# Localization Accuracy of Range-Only Sensors with Additive and Multiplicative Noise

Authors: Xiufang Shi; Rong Zhang; Zaiyue Yang
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

The localization accuracy of range-only sensors with both additive and multiplicative noises is investigated. The main contributions consist of three parts. First, the Cramer-Rao Lower Bound (CRLB) of range-only localization system is derived, based on which the area of the uncertain ellipse about estimation is proposed as the metric to evaluate the localization accuracy. In addition, the analytical relationships between target-sensor distance and localization accuracy and between noise and localization accuracy are derived. Second, the metric is utilized to evaluate the localization accuracy for three important regular patterns of sensor deployment, that is, triangle, square, and hexagon. Two aspects of localization accuracy have been examined and discussed, respectively, for three patterns by numerical methods, including (a) the geometric distribution of localization accuracy and (b) the average localization accuracy. Both theoretical and numerical results show that the multiplicative noise will influence significantly the localization accuracy. Third, on a lab-scale ultrasonic range-only sensor system the existence of additive and multiplicative noises is verified. It is also shown that the experimental localization accuracy is close to the previously analyzed theoretical accuracy.


---

## Body

## 1. Introduction

It is well known that target localization plays an important role in military and security [1], which is also the foundation of other advanced tasks, for example, target tracking. With the great development of electronics in recent years, wireless sensor networks (WSN) based target localization systems have attracted increasing research interests [1–9]. Several kinds of sensors are widely used to collect the necessary information, for example, range-only sensors [2, 10], angle of arrival (AOA) sensors [10–12], time of arrival (TOA) sensors [10], and time difference of arrival (TDOA) sensors [10]. In a WSN based localization system, the target localization problem is essentially an estimation problem. That is, a number of sensors are deployed to collect the position-related data of the target; then the data are sent to the leader node to estimate the position of the target.In the context of statistical signal processing, the Cramer-Rao Lower Bound (CRLB) is regarded as the most important benchmark to assess the estimation accuracy of an unbiased estimator [13–16]. Therefore, CRLB is widely used in the localization system to evaluate the localization accuracy [1, 17–21]. It is also of great significance in sensor selection and deployment, motion coordination, and dynamic sensor collaboration [7, 11, 22]. With regard to different measurement models, CRLB can be derived in various forms. For example, CRLB of the widely used range-only localization system is a matrix of target-sensor azimuth angle in previous literature [10].However, it is notable that this result is derived under the assumption that the measurements consist of only additive noise [10]. In other words, this assumption ignores the multiplicative noise that indeed reflects the effect of target-sensor distance on measurement error. However, [2] argues that in a lot of occasions multiplicative noise cannot be neglected, and the measurement error will increase significantly with the increase of target-sensor distance. In fact, the experimental results in [2] even show that multiplicative noise is 3 to 4 times larger than additive noise. Unfortunately, very few studies take multiplicative noise into consideration when deriving CRLB and assessing localization accuracy for the range-only localization system.In this paper, the localization accuracy problem of range-only sensors with both additive and multiplicative noises is investigated. The main contributions can be summarised as below.(1)
Theoretically, CRLB of range-only localization system is deduced based on a more general measurement model with both additive and multiplicative noise. Then, a metric is proposed based on CRLB to evaluate the localization accuracy, and three important properties of this metric are derived. The properties indicate, surprisingly, that shorter target-sensor distance does not always lead to better localization accuracy, and proper distance will yield the best accuracy under a certain situation.
(2)
Numerically, the localization accuracy for three regular patterns of sensor deployment, that is, triangle, square, and hexagon, has been evaluated by the proposed metric. For each pattern, two aspects of localization accuracy have been examined and discussed, including (a) geometric distribution of localization accuracy and (b) average localization accuracy. To the best of our knowledge, there is no such extensive analysis about the localization accuracy for regular deployment patterns. Numerical results show that the multiplicative noise will influence significantly the localization accuracy and also provide some important guidelines for optimal sensor deployment.
(3)
Experimentally, the existence of multiplicative noise on a lab-scale ultrasonic range-only sensor system is verified and the corresponding parameters are computed. Then, the experimental localization accuracy is calculated based on real data, which is very close to the theoretical accuracy.The rest of this paper is organized as follows. In Section2, the measurement model of range-only localization system with both additive and multiplicative noise is introduced. Then, CRLB, the metric, and its properties are derived in Section 3. Section 4 presents the numerical analysis and discussion about the localization accuracy in terms of the proposed metric in regular deployment patterns. The experiment is presented in Section 5 and the conclusion is drawn in Section 6.


## 2. Measurement Model

Without loss of generality, we will consider the localization problem for a single stationary target in two-dimensional field in this paper. The presented results can be readily extended to three-dimensional cases and are also the foundation of target tracking tasks.Let the localization system consist ofN homogeneous range-only sensors, whose positions are known a priori. Let the coordinate of the ith sensor be (
x
i
,
y
i
), and let the coordinate of the target be (
x
,
y
). Then, the true distance between the target and sensor i can be expressed as
(1)
r
i
=
(
x
i
-
x
)
2
+
(
y
i
-
y
)
2
.Being different from most previous studies [10], which adopt the sensor measurement with only additive noise, a more general model with both additive and multiplicative noise [2] is considered here:
(2)
z
i
=
(
1
+
ω
i
)
r
i
+
υ
i
,
where z
i denotes the measurement of the ith sensor and ω
i and υ
i are, respectively, multiplicative noise and additive noise in the ith sensor.Assume thatω
i and υ
i are independent white Gaussian noises, that is, ω
i
~
N
(
μ
ω
,
σ
ω
2
), υ
i
~
N
(
μ
υ
,
σ
υ
2
), which are also independent of those in the other sensors. From (2), the total measurement noise of sensor i is n
i
=
r
i
ω
i
+
υ
i. Clearly, it is still white Gaussian noise with the following mean and variance:
(3)
μ
i
=
r
i
μ
ω
+
μ
υ
,
(4)
σ
i
2
=
r
i
2
σ
ω
2
+
σ
υ
2
.Therefore, the measurementz
i also follows normal distribution:
(5)
z
i
~
N
(
μ
i
+
r
i
,
σ
i
2
)
.
Then, consider the entire localization system that consists of N sensors. Similar to (5), the measurement vector of N sensors follows an N-dimensional normal distribution:
(6)
Z
(
θ
)
~
N
(
μ
(
θ
)
,
C
(
θ
)
)
,
where θ
=
[
x
,
y
]
T, Z
(
θ
)
=
[
z
1
,
z
2
,
…
,
z
N
]
T, μ
(
θ
)
=
[
μ
1
+
r
1
,
μ
2
+
r
2
,
…
,
μ
N
+
r
N
]
T, and C
(
θ
)
=
diag
⁡
{
σ
1
2
,
σ
2
2
,
…
,
σ
N
2
}.Remark 1.
From (3) and (4), it can be found that both the mean μ
i and the variance σ
i
2 associated with sensor i are closely related to target-sensor distance r
i. In particular, the variance σ
i
2 will grow in a rapid speed with the increase of r
i.For a target at(
x
,
y
), the probability density function (PDF) of measurement of sensor i can be readily obtained according to (5):
(7)
p
(
z
i
∣
θ
)
=
1
2
π
σ
i
2
exp
⁡
[
-
(
z
i
-
r
i
-
μ
i
)
2
2
σ
i
2
]
.
Since the measurements of different sensors are independent, the joint PDF of the entire localization system with N sensors can be written as
(8)
p
(
Z
∣
θ
)
=
∏
i
=
1
N
‍
p
(
z
i
∣
x
,
y
)
.


