Fault Tolerant Matrix Pencil Method for Direction of Arrival Estimation

Signal & Image Processing : An International Journal (SIPIJ) Vol.2, No.3, September 2011
DOI : 10.5121/sipij.2011.2306 55
FAULT TOLERANT MATRIX PENCIL METHOD FOR
DIRECTION OF ARRIVAL ESTIMATION
Yerriswamy T.1
and S.N. Jagadeesha2
1
P.D. Institute of Technology, Visvesvaraya Technological University, Belgaum,
Karnataka, India.
2
J.N.N.College of Engineering, Visvesvaraya Technological University, Belgaum,
Karnataka, India.
1
swamy_ty@yahoo.com
2
jagadeesha_2003@yahoo.co.in
ABSTRACT
Continuing to estimate the Direction-of-arrival (DOA) of the signals impinging on the antenna array, even
when a few elements of the underlying Uniform Linear Antenna Array (ULA) fail to work will be of
practical interest in RADAR, SONAR and Wireless Radio Communication Systems. This paper proposes a
new technique to estimate the DOAs when a few elements are malfunctioning. The technique combines
Singular Value Thresholding (SVT) based Matrix Completion (MC) procedure with the Direct Data
Domain (D3
) based Matrix Pencil (MP) Method. When the element failure is observed, first, the MC is
performed to recover the missing data from failed elements, and then the MP method is used to estimate the
DOAs. We also, propose a very simple technique to detect the location of elements failed, which is required
to perform MC procedure. We provide simulation studies to demonstrate the performance and usefulness of
the proposed technique. The results indicate a better performance, of the proposed DOA estimation scheme
under different antenna failure scenarios.
KEY WORDS
Direction-of-Arrival, Matrix Completion, Matrix Pencil, Faulty Antenna Array, Singular Value
Thresholding.
1. INTRODUCTION
Antenna Array signal processing and estimation of Direction-of-Arrival (DOA) parameter of the
received signal of interest, is of practical interest in RADAR, SONAR and wireless radio
communication systems. There are many DOA estimation algorithms, these algorithms can be
classified into three broad categories; Maximum Likelihood approach, subspace based approach
and Direct Data Domain (D3
) approach [1] [2]. The Maximum likelihood (ML) method is optimal
in the maximum likelihood sense [1], but requires accurate initializations to ensure global
minimum and moreover, the method is highly computationally intensive. Expectation
Maximization (EM) methods [3] [4] for ML approach reduces the computational requirements.
However, the accurate initializations are still needed.
The subspace based algorithms like Multiple Signal Classification (MUSIC) [5] and Estimation
of Signal Parameters via Rotational Invariance Technique (ESPRIT) [6], overcome the high
computational requirement by exploiting the low rank structure of the noise free signal. These
methods rely on the statistical properties of the data, and thus, need a sufficient large number of
samples for accurate estimation. Furthermore, when the signal sources are highly correlated, the
correlation matrix of the data tends to lose rank. This leads to the performance degradation of the

