Source localization using compressively sampled vector sensor array

Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
30 – 31, December 2014, Ernakulam, India
221
SOURCE LOCALIZATION USING COMPRESSIVELY
SAMPLED VECTOR SENSOR ARRAY
Soniya Peter1
, Miji Jacob2
, Shiji Abraham3
1, 2, 3
Dept of ECE, SNGCE, Kadayiruppu, Ernakulam, India
ABSTRACT
Installation and maintenance of coastal surveillance system is the most complex and challenging task for modern
navy in the current turbulent global political climate. Traditionally, long sea bed array consisting of pressure sensors, is
deployed along the coastal belt, to protect the harbour resources by capturing the low frequency components emanating
from the subsurface targets. Deployment of the array along with its associated signal conditioning hardware and data
communication equipment (DCE) on the ocean bed is a challenging task and involves huge time and effort. In this paper,
a novel compressively sensed (CS) acoustic vector sensor (AVS) array architecture is proposed to estimate the direction
of arrival (DoA) of acoustic sources. The pursuit reconstruction algorithms are customized for AVS array processing.
This architecture guarantees an effective and highly potential surveillance system, with a substantial reduction in the
number of sensors, acquisition hardware, bandwidth, number of snapshots and the software complexity.
Keywords: Acoustic Vector Sensor Array, Compressive Sampling, Compressive Recovery, Doa Estimation, Fusion of
Matching Pursuit Algorithm, Subspace Pursuit Algorithm.
1. INTRODUCTION
Multiple threat scenarios exist to high value assets on or near ports and its surveillance have become one of the
most intricate and challenging problem faced by the Navy today. Traditionally, long ocean bed array, consisting of
hundreds of uniformly distributed pressure sensors, are installed on the ocean bed, to detect and localize the subsurface
target. The sensor output is conditioned and then digitized using multi- bit analog to digital converter (ADC). The
digitized channel data are send to the base station through a radio frequency or satellite link. The deployment of long
ocean bed array along with its signal conditioning hardware on the ocean bed is a challenging task involving huge cost
and effort. The investments can be greatly reduced if any reduction in number of sensors, front-end circuitry or data rate
can be achieved. It is well known that, the strategic performance of Waterside Security System(WSS) mainly depends
upon the capability of the system to localize an underwater threat with minimum number of snapshots or time samples. In
this paper, hardware efficient ocean bed array architecture is proposed which greatly reduces the complexity of the WSS
with superior strategic performance. The proposed system utilizes a compressively sensed acoustic vector sensor (AVS)
array along with high resolution compressive beamformer, suitable for localizing multiple acoustic sources.
An AVS [20] completely characterizes the acoustic field at a point in space through simultaneous measurements
of the three orthogonal components of the particle velocity along with the acoustic pressure. Though each AVS provides
the acoustic pressure and the tri-axial particle velocity components, in this work, the estimation of azimuth angle is
focused and hence the acoustic pressure and the particle velocity component along the horizontal plane are used. Particle
velocity component along the vertical axis is not considered. Throughout this paper, three measurements are used at
each AVS output. In the proposed architecture, an array consists of linearly distributed AVS, is installed on the ocean bed
for passive source localization. An AVS based ocean bed array requires 1/3rd
of the number of sensors [21] compared to
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the traditional pressure sensor ocean bed array to achieve the same localization performance. Though an AVS array
utilizes lesser number of sensors, the major issues to be addressed in conjunction with AVS array deployment for a given
localization performance are: (1) number of channels and the associated analog signal conditioning hardware remains
same as that of the pressure sensor array. This is due to the fact that, reduction in the number of sensors in the AVS based
array is compensated by increase in the number of measurements; (2) transmission data rate through RF link and the size
of the spatial correlation matrix remains same as that of pressure sensor array. Thus, it is not worthy that, though an AVS
array drastically reduces the number of sensors, the number of channels and the associated signal conditioning hardware,
transmission data rate, and spatial correlation matrix dimension remains same as that of pressure sensor array for
achieving a bearing performance.
Compressive sampling is a new signal acquisition technique which requires fewer measurements or front end
signal conditioning hardware chain to represent or reconstruct the signal, which are sparse in some basis vectors. In this
present context, the array signal vector is sparse in angular spectral basis. The applications of compressive sampling was
intitiated by Donoho [21] and Candes et al. [22] in the year 2006.A good review of compressive sensing is given in [23].
