I Introduction
Indoor tracking and positioning systems have been gaining traction among the research community after the huge success of outdoor tracking and positioning systems based on global positioning system (GPS) technology [1]. However, GPS is not suitable for indoor environments owing to its incapability to penetrate walls of a building. Other indoor positioning systems (IPS) based on ultrasound, radio frequency identification (RFID), and wireless local area network (WLAN) require additional resources for their operation, thus increasing the overall system cost [2, 3].
Visible light communication (VLC) with light emitting diodes (LEDs) is a relatively new technology that became the subject of exploration within academic and industrial research in the field of indoor positioning [4],[5]. Its advantages include high bandwidth, high positioning accuracy, license free operation and immunity to electromagnetic interference [6]. It can be integrated with the existing lighting infrastructure for its operation. VLC can be used for various applications ranging from navigation and emergency rescue to object localization.
There are two major components of the VLCbased IPS: LED transmitters and photodiode (PD) optical receivers. LED transmitters are installed at fixed locations inside a room and continuously transmit their position coordinates encoded as an optical signal. When a mobile optical receiver comes in the vicinity of a LED transmitter, it receives the optical signal and decodes the LED position to estimate its own position. The transmitted optical signals are affected by multipath propagation, shadowing, and interference from various noise sources [7]. Thus, considerable amount of research has been done to find suitable implementation techniques for the VLCbased IPS which helps in achieving high positioning accuracy.
Most of the research on VLCbased IPS investigates techniques based on triangulation/trilateration, fingerprinting or proximity methods [5]
. These approaches usually use features extracted from signal such as received signal strength (RSS), angle of arrival (AOA)
[8], or time of arrival (TOA) [9]. In most of the methods, authors assume that power from two or more different light sources can be easily separated. Also, to remove angular dependency in RSS based localization, some studies assume fixed height of the receiver and the receiver surface being parallel to the transmitter [10]. In practice, such assumptions are questionable as a mobile receiver cannot be constrained by a fixed angle or height. Additional problems such as system complexity, computational and implementation costs makes it difficult to implement these techniques.When the light beam regions of multiple LEDs in the IPS intersect, given the underlying optical wireless communication channel model, the problem of estimating signal strength can be translated to finding the channel gains of the set of received signals. Considering that most of the channel gains at the optical receiver will be zero, as the receiver can receive signal from only a limited number of nearby LEDs, the resulting problem reduces to an equivalent compressed sensing (CS) problem and can be efficiently solved using sparse signal recovery methods [11]. In this letter, the feasibility of using CS for signal separation is explored and a new indoor position estimation approach based on proximity method is devised following the intuition that overlapping light beam regions improve the accuracy of VLC based IPS due to increased number of reference positions. The proposed CSbased method is investigated in a realistic LEDilluminated largescale indoor openplan office space environment. For various system parameters, the position accuracy of the proposed method is evaluated and compared against the performance limits achievable using the proximity method.
Ii Indoor Positioning via Multiple LEDs
For the purpose of analysis, a large indoor LEDilluminated environment (e.g., an openspace office or an industrial hall) with LED sources attached to the ceiling is considered. LEDs are placed on a grid of known positions , where and are the th LED coordinates of the floor/ceiling plane, while is the height of all LEDs from the ground. It is assumed that the th LED illuminates a certain coverage area of the floor plane , as shown in Fig. 1. A user device (UD) equipped with a PD is placed at an unknown location (for simplicity, we assume ). Focusing on the lineofsight (LOS) communication and ignoring the signal reflections, we assume the PD is able to detect the signal from the th LED if . In general, the coverage areas of different LEDs overlap, thus leading to a positioning problem via multipleLED estimation model (MLEM) recently considered in [12].
For the purpose of positioning, we propose that, at a given time instant (e.g., periodically or upon request), all LEDs simultaneously transmit their binary ID sequences , of length bits. Each bit of the ID sequence is generated independently and uniformly at random and we assume that the UD knows the ID sequences of all LEDs. The LED signal is transmitted using a simple onoff keying (OOK) modulation with the symbol period selected sufficiently large so that we can: i) assume all LED transmissions are perfectly synchronized, and ii) ignore all distancedependent delays. At an unknown position, the received signal at the UD represents a superposition of signals arriving from surrounding LEDs:
(1) 
where is the gain of the optical channel^{1}^{1}1For the optical LOS channel gain, we adopt the following model:
is a vector of additive i.i.d. Gaussian noise samples, and
is a coverage indicator variable equal to one if the UD is covered by the th LED. Representing (1) in the matrix form, we get:(4) 
where is an binary matrix containing all LED ID sequences organized into columns. In addition, we compactly define:
(5) 
where is the coverage indicator vector and is the optical channel gain vector.
The problem we consider is that of estimating the unknown coordinates of the UD based on the signal received from LED transmitters and knowledge of the matrix of LED ID signatures. We split this problem into two steps: i) LED ID detection and channel estimation step, where we recover the vector from (4) using ideas from CS, and ii) UD position estimation step, where, based on the recovered vector , we estimate the UD position .