Signal & Image Processing : An International Journal (SIPIJ) Vol.2, No.3, September 2011
56
DOA estimation algorithms based on the subspace approach. However, preprocessing scheme
called spatial smoothing [7] is used to estimate the DOAs of the received highly correlated signal
sources. Furthermore, there are recent algorithms derived for Nonuniform Antenna Arrays based
on the above approaches [8] [9] and Ubiquitous Positioning [10].
Direct Data Domain (D3
) based algorithm namely, the Matrix Pencil (MP) Method [11] [12] is a
practically attractive algorithm. The algorithm needs only one snapshot to estimate the DOAs and
further, there is no need to form the correlation matrix. Correlated signal sources have no
significant impact on the performance. The technique can also be applied to Nonuniform Antenna
Array (NUA) without much modification [13]. However, the algorithm is sensitive to
perturbation and measurement errors, resulting in a low Signal-to-Noise (SNR) threshold.
It is a common practice to use large number of elements in an antenna array, and hence, failure of
a few elements will be disastrous in critical applications. Replacing the malfunctioning antenna
array elements will be time consuming and costly. To deal with such problems, redundant
antenna array is employed, which is wastage of the hardware and is a costly affair. The failure of
elements results in incomplete data observations or sparse data observations. In such cases,
conventional DOA estimation algorithms will have a problem to estimate the DOAs, because the
structure and the statistical properties of the data cannot be found directly. If it is possible to
reconstruct the complete data from the observed incomplete data, we can continue to estimate the
DOA, from the ULA, where a few elements have failed to work, this will avoid redundant
hardware.
For handling sensor failures, many modifications for conventional methods are proposed [14]
[15]. Larson and Stoica [14], proposed a technique for estimating the correlation matrix of the
incomplete data using the ML approach and has shown improvement in MUSIC for handling
sensor failure. However, it increases the complexity. A method for DOA estimator to handle
sensor failure based on neural network approach is proposed by Vigneshwaran et al [15]. The
technique can handle correlated signal sources, avoids the Eigen decomposition. The drawback
with these techniques is initialization of the network, and is performed by trial and error method.
The authors in [16] proposed a DOA estimation technique by combining the EM algorithm with
the MP method. The EM algorithm expects the missing data and maximizes the performance
using the MP method. However, the method suffers from the drawback of increased complexity
and requires good initialization.
Matrix Completion (MC) [17] is a process of completing the data matrix from incomplete data
matrix. This problem arises in a variety of situations like system identification [18], collaborative
filtering [19], DOA estimation [20], etc. The singular Value Threshold (SVT) algorithm proposed
by Candes [21], can exactly recover the missing data from the knowledge of the location of the
data elements in the matrix by solving a simple convex optimization problem [22], i.e.,
minimizing the nuclear norm, which is the sum of singular values of the data matrix.
This paper proposes a technique to deal with problem of DOA estimation when a few elements of
the antenna array fail. The technique uses two procedures when the element failure is observed at
time instance ‘’. The first procedure, is to form the complete data matrix from the incomplete
data matrix generated by the faulty ULA, using the SVT based MC algorithm. Later in the second
procedure, DOAs are estimated directly from the obtained complete data using the MP method.
The MC procedure first forms the location matrix. The location matrix describes the location of
the elements that are functioning. The location matrix is formed by finding the distance between
the samples at time instance ‘ − 1’ and at time instance ‘’. At ‘ − 1’ all the elements are
assumed to be working. The information from the location matrix, forms one of the input to the
MC procedure using SVT.
The novelty in our algorithm is therefore present only when element malfunctioning is observed.
The SNR and RMSE performance of the algorithm is evaluated in terms of varying percentage of
element failures. The algorithm is compared with the standard MP Method. For the proposed

Signal Image Processing : An International Journal (SIPIJ) Vol.2, No.3, September 2011
57
technique, we consider the faulty ULA, whereas for the latter all the elements are assumed to be
functioning.
The rest of the paper is organized as follows. The following section discusses the signal model for
DOA estimation. In section 3, proposed technique is discussed followed by simulation results in
section 4. Finally, conclusions are drawn in section 5.
2. SIGNAL MODEL
Figure 1. Uniformly spaced linear antenna array.
The DOA estimation problem is to estimate the directions of plane wave incident on the antenna
array in presence of errors. The problem can be looked as parameter estimation. We here, mainly
introduce the model of a DOA estimator. Consider a Uniform Linear Antenna Array (ULA) of
‘’ elements as shown in Figure 1. The elements are spaced equidistant ‘’ equal to ‘ 2
⁄ ’ from
each other, where, ‘’ is the wavelength of the signal received. The plane waves arrive at the
array from direction ‘
’ off the array broadside. The angle
is called direction-of-arrival (DOA)
of the received signal. Let , narrowband signals
,
, ⋯ ,
impinge on the array with
DOAs
, = 0,1,2, ⋯ − 1. Therefore, the signal xm received at the mth
element at time instance
‘t’ is
=

!
#
$%'

+ )

*+
, = 1,2, ⋯ ,
(1)
Eq. (1) can be written in a compact form as
- = . / + 0 = 1,2, ⋯ , 1 (2)
Where 1 is the number of snapshots, considering only one snapshot i.e. T=1, / is × 1
vector of signal sources, 0 is × 1 noise matrix which is assumed to be Additive White
Gaussian noise (AWGN), . is the × array steering matrix and - is the × 1
received signal vector.
M j 2 1
d
d sinθ
Reference
Element
Plane
wavefront
Incident wavefront
θi

