2016 6th International Conference on Information Technology for Manufacturing Systems (ITMS 2016) ISBN: 978-1-60595-353-3
1 INTRODUCTION
Location estimation of a mobile station (MS) in wireless communication systems has gained considerable attention since the Federal Communication Commission passed a mandate requiring cellular providers to generate accurate location estimates for Enhanced-911 services [2]. Wireless location as an important public safety feature has created many potential applications to future cellular systems such as: location-sensitive billing, fraud protection, person/asset tracking, fleet management, mobile yellow pages, wireless network design, radio resource management, and intelligent transportation systems [3].
An introduction to the basics of MS localization is given in [4-5]. Conventional geolocation techniques include of-arrival (TOA), time-difference-of-arrival (TDOA), angle-of-arrival (AOA), signal strength (SS) based methods, or a combination of these. Since the hybrid measurements can help to improve the positioning accuracy, these methods such as TOA/SS and TOA/AOA receive more attention.
One of main problems for accurate location estimates in cellular wireless location systems is a Non-Line-of-Sight (NLOS) propagation, when the signal arrives at a base station (BS) from reflections. There is no direct, or line-of-sight (LOS), path. This often happens in an urban environment and may lead
to severe degradations. Several location algorithms and performance analysis for mobile location in a NLOS environment have been addressed in the literature [6-10] based on different assumptions of NLOS propagation. This paper mainly focuses on performance analysis of mobile location in NLOS environments. Since Cramer-Rao lower bound (CRLB) sets a lower limit for the covariance matrix of any unbiased estimate of parameters and determines the physical impossibility of the variance of an unbiased estimator being less than the bound, it is widely used for performance evaluation in signal processing. Although many CRLBs for TOA, TDOA, SS, and TOA/SS localization techniques in NLOS environments have been addressed in the literature [11-14], those researches are based on mathematical models rather than measurement data obtained from field test. Cost259 channel model is developed based on measurements and has a wide application in mobile location [15]. Based on Cost259 channel model, the authors in [1] analyzed the performance of TOA/AOA location technique in a NLOS environments. Unfortunately, there are some mistakes in the CRLB derived in [1]. In this paper, the correct version of the CRLB is provided, and the corresponding simulation results are given.
CRLB of TOA/AOA Based Localization Method in Cost259 Channel
Junhui Zhang
North Engineering Co., LTD of the Electrification Bureau Group, Crcc, Taiyuan, China
Zheng Liu, Jiyan Huang
University of Electronic Science and Technology of china (UESTC), Chengdu, China
ABSTRACT: The determination of Cramer-Rao lower bound (CRLB) as an optimality criterion for the problem of mobile location is a very important issue. Although many CRLBs for TOA, TDOA, SS, and TOA/SS in NLOS environments have been addressed in the literature, those researches are based on mathematical models rather than measurement data obtained from field test. Cost259 channel model is developed based on measurements and has a wide application in mobile location. Based on Cost259 channel model, the authors in [1] analyzed the performance of TOA/AOA localization technique in a NLOS environments. Unfortunately, there are some mistakes in the CRLB derived in [1]. In this paper, the correct version of the CRLB is provided, and the corresponding simulation results are given.
2 SYSTEM MODEL
2.1 Modeling of Signals Measurements
The basic TOA and AOA models are briefly introduced in this section. Assuming that ( , , )x y z is
the position of a MS and ( , , )x y zi i i is the position of
the i th BS in a N -BSs system. The range
measurement ςl
) from the corresponding TOA
measurement tl
) is modeled as:
(
)
(
)
(
)
, ,
2 2 2
, ,
l l l l m l nl
i i i l m l nl
ct n n
x x y y z z n n
ς = =ς + +
= − + − + − + +
) )
(1)
where c is the speed of light, ςl is the true
distance between the MS and BS l, nl m, is range measurement noise subjected to zero-mean Gaussian distribution, and nl nl, is the NLOS error with positive value.
