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IN PREPARATION FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2019 1
Joint User Location and Orientation Estimation for
Visible Light Communication Systems with
Unknown Power Emission
Bingpeng Zhou†‡ , An Liu∗, and Vincent Lau‡
†Shenzhen Research Institute, Hong Kong University of Science and Technology, Shenzhen, China.
‡Department of ECE, Hong Kong University of Science and Technology, Hong Kong 999077, China.
∗College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, China.
eebzhou@ust.hk, anliu@zju.edu.cn, eeknlau@ee.ust.hk,
Abstract—In this paper, we are interested in the joint estimate
of user equipment (UE) position and orientation for visible light
communication (VLC) with uncertain emission power. This joint
estimation is a non-convex problem with a huge search space.
To address this challenge, a novel VLC localization algorithm
is proposed, which converges to the stationary solution of the
joint estimation problem at an asymptotic quadratic convergence
rate due to our problem-specific surrogate function design. A
closed-form update rule is obtained via exploiting the hidden
convex structure of the non-convex optimization problem. Hence,
our algorithm has low complexity compared with the particle
swarm-based optimization methods. In addition, the closed-form
Cramer-Rao lower bounds (CRLBs) on the estimation errors of
the respective UE location, orientation and LED emitting power
are derived. Moreover, the effect of critical parameters such as
signal-to-noise ratio (SNR), the number of LED transmitters,
transmission distance and non-line-of-sight propagation on the
VLC localization performance are revealed. The simulation result
verifies that the proposed VLC localization algorithm under
unknown VLC emitting power can achieve a huge performance
gain over the state-of-the-art localization baselines.
Index Terms—LiFi localization, visible light communication,
Cramer-Rao lower bound, unknown emitting power.
I. INT ROD UC TI ON
VISIBLE light communication (VLC), also known as LiFi,
is envisioned to be an important technique to improve
indoor communication quality with the widespread use of light
emitting diodes (LEDs) for illumination. Hence, visible light-
based positioning (VLP) has attracted increasing attention in
industrial and academic societies [1], [2], [3].
A. Technical Challenges
The knowledge of user equipment (UE) position and orien-
tation are indispensable for indoor location-based applications
such as robotic navigation [4], autonomous parcel sorting [5]
and automatic parking [6]. Conventional WiFi-based position-
ing methods [7], [8], [9] cannot satisfy the requirements of the
above applications, since they cannot provide the orientation
estimate for UEs, and their localization performance suffers
from the limited number of available access points in their
This work was supported by Science and Technology Plan of Shenzhen
under Grant No. JCYJ20170818114014753.
Fig. 1. Illustration of the non-convex mean squared error (cost function) of
a two-dimensional VLP problem.
local area. Hence, an accurate VLP solution is demanded.
However, VLP is challenging due to the following reasons.
•Non-Convex Optimization: The received signal strength
(RSS) of VLCs is a nonlinear function with respect to
(w.r.t.) the UE location [10], [11], which leads to a non-
convex optimization problem with a lot of local optima,
as illustrated in Fig. 1. The brute-force application of the
conventional gradient-descent-based optimization (GDO)
methods, e.g., [12] and [13], for VLP will result in a slow
convergence rate, which is not desirable in practice.
•Unknown Power Emission and UE Orientation: In addi-
tion to the UE location, the UE orientation and the LED
transmit power may be unknown in practice.1This ex-
tends the unknown-parameter set of the VLP problem. As
a result, the associated non-convex optimization problem
becomes much more challenging, since the dimensions of
the search space and hence the number of local optima
will be enlarged. Conventional VLP solutions [14]–[20]
only consider the estimate of UE location, and hence the
estimates of UE orientation and LED power emission
remain unresolved. In addition, the imperfect estimate
of these uncertain parameters will result in a serious
performance loss in the achieved VLP solution.
•Performance Limits of VLP: The performance limits of
1The knowledge of UE orientation may be unavailable, for instance, when
the UE has no inertial measurement unit (IMU). In addition, the LED emitting
power or the receiver optical filter gain may also be unknown or erroneous,
since there will be an inevitable error in their calibration results.
2 IN PREPARATION FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2019
VLP are not fully understood. The impact of unknown
emission power, non-line-of-sight (NLOS) propagation
and LED transmitter deployment on the UE location
and orientation estimation performances is also unknown.
The closed-form Cramer-Rao lower bound (CRLB) anal-
ysis on the VLP performance is challenging due to the
coupling of the UE location, UE orientation and LED
transmit power errors (namely, the estimation errors of
these parameters are impacted by each other).
In the following, we provide a review of the existing works
related to VLP and explain their strengths and weaknesses in
overcoming the above challenges, which motivates our work.
B. Related Works
1) Existing VLP Methods: A number of VLP methods, e.g.,
[14]–[20], have been proposed. Specifically, in [17], the nar-
row field-of-view (FOV) of transmitters in a two-dimensional
scenario is assumed to exploit the angular knowledge for UE
location estimate. In [19], both the LED transmitters and the
UE receiver are assumed to have perpendicular orientations to
the room ceiling, and the height of the UE is assumed to be
known. Similarly, an upward orientation of the UE is required
in [20]. An angle-of-arrival (AOA)-based VLP method is
proposed in [14], which uses a receiver array with known
orientation angle differences between receivers. In addition, an
inertial measurement unit (IMU) is required in [10] to measure
the UE tilt angle for position estimate.
Despite these efforts, there are still a number of limitations
in these approaches. First, some of them have special require-
ments on VLC systems, such as perfect alignment between
the UE orientation and the LED transmitter orientation [19], or
narrow transceiver FoV angles [17]. The requirements on ideal
system settings will limit the application of these UE location
estimate methods. Second, some of these existing approaches
leverage the geometric trilateration-based indirect localiza-
tion method [17]–[19], which first derives the transmission
distances/angles and then estimates the UE location. These
geometric-based VLP approaches are intuitive; however, the
imperfect estimate of transmission distance/angle will degrade
the UE location estimation performance. Third, the above VLP
methods [10], [11], [14]–[20] require perfect knowledge of
the UE pose, the UE height or the LED emission power [21].
Furthermore, only UE position estimate is considered. Hence,
the estimate of the UE orientation remains unresolved in their
solutions, and the adoption of an imperfect UE orientation
estimate will degrade the UE location estimation performance.
Unlike the above signal prorogation model-based solutions,
a data matching-based VLP method is proposed in [22], where
a Gaussian model is used to learn the correlation among the
location grids, and the location grid whose measurement is in
the minimum mismatch with the instantaneous RSS measure-
ment is viewed as the UE location. This VLP scheme gets
around the first challenge (i.e., the non-convex optimization)
via using a data-matching method. However, it requires prior
knowledge of both the UE orientation and the LED emission
power, and hence the second challenge is still unresolved.
Moreover, this data matching-based VLP method needs a fixed
UE orientation and a fixed LED emission power for different
UEs to preserve the consistency of measurement conditions.
However, it is difficult to satisfy in practice.
In a nutshell, an accurate VLP solution under unknown LED
power emission and without any requirement on the ideal VLC
system settings or perfect knowledge of UE states is highly
desirable, in spite of the existing VLP algorithm designs.
2) Existing VLP Performance Analysis: There are only a
few works on the performance analysis of RSS-based VLP.
In [23] and [24], the CRLB on the transmission distance
estimation error is studied, where it is assumed that all LEDs
have a downward orientation direction and the UE hight is
known. In [25], the CRLB for the UE location estimate with
a known UE height is derived. In addition to visible light RSS,
time-of-arrival-based VLP performance is studied in [26] and
[27]. However, the required assumptions in [23]–[25] restrict
the application of their results in general cases. In addition,
all these analyses focus on the error performance of the UE
location estimate or the transmission distance estimate, rather
than a straightforward result of the UE location and orientation
errors. Hence, the extension of performance analysis to the
joint UE location and orientation estimate with unknown LED
power emission is not trivial. In addition, the impacts of critical
parameters such as the transmission distance, LED emitting
power, NLOS propagation and the number of LED transmitters
on the VLP performance limits are not understood.
C. Contribution of This Paper
In this paper, we are interested in the simultaneous position
and orientation (SPAO) estimate for VLC users with unknown
LED emission power. Unlike the above VLP methods in [14]–
[20], we have no requirements on the prior knowledge of UE
pose or ideal system settings. To address the second challenge
(i.e., uncertain dependent parameters), we consider the joint
optimization of all unknown parameters, including the UE
location, UE orientation and LED emitting power. To address
the first challenge (i.e., the non-convex optimization), we
propose a novel successive linear least square (SLLS)-based
VLP algorithm, which can achieve a robust estimate of the UE
location and orientation under unknown LED emitting power.