Compressive beamforming is a method to recover the sparse angular spectral vector from very few non-adaptive, linear
measurements.
The outline of the paper is as follows. In section II, we present the AVS array data model. The details of
compressive sampling is discussed in section III. The DoA estimation techniques are presented in section IV. Simulation
results are presented in section V. Section VI concludes the paper.
2. AVS ARRAY DATA MODEL
Consider a uniform horizontal AVS array of N equispaced sensors positioned in a horizontal plane along the x-
axis as shown in Fig.1. Consider J mutually uncorrelated narrowband point sources of centre frequency
ఠ
ଶగ
located at (rj ,
θj) with respect to the first sensor of the array. The sources are positioned at azimuth angles rj , θj = 1, 2,. . . . . , J are
measured with respect to the axis of the array. The complex amplitude of acoustic pressure at the n-th sensor due to the
j-th source, is given by :
‫݌‬௡௝ = ݁௜(௡ିଵ)௞ௗ௖௢௦ఏ௝
ߟ௝(‫,)ݐ‬ (1)
where k is the wave number, d is the inter-sensor spacing and ߟ௝(‫)ݐ‬ is the slowly varying complex envelope of the
analytic signal from the j-th source.The amplitude ߟ௝(‫)ݐ‬ is a random process with mean zero and variance ߪ௝
ଶ
=
‫ߟ[ܧ‬௝
ଶ
(‫])ݐ‬ .
Fig.1: Source array geometry

223
The relation between the acoustic pressure p and the particle velocity v at a point r = (x,y,z) and time t is
governed by the law of conservation of momentum which is given by :
ߩ
డ௩(௥,௧)
డ௧
+ ߘ‫,ݎ(݌‬ ‫)ݐ‬ = 0 (2)
Using equations (1) and (2) and invoking the plane wave and the far-field approximation, the complex
amplitudes of x and y components, respectively, of the particle velocity at the n-th sensor due to the j-th source are given
by:
‫ݒ‬௫೙ೕ
=
௖௢௦ ఏೕ
ఘ௖௞
݁௜(௡ିଵ)௞ௗ௖௢௦ఏ௝
ߟ௝(‫,)ݐ‬ (3)
‫ݒ‬௬೙ೕ
=
௦௜௡ ఏೕ
ఘ௖௞
݁௜(௡ିଵ)௞ௗ௖௢௦ఏ௝
ߟ௝(‫,)ݐ‬ (4)
where ߩ is the density of the water and c is the speed of sound in water.
The received signal at the sensor array at time t can be expressed as:
ܺ(‫)ݐ‬ = ‫)ݐ(ߟܣ‬ + ‫)ݐ(ݓ‬ ∈ ‫ܥ‬ଷே×ଵ
, (5)
where
ߟ(‫)ݐ‬ = [ߟଵ(‫,)ݐ‬ ߟଶ(‫,)ݐ‬ ߟଷ(‫,)ݐ‬ … . , ߟ௃(‫])ݐ‬்
‫ܥ‬௃×ଵ
(6)
is the source signal vector and A is the 3N×J array manifold matrix defined as :
‫ܣ‬ = [ܽ(ߠଵ) ܽ(ߠଶ) . . . ܽ(ߠ௃)], (7)
where ܽ(ߠ௝) is the steering vector corresponding to the source direction ߠ௝.