## 3. CRLB and Metric

In this section, CRLB of range-only localization system with both additive and multiplicative noise is derived and a metric is proposed to evaluate the localization accuracy. Then, three important analytical properties about the relationships between target-sensor distance and localization accuracy and between noise and localization accuracy are derived.


### 3.1. Derivation of CRLB

It is well known that CRLB is the lower bound for the variance of any unbiased estimate. Thus, it is widely utilized in practice as an important benchmark to evaluate the estimation accuracy of an estimator [13–16, 23, 24]. The inverse of CRLB is the so-called Fisher Information Matrix (FIM), which stands for the amount of information contained in data.By referring the definition of FIM in [13], an entry of FIM denoted as F
l
k is
(9)
F
l
k
=
-
E
[
∂
2
ln
⁡
⁡
p
(
Z
∣
θ
)
∂
θ
l
∂
θ
k
]
,
where l
,
k
=
1,2, θ
1
=
x, and θ
2
=
y. Therefore, the corresponding FIM can be expressed as
(10)
F
=
[
F
11
F
12
F
21
F
22
]
=
-
E
[
∂
2
⁡
ln
⁡
⁡
(
p
(
Z
∣
x
,
y
)
)
∂
x
2
∂
2
ln
⁡
⁡
(
p
(
Z
∣
x
,
y
)
)
∂
x
∂
y
∂
2
ln
⁡
⁡
(
p
(
Z
∣
x
,
y
)
)
∂
x
∂
y
∂
2
ln
⁡
⁡
(
p
(
Z
∣
x
,
y
)
)
∂
y
2
]
.Alternatively, since the measurement given in (6) is Gaussian with nonzero mean, the entry of FIM can also be computed as below [13]:
(11)
F
l
k
=
[
∂
μ
(
θ
)
∂
θ
l
]
T
C
-
1
(
θ
)
[
∂
μ
(
θ
)
∂
θ
k
]
+
1
2
t
r
[
C
-
1
(
θ
)
∂
C
(
θ
)
∂
θ
l
C
-
1
(
θ
)
∂
C
(
θ
)
∂
θ
k
]
.
Because the measurements of different sensors are independent, the complete FIM of the entire localization system is the sum of the individual FIM associated with each sensor [13]. Then, taking (1), (3), and (4) into (11), we have
(12)
F
11
=
∑
i
=
1
N
1
σ
i
4
{
σ
i
2
[
(
1
+
μ
ω
)
x
i
-
x
r
i
]
2
+
1
2
[
2
σ
ω
2
(
x
i
-
x
)
]
2
}
,
F
12
=
F
21
=
∑
i
=
1
N
1
σ
i
4
[
σ
i
2
(
1
+
μ
ω
)
2
(
x
i
-
x
)
r
i
(
y
i
-
y
)
r
i
h
h
h
h
h
h
+
2
σ
ω
4
(
x
i
-
x
)
(
y
i
-
y
)
(
x
i
-
x
)
r
i
]
,
F
22
=
∑
i
=
1
N
1
σ
i
4
{
σ
i
2
[
(
1
+
μ
ω
)
y
i
-
y
r
i
]
2
+
1
2
[
2
σ
ω
2
(
y
i
-
y
)
]
2
}
.Letφ
i denote the azimuth angle of sensor i relative to the target. Obviously, we have cos
⁡
⁡
φ
i
=
(
x
i
-
x
)
/
r
i, sin
⁡
φ
i
=
(
y
i
-
y
)
/
r
i. As a result, the complete FIM of N sensors can be rewritten into the following compact form:
(13)
F
=
∑
i
=
1
N
g
i
[
cos
⁡
2
⁡
φ
i
cos
⁡
⁡
φ
i
sin
⁡
φ
i
cos
⁡
⁡
φ
i
sin
⁡
φ
i
sin
2
⁡
φ
i
]
,
where g
i is a function of noise and target-sensor distance of sensor i
:
(14)
g
i
=
1
σ
i
4
[
σ
i
2
(
1
+
μ
ω
)
2
+
2
σ
ω
4
r
i
2
]
.
At last, the corresponding CRLB can be expressed as
(15)
CRLB
=
F
-
1
.Remark 2.
Similar to previous studies, for example, [1, 2, 12], in this paper we will assume that all estimates of the target position are unbiased. Therefore, the derived CRLB corresponds to the lower bound that the variance of position estimate can achieve. That is, it can be regarded as the representation of localization accuracy. However, since CRLB is a 2
×
2 matrix with very complex entries, it is difficult to use and to analyze in the evaluation of localization accuracy. Therefore, a scalar metric will be proposed in the next subsection, which can replace the complicated CRLB and approximately represent the localization accuracy.


### 3.2. A Novel Metric

Geometrically, CRLB defines an ellipse of estimation errors in bothx-axis and y-axis [10]. Let λ
1 and λ
2 denote the eigenvalues of CRLB, and consequently λ
1 and λ
2 represent the length of x-axis and y-axis of the ellipse [10]. Then, the area of the ellipse is π
λ
1
λ
2, which reflects the localization accuracy. That is, a smaller ellipse area indicates a more accurate estimation of target position. For simplicity, the constant π is dropped and let
(16)
ℜ
=
λ
1
λ
2
be the metric of localization accuracy. Note that
(17)
λ
1
λ
2
=
det
⁡
{
CRLB
}
=
1
det
⁡
{
F
}
,
(18)
det
⁡
{
F
}
=
1
2
∑
i
=
1
N
∑
j
=
1
N
g
i
g
j
sin
2
(
φ
i
j
)
,
where φ
i
j
=
φ
i
-
φ
j is the intersection angle between the line of sensor i to target and the line of sensor j to target. Consequently, taking (17) and (18) into (16) yields
(19)
ℜ
=
1
det
⁡
{
F
}
=
2
∑
i
=
1
N
∑
j
=
1
N
g
i
g
j
sin
2
(
φ
i
j
)
.Remark 3.
In some previous studies [10, 19], λ
1
λ
2 is also used as a metric, which is the so-called D-Criterion. Since it is monotonic to ℜ, it can reflect the variation trend of localization accuracy to some extent. However, it is important to notice that the D-Criterion is nonlinear to the area of ellipse of estimation error; thus, the scale to quantify the localization accuracy will be distorted. In assessing the average localized accuracy, it would be useful to have a scale which is linear to the area of ellipse. This is achieved by using ℜ as the metric.Remark 4.
From [10], ℜ of additive noise measurement model is
(20)
ℜ
=
2
∑
i
=
1
N
∑
j
=
1
N
(
1
/
σ
υ
4
)
sin
2
(
φ
i
j
)
.
It is clear that (20) is a special case of (19). For a localization system with only additive noise, multiplicative noise does not exist; that is, σ
ω
2
=
0, μ
ω
=
0. Therefore, g
i
=
1
/
σ
υ
2 from (14) and ℜ retreats to a more simple form that only depends on the intersection angle φ
i
j.