For the third challenging issue (i.e., the VLP performance
limits), we obtain a closed-form CRLB on each unknown
parameter, and we also conduct a quantitative analysis on
the performance limits of the proposed VLP algorithm. The
main contributions of this paper are two-fold: the design of
the novel VLP algorithm and the closed-form analysis of the
VLP performance limit, which are elaborated as follows.
•The Proposed VLP Algorithm: Unlike the indirect VLP
methods [14]–[20], a joint optimization method is con-
sidered in our VLP algorithm design to straightforwardly
estimate the unknown UE location, UE orientation and
LED emitting power. To overcome the non-convex opti-
mization challenge with an extended uncertain-parameter
set, we observe that there is a convex sub-structure in the
SPAO problem as the system model is linear w.r.t. the
UE orientation and LED emitting power. We then exploit
this hidden convex sub-structure to obtain a closed-form
BINGPENG ZHOU, AN LIU & VINCENT LAU: LIFI-BASED SPAO SCHEME UNDER UNKNOWN POWER EMISSION 3
update rule for these two parameters. For the non-convex
optimization of the UE location, we employ the SLLS
method to design a convex surrogate function and derive
a closed-form update equation. Then, the unknown UE
location as well as the UE orientation and the LED
emitting power will be iteratively optimized as per the
obtained closed-form update equations until the proposed
VLP algorithm converges to a stationary point of the
non-convex SPAO problem. The convergence of the pro-
posed VLP algorithm is established, which can achieve
a quadratic convergence rate faster than the conventional
GDO methods [12], [13]. In addition, due to our problem-
specific surrogate function design, a closed-form update
rule is obtained for our VLP algorithm, which renders a
low-cost SPAO solution. It is shown in simulations that
the proposed SLLS-based VLP algorithm can achieve a
considerable localization performance gain over the state-
of-the-art VLP baselines.
•Closed-Form VLP Performance Analysis: We have quan-
tified the performance limits of the proposed VLP algo-
rithm, where a closed-form CRLB on the estimation error
of each individual variable is obtained via using the Schur
complement for the associated Fisher information matrix.
Unlike the results in [23]–[25], our CRLB is a direct
result on the UE location and orientation, and it has no
restrictive assumption on the system settings. In addition,
the effect of critical factors such as SNR, transmission
distance, NLOS propagation and the number of LEDs
on the VLP performance is quantitatively analyzed. This
closed-form performance analysis can not only provide a
uniform benchmark for the performance of various VLP
solutions, but also build a theoretical basis for the design
of the VLP system optimization strategies (e.g., power
allocation and LED transmitter selection).
The remainder of this paper is organized as follows. Section
II gives the system model. The proposed VLP algorithm is
given in Section III. Convergence and CRLB is analyzed in
Section IV and V, respectively. Simulations are presented in
Section VI. We conclude the paper in Section VII.
II. SY ST EM MO DE L
We first elaborate the VLC-based SPAO system setup and
then explicate the relationship between the RSS measurement
with the UE geometric location parameters.
A. System Setup
We consider a VLP system with MLED emitters and one
static UE receiver equipped with photodiode, as illustrated in
Fig. 2. We use pm∈R3and vm∈R3to denote the location
and orientation vector2, respectively, of the mth LED trans-
mitter, for m= 1,· · · , M . These LEDs will act as anchors for
UE localization, and we assume their locations and orientation
vectors are known. In addition, let xR∈R3and uR∈R3
denote the UE position and orientation vector, respectively,
which are unknown. Moreover, we assume ∥uR∥2= 1 without
loss of generality, where ∥•∥2is the ℓ2-norm on a vector.
2LED orientation means its main emitting direction, as shown in Fig. 2.
Fig. 2. Illustration of the VLC-based localization system.
When LED emitters turn on for illumination, they periodi-
cally communicate with UE for data transmission. During this
period, if the UE attempts to determine its own location xR
and orientation vector uR, it will first acquire the identifica-
tions (IDs) of observed LED transmitters and the strength of
received VLC signals. Then, based on the LED IDs, the RSS
measurements, LED locations, orientations and the RSS model
knowledge (elaborated later), the UE location and orientation
vector will be simultaneously determined.
We assume that some VLC protocols [28] (e.g., IEEE
802.15.7) and multiple access methods [29] (e.g., the time-
division access method) have been well defined by the VLC
system such that the received signals from different LED
sources are distinguishable.
B. Measurement Model
The visible light RSS is dependent on the LED emitting
power, radiation, transmission distance, incidence angle gain
and characteristic constants of the UE. We assume all LEDs
have the same emitting power (but it is unknown to the UE)
denoted by WT, and it is subject to the maximum power
Wmax. The radiation of LEDs is usually described by a
Lambertian pattern [30] characterized by a Lambertian order
r=−ln 2
ln cos(A1
2), where A1
2is the semi-angle at half power
of LEDs [31]. For a typical LED with an illumination range
within [−π/3, π/3] (i.e., A1
2=π/3), usually r= 1 [32].
For the photodiode of the UE, we assume its aperture, optical
filter gain and optical concentrator gain is ΦR,GRand ΓR,
respectively, where ΓR=ζ2
R
(sin(θFOV))2if the UE incidence
angle θm∈[0, θFOV], and zero otherwise, in which ζRis the
refractive index of the UE optical concentrator and θFOV is
the UE’s FoV [10].
Given the mth LED transmitter with location pmand ori-
entation vm, the receiver with location xRand orientation uR
will be able to detect the VLC signal from this LED transmitter
4 IN PREPARATION FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2019
if the incidence angle θmis within the receiver FoV θFOV and
the irradiance angle ϕmis within the FOV range ϕFOV of the
mth transmitter, i.e.,
θm
θFOV≤1 and
ϕm
ϕFOV≤1,where
ϕmis the angle between the mth LED orientation vmand the
incidence vector emwhich is given by em=xR−pm
∥xR−pm∥2
,
while θmis the angle between the UE orientation uRand
the incidence vector em, as shown in Fig. 2. In addition, |•|
denotes the absolute value. Let ΘRbe the index set of LED
sources, which are visible and acting as anchors for the UE,
given by ΘR=m|
θm
θFOV≤1,
ϕm
ϕFOV≤1,∀m= 1 : M.
Then, the RSS (i.e., the direct current gain) of this receiver
can be given by the following general form:
zm=hmxR,uR+ςnlos,m +ϵm,∀m∈ΘR,(1)
where ϵmis the measurement noise, arising from the thermal
noise, short noise, dark current, etc. [30], [33], and ςnlos,m is
the strength of the NLOS component, which is deterministic
but unknown to the UE.
We assume the relationship between ςnlos,m and {xR,uR}
is unknown due to the complex dependency on each other,
and thus we consider ςnlos,m as a disturbance source (without
any information contribution to UE location estimation). We
also consider a general case that the distribution of the NLOS
component ςnlos,m is unavailable.
In addition, hmxR,uRis the measurement function asso-
ciated with the line-of-sight (LOS) component, given by [42]
hmxR,uR= Ψ
RW1
2
T
(r+ 1)cos ϕmrcos θm
∥xR−pm∥2
2
,(2)
where ΨRis a constant dependent on the aperture, optical
filter gain and optical concentrator gain of the VLC receiver:
ΨR=Γ
RGRΦR
2π.(3)
The angular parameters θmand ϕmdepend on the UE
location xRand orientation vector uRas elaborated below,
ϕm= arccos (xR−pm)⊤vm
∥xR−pm∥2,(4)
θm= arccos (pm−xR)⊤uR
∥xR−pm∥2,(5)
where •⊤denotes the transpose. Let µRbe the product of UE
orientation uRand emitting power WT,
µR=W1
2
TuR.(6)
Then, hmxR,uRcan be rewritten as [34]
hmxR,uR=g⊤
m(xR)µR,(7)
where gm(xR)is given by
gm(xR) = −ΨR
(r+ 1) (xR−pm)⊤vmr
(∥xR−pm∥2)r+3 (xR−pm).
Let z= vec[zm|∀m∈ΘR]∈R|ΘR|denote the measure-
ment vector, where |ΘR|is the size of set ΘRand vec[· · · ]
yields a column vector by sequentially stacking the elements
specified. Then, zcan be modeled as a linear function of µR:
z=G(xR)µR+ςnlos +ϵ,(8)
where ϵis noise vector, ςnlos is the NLOS signal vector, and
G(xR)∈R|ΘR|×3is the coefficient matrix of µR, given by
G(xR) = mat[g⊤
m(xR)|∀m∈ΘR],(9)
where mat[•⊤
m|∀m]yields a matrix stacked by all row vectors
{•⊤
m:∀m∈ΘR}.
III. THE PRO PO SE D SPAO AL GO RI TH M
In this section, we shall formulate the SPAO problem, unveil
its challenges, and then elaborate the proposed algorithm and
explain how it addresses the associated challenges.