58
Assume that at random locations a few elements are malfunctioning. In such case, the outputs
from these elements are not available, resulting in the incomplete data vector -3 of size 3
×
1, where 3
is the number of elements functioning.
The objective in this paper is to continue the DOA estimation of source signals even when a
few elements of a given ULA are malfunctioning.
3. PROPOSED DOA ESTIMATION ALGORITHM
The various steps of the proposed algorithm is described in Table 1 and illustrated in Figure 2. If
there are element failures, we follow three important steps. First, we form the location matrix of
the elements that are functioning. In second step, the information from the location matrix is
Figure 2. Basic idea of the proposed method
Table 1. Proposed Algorithm for DOA Estimation
Assumption:
- All the elements are functioning at time instance − 1, the data from all the
elements are collected
- Some number of elements are failed at time instance
- Number of signals impinging on the array
Begin
Step1: Collect the data snapshots at time interval t from all the elements
Step
2:
If length - ≠ then
Form the location matrix Ω
Apply SVT algorithm to complete the incomplete data matrix
Go to step 3.
Else
Step
3:
Apply MP method to estimate the DOA
used to form the complete data matrix using the SVT algorithm [21] (MC Procedure) and once
the complete data is recovered, in the third step, the MP method is used to estimate the DOA. In
the following section, the details of these steps are discussed.

59
3.1 Array element location matrix
The matrix completion using the SVT algorithm requires the knowledge of the indices of the
observed entries in the data matrix, which are the location of the functioning elements in the
ULA. We propose a simple technique to locate the location of the working elements using
previous sample and the present sample and comparing them to detect whether the particular
element has failed or not. The algorithm to form a location matrix Ω is described in Table 2. and
is based on finding the distance between the respective samples.
Table 2. Algorithm for forming Location Matrix
Initialization :
Index j = 0
Begin:
Step1: Collect and store snapshot
− 1, i is the location value of the element
Step 2: Collect and store snapshot

Step 3: Calculate

6
− 1,
7 = |
− 1 −
|
If

6
− 1,
7 : where : is some threshold = 10-2
then
Ω; = i; Element i is functioning
else
Ω; = i; Element i is malfunctioning
End if;
j=j+1
3.2 Algorithm for DOA estimation
At time instance , a few of the elements will fail to work, resulting in incomplete data. To
recover the complete data from the observed incomplete data which resulted from failure of a few
elements, we apply the SVT matrix completion algorithm given in Table 3.
Recovering a matrix from a sample of its entries is known as the matrix completion problem. In
[16] [20], Candes and Recht, proved that the most of the low rank ‘r’ matrices can be recovered
from its partial set by solving a simple convex optimization problem.
min
-@*AB-
‖-‖∗ (3)
Where, ‖-‖∗ is called the nuclear norm and is defined as the sum of its singular values, - is the
matrix to recovered, Ω is the set of indices’s of the sampled entries, PF. is a masking operator
which selects the entries of - that are within Ω and -3
is the collected partial samples. Provided
that the number of samples obey ≥ I.
JKLM for some positive numerical constant I. The
SVT algorithm developed by Cai, Candes and Shen in [21] is used to solve the norm
minimization problem in (3). However, the entries must be selected randomly and we cannot
hope to complete the matrix if some of the singular vectors of the matrix are extremely sparse.
If the singular vectors of -3
are sufficiently spread, then there is a unique low rank matrix which
is consistent with the observed entries. In such cases, one could, in principle, recover the
unknown matrix by solving
min
-@*AB-
rank- (4)
Unfortunately this is NP-hard. A popular alternative to the NP hard problem is the convex
relation [17] given in (3) where, nuclear norm is the tightest convex relation of the NP-hard rank
minimization problem.

60
The inputs to the SVT algorithm are, a parameter Q, step size R, sampled set Ω, sampled entries
-Ω and initializing vector S+
= 0. The algorithm is
-T =
ℎJVWST, Q (5)
ST
= ST
+ RTΩX
-3
− -T

Repeat steps in (5) until convergence. shrink(.) is a nonlinear function which applies soft
thresholding rule at level τ, to the singular values of the input matrix. The key property here is
that for large values of τ, the sequence Z-T[ converges to a solution which very nearly minimizes
(3). Hence, at each step, one needs to compute only atmost one singular value decomposition and
perform a few elementary matrix additions.
For shrinkage operator, consider the SVD of - of rank r,
- = ]^_
, ] = `MZa
[, 1 J (6)
Where and ^ are the right and the left singular vectors and ] is the singular value matrix. For
each Q ≥ 0, a soft thresholding operator bc defined as follows
bc- = bc]^_, bc ] = `MZa
− Qd[, 1 J (7)
Where ∗d, is the positive part of the *. In other words, this operator simply applies the soft
thresholding rule to the singular values of -, effectively shrinking these towards zero.
The problem of estimating the DOA from incomplete observations in our case can be written as
min
-@*-F
‖-‖∗ (8)
-3
, is the data collected from the working elements and Ω is the location matrix whose entries are
the location of the elements that are working in a ULA. The operator -Ω represents the data
collected from only the working elements. Once the complete data is recovered, MP method is
followed to estimate the DOAs [11] [12]. The SVT algorithm for recovering the missing data is
shown in Table 3.
Table 3: Algorithm of SVT
Input: sampled set or location vetor Ω, incomplete data x3
, step size δ,
tolerence ϵ, parameter τ, kmax maximum iterations.
Output: Recovered data -
For k = 1 to kmax
Let sr = sσ
r
, ⋯ σu
r
v are s singular values and rr = rank-r

