq \frac{ I^{{\ast}}}{\mathcal{F}^{{\ast}}}.$$

Hence, posing a limitation on minimum fairness and minimum illumination is an implicit requirement of bounding also the maximum illumination. This is worth especially for eye safety issues/regulation.

Let us consider a room of area $\mathcal {L}_x \times \mathcal {L}_y$ [m$^2$], where $\mathcal {L}_x$ and $\mathcal {L}_y$ are the lengths in the horizontal and vertical direction, respectively. In this scenario, we assume a grid structure where different tiles on the floor are along the horizontal and vertical direction of the room, whose dimensions are defined as $\tau _x$ [m] and $\tau _y$ [m], respectively. Then, a given position $(x,y)$ is identified by a given tile of size $\tau _x \times \tau _y$ [m$^2$]. Additionally, we consider a similar structure on the ceiling, with tiles of size $q_x \times q_y$ [m$^2$] that determine the possible LEDs positions on the horizontal and vertical directions, respectively. It is important to notice that in general we can have $\tau _x \neq q_x$ and $\tau _y \neq q_y$. Figure 1 depicts the grid structure of the room environment.

Fig. 1. Room grid-structure model, with $\tau _x=q_x$ [m] and $\tau _y=q_y$ [m].

LED angle of irradiance, $ϕ_{j}$	$70^{\circ}$
PD field of view (FOV), $ψ_{C}$	$60^{\circ}$
PD physical area, $A$	$1 {cm}^{2}$
Communication bandwidth, $B$	$10$ MHz
Refractive index, $n$	$1.5$
Optical filter gain, $T_{s} (ψ)$	$1$
Noise power spectral density, $N_{0}$	$10^{- 21}$ $A^{2} /$ Hz
Localization Bandwidth, $B_{L}$	$50$ KHz
Room size (small)	$4 \times 5 \times 3$ m $^{3}$
Room size (large)	$9 \times 10 \times 3$ m $^{3}$
Vertical distance, $d_{v}$	$2$ m
1st use-case	$Θ^{} = 0.01$ m $^{2}$ , $R^{} = 30$ Mbps,
	$I^{} = 375$ lux, $F^{} = 0.25$
2nd use-case	$Θ^{} = 0.1$ m $^{2}$ , $R^{} = 50$ Mbps,
	$I^{} = 250$ lux, $F^{} = 0.4$

Tilting angle	$Θ^{*} = 0.01$ m $^{2}$	$R^{*} = 30$ Mbps	$I^{*} = 250$ lux	$F^{*} = 0.4$
$0^{\circ}$	$0.006$	$38.2$	$372$	$0.5$
$10^{\circ}$	$0.0076$	$36.5$	$361$	$0.47$
$30^{\circ}$	$0.0092$	$32.7$	$336$	$0.42$
$50^{\circ}$	$0.013$	$30.2$	$307$	$0.37$

Block-obj distance	$Θ^{*} = 0.01$ m $^{2}$	$R^{*} = 30$ Mbps	$I^{*} = 250$ lux	$F^{*} = 0.4$
$170$ cm	$0.0078$	$35.4$	$308$	$0.48$
$120$ cm	$0.0081$	$34.7$	$297$	$0.46$
$70$ cm	$0.0096$	$32.2$	$266$	$0.41$
$20$ cm	$0.012$	$12.6$	$192$	$0.23$

$N$	Solutions @ $R^{*} = 30$ Mbps	Solutions @ $R^{*} = 37$ Mbps
$4$	$1920$	$1520$
$5$	$2156$	$1890$
$6$	$2400$	$2240$

$N$	Solutions @ $R^{*} = 30$ Mbps	Solutions @ $R^{*} = 37$ Mbps
$4$	$1920$	$1520$
$5$	$2156$	$1890$
$6$	$2400$	$2240$

Abstract

1. Introduction

2. Related works

3. Proposed system and network architecture

3.1 System signaling

3.1.1 Phase I

3.1.2 Phase II

4. Design constraints

4.1 Localization error

4.2 Communication/access power constraint

4.3 Illumination constraint

4.4 Regarding blockage

5. LEDs-constrained optimal placement

5.1 About computational cost

6. Simulation results

6.1 Performance for Problem $1$

6.2 Space of solutions for Problem $1$

6.3 Robustness with respect to tilting and blocking

6.4 Achieved performance for Problem $2$ and Problem $3$

7. Conclusions

Acknowledgment

Disclosures

Data availability

References

Data availability

Cited By

Figures (16)

Tables (5)

Equations (30)

Optics Express