A. Problem Formulation of SPAO
The goal of SPAO is to estimate the UE position parameters
xRand uRfrom zwith an unknown emitting power WT.
It should be noted that the unknown parameter WThas been
merged with uRinto µRin (6). As such, we are interested
in estimating xRand µR, which can be solved as per the
following minimization problem:
PSPAO : (ˆ
xR,ˆ
µR) = arg min
xR,µR
∥z−G(xR)µR∥2
2,(10)
s.t.∥µR∥2
2≤Wmax.(11)
Then, once ˆ
µRis obtained, the emitting power and UE
orientation vector can be determined, respectively, by
ˆ
WT=∥ˆ
µR∥2
2,(12)
ˆ
uR=ˆ
µR
∥ˆ
µR∥2
.(13)
Challenge. The above SPAO problem PSPAO is non-convex
in (xR,µR), due to the nonlinear model G(xR)w.r.t. xR.
To overcome the non-convexity challenge, we observe that
PSPAO has hidden convexity in µR. In the following, we shall
exploit such a structure to derive an efficient SPAO algorithm.
B. The SLLS-Based SPAO Algorithm
The proposed algorithm exploits the hidden linear structure
of the measurement model w.r.t. the UE location and orienta-
tion angle to achieve a successive linear least square (SLLS)
solution. We adopt an alternating optimization technique to
iterate between xRand µR, which will achieve a joint position
and orientation estimate for a VLC user under unknown emit-
ting power. Specifically, starting from an initial point ˆ
x[0], the
proposed SLLS algorithm will alternately update ˆ
x[i]and ˆ
µ[i]
(corresponding to positioning and orientating, respectively) at
each iteration i. The iteration is repeated until ˆ
x[i]and ˆ
µ[i]
converge. It will be shown that the proposed SLLS algorithm
will converge at a quadratic rate to a stationary solution to
the non-convex SPAO problem. The principle diagram of the
proposed SLLS algorithm is illustrated in Fig. 3.
In the following, we shall elaborate the SLLS algorithm.
Since the measurement model (8) is linear w.r.t. µRand
BINGPENG ZHOU, AN LIU & VINCENT LAU: LIFI-BASED SPAO SCHEME UNDER UNKNOWN POWER EMISSION 5
Fig. 3. The diagram of the SLLS-based SPAO algorithm.
nonlinear w.r.t. xR, the SPAO problem is convex w.r.t. µRbut
non-convex w.r.t. xR. By exploring this hybrid convex/non-
convex nature, we decompose the SPAO problem into two
subproblems: the convex orientating and the non-convex po-
sitioning subproblems, which are elaborated as follows.
1) Orientation Estimate: Given ˆ
x[i]obtained at the ith
iteration, the estimate of µRwill be determined by solving
the following orientating subproblem (depending on ˆ
x[i]):
PO:ˆ
µ[i]= arg min
µR
∥z−G(ˆ
x[i])µR∥2
2.(14)
At each iteration, POis strictly convex due to the linear
dependency of zon µR. Thus, the optimal estimate ˆ
µ[i]can
be given in a closed form by
ˆ
µ[i]= (G(ˆ
x[i]))†z,(15)
where †is the pseudo-inverse, and G(ˆ
x[i])depends on ˆ
x[i].
Unlike the successive convex approximation [35], [36] and
majorization minimization [37], [38] for non-convex compo-
nent variables, no approximation in µRis introduced in the
cost function of PO. Thus, POretains the entire structure w.r.t.
orientation in the original problem PSPAO in (10), leading
to the fastest convergence in ˆ
µ[i]given ˆ
x[i](i.e., the optimal
solution of POis directly obtained without iteration).
2) Location Estimate: Once ˆ
µ[i]is determined as above,
we estimate the UE location ˆ
x[i+1] based on the following
positioning subproblem,
PP:ˆ
x[i+1] = arg min
xR
∥z−G(xR)ˆ
µ[i]∥2
2,(16)
which is non-convex due to the nonlinear function G(xR)
w.r.t. xR. To address this issue, we leverage the successive
linear least square method to achieve a stationary solution of
ˆ
x[i+1] to PP, by exploiting a convex approximation to the cost
function in (16), as elaborated below.
To be specific, at each iteration, the UE location estimate
ˆ
x[i+1] is updated based on the previous result ˆ
x[i]as follows:
ˆ
x[i+1] =ˆ
x[i]+γ[i]x♯
[i+1] −ˆ
x[i]
d[i],(17)
where γ[i]∈(0,1] is the step length subject to the Armijo
rule elaborated later, d[i]denotes the evolution direction, and
x♯
[i+1] is the suggested update, which is determined by solving
a convex optimization subproblem as follows,
P′
P,[i+1] :x♯
[i+1] = arg min
xR
φSxR;ˆ
x[i],ˆ
µ[i],(18)
where φSxR;ˆ
x[i],ˆ
µ[i]is the convex approximation of the
original cost function in (16) (this will be explained in Section
III-B-3), given by (19), where λ[i]>0is a regularization
constant (elaborated later), and ∇xR(G(ˆ
x[i])ˆ
µ[i])∈R3×|ΘR|
is the derivative of G(xR)ˆ
µ[i]w.r.t. xRaround xR=ˆ
x[i],
given by
∇xR(G(ˆ
x[i])ˆ
µ[i]) = ∥ˆ
µ[i]∥2[em,ˆ
u[i],vm]sm
∥ˆ
x[i]−pm∥3
2
:∀m∈ΘR,
where ˆ
u[i]=ˆ
µ[i]
∥ˆ
µ[i]∥2
and sm∈R3is given by
sm= ΨR
(r+ 3)(r+ 1)e⊤
mvmre⊤
mˆ
u[i]
−(r+ 1)e⊤
mvmr
−r(r+ 1)e⊤
mvmr−1e⊤
mˆ
u[i]
.(20)
At each iteration, sub-problem P′
P,[i+1] is strictly convex,
and thus the closed-form expression of x♯
[i+1] is given by (21),
where λ[i]is given by λ[i]= min{λ0νi
0, λmin},in which λ0>
0is its (large) initial value, ν0∈(0,1), and λmin >0is its
minimum value.
This regularization constant λ[i]is designed to allow a fast
initial convergence rate when the initial point is potentially
far from the stationary point. Specifically, λ[i]decreases as
the iteration progresses to achieve an asymptotically quadratic
convergence rate (as proved in Theorem 2). It is well known
that the second-order optimization algorithm (e.g., our SLLS-
based location update) has a fast convergence in the area near a
stationary solution, but suffers from a slow convergence in the
far-area. In contrast, the first-order algorithm (e.g., the gradient
descent-based one) has a slow convergence in the near-area
but a fast convergence in the far-area. Thus, a decreasing
λ[i]can render a first-order update at initial iterations and a
second-order update at later iterations,3ensuring a relatively
fast convergence throughout the whole iteration process.
In addition, the step length γ[i]at each iteration is deter-
mined by the Armijo rule for some a > 0, i.e.,
φˆ
x[i]+γ[i]d[i];ˆ
µ[i]≤φˆ
x[i];ˆ
µ[i]+aγ[i]∇⊤
xRφˆ
x[i];ˆ
µ[i]d[i]
(22)
where φˆ
x[i];ˆ
µ[i]denotes the cost function of PPin (16),
and ∇xRφˆ
x[i];ˆ
µ[i]is the derivative of φxR;ˆ
µ[i]w.r.t. xR
around xR=ˆ
x[i], given by
∇
xRφˆ
x[i];ˆ
µ[i]=2∇
xRG(ˆ
x[i])ˆ
µ[i]G(ˆ
x[i])ˆ
µ[i]−z.(23)
To be specific, starting with a certain step size γ[i]>0, the
Armijo rule repeatedly decreases γ[i]as γ[i]=ν1γ[i]for some
ν1∈(0,1) until the condition in (22) is satisfied.
3) Summary of the SLLS Algorithm: Given an initial point
of ˆ
x[0], the proposed SPAO algorithm will iteratively update
ˆ
x[i]and ˆ
µ[i]. The pseudo-codes of the proposed algorithm are
3A large λ[i]at initial iterations leads to a first-order update, while a
small λ[i]approaching zero at final iterations renders a second order update.