Set sr = rr + l
Repeat
Compute [r
, ]r
, ^r
]uy
Set sr = sr + l
until σuyz
r
≤ τ
Set rr = maxsj: σ;
r
Qv
Set -r
= ∑ 6σ;
r
− τ7€;
r
;
r
‚y
;
if
ƒΩX
-3
− -r
ƒ„
/‖-3‖„ ≤ ϵ then break
set
Sr
= †
0 if i, j ∉ Ω
Sr
+ δ-3
− - if i, j ∈ Ω
Š
end for k
- = -r

61
For MP method, the signal model given in (1) is rewritten as,
=
$‹$

+ )$

$*+
, = 1,2, ⋯ ,
(9)
Where,
‹ = Œ
2Ž

V $
(10)
For simplicity the index t is eliminated. The MP method for noiseless data the algorithm begins
by choosing a parameter , known as pencil parameter. A good choice of this parameter is in
between M/3 and 2M/3 [12].
Now we construct a matrix ‘ from the data samples of (9)
‘ = ’
⋯ “d
” ⋯ “d
⋮ ⋮ ⋱ ⋮
—“ —“d ⋯ —
˜
(11)
The two matrices ‘™ and ‘š are defined as
‘™ = ’
⋯ “
” ⋯ “d
⋮ ⋮ ⋱ ⋮
—“ —“d ⋯ —
˜
(12)
‘š = ’
” ⋯ “d
” › ⋯ “d
⋮ ⋮ ⋱ ⋮
—“d —“d ⋯ —
˜
(13)
The matrix pencil for two matrices ‘™ and ‘š are defined as the linear combination of the two
matrices with scalar β described by
‘™ − β‘š (14)
In the absence of noise, it is easy to verify that ‘™ and ‘š can be decomposed in to ‘™ = .ž
and ‘š = .ž such that
. = ’
1 1 ⋯ 1
‹+ ‹ ⋯ ‹
⋮ ⋮ ⋱ ⋮
‹+
—“
‹
—“
⋯ ‹
—“
˜ × ’

+ 0 ⋯ 0
0
⋯ 0
⋮ ⋮ ⋱ ⋮
0 0 ⋯

˜
(15)
ž =
Ÿ

¡
‹+
“
‹+
“
⋯ 1
‹
“
‹
“
⋯ 1
⋮ ⋮ ⋱ ⋮
‹
“
‹
“
⋯ 1¢
£
£
¤
(16)
= ’
‹+ 0 ⋯ 0
0 ‹ ⋯ 0
⋮ ⋮ ⋱ ⋮
0 0 ⋯ ‹
˜
(17)
provided that

62
≤ ≤ − , even
≤ ≤ − + 1, odd
(18)
the pencil is of rank . Under this condition each value of ¥ = ‹$ is a rank reducing number of
the pencil. However if is not within the above range, then none of ‹$’s are a rank reducing
number of a matrix pencil. This implies that the values of ‹$’s are the generalized eigenvalues of
the matrix pair [‘™‘š]. Furthermore, it can be shown that the generalized eigenvalues of ‘™ −
β‘š can be found from the non-zero eigenvalue of ‘š
¦
‘™ where † is the Moore-Penrose
pseudoinverse. Finally the DOAs are estimated by using equation.
$ = −`JM
V ¨
©