### 3.3. Analytical Properties ofℜ

Analytical properties ofℜ will be derived in this part, which indeed reveal very interesting results about the relationships between ℜ and the target-sensor distance, and between ℜ and the noise. In the following paragraphs, we will first derive the relationship between ℜ and g
i and then the relationship between g
i and r
i and at last the relationship between g
i and σ
ω
2. The first property can be directly observed from (19).Property 1.
ℜ is a strictly monotonically decreasing function of g
i.Property 2.
The relationship betweeng
i and target-sensor distance r
i depends on the mean and variance of multiplicative noise as follows.(i)
If2
σ
ω
2
<
(
1
+
μ
ω
)
2, g
i is a strictly monotonically decreasing function of r
i.
(ii)
If2
σ
ω
2
≥
(
1
+
μ
ω
)
2, g
i is a strictly monotonically increasing function of r
i when r
i
2
<
T
r, and g
i is a strictly monotonically decreasing function of r
i when r
i
2
>
T
r. g
i achieves the maximum value g
max
⁡
=
(
2
σ
ω
2
+
1
)
2
/
8
σ
υ
2
σ
ω
2 at r
i
2
=
T
r, where
(21)
T
r
=
σ
υ
2
(
2
σ
ω
2
-
(
1
+
μ
ω
)
2
)
σ
ω
2
(
2
σ
ω
2
+
(
1
+
μ
ω
)
2
)
.Proof.
The proof of Property2 is straightforward. The partial derivative of g
i with respect to r
i can be written as follows:(22)
∂
g
i
∂
r
i
=
(
{
-
σ
ω
2
[
2
σ
ω
2
+
(
1
+
μ
ω
)
2
]
r
i
2
h
h
h
h
+
σ
υ
2
[
2
σ
ω
2
-
(
1
+
μ
ω
)
2
]
}
2
σ
ω
2
r
i
)
×
(
(
σ
υ
2
+
σ
ω
2
r
i
2
)
3
)
-
1
.
The denominator of ∂
g
/
∂
r
i is always positive. If 2
σ
ω
2
<
(
1
+
μ
ω
)
2, the numerator of ∂
g
/
∂
r
i is negative; then ∂
g
/
∂
r
i
<
0 and g
i is a strictly monotonically decreasing function of r
i. Otherwise, if 2
σ
ω
2
≥
(
1
+
μ
ω
)
2, the monotonicity of g
i is related to r
i. When r
i
2
<
T
r, ∂
g
/
∂
r
i
>
0, thus g
i is a strictly monotonically increasing function of r
i; when r
i
2
>
T
r, ∂
g
/
∂
r
i
<
0, thus g
i is a strictly monotonically decreasing function of r
i. g
i achieves the maximum at T
r. The proof is completed.Figure1 illustrates the profiles of g
i over r
i with different σ
ω
2. In accordance with the two aforementioned properties, the relationship between r
i and ℜ can be readily conducted.The profiles ofg
i over r
i with different variance of multiplicative noise. Let additive noise be υ
~
N
(
0
,
1
) and let multiplicative noise satisfy (a) 2
σ
ω
2
<
(
1
+
μ
ω
)
2; (b) 2
σ
ω
2
≥
(
1
+
μ
ω
)
2. The x and y axes are r
i and g
i.
(a)
σ
ω
2
=
0.25,  μ
ω
=
0
(b)
σ
ω
2
=
1,  μ
ω
=
0Remark 5.
Due to the existence of multiplicative noise,ℜ becomes more complicated and is affected by both target-sensor distance and intersection angle. For the localization system with only additive noise, it is widely accepted only that the good angular diversity can lead to a low ℜ, namely, an accurate estimation of target position [10]. However, for the localization system with both additive and multiplicative noises according to Property 2, both a good angular diversity and a proper distance are necessary to yield the best localization accuracy. Surprisingly, a shorter distance does not necessarily lead to a better accuracy, and r
i
2
=
T
r will yield the best accuracy under a certain circumstance.Property 3.
(i) The relationship betweeng
i and additive noise: g
i is a strictly monotonically decreasing function of σ
υ
2.
(ii) The relationship betweeng
i and multiplicative noise:(a)
ifr
i
2
>
4
σ
υ
2
/
(
1
+
μ
ω
)
2, g
i is a strictly monotonically decreasing function of σ
ω
2;
(b)
ifr
i
2
≤
4
σ
υ
2
/
(
1
+
μ
ω
)
2, g
i is a strictly monotonically increasing function of σ
ω
2 when σ
ω
2
>
T
σ
ω
2, and g
i is a strictly monotonically decreasing function of σ
ω
2 when σ
ω
2
≤
T
σ
ω
2, where
(23)
T
σ
ω
2
=
(
1
+
μ
ω
)
2
σ
υ
2
4
σ
υ
2
-
(
1
+
μ
ω
)
2
r
i
2
.Proof.
The proof of Property3 is straightforward.(i)
The partial derivative ofg
i with respect to σ
υ
2 is
(24)
∂
g
i
∂
σ
υ
2
=
-
(
1
+
μ
ω
)
2
σ
i
4
-
4
σ
ω
4
r
i
2
σ
i
6
<
0
.
Then g
i is a strictly monotonically decreasing function of σ
υ
2.
(ii)
The partial derivative ofg
i with respect to σ
ω
2 can be written as follows:
(25)
∂
g
i
∂
σ
ω
2
=
r
i
2
σ
i
6
[
σ
ω
2
(
4
σ
υ
2
-
(
1
+
μ
ω
)
2
r
i
2
)
-
(
1
+
μ
ω
)
2
σ
υ
2
]
.In (25), r
i
2
/
σ
i
6 is always positive. If r
i
2
>
4
σ
υ
2
/
(
1
+
μ
ω
)
2, σ
ω
2
(
4
σ
υ
2
-
(
1
+
μ
ω
)
2
r
i
2
) is negative; then ∂
g
i
/
∂
σ
ω
2
<
0 and g
i is a strictly monotonically decreasing function of σ
ω
2. Otherwise, if r
i
2
≤
4
σ
υ
2
/
(
1
+
μ
ω
)
2, the monotonicity of g
i is related to σ
ω
2. When σ
ω
2
>
T
σ
ω
2, ∂
g
i
/
∂
σ
ω
2
>
0, thus g
i is a strictly monotonically increasing function of σ
ω
2; when σ
ω
2
≤
T
σ
ω
2, ∂
g
i
/
∂
σ
ω
2
≤
0, thus g
i is a strictly monotonically decreasing function of σ
ω
2.
The proof is completed.Figure2 illustrates the profiles of g
i over σ
ω
2 with different target-sensor distance r
i. In accordance with Properties 1 and 3, the relationship between σ
ω
2 and ℜ can be readily conducted.The profiles ofg
i over σ
ω
2 with different target-sensor distance r
i. Let additive noise be υ
~
N
(
0,1
) and let the target-sensor distance r
i satisfy (a) r
i
2
>
4
σ
υ
2
/
(
1
+
μ
ω
)
2; (b) r
i
2
≤
4
σ
υ
2
/
(
1
+
μ
ω
)
2. The x and y axes are σ
ω
2 and g
i.
(a)
r
i
=
10
(b)
r
i
=
1Remark 6.
From Property3(ii), surprisingly, we can conclude that the increase of multiplicative noise may not necessarily reduce the localization accuracy. When r
i
2
≤
4
σ
υ
2
/
(
1
+
μ
ω
)
2, with the further increase of σ
ω
2, g
i will also increase and the accuracy will be improved. However, two points should be noticed. First, lim
⁡
σ
ω
2
→
∞
g
i
=
2
/
r
i
2; therefore, g
i is upper bounded by the constant 2
/
r
i
2 even if σ
ω
2 is infinite. Second, in practice the target-sensor distance r
i is much larger than σ
υ
2; therefore, r
i
2
>
4
σ
υ
2
/
(
1
+
μ
ω
)
2 always holds and case (b) barely exists in applications.