Iii VLCbased Positioning via Compressed Sensing
Iiia LED ID Detection and Channel Estimation
In a large indoor environment, we assume that any LED coverage area illuminates only a small fraction of the floor plane . Thus, at any location , the UD will be covered by a small number of surrounding LEDs, where is the cardinality of the support set (i.e., the set of indices of nonzero elements) of the coverage vector . Consequently, the vector is a sparse vector, meaning that it contains nonzero components. Assuming , equation (4) represents a canonical noisy CS problem [11]. The theory of CS provides conditions under which the vector can be recovered at the UD from the received signal and knowledge of . In general, the sparse vector of length can be recovered from the vector of length , if the matrix satisfies certain properties (e.g., restricted isometry property), which are known to hold for random Bernoulli matrices [11].
From and , the UD applies suitable CS recovery method to generate an estimate of the vector . From , the UD estimates the support set of , which completes the LED ID detection part. The components of that correspond to the estimated support set provide estimated channel gains: We also note the groupsparsity property of the LED ID detection problem: the set of nonzero coordinates of is highly spatially correlated.
For the range of the number of LEDs in our setup, classical methods such as basis pursuit denoising (BPDN), or its lowcomplexity alternatives such as orthogonal matching pursuit (OMP), can be applied [11]. In this paper, we apply OMP as it is wellunderstood and investigated. OMP recovers via fast, greedy iterative procedure, by identifying sequentially the columns of that are most strongly correlated with the remaining part of . Proper sequence of column indices minimizes the residual which eventually leads to the convergence of OMP, as detailed in [14].
IiiB UD Position Estimation
After recovering vector using CS, the next step is to estimate the UD position. Due to complexity of taking the unknown angles into account, we resort to a simplified proximity method that uses only the estimated support set for position estimation, thus effectively disregarding the channel gains [15]. The estimated UD position is obtained as the centroid of an area obtained as a union of LED coverage areas of all detected LEDs, i.e., . With the sufficient LED density and overlapping of LED beams, this lowcost method provides high positioning accuracy in indoor environment, as detailed next.
IiiC VLCbased positioning using CS: The proposed algorithm
The proposed lowcomplexity CSbased VLC positioning algorithm is presented below. Besides the received signal and the matrix , the UD requires knowledge of system geometry: LED locations and coverage areas , maximum signal sparsity at any point of the floor plane: , and suitably selected distance threshold .
After recovering using OMP, the proposed method processes the strongest channel gains, and accepts LED ID as nearby only if it is sufficiently close, i.e., within to the running position estimate . The subroutine PROX() estimates the UD position using proximity method, i.e., as an average point of the set of LED positions .
Iv Results and discussion
In this section, we present numerical experiments for an openplan office space with squareshaped floor area of side length m and height m. LED sources are fixed on an equidistant grid on a ceiling plane, providing for LEDs. LED beams are shaped to provide circular coverage areas of radius . Each LED transmits a random binary LED ID signature of length , thus is an matrix of i.i.d. equallylikely entries. LED and PD parameters are: = 30, = 1, = 1.5, = 80 and = 1 , providing the channel gains based on the model described in (2). Optical power is adjusted to provide average horizontal illuminance of lx. The noise power is obtained from the fixed and preselected received signaltonoise ratio (SNR).
We first evaluate the case with the fixed geometry: i) fixed number of LEDs: (), and ii) fixed radius m of . For each simulation run, we randomly distribute UDs on the floor plane. For each UD, we generate its received signal corrupted by an additive Gaussian noise of a given SNR. The UD applies Algorithm 1 to estimate the set of nearby LEDs and its own position . From the obtained estimates and known true values, we calculate the Euclideandistance position error between position estimate and true position , and support recovery error (SRE) defined as the cardinality of symmetric difference between true and estimated support set . The results are averaged over random UD instances.
In the first set of experiments (Fig. 2), we fixed the length of LED ID signatures to bits, and evaluated the Mean (Euclidean distance) Position Error (MPE) and SRE as a function of SNR. We note that the MPE of CSbased positioning gradually approaches the lower bound (min MPE), i.e., the precision of proximity method assuming that the set of LEDs covering the UD is correctly recovered: . Similarly, the reconstructed becomes correct, i.e., SRE, as SNR increases. For SNRdB, both MPE/SRE achieve minimal values. For a given geometry, the min MPE of proximity method m, while sparsity is positiondependent and varies between and (note that the number of LEDs covering a given randomly placed point is a nontrivial generalization of a classical Gaussian circle problem).
In the next set of experiments (Fig. 3), we fixed SNR dB and varied LED ID signature length . We note that can be reduced down to without almost no loss in MPE and slight loss in SRE. Note also that, at SNR dB, MPE cannot go below MPE m, thus further decrease towards MPE is possible only by increasing SNR.
We continue experiments by fixing and SNRdB, but changing the system geometry by varying the LED coverage radius . Fig. 4 reveals interesting nonmonotone behavior of min MPE of proximity method as a function of . Our positioning method is able to match the min MPE up to m for a given set of parameters. Extending this behavior to larger values of would require increase of SNR.
Finally, in Fig. 5, for a fixed , m, and two values of SNRdB, we change the geometry via changing the LED density Due to space constraints, we present only MPE results, and note the similar behaviour as in the previous cases, where increased SNR shifts the performance towards min MPE bound.
V Conclusion
In this work, we presented a VLCbased positioning approach using compressed sensing for signal separation and proximity method for position estimation. The presented solution is of lowcomplexity while performing close to the bounds achievable using proximity method for a range of parameters.
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