`JM‹$ª
(19)
Where, c is the propagation velocity. In the absence of noise, the pencil will have rank , which
is not satisfied when the signal is corrupted by noise. To mitigate the effect of noise Total Least
Squares (TLS) [23] approach is applied by taking Singular Value Decomposition (SVD) of ‘ in
(11). This method is known as Total Least Squares – Matrix pencil (TLS-MP) method.
We start with taking the SVD of ‘ in (11). Only of singular values of this matrix corresponds
to signals, while the rest corresponds to the noise. We generate a new filtered version of the data
matrix.
«‘
¬ = «
¬ «®
¬ «^
¬ (20)
Where, «
¬ are the first left singular vectors, «^
¬ are the first right singular vectors and «®
¬
are the first singular values. Now ‘ is replaced by ‘
¬ in (11). The above steps are followed to
estimate the DOA of the received signals.
4. SIMULATION RESULTS
In this section, we examine the performance of the proposed DOA estimation technique using
several simulations and compare with the standard TLS-MP method for various noisy conditions
and elements failure scenario. The simulation is carried on Intel Core 2 Duo, 2.8 GHz processor,
2 Mb RAM, running on Windows XP SP2 16 bit operating system. We consider a ULA of 100
elements, the spacing between the elements is 2
⁄ , where, is the wavelength of the signal of
interest. The signals are assumed to be impinging from the direction
off the broadside of the
array.
The signals are assumed to be complex exponential sequence given by

= 6¯°7 (21)
Where ± is the random phase uniformly varying between «– Ž, Ž. The SNR in dB at each
element is defined as
³´ = 10 KLM+6
a#
aµ
¶ 7 (22)
Where a#, is the signal power and aµ, is the noise power. The Root Mean Square Error is defined
as
´³· = ¸· ¹6
−
º
7

»
(23)
Where,
is actual DOA and
º
is the estimated values. The failure of elements is considered to
be at random locations. Six signals of equal magnitude are assumed to be impinging on the array
from the directions
= Z0¼
, 5¼
, 10¼
, 15¼
, 20¼
, 30¼[.

63
In our first example, we compare the TLS-MP method and the proposed modified MP method for
functioning ULA. The SNR in this case is taken as 24 dB, the array size is 100 elements. It can be
observed form Table 4, the proposed technique performs better than the TLS-MP method. The
improvement is due to iteratively using the SVD. However, the improvement is obtained at the
cost of slightly increased complexity.
Table 4. Estimated DOA in degrees using MP method and Modified MP method and all the
100 elements are functioning
Assumed DOAs in
degrees
Algorithm
MP method Modified MP method

º RMSE
º RMSE
0 0.0067
0.0022
0.0004
0.0019
5 5.0024 5.0001
10 9.9990 9.9948
15 15.0019 15.0014
20 20.0060 20.0045
30 29.9967 29.9984
In our next example, we compare TLS-MP method and the proposed MP technique. It is assumed
that, all the elements are functioning properly in case of TLS-MP method, however, 5 elements at
random locations are assumed to have failed for the case of modified MP technique. We also
assume only one snapshot to estimate the DOA and 20 iterations are considered for the
simulation. From RMSE plot shown in Figure 3, we observe that the performance of the proposed
technique with 5 faulty elements is comparable with the actual MP method, where all the
elements are functioning. The results are also tabulated in Table 5, for the SNR of 24 dB.
In our third example, we consider the performance of the algorithm for varying number of
working elements. The numbers of elements that are working in each experiment are 95, 90, 85,
80 and 70 elements respectively and the results are plotted in RMSE plot, shown in Figure 4. It is
observed that the algorithm is consistent in its performance until 80 elements are functioning.
Furthermore, when only 70 elements are functioning the performance degrades. The time taken to
perform the matrix completion is observed to be 0.48 seconds and the error in reconstruction is
10-2
, using 50 iterations for the matrix completion
Table 5. Estimated DOA in degrees using MP method and Modified MP method. For
modified MP method 5 elements are malfunctioning
Assumed DOAs in
degrees
Algorithm
MP method Modified MP method

º RMSE
º RMSE
0 0.0067
0.0022
0.0027
0.0023
5 5.0024 5.0014
10 9.9990 10.0014
15 15.0019 14.9976
20 20.0060 19.9965
30 29.9967 29.9991
4.1 Computational complexity
The computational complexity of the algorithms is shown in Table 6. It is observed that even
though the proposed technique has more complexity, the tradeoff is that on the cost of

64
computations, we can save the use of redundant hardware when there is a failure of elements. The
increase in the complexity when compared with the TLS-MP method is only due to the iterative
application of SVD in MC procedure. The computational complexity in SVT is just the number of
iterations multiplied by the complexity of finding the largest singular vectors.
Figure 3: RMSE plot for MP method and the Modified MP method. Modified MP method is
considered for faulty array, where 5 elements are failed to work.
Figure 4: RMSE plot for Modified MP method for varying number of functioning elements.