## 3.1. Derivation of CRLB

It is well known that CRLB is the lower bound for the variance of any unbiased estimate. Thus, it is widely utilized in practice as an important benchmark to evaluate the estimation accuracy of an estimator [13–16, 23, 24]. The inverse of CRLB is the so-called Fisher Information Matrix (FIM), which stands for the amount of information contained in data.By referring the definition of FIM in [13], an entry of FIM denoted as F
l
k is
(9)
F
l
k
=
-
E
[
∂
2
ln
⁡
⁡
p
(
Z
∣
θ
)
∂
θ
l
∂
θ
k
]
,
where l
,
k
=
1,2, θ
1
=
x, and θ
2
=
y. Therefore, the corresponding FIM can be expressed as
(10)
F
=
[
F
11
F
12
F
21
F
22
]
=
-
E
[
∂
2
⁡
ln
⁡
⁡
(
p
(
Z
∣
x
,
y
)
)
∂
x
2
∂
2
ln
⁡
⁡
(
p
(
Z
∣
x
,
y
)
)
∂
x
∂
y
∂
2
ln
⁡
⁡
(
p
(
Z
∣
x
,
y
)
)
∂
x
∂
y
∂
2
ln
⁡
⁡
(
p
(
Z
∣
x
,
y
)
)
∂
y
2
]
.Alternatively, since the measurement given in (6) is Gaussian with nonzero mean, the entry of FIM can also be computed as below [13]:
(11)
F
l
k
=
[
∂
μ
(
θ
)
∂
θ
l
]
T
C
-
1
(
θ
)
[
∂
μ
(
θ
)
∂
θ
k
]
+
1
2
t
r
[
C
-
1
(
θ
)
∂
C
(
θ
)
∂
θ
l
C
-
1
(
θ
)
∂
C
(
θ
)
∂
θ
k
]
.
Because the measurements of different sensors are independent, the complete FIM of the entire localization system is the sum of the individual FIM associated with each sensor [13]. Then, taking (1), (3), and (4) into (11), we have
(12)
F
11
=
∑
i
=
1
N
1
σ
i
4
{
σ
i
2
[
(
1
+
μ
ω
)
x
i
-
x
r
i
]
2
+
1
2
[
2
σ
ω
2
(
x
i
-
x
)
]
2
}
,
F
12
=
F
21
=
∑
i
=
1
N
1
σ
i
4
[
σ
i
2
(
1
+
μ
ω
)
2
(
x
i
-
x
)
r
i
(
y
i
-
y
)
r
i
h
h
h
h
h
h
+
2
σ
ω
4
(
x
i
-
x
)
(
y
i
-
y
)
(
x
i
-
x
)
r
i
]
,
F
22
=
∑
i
=
1
N
1
σ
i
4
{
σ
i
2
[
(
1
+
μ
ω
)
y
i
-
y
r
i
]
2
+
1
2
[
2
σ
ω
2
(
y
i
-
y
)
]
2
}
.Letφ
i denote the azimuth angle of sensor i relative to the target. Obviously, we have cos
⁡
⁡
φ
i
=
(
x
i
-
x
)
/
r
i, sin
⁡
φ
i
=
(
y
i
-
y
)
/
r
i. As a result, the complete FIM of N sensors can be rewritten into the following compact form:
(13)
F
=
∑
i
=
1
N
g
i
[
cos
⁡
2
⁡
φ
i
cos
⁡
⁡
φ
i
sin
⁡
φ
i
cos
⁡
⁡
φ
i
sin
⁡
φ
i
sin
2
⁡
φ
i
]
,
where g
i is a function of noise and target-sensor distance of sensor i
:
(14)
g
i
=
1
σ
i
4
[
σ
i
2
(
1
+
μ
ω
)
2
+
2
σ
ω
4
r
i
2
]
.
At last, the corresponding CRLB can be expressed as
(15)
CRLB
=
F
-
1
.Remark 2.
Similar to previous studies, for example, [1, 2, 12], in this paper we will assume that all estimates of the target position are unbiased. Therefore, the derived CRLB corresponds to the lower bound that the variance of position estimate can achieve. That is, it can be regarded as the representation of localization accuracy. However, since CRLB is a 2
×
2 matrix with very complex entries, it is difficult to use and to analyze in the evaluation of localization accuracy. Therefore, a scalar metric will be proposed in the next subsection, which can replace the complicated CRLB and approximately represent the localization accuracy.


## 3.2. A Novel Metric

Geometrically, CRLB defines an ellipse of estimation errors in bothx-axis and y-axis [10]. Let λ
1 and λ
2 denote the eigenvalues of CRLB, and consequently λ
1 and λ
2 represent the length of x-axis and y-axis of the ellipse [10]. Then, the area of the ellipse is π
λ
1
λ
2, which reflects the localization accuracy. That is, a smaller ellipse area indicates a more accurate estimation of target position. For simplicity, the constant π is dropped and let
(16)
ℜ
=
λ
1
λ
2
be the metric of localization accuracy. Note that
(17)
λ
1
λ
2
=
det
⁡
{
CRLB
}
=
1
det
⁡
{
F
}
,
(18)
det
⁡
{
F
}
=
1
2
∑
i
=
1
N
∑
j
=
1
N
g
i
g
j
sin
2
(
φ
i
j
)
,
where φ
i
j
=
φ
i
-
φ
j is the intersection angle between the line of sensor i to target and the line of sensor j to target. Consequently, taking (17) and (18) into (16) yields
(19)
ℜ
=
1
det
⁡
{
F
}
=
2
∑
i
=
1
N
∑
j
=
1
N
g
i
g
j
sin
2
(
φ
i
j
)
.Remark 3.
In some previous studies [10, 19], λ
1
λ
2 is also used as a metric, which is the so-called D-Criterion. Since it is monotonic to ℜ, it can reflect the variation trend of localization accuracy to some extent. However, it is important to notice that the D-Criterion is nonlinear to the area of ellipse of estimation error; thus, the scale to quantify the localization accuracy will be distorted. In assessing the average localized accuracy, it would be useful to have a scale which is linear to the area of ellipse. This is achieved by using ℜ as the metric.Remark 4.
From [10], ℜ of additive noise measurement model is
(20)
ℜ
=
2
∑
i
=
1
N
∑
j
=
1
N
(
1
/
σ
υ
4
)
sin
2
(
φ
i
j
)
.
It is clear that (20) is a special case of (19). For a localization system with only additive noise, multiplicative noise does not exist; that is, σ
ω
2
=
0, μ
ω
=
0. Therefore, g
i
=
1
/
σ
υ
2 from (14) and ℜ retreats to a more simple form that only depends on the intersection angle φ
i
j.


