Probabilistic based Optimal Node Localization in Wireless Sensor Networks

International Journal of Computer Networks & Communications (IJCNC) Vol.14, No.3, May 2022
DOI: 10.5121/ijcnc.2022.14306 91
PROBABILISTIC BASED OPTIMAL NODE
LOCALIZATION IN WIRELESS SENSOR NETWORKS
Souparnika Jadhav and Dr. Nagesh K. N
Department of Electronics & Communication Engineering,
Nagarjuna College of Engineering & Technology Bangaluru, India
ABSTRACT
Localization is one of the most important technologies for many applications in wireless sensor networks
(WSNs). Node localization is the process of discovering the exact location of the node. If the number of
nodes and network size increase, it becomes very arduous to localize the nodes whose result leads to
complexity and path loss. In this paper, we proposed an approach called probabilistic based optimal node
localization to obtain the location of node in the WSNs. This approach provides an enhanced channel path-
loss model by capturing the features of the additive noise in WSN. In addition, the complexity has been
minimized by discovering a lower bound of the non-convex function. The problem of non-convex
optimization and subsequent nonlinear is solved with the help of relaxation to achieve a sub-optimal
solution. Simulation results show that our proposed localization approach has got better performance for
considered scenario settings.
KEYWORDS
Wireless Sensor Networks (WSN), Sensor Nodes (SNs), Node localization, Anchor Nodes (ANs), Root-
Mean-Square Error (RMSE).
1. INTRODUCTION
WSN contains a set of reasonable and mini sensors that are semantically distributed over an area
to evaluate some physical parameters and monitor different conditions and have several
pragmatic implementation areas like target tracking, agriculture, and precision [1-3]. These SNs
(sensor nodes) require evaluating their coordinates with fewer resource necessities for most of the
applications. The SNs can place their coordinates by utilizing an integrated GPS (Global-
Positioning system). Anyways, it’s practically not feasible to incorporate the GPS in entire
sensors because of their cost and size. Another method is utilizing the localization algorithm in
many ANs (anchor nodes) with incorporated GPS that will not know to define their coordinates.
Node localization is the process of detecting the accurate location of the node. SN’s are deployed
randomly with the help of an airplane in an AOI (Area of interest) called a forest. If there any
number of anchors are available in a network, the given inputs are anchor locations while other
inputs are based on some measurement method, and the process of node localization is shown in
figure1. The most significant method is to determine the location of a node in GPS. When a
network consists of a larger number of nodes, this technique becomes costliest and power-
consuming. Node localization methods [4] for WSNs are based on the measurement of more than
one and one physical-parameters of transmitted radio-signal like BNs and RNs. Their parameter
types includes RSS (received-signal strength), AOA (Angle of Arrival), TOA (time-of Arrival),
and TDOA (time-difference of arrival). There exists a trade-off between the accuracy of
localization and complexity of the implementation method and the RSS-based method gives