65
Table 6. Computational Complexity
MP Method Proposed Modified MP Method
SVD ¿4 −
+ 1
+ 8
− + 1
+ 9 + 1”
SVT # of Iteration × ¿ 4√ + 86√7
”
'
EVD ¿ + 1” TLS-
MP
4 − + 1 + 8 − + 1
+ 10 + 1”
Total ¿4 −
+ 1
+ 8
− + 1
+ 10 + 1”
Total 4 − + 1 + 8 − + 1
+ 10 + 1” + 4√ + 86√7
”
5. CONCLUSION
Failure of a few elements in the antenna array is not tolerated in DOA estimation applications.
This paper discusses a technique to estimate the DOAs, when a few elements in the ULA fail to
work. The technique uses a preprocessing scheme namely, singular value thresholding based
matrix completion procedure for recovering the missing data from the failed elements. Later the
MP method is employed to estimate the DOA from the recovered complete data matrix. We have
also proposed a very simple technique to know the location of the elements that are working,
which is required for matrix completion procedure. We evaluated the performance of the
proposed modified MP method under various noisy conditions and element failure scenarios.
Before considering the performance evaluation for the faulty ULA, from the simulation results,
the algorithm is observed to perform better than the conventional TLS-MP method. When the
elements are failed the proposed algorithms continue to estimate the DOAs until certain number
of elements malfunctioning. When a large number of elements have failed the performance of the
algorithm degrades. The advantage of the proposed modified MP method is usage of redundant
hardware is avoided with increase in the complexity of the algorithm due to the repetitive usage
of SVD. Further work has to be done to build more robust DOA estimation algorithms to tolerate
the element failure, as in many applications, all the elements in the underlying ULA can rarely be
expected to function properly. Direct data domain methods like Matrix Pencil method and matrix
completion from the convex optimization theory offer elegant possibilities for developing robust
DOA estimation algorithms.
Acknowledgments
The useful comments made by the reviewers and Mr. Ravindra. S of Jawaharlal Nehru national
college of Engineering, Shivamoga, Karnatka, India are gratefully acknowledged.
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Authors
Yerriswamy T. received B.E. in Electronics and Comm
University, Karnataka, India in 2000. He received his M.Tech. in Network and Internet
Engg. from Visvesvaraya Technological University, India in 2006. He is currently
working towards PhD from Visvesvaraya Technological University, India.
he is an Asst. Proff. at Proudhadevaraya Institute of
Visvesvaraya Technological University)
S.N. Jagadeesha received his B.E.
College of Engg., Davanagere affiliated to
1979, M.E. from Indian Institute of
Commn. Engg., India in 1987 and Ph
University of Roorkee, Roorkee, India
interest includes Array Signal Processing, Wireless Sensor Networks and Mobile
Communications. He has published and presented many papers on Adaptive Array
Signal Processing and Direction-of-
department of Computer Science and Eng
Visvesvaraya Technological University), Shivamog
Signal Image Processing : An International Journal (SIPIJ) Vol.2, No.3, September
Candes and Z. Shen, (2008) “A Singular Value Thresholding Algorithm for Matrix
v:0810.3286.
S. P. Boyd and L. Vandenberghe, (2004) “Convex optimization”, Cambridge University
Van Loan, (1980) “An Analysis of the Total Least Squares Problem
–83.
. in Electronics and Commn. Engg. from Gulbarga
University, Karnataka, India in 2000. He received his M.Tech. in Network and Internet
from Visvesvaraya Technological University, India in 2006. He is currently
PhD from Visvesvaraya Technological University, India. At present
at Proudhadevaraya Institute of Technology (affiliated to
Technological University), Hosapete, Karnataka, India.
B.E. in Electronics and Commn. Engg. from B.D.T
Davanagere affiliated to Mysore University, Karnataka, India in
Indian Institute of Science, Bangalore specializing in Electrical
and Ph.D in Electronics and Computer Engg. from
India in 1996. He is an IEEE member. His research
interest includes Array Signal Processing, Wireless Sensor Networks and Mobile
He has published and presented many papers on Adaptive Array
-Arrival estimation. Currently he is professor in the
Computer Science and Engg, Jawaharlal Nehru National College of Engg.
Visvesvaraya Technological University), Shivamogga, Karnataka, India.
67
A Singular Value Thresholding Algorithm for Matrix
ersity Press.
Squares Problem”,
g. (affiliated to

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