The probability density function (PDF) of nl m, is:
( )
( )
,
2
2 1
exp 2 2
l m m
n n
m m
v
f v f v
σ π σ
= = −
(2) where σm is the standard derivation of ,
l m
n .
Based on the measurements obtained from field test, the PDF of nl nl, in COST259 channel is modeled as:
( )
,
1
exp 0
0 0
l nl
n l l
v v
f v
v
λ λ
− >
=
≤
(3)
where λl =cτl =cτ ς ρm lε , τl is the root mean square
deplay spread from the lth BS to the MS, and τm
is the median value of τl, whose value depends on
various environments. ε is the path loss exponent,
and ρ is the factor of shadow fading.
For AOA measurement, the horizontal AOA measurement θ and the vertical AOA measurement φ can be obtained by:
1 1
, ,
1
tan
t m m
y y
n n n n
x x
θ θ θ θ
θ θ − −
= + + = + +
−
(4)
(
)
(
)
1 1
, 2 2 ,
1 1
tan
t m m
z z
n n n n
x x y y
φ φ φ φ
φ φ −
−
= + + = + +
− + −
where nθ and nφ are NLOS errors caused by
propagation. The PDFs of nθ and nφ are assumed
to be uniformly distributed [1]:
( )
( )
1 2
0
a a
a
n n
v
f v f v
others
θ φ
λ λ
λ
− ≤ ≤
= =
(5)
The model for the measurement noise nθ,m and
,m
nφ of the AOA signal is assumed to be Gaussian
distributed [16]. Thus, the PDFs for the combined noiseless angles ( θt and φt ) and the AOA
measurement noises (nθ,m and nφ,m) can be obtained
by:
(
)
,
2
2
1
( ) exp
2 2
t m
t n
fθ θ
θ θ
υ θ
υ
σ π σ
+
−
= −
(6)
(
)
,
2
2
1
( ) exp
2 2
t m
t n
fφ φ
φ φ
υ φ
υ
σ π σ
+
−
= −
(7)
where σθ and σφ are the standard deviation of the
horizontal AOA measurement and the vertical AOA measurement, respectively.
2.2 Cramer–Rao Lower Bound
It is well known that the CRLB sets a lower limit for the variance or covariance matrix of any unbiased estimate of unknown parameters [17].
The CRLB matrix is defined as the inverse of the Fisher information matrix (FIM) J [17]:
1
CRLB −
=J (8)
The FIM is determined by [3]:
ln ( ; ) ln ( ; ) T
P P
P P
f z f z
E
z z
υ υ
∂ ∂
= ⋅
∂ ∂
r,θ r,θ
J (9)
where fr,θ( ; )υ zP is the joint PDF of TOA and
AOA measurements, and
[
, ,]
T[
, , , ,]
Tp t t t t
z = xθ φ = x y zθ φ is a unknown vector to
be estimated. CRLB is used as an optimality criterion for the problem of parameter estimation. It provides a benchmark to evaluate the performance of any unbiased estimator and determines the physical impossibility of the variance of an unbiased estimator being less than the bound. Assuming that
p z
) is an estimate of
p
z from any unbiased
estimator, the following constraint must be satisfied:
(
)(
)
(
T)
p p p p
E z) −z z) −z ≥CRLB
3 PROPOSED CRLB
The Cramer-Rao Lower Bound (CRLB) in [1] should be derived correctly. It can be seen from (23)[1], (24)[1] and (25)[1] that the probability density functions for the AOA measurements do not consider the AOA measurement noise. In fact, there always exists AOA measurement noise in the real system. Another problem is that the authors let
/ 0
l zP
λ
∂ ∂ = in (27)[1]. However, / 0
l zP
λ
∂ ∂ ≠ since
l
λ contains
P
z . Thus, the CRLB, as given by
(27)[1]~(30)[1], is incorrect. We derive the correct version of the CRLB as follows.