## 3.3. Analytical Properties ofℜ

Analytical properties ofℜ will be derived in this part, which indeed reveal very interesting results about the relationships between ℜ and the target-sensor distance, and between ℜ and the noise. In the following paragraphs, we will first derive the relationship between ℜ and g
i and then the relationship between g
i and r
i and at last the relationship between g
i and σ
ω
2. The first property can be directly observed from (19).Property 1.
ℜ is a strictly monotonically decreasing function of g
i.Property 2.
The relationship betweeng
i and target-sensor distance r
i depends on the mean and variance of multiplicative noise as follows.(i)
If2
σ
ω
2
<
(
1
+
μ
ω
)
2, g
i is a strictly monotonically decreasing function of r
i.
(ii)
If2
σ
ω
2
≥
(
1
+
μ
ω
)
2, g
i is a strictly monotonically increasing function of r
i when r
i
2
<
T
r, and g
i is a strictly monotonically decreasing function of r
i when r
i
2
>
T
r. g
i achieves the maximum value g
max
⁡
=
(
2
σ
ω
2
+
1
)
2
/
8
σ
υ
2
σ
ω
2 at r
i
2
=
T
r, where
(21)
T
r
=
σ
υ
2
(
2
σ
ω
2
-
(
1
+
μ
ω
)
2
)
σ
ω
2
(
2
σ
ω
2
+
(
1
+
μ
ω
)
2
)
.Proof.
The proof of Property2 is straightforward. The partial derivative of g
i with respect to r
i can be written as follows:(22)
∂
g
i
∂
r
i
=
(
{
-
σ
ω
2
[
2
σ
ω
2
+
(
1
+
μ
ω
)
2
]
r
i
2
h
h
h
h
+
σ
υ
2
[
2
σ
ω
2
-
(
1
+
μ
ω
)
2
]
}
2
σ
ω
2
r
i
)
×
(
(
σ
υ
2
+
σ
ω
2
r
i
2
)
3
)
-
1
.
The denominator of ∂
g
/
∂
r
i is always positive. If 2
σ
ω
2
<
(
1
+
μ
ω
)
2, the numerator of ∂
g
/
∂
r
i is negative; then ∂
g
/
∂
r
i
<
0 and g
i is a strictly monotonically decreasing function of r
i. Otherwise, if 2
σ
ω
2
≥
(
1
+
μ
ω
)
2, the monotonicity of g
i is related to r
i. When r
i
2
<
T
r, ∂
g
/
∂
r
i
>
0, thus g
i is a strictly monotonically increasing function of r
i; when r
i
2
>
T
r, ∂
g
/
∂
r
i
<
0, thus g
i is a strictly monotonically decreasing function of r
i. g
i achieves the maximum at T
r. The proof is completed.Figure1 illustrates the profiles of g
i over r
i with different σ
ω
2. In accordance with the two aforementioned properties, the relationship between r
i and ℜ can be readily conducted.The profiles ofg
i over r
i with different variance of multiplicative noise. Let additive noise be υ
~
N
(
0
,
1
) and let multiplicative noise satisfy (a) 2
σ
ω
2
<
(
1
+
μ
ω
)
2; (b) 2
σ
ω
2
≥
(
1
+
μ
ω
)
2. The x and y axes are r
i and g
i.
(a)
σ
ω
2
=
0.25,  μ
ω
=
0
(b)
σ
ω
2
=
1,  μ
ω
=
0Remark 5.
Due to the existence of multiplicative noise,ℜ becomes more complicated and is affected by both target-sensor distance and intersection angle. For the localization system with only additive noise, it is widely accepted only that the good angular diversity can lead to a low ℜ, namely, an accurate estimation of target position [10]. However, for the localization system with both additive and multiplicative noises according to Property 2, both a good angular diversity and a proper distance are necessary to yield the best localization accuracy. Surprisingly, a shorter distance does not necessarily lead to a better accuracy, and r
i
2
=
T
r will yield the best accuracy under a certain circumstance.Property 3.
(i) The relationship betweeng
i and additive noise: g
i is a strictly monotonically decreasing function of σ
υ
2.
(ii) The relationship betweeng
i and multiplicative noise:(a)
ifr
i
2
>
4
σ
υ
2
/
(
1
+
μ
ω
)
2, g
i is a strictly monotonically decreasing function of σ
ω
2;
(b)
ifr
i
2
≤
4
σ
υ
2
/
(
1
+
μ
ω
)
2, g
i is a strictly monotonically increasing function of σ
ω
2 when σ
ω
2
>
T
σ
ω
2, and g
i is a strictly monotonically decreasing function of σ
ω
2 when σ
ω
2
≤
T
σ
ω
2, where
(23)
T
σ
ω
2
=
(
1
+
μ
ω
)
2
σ
υ
2
4
σ
υ
2
-
(
1
+
μ
ω
)
2
r
i
2
.Proof.
The proof of Property3 is straightforward.(i)
The partial derivative ofg
i with respect to σ
υ
2 is
(24)
∂
g
i
∂
σ
υ
2
=
-
(
1
+
μ
ω
)
2
σ
i
4
-
4
σ
ω
4
r
i
2
σ
i
6
<
0
.
Then g
i is a strictly monotonically decreasing function of σ
υ
2.
(ii)
The partial derivative ofg
i with respect to σ
ω
2 can be written as follows:
(25)
∂
g
i
∂
σ
ω
2
=
r
i
2
σ
i
6
[
σ
ω
2
(
4
σ
υ
2
-
(
1
+
μ
ω
)
2
r
i
2
)
-
(
1
+
μ
ω
)
2
σ
υ
2
]
.In (25), r
i
2
/
σ
i
6 is always positive. If r
i
2
>
4
σ
υ
2
/
(
1
+
μ
ω
)
2, σ
ω
2
(
4
σ
υ
2
-
(
1
+
μ
ω
)
2
r
i
2
) is negative; then ∂
g
i
/
∂
σ
ω
2
<
0 and g
i is a strictly monotonically decreasing function of σ
ω
2. Otherwise, if r
i
2
≤
4
σ
υ
2
/
(
1
+
μ
ω
)
2, the monotonicity of g
i is related to σ
ω
2. When σ
ω
2
>
T
σ
ω
2, ∂
g
i
/
∂
σ
ω
2
>
0, thus g
i is a strictly monotonically increasing function of σ
ω
2; when σ
ω
2
≤
T
σ
ω
2, ∂
g
i
/
∂
σ
ω
2
≤
0, thus g
i is a strictly monotonically decreasing function of σ
ω
2.
The proof is completed.Figure2 illustrates the profiles of g
i over σ
ω
2 with different target-sensor distance r
i. In accordance with Properties 1 and 3, the relationship between σ
ω
2 and ℜ can be readily conducted.The profiles ofg
i over σ
ω
2 with different target-sensor distance r
i. Let additive noise be υ
~
N
(
0,1
) and let the target-sensor distance r
i satisfy (a) r
i
2
>
4
σ
υ
2
/
(
1
+
μ
ω
)
2; (b) r
i
2
≤
4
σ
υ
2
/
(
1
+
μ
ω
)
2. The x and y axes are σ
ω
2 and g
i.
(a)
r
i
=
10
(b)
r
i
=
1Remark 6.
From Property3(ii), surprisingly, we can conclude that the increase of multiplicative noise may not necessarily reduce the localization accuracy. When r
i
2
≤
4
σ
υ
2
/
(
1
+
μ
ω
)
2, with the further increase of σ
ω
2, g
i will also increase and the accuracy will be improved. However, two points should be noticed. First, lim
⁡
σ
ω
2
→
∞
g
i
=
2
/
r
i
2; therefore, g
i is upper bounded by the constant 2
/
r
i
2 even if σ
ω
2 is infinite. Second, in practice the target-sensor distance r
i is much larger than σ
υ
2; therefore, r
i
2
>
4
σ
υ
2
/
(
1
+
μ
ω
)
2 always holds and case (b) barely exists in applications.