ܽ(ߠ௝) = ܿ(ߠ௝) ⊗ ݀(ߠ௝), (8)
where ⊗ denotes Kronecker product and
ܿ(ߠ௝) = [1, ݁௜௞ௗ௖௢௦ ఏೕ
, . . . , ݁௜(ேିଵ)௞ௗ௖௢௦ ఏೕ
]T
, (9)
where k=
ଶగ
ఒ
, λ being the wavelength of received signal and
݀(ߠ௝) = [1, √2ߩܿܿ‫ߠ ݏ݋‬௝ , √2ߩܿ‫ߠ ݊݅ݏ‬௝ ] , (10)
The array noise vector w(t) is given by :
‫)ݐ(ݓ‬ = [‫ݓ‬ଵ(‫,)ݐ‬ … , ‫ݓ‬ଷே(‫])ݐ‬T
, (11)
where ‫ݓ‬ଵ(‫,)ݐ‬ … , ‫ݓ‬ଷே(‫)ݐ‬ are the i.i.d circular complex random variables with variance ߪଶ
and
‫ݓ‬௡(‫)ݐ‬ = [‫ݓ‬௣,௡(‫,)ݐ‬ ‫ݓ‬௩ೣ,௡(‫,)ݐ‬ ‫ݓ‬௩೤,௡(‫.])ݐ‬ (12)
The SNR for the j-th source is defined as :
(SNR)j =
ఙೕ
మ
ఙమ. (13)
The correlation matrix of the data vector y(t) is defined as :
ܴଷே = ‫)ݐ(ݕ)ݐ(ݕ[ܧ‬ு
], (14)

224
In practical calculations, considering the received data is finite, the true correlation matrix can be estimated as
the following sample correlation matrix:
ܴ෠ଷே =
ଵ
௅
∑௅
௧ୀଵ [‫])ݐ(ݕ)ݐ(ݕ‬H
, (15)
where L is the number of snapshots.
3. COMPRESSIVE SAMPLING
Compressive Sampling (CS) is a new sampling strategy that uses a much fewer set of linear measurements than
suggested by conventional Nyquist sampling theory together with a nonlinear recovery process. Compressive sampling
enables sparse or compressible signals to be captured and stored at a rate much below the Nyquist rate. The
reconstruction of the original signal from its random projections is possible by means of an optimization process as long
as the measurements satisfy reasonable conditions such as incoherence and Restricted Isometry Property [25]. In
compressive sampling, the conventional two step process of data acquisition and compression can be integrated into a
single step.
Consider a real-valued signal x in an N -dimensional vector space. Using the N × N basis matrix Ψ = [ψ1 , ψ2 , . .
. , ψN ] signal x can be expressed as :
‫ݔ‬ = ∑ே
௜ୀଵ ‫ݏ‬௜߰௜ (16)
where ‫ݏ‬௜ is the weighting coefﬁcients. The signal x is K- sparse if it is well approximated by the K most significant
coefficients in the expansion. When K << N, the signal x is compressible and the representation in Eq. (16) need only a
few coefficients. The signal is sampled using H =KO(log N) measurements with the measurement vectors {߶}௝ୀଵ
ு
as :
‫ݕ‬௝ =< ‫,ݔ‬ ߶௝ > (17)
We can compactly denote measurement vector y using :
y = Φx = ΦΨs = Ωs, (18)
where Ω = ΦΨ is an H × 3N matrix. The choice of measurement matrix Φ is important since it decides the
stability and reliability of the compressive sensing process. The measurements should satisfy certain conditions such as
incoherence with respect to the original basis and restricted isometric property (RIP) for the recovery of the original
signal from its randomized projections by means of an optimization process. RIP means that the most of the energy of the
sparse signal is captured by these compressive measurements. Both the RIP and incoherence can be achieved with high
probability simply by selecting Φ as a random matrix.
3.1. Compressive sensing recovery
Compressed sensing recovery reconstructs a high resolution signal from low dimensional measurements. The
signal reconstruction algorithm must take the measurement vector y, the random measurement matrix Φ, and the basis Ψ.
We have to reconstruct the signal x or, equivalently, its sparse coefficient vector s. For K-sparse signals, since H < N,
there are infinitely many ‫ݏ‬′
that satisfy Ω‫ݏ‬′
= y. Thus recovery can be cast as an optimization problem where the objective
is to find the signals sparse coefficient vector while simultaneously satisfying the constraint y = Ωs. This can be
expressed mathematically as :
‫̂ݏ‬ = ܽ‫ݏ‖ ݊݅݉ ݃ݎ‬′
‖௣ subject to Ω‫ݏ‬′
= y (19)
where ‖‫‖ݏ‬௣ is the pth
norm of s. A least squares solution is obtained with p = 2, but it does not give a global minimum
value. That is, l2 norm measures only the signal energy and not signal sparsity. An alternate method is to use the l0 norm.