The pdf of the measured horizontal angles from the AOA signal can be acquired by convoluting (5) and (6) as:
(
)
, 2 2 ( ) ( ) ( ) 1 exp 2 2 2 1 2 t m a a n n t aa t a t
a
f f w f w dw
w dw θ θ θ θ υλ υ λ θ θ θ θ υ υ θ σ
λ π σ
υ λ θ υ λ θ
λ σ σ
+∞ + −∞ + − = − − = − + − − − = −
∫
∫
Φ Φ (11)where ( ) 1 exp 2
2 2 w dw υ υ π −∞ = −
∫
Φ . Similarly, the
PDF of the measured vertical angles can be acquired by convoluting (5) and (7) as:
,
( ) ( ) ( )
1 2
t nm n
a t a t
a
fφ fφ φ w fφ w dw
φ φ
υ υ
υ λ φ υ λ φ
λ σ σ
+∞ + −∞ = − + − − − = −
∫
Φ Φ (12)It can be seen from [1] that the PDF of TOA measurements is:
( )
2 2 1 exp 2 ll m l m
r
l l l m l
v v
f v ζ σ ζ σ
λ λ λ σ λ
− − = − + − Φ (13) Substituting (11-13) into (9), the logarithm of the joint conditional probability density function now becomes:
, 1
2
2
2
ln ( ; ) ln ( ; ) ( ; , ) ( ; , )
ln(4 ) ln
ln
ln ln
2 l N
P r l
l
a t a t
a
a t a t
l l m l
l
l l
f z f fθ θ fφ φ
θ θ
θ θ
φ φ
φ φ
υ υ υ θ υ φ
υ λ θ υ λ θ
λ
σ σ
υ λ φ υ λ φ
σ σ
υ ς σ υ
λ
λ λ
=
= Π ⋅ ⋅
+ − − − = − + − + − − − + − − + − + − + + Φ Φ Φ Φ Φ
rθ x x t x t
1
N
l m
l m l
ς σ σ λ = − −
∑
(14)The partial derivative of with lnfr,θ( ; )υ zP respect
to zP can be obtained by:
,
ln ( ; )P P f z z υ ∂ =
∂ rθ
1 2 2 1 1 2 1 1 exp exp 2 2 1 2 1 exp 2
t a t a t
P
a t a t
a t a t
t P z z θ θ θ θ θ θ θ θ θ φ φ φ φ φ φ
θ υ λ θ υ λ θ
σ σ
π σ
υ λ θ υ λ θ
σ σ
υ λ φ υ λ φ
φ σ σ π σ υ − − ∂ + − − − = − ∂ − − + − ⋅ − − − + − − − ∂ + − ∂ − ⋅ − Φ Φ Φ Φ 2 2 1 1 2 2 3 1 exp 2 1 1
a t a t
N
l l m l l m
l m l P m l
l l m l
l P l P
l l
z
z z
φ
φ φ
λ φ υ λ φ
σ σ
υ ς σ υ ς σ
σ λ σ λ
υ ς σ λ ς
λ λ λ λ − = − + − − − − ∂ − + − ⋅ − ∂ − ∂ ∂ + − − + ∂ ∂
∑
Φ Φ(15)
where
l l m
P m l
z
υ ς σ
σ λ − ∂ − ∂ Φ 2 2
1 1 1
exp 2 2
l l m l m l
m l m zP l zP
υ ς σ ς σ λ
σ λ σ λ
π − ∂ ∂ = − − − + ∂ ∂
(
/1000)
11000 m l l l P P c z z ε
τ ε ς ρ
λ − ς
∂ ∂ = ∂ ∂
[
]
1 T ll l l
P l
x x y y z z
z ς ς ∂ = − − − ∂
(
)
(
)
(
)
(
)
2 1 12 2 2
1
1 1 1
1 0
T t
P
x x y y
z x x y y x x x x
θ −
∂ − = − ∂ − + − − −
(
)
(
)
(
)
(
)
(
)
(
) (
)
(
)
(
)
(
)
2 21 1 1 1
2 2 2 2 2 3/ 2
1 1 1 1 1
t P
x x y y z z x x
z x x y y z z x x y y
φ − + − − − −
∂ = ∂ − + − + − − + −
(
) (
)
(
)
(
)
(
)
(
)
(
)
1 13/ 2 2 2
2 2
1 1
1 1
1
T
z z y y
x x y y
x x y y
− − −
− + −
− + −
Substituting (15) into (9), the CRLB for TOA/AOA in Cost259 channel can be obtained.