6 IN PREPARATION FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2019
φS(xR;ˆ
x[i],ˆ
µ[i]) = ∥z−G(ˆ
x[i])ˆ
µ[i]− ∇⊤
xRG(ˆ
x[i])ˆ
µ[i]xR−ˆ
x[i]∥2
2
Least square based on local linear approximation
+λ[i]∥xR−ˆ
x[i]∥2
2
ℓ2¯norm regularization
.(19)
x♯
[i+1] =ˆ
x[i]+∇xR
G(ˆ
x[i])ˆ
µ[i]∇⊤
xR
G(ˆ
x[i])ˆ
µ[i]+λ[i]I3−1∇xRG(ˆ
x[i])ˆ
µ[i]z−G(ˆ
x[i])ˆ
µ[i].(21)
given in Algorithm 1. Since PSPAO is non-convex, the initial
point ˆ
x[0] will affect the optimization result. To obtain a good
initial point ˆ
x[0], one can resort to the random location samples
in practice. Specifically, one generates multiple samples in the
location space at random, tries all location samples and then
chooses the best sample with the minimum cost (see (10)) as
the initial point. These location samples are only used in the
initial iteration and hence will not significantly increase the
associated computation cost.
C. Discussion of the SLLS Algorithm
In the following, we discuss the properties of the proposed
SLLS algorithm and explicate its difference from conventional
GDO methods, which will help understand the superiority of
our algorithm. The computational complexity of the proposed
SLLS algorithm is also explicated.
1) Properties of the SLLS Algorithm: As shown in (19), the
SLLS-based surrogate φSxR;ˆ
x[i],ˆ
µ[i]is obtained by using
the first-order expansion of G(xR)ˆ
µ[i]around xR=ˆ
x[i]and
by imposing an ℓ2-norm regularization λ[i]∥xR−ˆ
x[i]∥2
2. This is
different from the gradient-based surrogate φGxR;ˆ
x[i],ˆ
µ[i]
given by (24). In addition, φSxR;ˆ
x[i],ˆ
µ[i]satisfies the
following consistency conditions.
•[Local Tightness of φSxR;ˆ
x[i],ˆ
µ[i]]φSxR;ˆ
x[i],ˆ
µ[i]
is locally tight to φ(xR;ˆ
µ[i])around xR=ˆ
x[i], i.e.,
φSˆ
x[i];ˆ
x[i],ˆ
µ[i]=φ(ˆ
x[i];ˆ
µ[i]).
•[Consistency of Gradient Vector] The gradient vector
of surrogate function φSxR;ˆ
x[i],ˆ
µ[i]w.r.t. xRaround
xR=ˆ
x[i]is equal to that of φxR;ˆ
µ[i], i.e.,
∇xRφSˆ
x[i];ˆ
x[i],ˆ
µ[i]=∇xRφˆ
x[i];ˆ
µ[i].(25)
These conditions show that φSxR;ˆ
x[i],ˆ
µ[i]is an effective
surrogate function for subproblem PPfrom the perspective of
successive convex approximation. Hence, it will give rise to a
feasible update direction d[i](see (21)), i.e., a descent direction
satisfying ∇⊤
xRφ(ˆ
x[i];ˆ
µ[i])d[i]<0for any non-stationary ˆ
x[i],
as will be proved in Section IV. Hence, the closed-form update
equation (17) subject to the Armijo rule (22) will ensure a
sufficient descent of cost function φxR;ˆ
µ[i]at each iteration
such that the obtained solution ˆ
x[i+1] converges to a stationary
solution to PP(see theorem 1 in Section IV).
Furthermore, the SLLS-based location update direction d[i]
is related to the pseudo-inverse of the gradient matrix of the
(vector-valued) measurement function, which is different from
the conventional GDO methods in which the update direction
is usually the gradient vector of the cost function. As such,
even without the knowledge of the Hessian matrix, our SLLS
algorithm can achieve the second-order convergence speed,
Algorithm 1: The SLLS-based SPAO Algorithm
Input : The measurement vector z.
1Initialize ˆ
x[0].
2While not converge do (for i= 1 : K)
[Orientation Update: input {ˆ
x[i],z}and output ˆ
µ[i]]
3- Determine ˆ
µ[i]as per (15).
[Position Update: input {ˆ
µ[i],z}and output ˆ
x[i]]
4- Determine x♯
[i]as per (21).
5- Determine γ[i]as per (22).
6- Determine ˆ
x[i]as per (17).
7End
8Determine ˆ
WTand ˆ
uRas per (12) and (13), respectively.
Output:ˆ
xR=ˆ
x[i],ˆ
WTand ˆ
uR.
which is faster than conventional GDO methods [12], as will
be proved in Section IV and verified in Section VI.
2) Computational Complexity: The computational cost of
the proposed SLLS algorithm at one iteration step is O(D3)
where Dis the dimensions of the optimization space, while the
conventional GDO methods in [12] and [13] are O(D|ΘR|)
where |ΘR|(larger than D) is the number of LEDs. Thus, at
one iteration step, our SLLS algorithm has a larger cost than
the GDO methods. Yet, considering the overall iterations, our
SLLS algorithm needs a smaller computation time than the
GDO methods because SLLS has a quadratic convergence rate,
which is faster than the GDO methods. Thus, a fewer number
of iterations and hence a shorter computation time will be used
by SLLS. This is verified in the simulation section.
The conventional particle swarm-based optimization (PSO)
method [9], [34] leverages multiple particles throughout the
iterations. If NSparticles are used in PSO, its computational
cost will be approximately NStimes our SLLS algorithm,
since SLLS uses only one “particle” (ˆ
x[i],ˆ
µ[i]).
IV. CON VE RG EN CE ANALYS IS
We shall establish the convergence of our SLLS algorithm.
It will be shown that the SLLS-based estimate (ˆ
x[i],ˆ
µ[i])
converges to a stationary point of the non-convex problem
PSPAO , at an asymptotically quadratic convergence rate.
A. Challenges and Assumptions
Challenges. It is not easy to establish the convergence
of both UE positioning error and orientating error because
the system model is essentially nonlinear (and equivalently
non-convex). In addition, the updates of the UE location
and orientation angle are coupled with each other, which
BINGPENG ZHOU, AN LIU & VINCENT LAU: LIFI-BASED SPAO SCHEME UNDER UNKNOWN POWER EMISSION 7
φGxR;ˆ
x[i],ˆ
µ[i]=∥z−Gˆ
x[i]ˆ
µ[i]∥2
2−2z−Gˆ
x[i]ˆ
µ[i]⊤∇⊤
xRGˆ
x[i]ˆ
µ[i]xR−ˆ
x[i]+∥xR−ˆ
x[i]∥2
2.(24)
complicates the convergence rate analysis for UE location and
orientation estimates.
Assumptions. To establish the convergence, we first give the
following necessary assumptions on the SPAO system.
(A1) The coefficient matrix G(ˆ
x[i])is full-column-rank.
(A2) The measurement noise ϵis zero-mean.
(A3) The gradient ∇
xR(G(ˆ
x[i])ˆ
µ[i])is full-row-rank.
Assumptions A1 and A3 mean the number of measurements
should be not less than the number of unknown parameters
such that G(ˆ
x[i])and ∇⊤
xR(G(ˆ
x[i])ˆ
µ[i])are full-column-rank,
which is usually satisfied and verified by simulations.
B. Convergence Results
We first give the following Lemma to ensure the update
direction d[i]of ˆ
x[i]is feasible, which will ensure a sufficient
descent of cost function along this direction at each iteration.
Lemma 1 (Effectiveness of the SLLS’s Location Update):
If A1 and A2 are satisfied, at each iteration, the SLLS-based
location update direction d[i]satisfies ∇⊤
xRφ(ˆ
x[i];ˆ
µ[i])d[i]<0
for any non-stationary ˆ
x[i].
Proof: This can be proved through inner product calcu-
lation, and it should be noted that d[i]and ∇xRφ(ˆ
x[i];ˆ
µ[i])is
given by (17) and (23), respectively.
Theorem 1 (Convergence of the SLLS Algorithm): If A1 and
A2 are satisfied, any limiting point of (ˆ
x[i],ˆ
µ[i])generated by
Algorithm 1 is a stationary point of PSPAO .
Proof: This theorem can be proved by the following
lemma 1. Since the SLLS algorithm falls into the framework
of the feasible direction method [39], [40], the SLLS-based
location update in (17) subject to its Armijo rule will converge
to a stationary point(s) to PSPAO , and so is the SLLS-based
orientation update in (15).
This means, even when A3 is not satisfied in some extreme
cases, the proposed SLLS algorithm will still converge due to
the introduction of non-zero regularization λ[i](also λmin) in
the parameter update. The following theorem will show that,
if we set λmin →0with A3 holding, the decreasing λ[i]will
lead to an asymptotically quadratic convergence rate.
Theorem 2 (Asymptotically Quadratic Convergence Rate):
If A1–A3 are satisfied and (ˆ
x[0],ˆ
µ[0])is sufficiently close to
a stationary solution (x•
R,µ•
R)of PSPAO , then convergence
rates of the SLLS-based location and orientation estimation
errors are asymptotically quadratic, as λmin →0:
∥x•
R−ˆ
x[i+1]∥2=O(∥x•
R−ˆ
x[i]∥2
2),(26)
∥µ•
R−ˆ
µ[i+1]∥2=O(∥µ•
R−ˆ
µ[i]∥2
2).(27)
Proof: See APP EN DI X A.