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lower cost and easier implementation. Two commonly utilized parameters like LLS (Linearized-
least-squares) and ML estimator (Maximum-likelihood) [5-6]. When the measurement error is
known, then the estimator of MI is asymptotically optimal. Anyways, localization problems
formed as an estimator of ML that has a closed-form solution and an iterative solver is needed.
The formed ML-estimator is non-convex and its performance is dependent on the preliminary
given for the iterative solver. Afterward, poor initialization leads to a bad estimator. Furthermore,
because of non-convexity, discovering ML-estimator is difficult.
Figure 1. Node localization process
A huge number of localization algorithms have been represented to resolve various localization
issues [7]. These given algorithms are estimated more flexible so they can work in different
diverse outdoor and indoor topologies and scenarios. The localization algorithms are parted into
two types range free and range-based algorithms. The location of unidentified nodes is calculated
with distance among unknown and anchor SNs. They use range metrics like RSSI (Received-
Signal Strength-Indication), angle of arrival, and time of arrival [8-9]. In addition, the algorithms
like centroid [10] and Ad-Hoc positioning method [11] generate the utilization of simple
techniques related to some connectivity to localize the sensor node. They require the presence of
a beacon-signal by AN in a medium. Among all, range based-algorithms are utilized and desired
over range free-algorithms [12].
In order to design lower complicated algorithms, different types of bio-inspired algorithms have
been introduced for the range-based technique [13], whereas in paper [14], they rendered node-
based localization technique that formed on PSO (Particle-swarm-optimization) [15] that limited
for swarm optimization to discover the food. This type of algorithm represented better outcomes,
but implementation inclined to obtain the local optimum, which outcomes in the convergence of
premature. They implemented CS-based node localization in WSNs in [16]. This is due to tuning
parameters in the algorithm of CS that ease the computation process. Presently, the modified
version of CS was introduced by [17] that improvised the rate of the conventional-CS algorithm.
They use to improvise the search process of mutation probability.
This paper mainly intends to node localization algorithm in WSNs that is basically based on a
channel-path loss model with GD (Gaussian-Distribution). Anyways, the noise doesn’t follow
GD because of more than one heterogeneity source [18]. However, the Gaussian model can’t
represent measured noise which leads to inappropriate node localization algorithms in the WSNs.
Based on our knowledge; WSN doesn’t find the practical channel of a path-loss model with a
non-Gaussian node in node localization. Based on the above assumptions, we introduced an
improvised RSS-based node-localization algorithm called the probabilistic-based optimal node
localization (PONL) approach to obtain node location in the WSNs. Explicitly; noise is
introduced by a non-Gaussian that is followed by the parameter estimator of empirical noise.
Afterward, we formulate a node localization algorithm like an optimization problem to obtain the
performance of ML. The non-convex and subsequent non-linear optimization problems are
resolved with the help of semi-definite to achieve an optimal solution. Lastly, both experimental
outcomes and simulation demonstrate the performance of PONLover the other localization
algorithms. This paper represents in such a way that section-2 represents a literature survey
related to the previous paper based on node localization in WSN, section-3 represents the

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proposed methodology based on the PSDP model, section-4 represents our experimental outcome
that outperforms other existing node localization algorithms, and finally concludes our work.
2. LITERATURE SURVEY
In this section, different localization algorithms have been represented. In paper [19], the author
addressed the problem of positioning in WSNs as an adversarial situation. They also proposed a
method called VM (Verifiable-Multi-Iteration) for location verification. ANs are called verifiers.
This technique secures the confirmation and estimation of unknown location nodes in malicious
nodes. They devised a method for securing the positioning in SNs called SPINE. By utilizing this
node technique are capable to find it securely. The limitation of this technique has a larger
number of verifiers that are required to perform the verifiable multi-literation. Whereas in paper
[20], the author improvised a secure node localization method called SLM (secure-localization-
method) for a wide range of SNS. The SLM was more robust than the VM but the process of
SLM was more sophisticated than the VM that led to more power consumption.
In [21], the author introduced a new method to discover the node localization in a secure way that
utilized mobile-BS (base-station) and hidden concepts. Furthermore, by utilizing this technique,
one can validate the location of the unknown nodes, whereas [22], emphasized the requirement
for node localization and recommended the extension of existing SLM. The novel method is
called as secure-based enhanced localization method. This method was the understanding against
the attacks of distance reduction but also the attacks of distance enlargement and it gave a more
accurate location of nodes.
In paper [23], author introduced a method to minimize the localization problem with the help of
the BPSO algorithm for the node localization in the WSN. Each undefined node performs the
localization under distance measurement from 3 or more AN. The node achieves localization by
utilizing iteration that would be utilized for the other AN. Among BPSO, and PSO of localization
methods, modified-BPSO is represented in terms of the localization error. In [24], the author
introduced a localization issue in WSN, and to resolve this issue PSO was utilized. In order to
improvise the localization precision and efficiency of an algorithm, the author represented the
main function based on the distribution of the ranging error and obtains the search space of the
particles.
In [25], the author introduced node localization in WSN is very essential due to several
applications that need more SNs to know their exact place with a higher precision degree. The
optimal path for the mobile anchors was based on the node localization. The introduced path
planning technique is defined the exact location of individual SNs with the Mobile-AN. It ensures
that the trajectory of mobile-ANs reduced localization problems and ensures all SNs could define
their exact locations. Afterward, the PSO algorithm defined the trajectory of mobile-AN. In paper
[26], the author introduced applications of SNs that were enhanced for wireless device-location
and localization method has been improvised to meet their needs. WSNs prove very useful in
several applications such as military surveillance and environmental monitoring.
The author addressed the applications of various PSO and BBO migration-variants algorithms for
optimal node localization for deployment sensors [27]. Biogeography is the learning path of
biological-geographical particles. This algorithm has novel vigor that is based on biogeography
and it employs the migration operator for allocating the data among various habitats and places
like problem-solving. The PSO model has faster convergence but also has lower maturity. Hence,
the given nodes will obtain localization on repetition as AN and also compared with the
performance of PSO and various migration variants of the BBO with any number of localized
nodes like computation time and accuracy of localization. In [28], the presented RSSI range-