The l0 norm counts the number of non-zero samples in s and therefore can result in the sparsest solution. However,
solving Eq. (19) using l0 is both numerically unstable and NP hard. A much easier l1 optimization, based on linear
programming (LP) [14] techniques, yields an equivalent solution, if the sampling matrix satisfies the restricted isometry
property (RIP).The l1 optimization problem can be recast as a Linear Programming (LP) problem which can efficiently

225
solved using the algorithm Basis Pursuit [17]. Even though l1 optimization has high reconstruction accuracy, their
complexity is impractical for several applications. For a signal of length N, the complexity of l1 optimization is O(N3).
Several classes of low complexity reconstruction techniques were later put forward as alternatives to LP based
recovery [5]–[8], [10], [22], [24], [29]. These include the greedy algorithms including Orthogonal Matching Pursuit
(OMP). OMP computes the non-zero values in the signal in an iterative fashion. The basic idea behind these methods is to
find the support of the unknown signal sequentially. At each iteration of the algorithm, one or several coordinates of the
vector s are selected for testing based on the correlation values between the columns of the measurement basis Φ and the
regularized measurement vector. If deemed sufficiently reliable, the candidate column indices are subsequently added to
the current estimate of the support set of s. The Orthogonal matching pursuit (OMP) [16] algorithm iterates this procedure
until all the coordinates in the correct support set are included in the estimated support set. The complexity of such
algorithms is significantly smaller than that of LP methods especially when the signal sparsity level K is small. However,
its guarantees are not as strong as that of Basis Pursuit. Subspace pursuit (SP) and Fusion of matching pursuit (FuMP)
algorithms are two novel methods for reconstruction of sparse signals.
1) Benefits of compressive sampling on AVS array processing: Use of AVS array significantly reduces the number of
sensors to achieve a given performance factor in the form of estimation error and bearing resolution. However, the analog
signal conditioning hardware complexity and the data transmission rate increases by 3 times. The data transmission rate
of the AVS array is given by :
Transmission-‫݁ݐܽݎ‬஺௏ௌ = 3N‫ܨ‬௦G bits/second, (20)
where N is the number of sensors, ‫ܨ‬௦ is the sampling frequency in Hz, G is the number of bits/sample. Generally 16-bit or
24-bit sigma delta converters are used for digitization. The above issues are efficiently tackled by the proposed
compressive sampling or sensing receiver architecture. The proposed architecture uses only around Jlog(3N) analog front-
end signal conditioning hardware and two bits of phase precision [27] instead of 16 or 24 bits. The data rate of the
compressively sampled AVS array is given by :
Transmission-‫݁ݐܽݎ‬஼ௌି஺௏ௌ = J log (3N)‫ܨ‬௦2 bits/second (21)
where J is the maximum number of expected acoustic sources.
CS array has the effect of compressing a large sized array into a smaller sized array. This in turn reduces the
hardware complexity on account of the much smaller number of frontend circuit chains. Also, due to the smaller
dimension of the array data vector, the software complexity is greatly reduced. The high resolution DoA estimation
algorithms consist of inverse and eigen value decomposition of the spatial correlation matrix with computational
complexity O(݊ଷ
) for an n × n matrix. The CS array architecture with CS beamformers needs to work with the correlation
matrix ܴ෠yof size H × H only, where H<< 3N, while the conventional AVS array needs to work with ܴ෠N3of size 3N × 3N.
ܴ෡ ௬ =
ଵ
௅
∑௅
௧ୀଵ [‫])ݐ(ݕ)ݐ(ݕ‬H
(22)
Table.I summarizes the hardware requirement of the proposed CS-AVS array architecture and compares it with
that of the conventional scalar sensor array and the AVS array. It is clear that with greatly reduced complexity, the CS-
AVS array can still achieve similar results in DoA estimation as a conventional large size AVS array.
4. DoA ESTIMATION USING AVS ARRAY
The sensor array collects the spatial samples of the propagating wave field. The objective is to estimate the
direction of arrival of source from the signal in the presence of noise and interfering signals. The spatial spectrum of the
signal is a large discrete set of spatially distributed far field sources that induces a large set waves, with only a few of
them being of high power. Motivated by this physical structure of the spatial spectrum, we can viewed it as a sparse
signal in the angular spectral domain. This representation allows the DoA estimation problem to be formulated in terms of
a compressed sampling problem. Using CS technique a sensor array of large number of elements is compressed or
transformed into an array of much smaller number of elements. The sparsity pattern vector s can be related to the received
signal at the ith
sensor is given by:
‫ݔ‬௜ = ‫ݏ‬௜߰௜ (23)
The sparsifying basis Ψ, also called angle scanning matrix in this case. It is constructed by finding out the
steering vectors using Eq. (8) for each angle, θi from the set of angles B = θ1,...,θR, where R determines the resolution of
look angles. The steering vector at an angle θj is given in Eq. (8). The angle scanning matrix can be expressed as:

226
ߖ = [ܽ(ߠଵ) ܽ(ߠଶ) . . . ܽ(ߠோ)] (24)
The compressed array signal can be expressed as:
y = Φx = ΦΨs , (25)
Here we analyse various DoA estimation using CS array.