This means, even without the Hessian matrix, the proposed
SLLS algorithm can still achieve the quadratic convergence
rate asymptotically, faster than conventional GDO algorithms
(verified in Section VI). This is because the obtained closed-
form update equation (21) retains some second-order structure
of the cost function in (16) of the original positioning problem.
V. ERROR BOUND OFV LP
In this section, we derive the closed-form error bounds for
each of the UE location, orientation and LED emitting power,
and analyze the effect of SNR, NLOS and the number of LED
transmitters on the VLC-based SPAO error bounds.
A. Closed-Form CRLB
CRLB quantifies the minimum covariance of mean squared
errors that an unbiased estimator can achieve [44].4We adopt
it as a performance metric to evaluate the SPAO errors.
To facilitate the analysis, we assume the measurement noise
ϵmin (1) is zero-mean Gaussian, i.e., ϵm∼ N (ϵm|0, ω),∀m∈
ΘR, with precision ω(inverse variance). For ease of notation,
let ρmdenote the transmission distance given by
ρm=∥xR−pm∥2,(28)
and define the SNR as follows,
SNR =ω WTΨ2
R.(29)
In addition, let BxR(xR,µR)and BuR(xR,µR)∈R3and
BWT(xR,µR)∈Rdenote the CRLBs of ˆ
xR,ˆ
uRand ˆ
WT,
respectively, and also let υxR,υuRand υWTdenote the bias
of ˆ
xR,ˆ
uRand ˆ
WT, respectively, due to the NLOS effect, i.e.,
υxR=∥E{ˆ
xR} − xR∥2,(30)
υuR=∥E{ˆ
uR} − uR∥2,(31)
υWT=∥E{ˆ
WT} − WT∥2.(32)
We have theorem 3 to bound the VLC-based SPAO errors.
Theorem 3 (Closed-Form CRLB): VLC-based UE location,
orientation and emitting power estimation errors will be,
respectively, bounded from below as follows:
E{∥ˆ
xR−xR∥2
2} ≥ traceBxR(xR,µR)+E{υ2
xR},(33)
E{∥ˆ
uR−uR∥2
2} ≥ traceBuR(xR,µR)+E{υ2
uR},(34)
E{∥ ˆ
WT−WT∥2
2} ≥ BWT(xR,µR) + E{υ2
WT},(35)
where trace(•)denotes the matrix trace, and the closed-form
expression of their CRLBs is given by (36)–(38), respectively,
Proof: See APPENDIX B.
It is difficult to derive a closed-form expression for each
estimation bias υxR,υuRand υWT, due to the nonlinear
system model (1). Yet, the above closed-form CRLBs are
still meaningful for bounding the achieved SPAO errors. This
means, the obtained CRLBs BxR,BuRand BWTreveal the
best performance the SPAO algorithm can achieve, for given
system resources (such as LED emitter deployment and SNR).
In addition, we have the following theorem to reveal the es-
timation bias scaling w.r.t. the NLOS signal strength ∥ςnlos∥2.
4Considering the difficulty in deriving the exact localization error, it is
a common practice in literature (e.g., [23]–[27]) to seek a lower bound for
localization errors to reveal the associated localization performance limit.
8 IN PREPARATION FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2019
BxRxR,uR=SNR−1HxRUuRAxRU(uR)⊤HxR⊤−1,(36)
BuRxR,uR=SNR−1G′xR⊤VxR,uRG′xR−1,(37)
BWTxR,uR= 4WTΨ2
RωuRG′(xR)⊤V(xR,uR)G′(xRu⊤
R−1,(38)
where G′(xR) = Ψ−1
RG(xR), while H(xR)∈R3×3|ΘR|,U(uR)∈R3|ΘR|×|ΘR|,A(xR)∈S|ΘR|and V(xR,uR)∈S|ΘR|
are, respectively, given by
H(xR) = HmxR|∀m∈ΘR,(39)
HmxR∈R3×3=−(r+ 1)e⊤
mvmrI3+r(r+ 1)e⊤
mvmr−1vme⊤
m−(r+ 3)(r+ 1)e⊤
mvmreme⊤
m
ρ3
m
,(40)
UuR=I|ΘR|⊗uR,(41)
AxR=I|ΘR|−G′(xR)G′(xR)⊤G′(xR)−1G′(xR)⊤,(42)
VxR,uR=I|ΘR|−UuR⊤HxR⊤H(xR)U(uR)UuR⊤HxR⊤−1
HxRUuR,(43)
in which I|ΘR|denotes the |ΘR|-dimensional identity matrix, and ⊗denotes the Kronecker product.
Theorem 4 (Scaling Rule of Estimation Bias): The scaling
of each estimation bias w.r.t. the NLOS signal strength follows
lim
∥ςnlos∥2→0
υxR
∥ςnlos∥2
=W−1
2
TΨ−1
R∥HxRUuR∥−1
2,(44)
lim
∥ςnlos∥2→0
υuR
∥ςnlos∥2
=W−1
2
TΨ−1
R∥G′(xR)∥−1
2,(45)
lim
∥ςnlos∥2→0
υWT
∥ςnlos∥2
= 2W1
2
TΨ−1
R∥G′(xR)uR∥−1
2,(46)
where ∥ • ∥2denotes the ℓ2-norm on a matrix.
Proof: See APP EN DI X C.
This theorem shows that the estimation bias reduces with
the NLOS signal strength, and the reducing rate depends on
the LOS channel gain. In addition, combining theorems 3 and
4, we can see that UE location CRLB BxRand orientation
CRLB BuRare reducing with the SNR and hence the noise
power. In contrast, the NLOS signal-caused estimation bias
υxRand υuRare invariant with measurement noises. Thus,
as the SNR increases, the achieved SPAO error will gradually
reduce and finally hit an error floor due to the NLOS effect.
The NLOS effect will be further analyzed in Section VI-B-2
by combining with numerical results.
B. Asymptotic Analysis of CRLB
In the following, we reveal how system parameters (e.g.,
SNR, transmission distance and the number of LEDs) affect
the VLP performance via CRLB analysis.
Firstly, for the impact of the LED-UE transmission distance
on the localization CRLBs, we have the following conclusion.
Corollary 1 (Scaling Rule of CRLBs w.r.t. the Transmission
Distance): The VLC-based SPAO error bounds scale with the
distance between the LED and the UE as follows,
traceBxR(xR,µR)∼ O(ρ6
max),(47)
traceBuR(xR,µR)∼ O(ρ4
max),(48)
BWT(xR,µR)∼ O(ρ4
max),(49)
as ρmin → ∞, in which ρmin = min{ρm|∀m∈ΘR}and
ρmax = max{ρm|∀m∈ΘR}.
Proof: See APPENDIX D.
This means that the VLP error increases with the room size,
which can be used to evaluate how many LEDs are needed for
a ceratin room to achieve the desired localization performance.
Secondly, for the CRLB scaling rule w.r.t. the SNR and
LED emitting power, we have the following two corollaries.
Corollary 2-1 (Scaling Rule of UE Location and Orientation
CRLBs w.r.t. SNR): The CRLBs of VLC-based UE location
and orientation angle estimation errors scale with receiver-end
SNR in the following manner, as SNR → ∞,
traceBxR(xR,µR)∼ O(SNR−1),(50)
traceBuR(xR,µR)∼ O(SNR−1).(51)
Proof: It directly follows from theorem 3.
This means that the UE location and orientation estimation
error is linearly reducing with an increasing SNR. In addition,
for the scaling of LED emitting power estimation error, we
have the following corollary, where we assume the measure-
ment noise power is fixed.
Corollary 2-2 (Scaling Rule of LED Emitting Power CRLB
w.r.t. Actual Emitting Power): LED emitting power estimation
CRLB scales with the actual emitting power in the following
manner, as WT→ ∞,
BWT(xR,µR)∼ O(WT).(52)
Proof: It directly follow from theorem 3.
This shows that a large LED emitting power will lead to a
large estimation error of the LED emitting power, at a linear
increasing rate, for the fixed measurement noise power.
Thirdly, regarding the CRLB scaling rule w.r.t. the number
of LEDs (i.e., |ΘR|), we have the following conclusion.
Corollary 3 (Scaling of CRLBs w.r.t. the Number of LEDs):
BINGPENG ZHOU, AN LIU & VINCENT LAU: LIFI-BASED SPAO SCHEME UNDER UNKNOWN POWER EMISSION 9
Assume the LED transmitters are uniformly distributed within
the room. The VLC-based SPAO error bounds scale with |ΘR|
in the following manner, as |ΘR| → ∞,
traceBxR(xR,µR)∼ O(|ΘR|−1),(53)
traceBuR(xR,µR)∼ O(|ΘR|−1),(54)
BWT(xR,µR)∼ O(|ΘR|−1).(55)
Proof: See APP EN DI X E.