## 4. Numerical Analysis ofℜ under Different Sensor Deployment Patterns

In practice, it is usually desirable for target localization systems to deploy sensors in regular patterns. The regular deployment patterns outperform other irregular ones in many aspects, such as the convenience of deployment, a high degree of coverage, and connectivity [25]. Nevertheless, the localization accuracy under different regular patterns is rarely discussed. To this end, the localization accuracy of range-only localization system with both additive and multiplicative noises will be studied in this section. Three most important regular patterns will be investigated, that is, triangle, square, and hexagon, which indeed are the basic elements to form a large localization system. From (13), (18), and (19), it is clear that ℜ is a complicated function of noise, intersection angle φ
i
j, and target-sensor distance r
i. In addition, for a given deployment pattern the sensors are fixed; thus, both φ
i
j and r
i are complicated functions of the target position (
x
,
y
). That is, ℜ becomes an extremely complicated function of noise and (
x
,
y
). As a result, theoretical analysis of ℜ with respect to the variations of noise and (
x
,
y
) seems impossible. Therefore, numerical study will be conducted in this section. Due to page limitation, the numerical experiments will be carried out only for several typical parameter settings, which, however, are able to reveal the primary properties of ℜ.


### 4.1. Distribution ofℜ

In this subsection, the distribution ofℜ inside the three regular patterns, that is, triangle, square, and hexagon, will be investigated. Since the effect of additive noise on ℜ is relatively simple, we will mainly focus on the impact of multiplicative noise. To this end, let the variance of additive noise be constant for simplicity, that is, σ
υ
2
=
1, and assume that both additive and multiplicative noises are zero-mean white Gaussian noise; that is, μ
ω
=
μ
υ
=
0. The work in [2] indicates that in practice the distance between sensors is much larger than noise disturbance; therefore, let the side length of regular patterns be a
=
100. As the experimental results in [2] also suggest that multiplicative noise is 3 to 4 times larger than additive noise, let the variance of multiplicative noise be σ
ω
2
=
0
,
1
,
3, and the corresponding distributions of ℜ are shown in Figures 4, 5, and 6, respectively.In Figure4 with only additive noise, it is clear that the lowest ℜ always locates at the inner center of the three regular patterns. That is, the inner center achieves the best localization accuracy. On the other hand, the worst localization accuracy is located near the sensors for all patterns. This is because under additive noise, the target-sensor distance is irrelevant to localization accuracy, and the point with the best angular diversity, that is, the inner center, yields the best accuracy.Figures5 and 6 show that the distribution of ℜ under both additive and multiplicative noises is completely different from that under additive noise. For all patterns, the points with the best localization accuracy are located at the active sensors, while the points with poor accuracy are located at the inner centers and the midpoints on each side. It is clear that, due to the existence of multiplicative noise, the target-sensor distance significantly influences the distribution of ℜ. For more details, the profiles of ℜ along some representative lines will be examined next.Without loss of generality, the square will be studied and similar results can readily be obtained for triangle and hexagon. Let us look at the side lineA
B, the diagonal line A
C, and midline M
N of the square in Figure 3(b), and the corresponding profiles of ℜ are shown in Figures 7, 8, and 9. It is clear that the minimal points of ℜ are driven from the central points to sides and corners with the increase of σ
ω
2. In addition, the shape of profile will be maintained when σ
ω
2 is larger than a very small value, which also suggests that the shape of the distribution of ℜ with other σ
ω
2 should be similar to the shape of Figures 5 and 6.Three kinds of regular deployment patterns, each vertex means a sensor node.
(a)
Triangle
(b)
Square
(c)
HexagonDistribution ofℜ under three kinds of regular patterns with σ
ω
2
=
0 and σ
υ
2
=
1.
(a)
Triangle
(b)
Square
(c)
HexagonDistribution ofℜ under three kinds of regular patterns with σ
ω
2
=
1 and σ
υ
2
=
1.
(a)
Triangle
(b)
Square
(c)
HexagonDistribution ofℜ under three kinds of regular patterns with σ
ω
2
=
3 and σ
υ
2
=
1.
(a)
Triangle
(b)
Square
(c)
HexagonFigure 7
The profiles ofℜ under the effect of multiplicative noise with target moving along A
B. The x-axis and y-axis are the distance between target and sensor A and ℜ.Figure 8
The profiles ofℜ under the effect of multiplicative noise with target moving along A
C. The x-axis and y-axis are the distance between target and sensor A and ℜ.Figure 9
The profiles ofℜ under the effect of multiplicative noise with target moving along M
N. The x-axis and y-axis are the distance between target and M and ℜ.