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based localization scheme will rely on RSS measurement for evaluation of the distance. The
experimental has been shown indoor and outdoor environments for developing the model of path
loss.
This paper [29] represents the tracking system based on ARMA (auto-regressive moving-
average) method while the distribution in the signal processing framework. This framework act
as a peer that is performing the tracking, target detections, classifications, feature extraction in
case of the target localization that needed collaboration among WSNs for improving the
robustness and accuracy of the networks. Furthermore, the progressive multi-view localization
algorithm is prepared in the distributed P2P signals as a processing framework that will assume a
trade-off between power consumption and accuracy. Lastly, the real-world tracking experiment
has few illustrations. Few outcomes from implementations have represented the target tracking
method which is dependent on the P2P signal processing method is generating economic
utilization of the scarce energy and communication resources and also obtained the target
tracking. In paper [30], the author introduced localization models by utilizing linear intersections
and does few concerned experiments for evaluation of the location computation-algorithms. By
knowing the locations of the nodes in WSN it becomes most eminent for several useful benefits.
The nodes in WSN have many capabilities and exploitation of one and many capabilities that will
support resolving the localization issue. They also considered each node in WSN that has the
capability of the distance measurement and also represented the allocation of the computation
method known as the linear intersections for the node localization.
3. PRELIMINARIES
We begin by representing some notation that is used in this paper.𝐼𝑠
and 𝐽𝑠
represent set of 𝑠
vectors and symmetric matrix 𝑠 × 𝑠. Additionally, any symmetric matrix 𝑚, 𝑚 ≽ 0 means 𝑚 is
the or positive semi-definite.
Let’s represent defined coordinates of 𝑎𝑡ℎanchor-node 𝛼𝑎 = [𝛼𝑎1, 𝛼𝑎2]𝕋
(𝛼𝑎 ∈ 𝐼2
, 𝑎 =
1, … . , 𝑇)and undefined coordinates of 𝑏𝑡ℎ target node as 𝛽𝑏 = [𝛽𝑏1, 𝛽𝑏2]𝕋(𝛽𝑏 ∈ 𝐼2
, 𝑎 =
1, … . , 𝐴), where 𝐴 and 𝑇 are the total number of anchors and the targets. The power is received
at 𝑏𝑡ℎ target from 𝑎𝑡ℎ anchor node is demonstrated as:
ℙ𝑎,𝑏 = ℙ0 − 10𝛾𝑙𝑜𝑔10
𝑑𝑖(𝛼𝑎,𝛽𝑏)
𝑑𝑖
+ 𝑠𝑎,𝑏
(1)
ℙ0 is simplifies transmitted power from the receiver at the 𝑑𝑖 distance, 𝑑𝑖(𝛼𝑎,𝛽𝑏) =∥ 𝛼𝑎,𝛽𝑏 ∥2
is Euclidean distance among 𝑎𝑡ℎ and 𝑎𝑡ℎ, 𝛾 is defined as the exponent of path-loss with common
value, 𝑠𝑎,𝑏 additive noise follows GD (Gaussian-distribution) that shows the shadowing effect of
log-normal in the multipath-environments, respectively. One of the issues is addressed in (1). 𝑠𝑎,𝑏
always not follow the GDs in the real environment. We intend to improvise the channel path of
the loss model by taking the features of additive-noise.
3.1. System Model
Here, we introduce noise with the probabilistic model (PM) and resulting in joint PDF
(Probability distributed-function) at the detected power vector ℙ𝑏 =. ℙ1,𝑏,………ℙ𝐴,𝑏 at the target
𝑏𝑡ℎ:

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𝔭(ℙ𝑏|𝛽𝑏) = ∏ ∑ 𝔱𝑏,𝑐𝒩(𝑚𝑐,𝑣𝑐
2),
𝐶
𝑐
𝐴
𝑏=1
(2)
Where, 𝔱𝑏,𝑠 is defined as corresponding cluster weight 𝑐 with 𝑣𝑐
2
variance and 𝑚𝑐 mean of noise,
and 𝐶 is defined as the mixture component.
Note: the exponent of path-loss is known and it is fixed in our experiments and simulations in the
next section. By using experimental measurements, 𝛾 and ℙ0 can be evaluated as the logarithmic
fitting.
3.2. Problem Formulation
Here, mean 𝑚𝑐and variance 𝑣𝑐
2
and also 𝔱𝑏,𝑐 are required to be evaluated in the process of node
localization. In the criterion of ECM, both variables like mean and variance 𝑣𝑐
(𝓃)
and 𝑚𝑐
(𝓃)
are
evaluated on iteration 𝓃for informing the position 𝑣𝑐
(𝓃)
without 𝔱𝑏,𝑐
(𝓃)
would be equally
improved. The iteration index 𝓃 is not represented.
In the end, in order to design the main objective like node localization algorithm through GM-
Semi-definite programming (SDP) is to achieve ML to evaluate 𝑣𝑏
∗
by discovering the
parameter𝔱𝑏,𝑐. Afterward, the estimator of ML can be expressed:
maximum
𝛽𝑏,𝔱
𝜓(𝛽𝑏, 𝔱)
𝑠. 𝑡. 𝕂1: ∑ 𝔱𝑎,𝑐
𝐶
𝑐=1
= 1, ∀𝑎
𝕂2: 0 ≤ 𝔱𝑎,𝑐 ≤ 1, ∀𝐶, 𝑎
𝕂3: 𝑑𝑖(𝛽𝑏,𝑎𝑎)∀𝐶, 𝑎
(3)
Where, 𝜓(𝛽𝑏, 𝔱) is defined as the function of log-likelihood combined for conditional-pdf in
equation (2) such as 𝜓(𝛽𝑏, 𝔱) = 𝑙𝑛 𝔭(ℙ𝑏|𝛽𝑏) and 𝔱𝑎,𝑐 is defined as weights of a mixture, which
are constrained to sum-up to 1.
Anyways, the main function of combinatorial-nature in (3) those outcomes in the problem of NP-
hard. In order to minimize the complexity, an optimization issue is formulated by discovering the
LB function of the non-convex objective. Then, the formulated issue is resolved as the SDP issue
through relaxation to get an optimal solution. We begin to resolve an original issue in (3) by
getting the sub-optimal solution through the theorem.
Pairs have an optimal solution for the feasible pair (𝛽𝑏,𝔱) then it will follow optimization issue
and feasible pair (𝛽𝑏, 𝔱) is optional for the original issue in (3).
max
𝛽𝑏
∑ ∑ 𝔱𝑏,𝑐𝒩(𝑚𝑐,𝑣𝑐
2),
𝐶
𝑐=1
𝐴
𝑏=1
𝑠. 𝑡. 𝕂1, 𝕂2, 𝕂3.
(4)