This means that the VLP error is reduced with the number
of independent VLC signal sources, at the rate of O(|ΘR|−1).
The obtained CRLB cannot only be used as a performance
benchmark for VLP algorithms but also shed lights on the
performance limits of our VLP problem PSPAO . In addition,
the dependency of CRLB on system parameters also provides
theoretical foundations for VLP system optimization strategies
(such as resource allocation or LED deployment optimization),
which can be exploited to fundamentally improve the VLP
system performance.
VI. SI MU LATI ON RE SU LTS
In this section, simulation results are presented to examine
the performance of the proposed SLLS-based SPAO scheme.
A. Simulation Settings
We consider M= 20 LEDs which are uniformly installed
on the ceiling of a room of the size 9 [m] ×9 [m] ×4 [m]. In
addition, we consider a common case that all LED orientations
are downward, i.e., vm= [0,0,−1]⊤,∀m, although the
proposed algorithm has no restrictive requirement on LED
orientation angles. We set the UE to appear in the room at
a random location with a random orientation. In addition,
we assume ΦR= 1 [cm2],r= 1,GR= 1,ΓR= 2.25,
θFOV =ϕFOV =π/2. and we set WT= 2.2[Watt] and SNR
= 40 [dB], which follows from a typical LED setup and is
widely adopted in papers such as [10], [32], [41] and [42].
The number of location samples used at the initial iteration
for the UE location initialization of our SPAO algorithm is
set to be 100, unless clarified otherwise. In addition, we set
the NLOS component to be ςnlos,m =℘nlos
1−℘nlos
hm(xR,µR),
∀m∈ΘR, where ℘nlos ∈[0,0.5) is the ratio between the
NLOS signal strength and the overall RSS.5We use a typical
value ℘nlos = 0.1[30], [41], [43], unless specified otherwise.
We consider the following VLP baseline methods for the
performance comparison with our SPAO algorithm.
•Baseline 1: Gradient descent-based SPAO method [12].
•Baseline 2: Line search-based SPAO method [13].
•Baseline 3: Geometric trilateration-based UE localization
method [19], with a perfect alignment of LED and UE
orientations but with an unknown LED emitting power.
•Baseline 4: Nonlinear least square-based UE localization
algorithm [42], without the knowledge of UE orientation
angle and LED emitting power.
5We consider a general case that the NLOS signal strength is less than
the LOS signal strength.
Iteration number
0 100 200 300 400 500 600 700 800 900 1000
RMSE
10-4
10-3
10-2
10-1
100
101
102The proposed SLLS-based location estimate
The proposed SLLS-based emitting power estimate
Orientation vector CRLB
The proposed SLLS-based orientation estimate
Location CRLB
Emitting power CRLB
Fig. 4. The root mean square error (RMSE) of the proposed SLLS-based
UE location, orientation and emitting power estimates.
•Baseline 5: PSO-based SPAO algorithm [34] without the
knowledge of LED emitting power, where 10 location
samples (each with 9 detection particles) are used.
•Baseline 6: AOA-based UE location estimate method [14]
using a uniform squared array of 3×3receivers.
•Baseline 7: Data matching-based UE localization method
[22] with 32400 training location grids.
In addition, the closed-form CRLB obtained in Section V
is simulated, which is used as the performance benchmark.
B. Simulation Results
1) Overall SPAO Performance: The estimation errors of
UE location, orientation and LED emitting power achieved by
the proposed SLLS algorithm are given in Fig. 4, the location
estimation errors of various algorithms v.s. iteration number
are shown in Fig. 5, and the cumulative distribution functions
(CDFs) of localization errors are given in Fig. 6, where we
consider the LOS-only case (the NLOS effect will be analyzed
later). It is shown in Fig. 4 that, when the SNR is 40 [dB], the
proposed SLLS algorithm can achieve a localization error of
0.0048 [m] and an orientation vector error of 0.003 [m] (an
angle error of 0.17 [deg]). In addition, we can see from Fig. 5
that the proposed SLLS algorithm has faster convergence than
the gradient descent-based SPAO method (baseline 1) and line
search-based SPAO (baseline 2) due to our problem-specific
update rule, as proved in theorem 2. Moreover, it is shown in
Fig. 6 that the proposed SLLS algorithm can achieve a quite
lower localization error than those of the baseline method 3–7,
due to the gain from the joint optimization of unknown UE
orientation and LED emitting power in our SLLS algorithm.
2) The Effect of NLOS and SNR: The UE location, ori-
entation and LED emitting power estimation errors achieved
by the proposed SLLS algorithm under various NLOS power
settings are given in Fig. 7, where the ratio of the NLOS
signal strength over the overall RSS ranges from 0 to 0.4 and
the SNR is fixed at 40 [dB]. As we expected, a larger NLOS
power leads to a larger localization error. In particular, for a
typical case that the NLOS signal strength is 0.1 times the
overall RSS [30], [43], the associated NLOS signal will lead
to a 0.3 [m] localization error, as shown in Fig. 7.
10 IN PREPARATION FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2019
0 200 400 600 800 1000
10−3
10−2
10−1
100
101
102
Iteration number
UE localization RMSE [m]
Baseline 1
Baseline 2
The proposed SLLS algorithm
Localizaton CRLB
Baseline 4
Baseline 5
Performance gain from the joint optimization
of UE orientation and LED emitting power
Performance gain from the
optimization of emitting power
Fig. 5. The achieved location estimation errors of various algorithms.
012345
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Localization error [m]
CDF
The proposed SLLS algorithm
Baseline 1
Baseline 3
Baseline 4
Localization CRLB
Baseline 5
Baseline 2
Baseline 6
Baseline 7
Gain from the optimization of
LED emitting power
Gain from the joint optimization of LED
emitting power and UE orientation
Fig. 6. The CDFs of location estimation errors of various algorithms.
In addition, the achieved UE location error of our SLLS-
based SPAO algorithm (associated with various NLOS power
settings) v.s. SNR is shown in Fig. 8, where the SNR varies
from -20 [dB] to 80 [dB] via reducing the measurement noise
strength and keeping the emitting power fixed. We can see
that there will be a phenomenon of estimation error floor in
the high SNR region due to the presence of NLOS signals. It is
unsurprising that a lower NLOS signal power leads to a lower
localization error, as shown in Fig. 8. It can be imagined that
the UE orientation and LED emitting power estimation errors
will have a similar phenomenon, which complies with theorem
3. Moreover, the obtained UE location error scaling w.r.t. SNR
in Fig. 8 complies with corollary 2-1.
3) The Effect of Unknown Emitting Power: The effect of
actual unknown emitting power on VLC localization errors is
presented in Fig. 9, in which the measurement noise variance
is set to be fixed at ω−1= 10−7(which corresponds to a
SNR of around 40 [dB] when WT= 2.2[Watt]). In addition,
we consider the LOS-only case, i.e., ℘nlos = 0. It is shown that
the location and orientation vector estimation errors (in meters)
of the proposed SLLS algorithm as well as their CRLBs are
decreased with actual emitting power, whereas the estimation
error of emitting power is linearly increased with its actual
value. This result complies with corollary 2-1 and 2-2.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
The ratio of NLOS signal strength
10-3
10-2
10-1
100
101
RMSE
X: 0.1
Y: 0.3036
The proposed SLLS-based emitting power estimate
The proposed SLLS-based location estimate
The proposed SLLS-based orientation estimate
Fig. 7. The SLLS-based estimation errors v.s. the NLOS signal power.
-20 0 20 40 60 80
SNR [dB]
10-4
10-3
10-2
10-1
100
101
102
RMSE [m]
[The proposed SLLS-based SPAO]
The ratio of NLOS signal strength
[UE location CRLB]
Fig. 8. UE location errors (with various NLOS power settings) v.s. SNR.
4) The Effect of Localization Area: We assess the SLLS-
based localization performance v.s. the localization area to
reveal the impact of transmission distance. Specifically, we
consider a cubic room with a width ranging from 10 [m] to
100 [m], and the LED sources are uniformly distributed on
the room ceiling. In addition, we consider the LOS-only case
where ℘nlos = 0. It is shown in Fig. 10 that the proposed
SLLS-based localization error and its CRLB are increasing
with the room size. The obtained scaling rule shown in Fig.
10 complies with the result in corollary 1.
5) The Effect of the Number of LEDs: The achieved local-
ization error of our SLLS algorithm (considering ℘nlos = 0)
v.s. the number of LED emitters are given in Fig. 11, where the
number of LED emitters varies from 1 to 80 while the SNR is
fixed at 40 [dB]. It is shown that the achieved localization error
reduces with the number of LEDs, as revealed in corollary 3.