### 4.2. Evaluation of Averageℜ

In many localization tasks, it is required to monitor a certain area by a certain number of sensors. Consequently, there arises a question of what the best regular pattern is to deploy these sensors. To solve this problem, it is important to investigate the average localization accuracy of different patterns.Suppose that, in a given areaS, the total number of deployed sensors is N
a; then, the area that a single sensor can cover is S
0
=
S
/
N
a. For regular deployment patterns, S
0 satisfies [25]
(26)
S
0
=
S
p
N
p
·
N
s
,
where S
p is the area of a regular pattern, N
p stands for the number of sensors that a pattern contains, and N
s denotes the number of patterns that a sensor connects. As we can see in Figure 10, N
p
=
3, N
s
=
6 for triangle, N
p
=
4, N
s
=
4 for square, and N
p
=
6, N
s
=
3 for hexagon.The extension of three kinds of regular deployment patterns.
(a)
Triangle
(b)
Square
(c)
HexagonLet their side lengths bea
3,  a
4, and a
6, respectively. Then, we have
(27)
S
p
3
=
3
4
a
3
2
,
S
p
4
=
a
4
2
,
S
p
6
=
3
3
2
a
6
2
,
where S
p
3,  S
p
4, and S
p
6 denote the area of triangle, square, and hexagon, respectively. Consequently, the side length of each regular pattern can be derived as follows:
(28)
a
3
=
2
3
3
S
0
,
a
4
=
S
0
,
a
6
=
4
3
9
S
0
.Letℜ
- denote the average ℜ inside a certain regular pattern. Figure 11 illustrates the profiles of ℜ
- of three regular patterns under different situations, where σ
υ
2
=
1 and the x-axis is S
0 and y-axis is ℜ
-. The left-top subfigure shows the situation with only additive noise. It is clear that ℜ
- of each pattern remains constant despite the increase of S
0, because ℜ depends only on the angular diversity other than the target-sensor distance. On the other hand, other subfigures confirm that ℜ
- increases with the increase of S
0 under both additive and multiplicative noises, which is consistent with our common sense. Figure 11 also indicates that the hexagon and the triangle always yield the lowest and highest ℜ
-, respectively, under all situations. That is, deploying the sensors in a hexagon pattern will lead to the best localization accuracy.Figure 11
The profiles ofℜ
- with different σ
ω
2. The x-axis and y-axis are S
0 and ℜ
-.However, though the hexagon pattern obtains the lowestℜ
-, it does not mean that the hexagon pattern is the best way to deploy sensors in practice. It is notable that, for the three deployment patterns, the numbers of active sensors are different, that is, N
p
3
=
3, N
p
4
=
4, N
p
6
=
6, which leads to different cost, including energy, communication, and computation. Therefore, a new index is desired to represent the normalized average localization accuracy that a single active sensor can contribute. As the lower ℜ
- corresponds to the better accuracy, this index can be given as
(29)
ℑ
=
ℜ
-
·
N
p
.
The comparison of ℑ of these regular patterns is illustrated in Figure 12. From the left-top subfigure with only additive noise, the hexagon pattern has the smallest ℑ; that is, deploying the sensors in hexagon will yield the best normalized average accuracy. However, if both additive and multiplicative noises are considered and the variance of multiplicative noise is comparable with that of additive noise, the results are completely different. That is, the triangle pattern owns the best normalized average localization accuracy, while the hexagon pattern yields the worst.Figure 12
The profiles ofℑ with different σ
ω
2. The x-axis and y-axis are S
0 and ℑ.Remark 7.
In practice, the localization accuracy and other factors may be combined together for optimal sensor deployment. Different weights can be utilized to reflect the preference of designer under different situations. However, the same methodology can be applied to obtain the tradeoff between localization accuracy and other concerns.


## 4.1. Distribution ofℜ

In this subsection, the distribution ofℜ inside the three regular patterns, that is, triangle, square, and hexagon, will be investigated. Since the effect of additive noise on ℜ is relatively simple, we will mainly focus on the impact of multiplicative noise. To this end, let the variance of additive noise be constant for simplicity, that is, σ
υ
2
=
1, and assume that both additive and multiplicative noises are zero-mean white Gaussian noise; that is, μ
ω
=
μ
υ
=
0. The work in [2] indicates that in practice the distance between sensors is much larger than noise disturbance; therefore, let the side length of regular patterns be a
=
100. As the experimental results in [2] also suggest that multiplicative noise is 3 to 4 times larger than additive noise, let the variance of multiplicative noise be σ
ω
2
=
0
,
1
,
3, and the corresponding distributions of ℜ are shown in Figures 4, 5, and 6, respectively.In Figure4 with only additive noise, it is clear that the lowest ℜ always locates at the inner center of the three regular patterns. That is, the inner center achieves the best localization accuracy. On the other hand, the worst localization accuracy is located near the sensors for all patterns. This is because under additive noise, the target-sensor distance is irrelevant to localization accuracy, and the point with the best angular diversity, that is, the inner center, yields the best accuracy.Figures5 and 6 show that the distribution of ℜ under both additive and multiplicative noises is completely different from that under additive noise. For all patterns, the points with the best localization accuracy are located at the active sensors, while the points with poor accuracy are located at the inner centers and the midpoints on each side. It is clear that, due to the existence of multiplicative noise, the target-sensor distance significantly influences the distribution of ℜ. For more details, the profiles of ℜ along some representative lines will be examined next.Without loss of generality, the square will be studied and similar results can readily be obtained for triangle and hexagon. Let us look at the side lineA
B, the diagonal line A
C, and midline M
N of the square in Figure 3(b), and the corresponding profiles of ℜ are shown in Figures 7, 8, and 9. It is clear that the minimal points of ℜ are driven from the central points to sides and corners with the increase of σ
ω
2. In addition, the shape of profile will be maintained when σ
ω
2 is larger than a very small value, which also suggests that the shape of the distribution of ℜ with other σ
ω
2 should be similar to the shape of Figures 5 and 6.Three kinds of regular deployment patterns, each vertex means a sensor node.
(a)
Triangle
(b)
Square
(c)
HexagonDistribution ofℜ under three kinds of regular patterns with σ
ω
2
=
0 and σ
υ
2
=
1.
(a)
Triangle
(b)
Square
(c)
HexagonDistribution ofℜ under three kinds of regular patterns with σ
ω
2
=
1 and σ
υ
2
=
1.
(a)
Triangle
(b)
Square
(c)
HexagonDistribution ofℜ under three kinds of regular patterns with σ
ω
2
=
3 and σ
υ
2
=
1.
(a)
Triangle
(b)
Square
(c)
HexagonFigure 7
The profiles ofℜ under the effect of multiplicative noise with target moving along A
B. The x-axis and y-axis are the distance between target and sensor A and ℜ.Figure 8
The profiles ofℜ under the effect of multiplicative noise with target moving along A
C. The x-axis and y-axis are the distance between target and sensor A and ℜ.Figure 9
The profiles ofℜ under the effect of multiplicative noise with target moving along M
N. The x-axis and y-axis are the distance between target and M and ℜ.