4.1. DoA estimation using CS
The data model of AVS array used for is given in section 2. The CS-AVS array signal is obtained is given in Eq.
(25). DoA estimate of the CS array can then find out using some beamformers. In this paper, angle spectrum is calculated
by finding a sparse basis of the array signal and then solving it with CS recovery algorithms [26].
1) DoA estimation using CS recovery: In this approach the sparse vector s from y is recovered using CS reconstruction
algorithm. From the recovered solution, the angle spectrum is calculated by:
ܲ௬(ߠ) =
ଵ
்
∑்
௧ୀଵ ‖‫ݏ‬ఏ(‫‖)ݐ‬2
, (26)
for θ = θ1,...,θR, where T is the number snapshots. Here we use two efficient CS recovery algorithms Subspace Pursuit
algorithm (SP) [11] and Fusion of Matching Pursuits (FuMP) [12] for sparse signal recovery.
In the CS recovery algorithms, the major task is to identify in which subspace Ω, the measured signal y lies. The
SP algorithm starts by selecting the K columns of Ω that has the highest correlation with the measurement vector y. If the
distance of the received vector to the selected subspace is large, then the list is refined by retaining the reliable candidates
and discarding the unreliable ones while adding the same number of new candidates. The refinement of the solution set
continues as long as the l2-norm of the residue decreases. The main steps of the SP algorithm is summarized below.
• Input K,Φ,Ψ
• Initialization
1) T0
= {K indices corresponding to the largest magnitude in Ω∗∗∗∗y}.
2) ‫ݕ‬௥
଴
= residue(y,Ω்బ), columns of Ω்బ is same as the columns of Ω corresponds to the indices of T0
.
• At the lth
iteration,
1) ܶ௟
=ܶ௟ିଵ
U{ K indices corresponding to the largest magnitude in Ω∗‫ݕ‬௥
௟ିଵ
}.
2) ܶ௟
= {K indices corresponding to the largest magnitude in ‫ݏ‬ ̂௟
= Ω்෠೗
ற
y}, columns of Ω்෠೗corresponds to the indices of
ܶ෠௟
.
3) ‫ݕ‬௥
௟
= residue(y, Ω்೗).
4)‖‫ݕ‬௥
௟
‖2 >‖‫ݕ‬௥
௟ିଵ
‖2 , stop the iteration and ܶ௟
=ܶ௟ିଵ
.
• Output estimated signal ‫̂ݏ‬ satisfying ‫̂ݏ‬(ଵ,ଶ….ே)ି்೗=0 and ‫ݏ‬ ̂௟
= Ω்෠೗
ற
y,
where ‫ݕ‬௥= residue(y,Ωூ)=y − yp.
yp is projection on to span (Ω௟) and is given by :
yp=ΩூΩூ
ற
‫ݕ‬ , (27)
where pseudoinverse of Ω , Ωூ
ற
= (Ωூ
∗
Ωூ)-1
Ωூ
∗
TABLE I: Comparison of the different array architectures
ParameterScalar Sensor
Array
AVS- ArrayCS-AVS Array
No. of ChannelsN3NJlog(3N)
No. of signal conditioning hardwareN3NJlog(3N)
Data rate (bits/sec)N‫ܨ‬௦K3N‫ܨ‬௦KJlog(3N) ‫ܨ‬௦2
Spatial Correlation MatrixN×N3N× 3ܰJlog(3N) × Jlog(3N)
Minimum number of snapshotsN3NJlog(3N)
Note: The hardware requirement for the CS-AVS array is greatly reduced for large values of N and K

227
FuMP is a novel CS reconstruction method which fuses the information from two matching pursuit methods and
estimates the correct elements of its support set from the union of the support sets of that two matching pursuit methods.
