2015_Reduced-Complexity Super-Resolution DOA Estimation with Unknown Number of Sources

1070−9908 (c) 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See
http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/LSP.2014.2368987, IEEE Signal Processing Letters
1
Reduced-complexity super-resolution DOA
estimation with unknown number of sources
V. V. Reddy, Member, IEEE,, Mohamed Mubeen, Boon Poh Ng, Member, IEEE.
Abstract—When the number of sources is inaccurately esti-
mated, it is well-known that the conventional subspace-based
super-resolution direction-of-arrival (DOA) estimation techniques
provide inconsistent spatial spectrum, and hence the DOA esti-
mates. In this work, we present a novel technique which provides
resolution capability comparable with that of the super-resolution
techniques. While the working principle of the proposed tech-
nique is similar to that of the minimum-norm algorithm, the
algorithm is insensitive to the estimated model-order. Simulation
studies show that the proposed technique is advantageous over
the use of subspace-based techniques with the number of sources
estimated by well-known model order estimation techniques.
Index Terms—Super-resolution DOA estimation, model order
estimation, minimum-norm algorithm.
I. INTRODUCTION
DIRECTION-of-arrival (DOA) estimation of closely-
situated sources is of practical interest in radar, sonar,
wireless communications and other applications [1]. High-
and super-resolution techniques for DOA estimation have been
widely used in these applications. With superior capability
to resolve closely-situated sources over techniques such as
the Capon’s estimator [2] and the Maximum Entropy (ME)
method [3], the super-resolution techniques such as multiple
signal classification (MUSIC) [4], minimum-norm [5] (MN)
and estimation of signal parameters via rotational invariance
technique (ESPRIT) [6] have set the benchmark for direction-
of-arrival (DOA) estimation for several years.
For accurate DOA estimation, techniques with super-
resolution capabilities require precise estimation of the model
order. Although several model order estimation techniques
identify accurate number of sources present in the scenario
under low SNR conditions, their performance deteriorates
when the observations deviate from the presumed signal model
due to the medium or noise characteristics. In such scenarios,
techniques such as the Capon’s estimator or the ME method
are employed with a compromise in the resolution. In a
scenario with two closely-situated sources for instance, these
beamforming-based techniques yield a single compounded
peak corresponding to both the sources. The problem in hand
is to estimate the source directions with the number of sources
kept unknown.
DOA estimation with unknown number of sources has been
studied in [7]–[10]. Exploiting non-Gaussianity, a criterion
based on joint diagonalization of spatial cumulant matrices is
The authors are with the School of Electrical and Electronic
Engineering, Nanyang Technological University, Singapore. email:
vinodreddy@pmail.ntu.edu.sg, MOHAMEDM002@e.ntu.edu.sg,
ebpng@ntu.edu.sg
employed for DOA estimation in [7]. A similar non-parametric
criterion has been devised for DOA estimation of tempo-
rally correlated narrowband sources [10] and nonstationary
wideband sources [8]. Alternatively, a MUSIC-like algorithm
was presented in [9] with no such explicit assumptions on
the source signal characteristics. This algorithm solves a
beamforming-based optimization problem independent of the
estimated model order in each direction. While this technique
has superior capability to resolve closely-situated sources over
Capon’s estimator, the performance approaches that of the
MUSIC algorithm (when MUSIC is provided with actual
number of sources) asymptotically. This technique is limited
by its computation cost in solving a generalized eigenvalue
(GEV) problem in each look direction.
In this work, we present a novel technique for super-
resolution DOA estimation with unknown number of sources.
We derive this technique using the optimization problem simi-
lar to the MUSIC-like algorithm [9] while having the working
principle of the MN algorithm. Besides improved performance
over the MUSIC-like algorithm, the proposed technique is
advantageous with significantly reduced computational cost.
The analysis presented in Section III discloses the similarity
in its working principle with that of the MN algorithm.
The performance of the proposed algorithm is evaluated in
Section IV.