In addition, it indicates that 20 LED transmitters are sufficient
for achieving a satisfactory VLC localization performance.
6) Spatial Distribution of VLP Error: In this simulation, we
reveal the spatial distribution of the UE location error of our
proposed SPAO algorithm, where the NLOS signal strength
ratio is set to be ℘nlos = 0.1and SNR is 40 [dB]. In this
case, the NLOS effect will become the main error source, as
implied in Fig. 8. The achieved UE location errors in 2-D
BINGPENG ZHOU, AN LIU & VINCENT LAU: LIFI-BASED SPAO SCHEME UNDER UNKNOWN POWER EMISSION 11
Emitting power [Watt]
10-2 10-1 100101102103
RMSE
10-5
10-4
10-3
10-2
10-1
100
101The proposed SLLS-based location estimate
The proposed SLLS-based orientation vector estimate
The proposed SLLS-based emitting power estimate
Location CRLB
Emitting power CRLB
Orientation vector CRLB
Fig. 9. The achieved SPAO error v.s. unknown emitting power.
The length of room [m]
101102
RMSE
10-4
10-3
10-2
10-1
100
101
102
Emitting power CRLB
Location CRLB
The proposed SLLS-based location estimate
Orientation vector CRLB
The proposed SLLS-based orientation vector estimate
The proposed SLLS-based emitting power estimate
Fig. 10. The achieved SPAO error v.s. the length of the cubic room.
location space are shown in Fig. 12, where we assume the
UE height is fixed at 0.5 [m] and the UE orientation is set to
be a random direction. We can see that the corner area has a
poor localization performance, as expected, due to the limited
number of visible LED sources. The UE location errors v.s. the
height of the UE are given in Fig. 13, where the UE location
is set to be a random parameter. It is shown that a lower UE
height will lead to a lower localization error due to better UE
sight. When the UE height is larger than 3 meters for a room
of 4 meters height, the localization error will be dramatically
increased due to the bad UE sight.
7) The Impact of Initial Location Point: Since SPAO is a
non-convex optimization problem, the initial point ˆ
x[0] will
affect the result of the proposed SLLS algorithm (as well
as the gradient descent-based optimization baseline 1 and 2).
We shall set the initial point ˆ
x[0] at random within the room
area (rather than resorting to multiple location samples), to
reveal its impact on the achieved localization performance. It
is shown in Fig. 14 that the probability of poor localization
performance is around 0.2. This is because SPAO is essentially
a non-convex problem. If the initial point is far from the
globally optimal solution, the achieved stationary point will
probably be far from the globally optimal one, leading to poor
performance. This phenomenon indicates that an advanced
SPAO algorithm (e.g., resorting to multiple location samples
The number of LEDs
0 10 20 30 40 50 60 70 80
RMSE
10-5
10-4
10-3
10-2
10-1
100
101
102
Orientation vector CRLB
The SLLS-based orientation vector estimate
Location CRLB
The SLLS-based location estimate
The SLLS-based emitting power estimate
Emitting power CRLB
Fig. 11. The impact of the number of LEDs on SPAO performance.
0.2
0.4
8
0.6
UE location RMSE [m]
68
0.8
Y-axis [m]
1
6
4X-axis [m]
4
22
00
Fig. 12. The distribution of UE location errors.
for the initial point as we have adopted) is desired to improve
the reliability of the obtained solution.
8) Convergence Speed: The achieved UE localization error
(considering the LOS-only case) v.s. the CPU time is presented
in Fig. 15. We can see that the proposed SLLS algorithm is
faster in CPU time than the baseline method 1 and 5, due
to our problem-specific update rule design and its quadratic
convergence rate. Although at each iteration step our algorithm
is more slightly complex compared with the GDO baseline 1,
the number of iterations needed by our algorithm is small and
hence the overall CPU time is reduced.
VII. CONCLUSIONS
In this paper, a novel SLLS algorithm with an established
convergence rate is proposed for VLC localization. The pro-
posed SLLS algorithm can be used to achieve a simultane-
ous location and orientation estimate for VLC users under
unknown LED emitting power. It is shown that the proposed
SLLS algorithm outperforms the baseline methods, and it can
achieve a location and orientation error of 0.0048 [m] and
0.17 [deg], respectively, with a SNR of 40[dB].
In addition, the effect of SNR, transmission distance, NLOS
signal strength and the number of LED transmitters on the
VLC localization performance is quantitatively analyzed. It is
shown that for a regular scenario where the SNR is 40 [dB]
12 IN PREPARATION FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2019
0 0.5 1 1.5 2 2.5 3 3.5 4
The height of UE [m]
10-1
100
101
102
UE location RMSE [m]
Fig. 13. UE location errors v.s. the UE height.
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
UE location estiamte error [m]
CDF
Baseline method 1 using a random initial point.
The proposed SLLS algorithm using a random initial point
The proposed SLLS algorithm with a initial point
resorting to multiple location samples.
Baseline method 1 with the initial point resorting to
multiple location samples.
The probability of poor locally optimal solution
of our SLLS algorithm using a random initial point.
Fig. 14. The impact of initial point ˆ
x[0] on the UE localization performance.
and the NLOS signal power is 10% of the overall RSS, our
algorithm can achieve a localization error within (0.1 [m],
0.5 [m]). The obtained closed-form CRLBs on localization
errors cannot only be used as a performance benchmark for
VLP algorithms but also shed lights on the performance limits
of the VLP problem. It als provides theoretical foundations for
the design of VLC localization system optimization strategies
(e.g., resource allocation and LED deployment optimization).
APP EN DI X A
PROO F OF TH EO RE M 2
To prove theorem 2, we first establish the convergence rate
of location update ˆ
x[i], and following that, we establish the
convergence rate of orientation update ˆ
µ[i].
Let us start with the convergence analysis of ˆ
x[i]with
λmin ̸= 0. Let h(•) = vec[hm(•)|∀m∈ΘR]. Applying the
second-order approximation to h(xR,ˆ
µ[i])around xR=ˆ
x[i],
zcan be cast as (56), where τis the higher-order residual
error, Fm(ˆ
x[i],ˆ
µ[i])=∇xRhm(ˆ
x[i],ˆ
µ[i])∇⊤
xRhm(ˆ
x[i],ˆ
µ[i]). In
(56), we use ∇⊤
xRh(ˆ
x[i],ˆ
µ[i])∇xRh(ˆ
x[i],ˆ
µ[i])to approximate
the Hessian matrix for computational ease (only gradient is
needed). Ignoring the high-order residual error, we eventu-
ally arrive at (58). Thus, we can obtain ∥x•
R−x♯
[i+1]∥2=
O(∥x•
R−ˆ
x[i]∥2
2)+O(λmin∥x•
R−ˆ
x[i]∥2), for a sufficiently small
CPU time [s]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
UE location estimate error [m]
10-3
10-2
10-1
100
101Baseline 5
Baseline 1
The proposed SLLS algorithm
Fig. 15. The convergence speed of various VLP methods in CPU time lines.
∥x•
R−ˆ
x[i]∥2. Based on (22), we know ˆ
x[i+1] is a more efficient
update than x♯
[i+1], since it can lead to a more sufficient
decrease in cost function than x♯
[i+1]. Thus, the convergence
rate of ˆ
x[i+1] is asymptotically quadratic as λmin →0.
We shall analyze the convergence of ˆ
µ[i+1]. Based on (15)
we have ∥ˆ
µ[i+1] −µ•
R∥2=∥(G(ˆ
x[i+1]))†z−(G(x•
R))†z∥2.
As per A1, (G(ˆ
x[i+1]))†zis Lipschitz continuous w.r.t. ˆ
x[i+1].
For ∥ˆ
x[i+1] −x•
R∥2=O(∥ˆ
x[i]−x•
R∥2
2)+O(λmin∥x•
R−ˆ
x[i]∥2),
we will thus have that ∥(G(ˆ
x[i+1]))†z−(G(x•
R))†z∥2=
O(∥(G(ˆ
x[i]))†z−(G(x•
R))†z∥2
2) + O(λmin∥(G(ˆ
x[i]))†z−
(G(x•
R))†z∥2)as per continuous mapping theorem. Thus,
∥ˆ
µ[i+1] −µ•
R∥2=O(∥ˆ
µ[i]−µ•
R∥2
2) + O(λmin∥ˆ
µ[i]−µ•
R∥2).
Thus, ˆ
µ[i]is asymptotically quadratic as λmin →0and
i→ ∞, when A3 is satisfied. Hence, theorem 2 is proved.