Orthogonal Matching Pursuit (OMP) [16] and SP are used as two examples of Matching Pursuit (MP) in [12].
The steps for FuMP algorithm is explained below.
• Input K, Ω , y, support set of OMPܶ෠ைெ௉, support set of SP ܶ෠ௌ௉
• Initialization
1) Joint support set, Γ =ܶ෠ைெ௉ ∪ ܶ෠ௌ௉
2) Commom support set, Λ = ܶ෠ைெ௉∩ ܶ෠ௌ௉ 0 ≤|ܶ෠ைெ௉|, | ܶ෠ௌ௉|≤ K, therefore 0 ≤| Λ |≤ K,
K ≤| Γ |≤ 2K.
• Elements in Λ give more confidence since it agrees both with SP and OMP algorithm. Support set of FuMP includes
this. We have to identify K− | Λ | elements then and it is chosen from Γ. Least square solution is used for this purpose.
1) Intermediate signal estimate, ‫̂ݏ‬= Ω௰
ற
y, columns of ΩΓ is the columns of Ω corresponds to the indices of Γ.
2) ܶ෠= indices corresponding to (K− | Λ |) largest magnitude entries in ‫̂ݏ‬which are not in Λ.
• Actual support set for FuMP, ܶி௨ெ௉
෣ = ܶ෠∪ Λ
• Output estimated signal, ‫̂ݏ‬= Ω்෠೗
ற
y satisfying‫̂ݏ‬(ଵ,ଶ….ே)ି்ಷೠಾು෣ =0.
5. SIMULATION RESULTS
In this section, simulation results for various DoA estimation methods using compressive sampling for an AVS
array architecture are presented and are evaluated. The compression matrix Φ has entries drawn from an i.i.d. Gaussian
distribution with mean=0, variance=1/M. CS recovery algorithms are employed for DoA estimation.
We first evaluate the performance of CS-recovery algorithms, SP and FuMP in DoA estimation with CS-AVS
array architecture with H = 10. The pursuit algorithms, subspace pursuit and fusion of matching pursuit are implemented
in MATLAB. The data model that explained in section II is used. Three source signals are taken and are placed at the far-
field of a ULA (Uniform Linear Array) with 36 sensors placed on the x-axis. The number of scanning angles taken are
360, searching from 0◦
to 180◦
and 100 snapshots (L = 100) are taken for each case.
Fig. 2 shows the pseudo spectrum of CS-AVS array using Subspace Pursuit and Fusion of Matching Pursuit CS-
recovery algorithms. Three targets are placed at 87◦
, 92◦
and 130◦
with respect to the end-fire direction of the array. The
SNR of both being 10dB. From the fig. 2, we can see that, using CS-recovery algorithms, SP and FuMP algorithms, we
could accurately estimate DoA of the unknown sources.
Fig. 2: Pseudo-spectrum of the CS-AVS array (N = 36, H = 10, L = 100) using SP and FuMP algorithm. Three sources
are at 87◦
, 92◦
and 130◦
. SNR=10 dB each, f = 50 Hz

228
Fig. 3: RMSE vs. source bearing of SP and FuMP algorithm on the CS-AVS array (N = 36, H = 10, L = 200). SNR=10
dB, f = 50 Hz
Fig 3 show, the plot of root mean square error (RMSE) vs source bearing for a 36-element AVS array when a
single source is located at 60◦
. With CS-AVS array, the DoA estimate of the sources could be resolved efficiently using
CS recovery algorithm.
6. CONCLUSION
In this paper, a hardware efficient compressively sensed AVS array architecture for acoustic source localization
has been presented. The sparse angular spectral reconstruction pursuit algorithms were customized for passive source
localization. The sparse recovery algorithms performed very efficiently for the angular spectral reconstruction. The CS-
AVS array architecture signiﬁcantly reduced the number of sensors, signal conditioning hardware, transmission data rate,
number of snapshots and software complexity. In this work, the advantages of the inherent capabilities of acoustic vector
sensor and compressed sampling were ventured.
7. ACKNOWLEDGEMENTS
The authors express their gratitude to Dr. N Suresh Kumar, Scientist ’F’ of NPOL, Kochi for his time and effort
in reviewing the initial manuscript and providing valuable comments and suggestions.
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