Remark 1: It can be seen from the sentence below (19)[1] that λl =c⋅τ ς ρm lε . In fact, the unit of ςl in
(19)[1] are kilometers [17]. However, the unit of ςl
in [1] are meters. In order to hold consistent, we let
(
/1000)
l c m l
ε
λ = ⋅τ ς ρ with ςl in meters.
4 SIMULATION RESULTS
noted that Mean CRLB
1
1
10 log N ( )i
i trace
N =
= ϒ
∑
indecibels, where N =1000 indicates the number of
runs in the simulations and ϒi is the CRLB which
can be obtained by (30)[1]. In the three TOA with one AOA case, the true position of the MS is located at ( 160⋅Um , 160⋅Um , 220 50− ⋅Um ), where Um
represents a uniform distribution function within the interval [0 1]. In the two TOA with one AOA case, the true position of the MS is located at (500⋅Um, 500⋅Um ). We set the parameters of the AOA
[image:4.612.77.272.392.542.2]measurements as follow:
Table 1. The parameters of AOA measurements.
a
λ
θ
σ
φ
σ
Case1 5o 1o 1o
Case2 10o 1o 1o
Case3 15o 1o 1o
Case4 10o 0.1o 0.1o
The other parameters have the same value as the parameters in [1]. Fig.1 and Fig.2 show that the corrected CRLB changes for different levels of AOA measurement noise while the CRLB derived in [] always keeps the same value. This implies that the proposed CRLB can provide the more accurate performance evaluation.
1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-7 20
22 24 26 28 30 32 34 36 38 40
Median Value of NLOS Noise/s
M
e
a
n
C
R
L
B
/d
B
CRLB in [1] corrected CRLB-case1 corrected CRLB-case2 corrected CRLB-case3 corrected CRLB-case4
Figure 1. Comparison between the corrected CRLB and the CRLB in [1] for the three TOA and one AOA case.
1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-7 20
30 40 50 60 70 80
Median Value of NLOS Noise/s
M
e
a
n
C
R
L
B
/d
B
CRLB in [1] corrected CRLB-case1 corrected CRLB-case2 corrected CRLB-case3 corrected CRLB-case4
Figure 2. Comparison between the corrected CRLB and the
5 CONCLUSIONS
The performance of TOA/AOA estimation in Cost259 channel has been analyzed in this paper. The best estimation accuracy is evaluated in terms of CRLB. Although the authors in [1] analyzed the performance of TOA/AOA location technique in a NLOS environments. Unfortunately, there are some mistakes in the CRLB derived in [1]. In this paper, the correct version of the CRLB is provided, and the corresponding simulation results are given. Since the Cost259 channel is developed based on measurements, the derived CRLB in this paper can be widely used for performance evaluation of mobile location in a practice system.
ACKNOWLEDGEMENTS
This work was supported by the Open Research Fund of north engineering Co., LTD of the electrification bureau group.
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![Figure 1. Comparison between the corrected CRLB and the CRLB in [1] for the three TOA and one AOA case](https://thumb-us.123doks.com/thumbv2/123dok_us/303092.1031601/4.612.77.272.392.542/figure-comparison-corrected-crlb-crlb-toa-aoa-case.webp)