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The value of optional 𝛽𝑏 is described as 𝜓∗
= sup{𝜓(𝛽𝑏, 𝔱)} |𝕂1, 𝕂2, 𝕂3, and the solution of
feasible (𝛽𝑏, 𝔱) is defined as ∈ − optional if 𝜓(𝛽𝑏, 𝔱) ≥ 𝜓∗
−∈. Utilizing Jensen’s inequity, the
objective function of LB-non-convex by (3) is achieved and mentioned by:
𝜓(𝛽𝑏,𝔱) ≥ ∑ ∑ 𝔱𝑏,𝑐𝐼𝑛 𝒩(𝑚𝑐,𝑣𝑐
2) =
𝐶
𝑐=1
𝐴
𝑏=1
𝜓1(𝛽𝑏,𝔱) (5)
𝜓1(𝛽𝑏, 𝔱)assists as LB for the parameters of likelihood-function (𝛽𝑏, 𝔱). Anyways, ∈′
exists,
afterward, it generates (𝛽𝑏, 𝔱) that satisfies𝜓1(𝛽𝑏,𝔱) ≥ 𝜓∗
−∈′
. Therefore, (𝛽𝑏, 𝔱) defines ∈′
an as
optional issue in (3) and the theorem holds. Moreover, the optimization issue in (3) can be
formed as:
minimum
𝛽𝑏,𝔱
𝜓(𝛽𝑏, 𝔱)
𝑠. 𝑡. 𝕂1, 𝕂2, 𝕂3.
(6)
Where,
𝜓2(𝛽𝑏, 𝔱) = −𝜓1(𝛽𝑏, 𝔱)
∑ ∑ 𝔱𝑏,𝑐
𝐶
𝑐=1
[𝐼𝑛 √2𝑣𝑐 +
(𝑠𝑎,𝑏 − 𝑚𝑐)2
2𝑣𝑐
2 ]
𝐴
𝑏=1
(7)
The main objective of (6) is non-convex but its domain (𝛽𝑏|(𝛽𝑏) ≠ 𝛼𝑎 is not convex-domain and
𝑙𝑜𝑔10
𝑑𝑖(𝛼𝑎,𝛽𝑏)2
2𝑣𝑐
2 is non-convex on the domain of convex. Anyways, it is very difficult to discover
the solution of optimal in (6). In order to obtain the estimator of convex in (6), we represent the
following Proposition.
3.3. Probabilistic based Optimal Node Localization Approach
Proposition: if 𝑔 ∈ 𝐼𝑠
Then, we have ‖𝑔‖∞ ≤ ‖𝑔‖2 ≤ √𝑠‖𝑔‖∞and ‖𝑔‖∞ ≤ ‖𝑔‖1 ≤ √𝑠‖𝑔‖∞.
In order to replace𝔩2 in 𝜓2 with 𝔩∞ norm like Chebyshev-norm, then we have
(𝑠𝑎,𝑏 − 𝑚𝑐)2
2𝑣𝑐
2
(𝑥)
⇔ 𝑚𝑎𝑥 |
1
𝑣𝑐
𝑙𝑜𝑔10
𝑑𝑖(𝛽𝑏, 𝛼𝑎)
𝛾𝑖,𝑐
2 |
(𝑦)
⇔ 𝑚𝑎𝑥 |
𝑑𝑖2(𝛽𝑏,𝛼𝑎)
𝑣𝑐𝛾𝑖,𝑐
2 ,
𝛾𝑖,𝑐
2
𝑣𝑐𝑑𝑖2(𝛽𝑏, 𝛼𝑎)
|
(8)
Where, 𝛾𝑖,𝑐
2
= 𝑑𝑖0
2
10
ℙ0+𝑚𝑐−ℙ𝑎,𝑏
5𝜍
. In equation (8), (x) outcomes because of 1 in Proposition and
(y) outcomes based on𝑙𝑜𝑔10𝑔, which is a strictly monotonic function when maximize all
domains(0, ∞). Then, the optimization problem in (6) can be articulated as:
minimum
𝛽𝑏,𝔱,,𝜔
∑[𝒯𝓇(𝓉𝑎𝔫𝕋) + 𝒯𝓇(𝓉𝑎𝜛𝑎
𝕋)]
𝑇
𝑎=1