## 4.2. Evaluation of Averageℜ

In many localization tasks, it is required to monitor a certain area by a certain number of sensors. Consequently, there arises a question of what the best regular pattern is to deploy these sensors. To solve this problem, it is important to investigate the average localization accuracy of different patterns.Suppose that, in a given areaS, the total number of deployed sensors is N
a; then, the area that a single sensor can cover is S
0
=
S
/
N
a. For regular deployment patterns, S
0 satisfies [25]
(26)
S
0
=
S
p
N
p
·
N
s
,
where S
p is the area of a regular pattern, N
p stands for the number of sensors that a pattern contains, and N
s denotes the number of patterns that a sensor connects. As we can see in Figure 10, N
p
=
3, N
s
=
6 for triangle, N
p
=
4, N
s
=
4 for square, and N
p
=
6, N
s
=
3 for hexagon.The extension of three kinds of regular deployment patterns.
(a)
Triangle
(b)
Square
(c)
HexagonLet their side lengths bea
3,  a
4, and a
6, respectively. Then, we have
(27)
S
p
3
=
3
4
a
3
2
,
S
p
4
=
a
4
2
,
S
p
6
=
3
3
2
a
6
2
,
where S
p
3,  S
p
4, and S
p
6 denote the area of triangle, square, and hexagon, respectively. Consequently, the side length of each regular pattern can be derived as follows:
(28)
a
3
=
2
3
3
S
0
,
a
4
=
S
0
,
a
6
=
4
3
9
S
0
.Letℜ
- denote the average ℜ inside a certain regular pattern. Figure 11 illustrates the profiles of ℜ
- of three regular patterns under different situations, where σ
υ
2
=
1 and the x-axis is S
0 and y-axis is ℜ
-. The left-top subfigure shows the situation with only additive noise. It is clear that ℜ
- of each pattern remains constant despite the increase of S
0, because ℜ depends only on the angular diversity other than the target-sensor distance. On the other hand, other subfigures confirm that ℜ
- increases with the increase of S
0 under both additive and multiplicative noises, which is consistent with our common sense. Figure 11 also indicates that the hexagon and the triangle always yield the lowest and highest ℜ
-, respectively, under all situations. That is, deploying the sensors in a hexagon pattern will lead to the best localization accuracy.Figure 11
The profiles ofℜ
- with different σ
ω
2. The x-axis and y-axis are S
0 and ℜ
-.However, though the hexagon pattern obtains the lowestℜ
-, it does not mean that the hexagon pattern is the best way to deploy sensors in practice. It is notable that, for the three deployment patterns, the numbers of active sensors are different, that is, N
p
3
=
3, N
p
4
=
4, N
p
6
=
6, which leads to different cost, including energy, communication, and computation. Therefore, a new index is desired to represent the normalized average localization accuracy that a single active sensor can contribute. As the lower ℜ
- corresponds to the better accuracy, this index can be given as
(29)
ℑ
=
ℜ
-
·
N
p
.
The comparison of ℑ of these regular patterns is illustrated in Figure 12. From the left-top subfigure with only additive noise, the hexagon pattern has the smallest ℑ; that is, deploying the sensors in hexagon will yield the best normalized average accuracy. However, if both additive and multiplicative noises are considered and the variance of multiplicative noise is comparable with that of additive noise, the results are completely different. That is, the triangle pattern owns the best normalized average localization accuracy, while the hexagon pattern yields the worst.Figure 12
The profiles ofℑ with different σ
ω
2. The x-axis and y-axis are S
0 and ℑ.Remark 7.
In practice, the localization accuracy and other factors may be combined together for optimal sensor deployment. Different weights can be utilized to reflect the preference of designer under different situations. However, the same methodology can be applied to obtain the tradeoff between localization accuracy and other concerns.


## 5. Experimental Verification

A lab-scale experiment is set up to illustrate, first, the existence of additive noise and multiplicative noise and, then, the profiles ofℜ of a moving target. The ultrasonic range-only sensors are used in this experiment, which consists of three modules: core microprocessor Atmegal128L, wireless communication module CC2420, and ultrasonic sensor module URM37 V3.2.In order to show the existence of additive and multiplicative noises and compute the corresponding means and variances, a 0.05 m × 0.05 m × 0.1 m rectangular target is placed in the front of an ultrasonic range-only sensor from 0.1 m to 2 m for every 0.1 m. At each point, 600 distance readings are recorded. Based on (3) and (4) and following the least square fitting procedure detailed in [2], we can readily obtain for additive noise μ
υ
=
-
0.00072489, σ
υ
2
=
0.04401 and for multiplicative noise μ
ω
=
0.01597, σ
ω
2
=
0.07251. It is clear that the variance of multiplicative noise is even larger than that of additive noise. However, the sensor is not unbiased. Figure 13 shows 300 distance readings when the true target-sensor distance is 1.2 m. We can see that the existence of multiplicative noise and additive noise will bring biases to the distance measurements.Figure 13
Distance measurements, true target-sensor distance is 1.2 m.In order to show the profiles ofℜ of a moving target, we set up a 1.2 m × 1.2 m square deployment with 4 ultrasonic range-only sensors for the ease of manipulating, as shown in Figure 3(b). The experimental testbed is shown in Figure 14. Now, we will move a round target with diameter of 0.1 m along diagonal line A
C and midline M
N, respectively, for every 0.1 m. At each point, 250 sets of distance readings will be recorded. For each set of readings, it consists of the distance measurements of 4 sensors, which can be used to readily estimate the target location. Consequently, the empirical variance matrix of localization error can be obtained for each point, as well as its determinant that is plotted in Figures 15 and 16. Meanwhile, ℜ also can be computed based on (16) for each point and plotted in Figures 15 and 16, with σ
υ
2
=
0.04401 and σ
ω
2
=
0.07251. It is clear that the experimental profiles are close to the theoretical ones. It is also noticed that the determinant of variance matrix is constantly larger than ℜ for all points. Because ℜ is derived from CRLB that is the lower bound of any unbiased estimator, ℜ only reflects the lower bound of unbiased localization. In addition, the ultrasonic range-only sensor itself is biased, which may also increase the variance.Figure 14
Experimental testbed consisting of four ultrasonic range-only sensors and a round target with radius as 0.05 m.Figure 15
The profiles of experimental localization accuracy andℜ for a target moving along diagonal line A
C.Figure 16
The profiles of experimental localization accuracy andℜ for a target moving along midline M
N.


## 6. Conclusion

This paper studies the issue of localization accuracy for range-only sensors with both additive and multiplicative noises. The CRLB of the localization system and a novel metricℜ of CRLB are derived. Then, three analytical properties of ℜ are derived, which are helpful for us to understand the underlying relationships between target-sensor distance and localization accuracy, as well as between noise and localization accuracy. In particular, two unexpected results about the relationships are found.Then, the localization accuracy for three regular deployment patterns, including triangle, square, and hexagon, is evaluated numerically in terms ofℜ. It shows that the existence of multiplicative noise will significantly change the geometric distribution of localization accuracy. The average localization accuracy is also examined, and the results indicate that the hexagon always has the best average accuracy, no matter whether multiplicative noise exists or not. However, if the average accuracy is normalized by the number of active sensors, the results suggest that the triangle would be the best regular pattern to deploy the sensors.At last, lab-scale experiment is carried out to illustrate the existence of additive and multiplicative noises. It is also shown that the experimental localization accuracy is close toℜ, which nicely verifies the previous theoretical analysis.



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*Source: 956895-2014-03-25.xml*