APPENDIX B
PROO F OF TH EO RE M 3
Let αR= [xR;µR]be the joint variable. We shall first
derive the CRLB for ˆ
αR. Let υRbe the bias of ˆ
αRdue to
the NLOS effect, i.e., υR=E{ˆ
αR} − αR. Thus, the mean
squared error of ˆ
αRcan be formulated as
E{∥ ˆ
αR−αR∥2
2}=E{∥υR∥2
2}+E{∥ ˆ
αR−E{ˆ
αR}∥2
2}
cov( ˆ
αR)
(59)
where cov( ˆ
αR)is the covariance of the estimation error, which
is bounded from above as follows [44],
cov( ˆ
αR)≥traceB(αR),(60)
B(αR) = I(αR)−1,(61)
where B(αR)stands for the CRLB of αR, and I(αR)denotes
the Fisher information matrix (FIM) defined as [44]
I(αR) = −Ez|αR∇2
αRln p(z|αR),(62)
where ∇2
αR(•)is the 2nd-order derivative w.r.t. αR,p(z|αR)
∝exp −1
2ωz−G(xR)µR−ςnlos⊤z−G(xR)µR−ςnlos .
Based on the above model, B(αR)will be eventually
formulated in the following compact form [8], [9], [45],
B(αR) = ωΨ2
RR(αR)−1,(63)
BINGPENG ZHOU, AN LIU & VINCENT LAU: LIFI-BASED SPAO SCHEME UNDER UNKNOWN POWER EMISSION 13
z=h(ˆ
x[i],ˆ
µ[i]) + ∇⊤
xRh(ˆ
x[i],ˆ
µ[i])x•
R−ˆ
x[i]+w(x•
R;ˆ
x[i],ˆ
µ[i]) + τ,(56)
w(x•
R;ˆ
x[i],ˆ
µ[i])∈R|ΘR|= 0.5vec[(x•
R−ˆ
x[i])⊤Fm(ˆ
x[i],ˆ
µ[i])(x•
R−ˆ
x[i])|∀m∈ΘR].(57)
∇xRh(ˆ
x[i],ˆ
µ[i])∇⊤
xRh(ˆ
x[i],ˆ
µ[i]) + λminI3−1∇xRh(ˆ
x[i],ˆ
µ[i])z−h(ˆ
x[i],ˆ
µ[i])+ˆ
x[i]
x♯
[i+1]
−x•
R
2
=
∇⊤
xRh(ˆ
x[i],ˆ
µ[i])†wx•
R;ˆ
x[i],ˆ
µ[i]
O(∥x•
R−ˆ
x[i]∥2
2)
−λmin(x•
R−ˆ
x[i])
2=O∥x•
R−ˆ
x[i]∥2
2+Oλmin∥x•
R−ˆ
x[i]∥2.(58)
where R(αR)denotes the geometric location-domain resolu-
tion matrix of the measurement model [45], [46], given by
R(αR) =
WTJxR(xR,uR)W1
2
TJxR,µR(xR,uR)
W1
2
TJµR,xR(xR,uR)JµR(xR)
,
where FIMs are given by
JxRxR,uR=H(xR)U(uR)U(uR)⊤H(xR)⊤,(64)
JxR,µRxR,uR=H(xR)U(uR)G′(xR),(65)
JµR,xR(xR,uR) = J⊤
xR,µR(xR,uR),(66)
JµRxR=G′(xR)⊤G′(xR),(67)
and H(xR),U(uR),A(xR)and V(xR,uR)is given by (39),
(41), (42) and (43), respectively. Then, based on the structure
of joint variable αR’s CRLB and using Schur complement
[45], the CRLB of xRis eventually given by (36), while the
CRLB of µRis given by
BµRxR,µR=ωΨ2
RG′(xR)⊤V(xR,uR)G′(xR)−1.
In the following, we shall derive the CRLBs of uRand WT,
respectively, based on BµRxR,µR. Based on (6) we have
duR
dµR
=W−1
2
TI3,and dWT
dµR
= 2µR= 2W1
2
TuR.Thus, the
individual CRLBs for uRand WTare cast as
BuRxR,µR=duR
dµR⊤
BµRxR,µRduR
dµR
,(68)
BWTxR,µR=dWT
dµR⊤
BµRxR,µRdWT
dµR
,(69)
which is eventually cast as (37) and (38), respectively.
Combining (59), (60), (68) and (69), theorem 3 is proved.
APPENDIX C
PROO F OF TH EO RE M 4
Let ςlos be the LOS measurement associated with the unbi-
ased estimate of (xR,µR), and let (xtrue,µtrue )be the true val-
ue of (xR,µR). Based on the system model (8), we have z=
ςlos +ςnlos +ϵ, and G(xtrue)µtrue =ςlos . Let (ˆ
xR,ˆ
µR)be the
biased estimate based on z. Then we have G(ˆ
xR)ˆ
µR=ςlos +
ςnlos +ϵ. By applying the first-order expansion to G(ˆ
xR)ˆ
µR
around (xR,µR) = (xtrue,µtrue ), we have (70). It should be
noted that ∥E{ˆ
xR}−xtrue∥2→0and ∥E{ˆ
µR}−µtrue∥2→0
as ∥ςnlos∥2→0. Ignoring the high-order infinitesimal term, we
have ∇xRG(xtrue)µtrue
G(xtrue)⊤⊤ˆ
xR−xtrue
ˆ
µR−µtrue =ςnlos +ϵ. We
have assumed E{ϵ}=0. Combining with the singular-value-
decomposition of the left coefficient matrix, we can obtain
lim
∥ςnlos∥2→0
∥E{ˆ
xR} − xtrue∥2
∥ςnlos∥2
=∥∇xRG(xtrue)µtrue ∥−1
2,
lim
∥ςnlos∥2→0
∥E{ˆ
µR} − µtrue∥2
∥ςnlos∥2
=∥G(xtrue)∥−1
2.
Furthermore, for ˆ
uRand ˆ
WTwe have (71) and (72), where
duR
dµR
=W−1
2
TI3,and dWT
dµR
= 2µR= 2W1
2
TuR.Note that
υxR=∥E{ˆ
xR} − xtrue∥2, and so is υuRas well as υWT.
Thus, theorem 4 is proved.
APPENDIX D
PROO F OF CO ROLLARY 1
Let ρmax = max{ρm|∀m∈ΘR}be the maximum distance
and ρmin = min{ρm|∀m∈ΘR}be the minimum distance.
Then, based on (36) we have
Bmin
xRxR,uR≼BxRxR,uR≼Bmax
xRxR,uR,(73)
where the lower bound Bmin
xRxR,uRis the CRLB associ-
ated with ρmin, which is formed as
Bmin
xRxR,uR=ρ6
min ·SNR−1KxR(xR,uR)−1,(74)
where KxR(xR,uR)denotes the direction information matrix
of xR, and it is obtained via extracting the distance parame-
ter ρmin out of HxRUuRAxRU(uR)⊤HxR⊤.
Note that KxR(xR,uR)is independent of ρmin. Therefore,
Bmin
xRxR,uR∼ O(ρ6
min). Similarly, for Bmax
xRxR,uRwe
have Bmax
xRxR,uR∼ O(ρ6
max). Combining with (73), we
thus have BxRxR,uR∼ O(ρ6
max), as ρmin → ∞.
For the CRLBs of uRand WT, we can obtain (48) and (49)
by following the above process. Thus, corollary 1 is proved.
APPENDIX E
PROO F OF CO ROLLARY 3
As per the structure of H(xR)and U(uR)in (39) and (41),
we known that HxRUuRAxRU(uR)⊤HxR⊤
14 IN PREPARATION FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2019
G(ˆ
xR)ˆ
µR=G(xtrue)µtrue
ςlos
+∇xRG(xtrue)µtrue
G(xtrue)⊤⊤ˆ
xR−xtrue
ˆ
µR−µtrue +O(∥ˆ
xR−xtrue∥2
2+∥ˆ
µR−µtrue∥2
2).(70)
∥E{ˆ
uR} − utrue∥2
2= traceduR
dµR⊤E{ˆ
µR} − µtrueE{ˆ
µR} − µtrue⊤duR
dµR,(71)
∥E{ˆ
WT} − WT∥2
2= tracedWT
dµR⊤E{ˆ
µR} − µtrueE{ˆ
µR} − µtrue⊤dWT
dµR.(72)
can be formed as in the following form,
HxRUuRAxRU(uR)⊤HxR⊤
=
m∈ΘR
HmxRuRAmxRu⊤
RHmxR⊤,(75)
where AmxRis a certain non-singular matrix dependent on
AxR. Hence, if the LEDs are uniformly distributed, we have
HxRUuRAxRU(uR)⊤HxR⊤∼ O(|ΘR|),and
thus (53) is obtained. We will also have (54) and (55) by
following the above process, and thus corollary 4 is proved.
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