97
𝑠. 𝑡. 𝕂1, 𝕂2, 𝕂3.
𝕂4: ‖𝛽𝑏 − 𝛼𝑎‖2
2
≤ 𝜆𝑎,𝑐
2
𝑣𝑐𝜛𝑎,𝑐
𝕂5:‖𝛽𝑏 − 𝛼𝑎‖2
2
≥ 𝜆𝑎,𝑐
2
𝑣𝑐
−1
𝜛𝑎,𝑐
−1
, ∀𝑎,𝑐
Where,
𝓉𝑎 = [𝓉𝑎,1, … . . , 𝓉𝑎,𝐶]
𝕋
𝔫 = [𝐼𝑛√2𝜋𝑣𝑐, … … , 𝐼𝑛√2𝜋𝑣𝑐]
𝕋
𝜛𝑎 = [𝜛𝑎,1, … . . , 𝜛𝑎,𝐶]
𝕋
(9)
‖𝑣𝑐 − 𝛼𝑎‖2
2
= 𝒯𝓇(Ψ) − 2𝑣𝑐
𝕋
𝛼𝑎 + ‖𝛼𝑎‖2
2
. Hence, we write this in the norm as a formulated
problem:
minimum‖g‖1 − ‖h‖1
𝛽𝑏,𝔱,,𝜔
𝑠. 𝑡. 𝕂1, 𝕂2, 𝕂3.
𝕂4:𝒯𝓇(Ψ) − 2𝑣𝑐
𝕋
2
≤ 𝜆𝑎,𝑐
2
𝑣𝑐𝜛𝑎,𝑐
𝕂5: 𝒯𝓇(Ψ) − 2𝑣𝑐
𝕋
2
≥ 𝜆𝑎,𝑐
2
𝑣𝑐
−1
𝜛𝑎,𝑐
−1
, ∀𝑎,𝑐
Where,
𝕂6: Ψ = (𝛽𝑏𝛽𝑏
𝕋
)=, Ψ ∈ ℂ2
𝕂7: 𝑔𝑖 = 𝒯𝓇(𝓉𝑎𝔫𝕋
)
𝕂8: ℎ𝑖 = 𝒯𝓇(𝓉𝑎𝜛𝑎
𝕋),∀𝑎
(10)
Anyways, the issue in (10) is not convex, since the similarity constraint is not affine as 𝕂7and
𝕂8. In order to achieve the problem of convex optimization in (10), 𝕂6 is defined as not
similarity constraint Ψ ≥ 𝛽𝑏𝛽𝑏
𝕋
(semidefinite relaxation) and 𝕂8 is defined into ℎ𝑖 = ∑ 𝜛𝑎,𝑐
𝐶
𝑐=1
like Jensen’s inequality. Furthermore, the given constraints 𝕂5 and 𝕂6 utilizing Schur
complement in the linear matrix, which is represented in the optimization problem is:
minimum‖g‖1 − ‖h‖1
𝛽𝑏,𝔱,,𝜔
𝑠. 𝑡. 𝕂1, 𝕂2, 𝕂3.
𝕂5: [
𝒯𝓇(Ψ) − 2𝑣𝑐
𝕋
2
Υ𝑎,𝑐/√𝑣𝑐
Υ𝑎,𝑐/√𝑣𝑐 𝜛𝑎,𝑐
] ≥ 0, ∀𝑎,𝑐
𝕂6: [
Ψ 𝛽𝑏
𝛽𝑏
𝕋
1
] ≥ 0, Ψ ∈ ℂ2
𝕂7: 𝑔𝑖 ≥ 𝒯𝓇(𝓉𝑎𝔫𝕋
)
𝕂8: ℎ𝑖 ≥ ∑ 𝜛𝑎,𝑐
𝐶
𝑐=1
, ∀𝑎
(11)
Equation (11) is the problem of optimization like convex that can be reconciled with numerical
process to achieve the optimal solution 𝛽𝑏
∗
of the optimization problem, which is defined in (3).

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It is very easy to discover PONL estimator variance 𝛽
̂(ℙ) is not biased such as ℰ[𝛽
̂(ℙ)] = 𝛽(ℙ).
An estimator's accuracy may be demonstrated by costs that are related to MSE (mean-square-
error), or the variance of the estimator for the unbiased estimate. CRB is the FI (Fisher
Information) inverse defines LB on estimators of unbiased variance. The variance of the PONL
estimator will be:
[𝛽
̂(ℙ) − ℙ][𝛽
̂(ℙ) − ℙ]
𝕋
≥ 𝜇−1 (12)
Where, |𝜇|𝑒,𝑓 element of 𝜇 FIM is described by:
|𝜇|𝑒,𝑓 = ℰ [
𝜁𝐼𝑛𝔭(ℙ𝑏|𝛽𝑏)
𝜁𝛽𝑏,𝑒
.
𝜁𝐼𝑛𝔭(ℙ𝑏|𝛽𝑏)
𝜁𝛽𝑏,𝑓
] , 𝑒, 𝑓 ∈ 𝒱
(13)
Where 𝒱 is defined as a set of the dimensions in the coordinated axis. For a localized node in 2D-
scenario |𝒱| = 2, we have,
|𝜇|𝑒,𝑓 = [
10𝐵
𝐼𝑛 10
]
2
𝐴𝔫 ∑
(𝛽𝑏,𝑒 − 𝛼𝑎,𝑓),(𝛽𝑏,𝑓 − 𝛼𝑎,𝑓)
‖Φ𝑏 − 𝛼𝑎‖4
2
𝑇
𝑎=1
, 𝑒, 𝑓 ∈ 𝒱 (14)
Where,
𝐴𝔫 = ℰ {[
∆𝔫𝔭(𝔫)2
𝔭(𝔫)
]} = ∫
[∆𝔫𝔭(𝔫)2]
𝔭(𝔫)
𝑑𝑖𝔫
(15)
√ℰ(𝕖2) ≥ √𝒯𝓇[𝜇−1] ≜ 𝐶𝑅𝐵(𝛽) (16)
In equation (15), 𝔭(𝔫)~ ∑ 𝔱𝑐𝒩(𝛽𝑐 − 𝑣𝑐
2
)
𝐶
𝑐=1 and 𝐴𝔫 is estimated by Monte-Carlo. Furthermore,
by describing the location EE (estimator error) as 𝕖 = ‖𝛽
̂ − 𝛽‖2
. CRB is LB on its MSE (mean-
square error) for any type of unbiased-estimator.
4. RESULT AND ANALYSIS
In this section, numerical simulations are conducted to do a performance analysis of our proposed
PONL approach. The system configuration; Intel i7 windows-based operating system, 12GB
RAM and the complete simulation is done using Matlab 2018a software.
Simulations are conducted to evaluate the performance of our proposed method through
comparing with the weighted least squares (WLS) [31] approach, which is referred to as an
existing system (ES) incomplete result analysis. In addition, the performance is measured using
the Root-Mean-Square Error (RMSE), obtained throughput and Cumulative Distribution
Function (CDF). All these parameters are considered, which are computed based on 100 Monte
Carlo runs with cell size 10x10 m2
and 20x20m2
. Therefore, the total numbers of nodes are 104,
112 and 120, the transmitted power at reference distance is -55dB with reference distance of 1
meter, and cluster weight is considered to be 0.37 and 0.63. Table 1 provides the complete
parameters details for simulation.

99
Table 1. Parameters Details for Simulation
Monti Carlo Simulation 100
Cell size 10x10 m2
and 20x20m2
Total Number of Nodes 104, 112 and 120
Anchor Nodes 4, 12 and 20
The transmitted power in dB at reference distance -55
Path-loss exponent in dB 2
Reference distance in meter 1
Cluster weight 0.37 and 0.63
There are two types of scenario has considered with varying the cell size; the first scenario we
have a cell size of 10x10 m2
and in the second scenario we have cell size of 20x20 m2
. Other than
that we have the same parameters for both the scenarios and performance evaluation is done
using RMSE, obtained throughput and CDF. The assumed noise follows Gaussian distribution/
white noise.
4.1. Scenario-A
In scenario-A, we have considered the cell size of 10x10 m2
, and other than that we have
considered various anchor nodes (ANs) 4, 12 and 20. As a validation parameter RMSE provides
us information on the variance between the estimated and actual sensor position and it is
evaluated through the Monte Carlo runs (i.e., for a number of runs).
Figure 2. Obtained RMSE w.r.t Anchor Nodes

100
Figure 3. Cumulative throughput with varying Monti Carlo run (Ans=12)

101
Figure 5. Obtained CDF w.r.t Error (m)
Figure 2 is showing the obtained RMSE with respect to a number of anchor nodes. In each ANs,
we have completed the simulation of 100 runs. From figure 2 it is clearly seen that the increasing
number of ANs decreases the RMSE value in ES and our proposed system, but PONL approach
computed RMSE is very less compared to ES. In 4 ANs, our proposed approach got 2.3m RMSE
which is 71% less compared to ES (i.e., RMSE=8.1m). Similarly, when compared at other ANs
such as 12 and 20, we observed that our proposed system RMSE is 56% and 55% less compared
to the existing system. Further, we also checked with cumulative throughput values that are
represented in figures 3 and 4. Having successful node localization, we computed a cumulative
throughput, which is increasing as per the Monte Carlo run at ANs 4 and 20. Our proposed model
has able to achieve better throughput/successful packet compared to ES. Figure 5 shows the
obtained CDF with respect to an error in meter, where our proposed system at 1 meter has got
0.72 increments in CDF, whereas ES has achieved 0.01 CDF value. The 1 CDF value is achieved
at 4m using PS and 6.5m using ES.
4.2. Scenario-B
In scenario-B, we have considered the cell size of 20x20 m2
, and rest of the parameters value is
same as scenario-A with ANs; 4, 12 and 20.

102
Figure 6. Obtained RMSE w.r.t Anchor Nodes

103
Figure 8. Cumulative throughput with varying Monte Carlo run (Ans=20)
Figure 9. Obtained CDF w.r.t Error (m)

104
Table 2. Obtained RMSE values with Anchor Nodes at various cell sizes
Cell size 10x10m2
20x20m2
Methodology ES PS ES PS
4 8.101673 2.323195 12.93609 4.384795
12 2.939994 1.287258 6.090077 2.624638
20 2.218508 0.992214 3.817957 2.214256
In figure 2 is presenting the obtained RMSE with respect to a number of anchor nodes and for
considered ANs we have completed 100 simulation runs. In figure 6 at 4 ANs, our proposed
approach got 4.3m RMSE which is 66% less compared to ES with RMSE of 12.9m. Similarly,
with considered ANs such as 12 and 20, our proposed system has got 56% and 42% less RMSE
compared to an existing system. Therefore, figure 6 shows an increase in number of ANs and
decrease RMSE value in ES and our proposed system, but our proposed approach computed
RMSE is very less compared to ES. In addition, the cumulative throughput values have shown in
figures7 and 8. Our proposed model has got better throughput compared to ES. The obtained
CDF with respect to error is shown in figure 9, where our proposed system at 2meters has got
0.65increments in CDF and ES has achieved a 0.04 CDF value.1 CDF value is achieved at 5m
using PS and 15 m using ES. The graph shows the significant improvement in CDF values using
PONL approach. In addition, more numeric information of the obtained RMSE values with
anchor nodes at various cell sizes as shown in table 2.
5. CONCLUSION
WSNs are key components of the cyber-physical systems, the internet of things, etc. and they
consisted of a large number of SNs which causes difficulty in node localization. The localization
problems formed as ML-estimator is non-convex and its performance was totally dependent on
the preliminary given for the iterative solver. Therefore, in this paper; we proposed the
improvised RSS-based node-localization algorithm called the PONL approach. The proposed
localization approach provides a better and more stable result in considered scenarios. The
proposed method was compared through WLS approach incomplete result analysis. The
performance is measured using the RMSE, throughput and CDF at two different network
scenarios, where our proposed approach had shown significant results as compared to the existing
approach.
CONFLICTS OF INTEREST
The authors declare no conflict of interest.
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AUTHORS
Mrs. Souparnika Jadhav
She is currently working as Assistant Professor in Electronics and Communication
Engineering, Nagarjuna College of Engineering and Technology (NCET), Bangalore.
Her research interest includes Wireless Communication, Image Processing and Antennas.
She has an overall teaching experience of 6 years where she taught modules related to
Electronics & Communication
Dr. Nagesh K.N
He is currently working as Professor and HOD in Electronics and Communication
Engineering, Nagarjuna College of Engineering and Technology (NCET), Bangalore. His
research interest includes Wireless Communication, Communication System, IOT and
Antennas. He has a overall teaching experience of 15 years where he taught modules
related to Electronics & Communication

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