II. PROPOSED ALGORITHM
A. Signal Model
Consider a scenario with M far-field sources. The signal
acquired by an L−sensor array can be modelled as
x(n) = A(Θ)s(n) + v(n), (1)
where s(n) ∈ CM×1
and v(n) ∈ CL×1
are the source and
additive uncorrelated sensor noise vectors, respectively, and
A(Θ) = [a(θ1), . . . , a(θM )] ∈ CL×M
is the array manifold
matrix formed by the M steering vectors corresponding to the
sources from directions Θ = [θ1, . . . , θM ]. The steering vector
for a direction θ is given by
a(θ) =
1
√
L
exp kT
θ r1 , . . . , exp kT
θ rL
T
, (2)
where ri is the ith
sensor position vector, kθ = (2πf/ν)uθ
is the wavenumber vector along the direction of wave prop-
agation denoted with a unit vector uθ and f and ν are
the signal frequency and propagation speed, respectively. De-
noting the source and noise covariance matrices as Rss =

2
E s(n)sH
(n) and Rvv = E v(n)vH
(n) , the array co-
variance matrix can be expressed as
R = A(Θ)RssAH
(Θ) + Rvv. (3)
With the additive noise assumed to be uncorrelated across the
array, the noise covariance and the array covariance matrices
are positive definite matrices. The problem in hand is to
estimate the parameter set Θ with unknown number of sources.
B. Optimization Problem
The MN algorithm estimates a weight vector residing in the
noise subspace of R with its first element constrained to be
unity and its Euclidean length is required to be minimum [5].
The dependence on the estimated model order can be cir-
cumvented by modifying the optimization problem such that
the weight vector minimizes the beamformer output power to
ensure that the vector resides in the noise subspace in line with
that of the MUSIC-like algorithm [9]. The norm of the weight
vector solution can be constrained with the first element of the
vector provided additional emphasis. The new DOA estimator
solves the following optimization problem:
minimize
w
wH
Rw
subject to wH
e1eT
1 + βIL w = c,
(4)
where c, β > 0 are constants, w is the beamformer weight and
e1 = [1 0 . . . 0]
T
∈ RL×1
. The optimization problem in (4)
is a special case of the optimization problem presented in [9]
obtained by replacing the steering vector in the constraint by
e1. The Lagrangian function for (4) can be constructed as
L(w, χ) = wH
Rw − χ(wH
e1eT
1 + βIL w − c), (5)
where χ is the Lagrangian multiplier. Equating the gradient of
L(w, χ) w.r.t. the weight vector to zero, the solution is given
by the eigenvector corresponding to the minimum eigenvalue
χmin of the generalized eigenvalue (GEV) problem
Rw = χmin e1eT
1 + βIL w. (6)
Since R and e1eT
1 + βIL are positive definite matrices for
any β > 0, all the generalized eigenvalues of the matrix pencil
R, e1eT
1 + βIL are real and positive [11]. We also note
that the solution w in (6) is independent of the value of c.
The spatial spectrum is given by
P(θ) = 10 log10
1
|wHa(θ)|2
. (7)
For an L−sensor uniform linear array (ULA), the weight vec-
tor solution can be interpreted as an L−order polynomial [5].
The proposed algorithm can therefore resolve a maximum
of L − 1 sources similar to the MN algorithm. In contrast
to the MUSIC-like algorithm, we note that the optimization
problem is independent of the scan direction. The algorithm
therefore requires to solve for the weight vector only once.
This greatly reduces the computational complexity of the
proposed algorithm over that of the MUSIC-like algorithm.
III. DETAILED ANALYSIS
A. Working principle
In order to gain insights into the optimization problem (4),
we assume that the additive noise is uncorrelated across the
array with variance σ2
v. We express the array covariance matrix
by its eigen decomposition as
R = UΛUH
= UsΛsUH
s + UnΛnUH
n ,
(8)
where U and Λ are the eigenvector and eigenvalue matrices of
R, {Us, Λs} and {Un, Λn} are the eigenvector and eigenvalue
matrices corresponding to the signal and noise subspaces,
respectively. For uncorrelated noise, we have Rvv = σ2
vIL =
σ2
vUsUH
s + UnUH
n . Reorganizing (8), we obtain
R = Us Λs − σ2
vIM UH
s + σ2
vUsUH
s + UnUH
n
= UsΛsUH
s + σ2
vIL, (9)
where Λs = Λs − σ2
vIM . Substituting (9) in (6), we have
UsΛsUH
s + σ2
vIL w = χmin e1eH
1 + βIL w. (10)
We simplify this analysis by interpreting the solution for the
GEV problem as the one obtained by simultaneously solving
two symmetric eigenvalue problems while satisfying the unit
weight vector norm constraint. The resulting problem can then
be written as
UsΛsUH
s + σ2
vIL w =
√
χminw (11)
e1eH
1 + βIL w =
1
√
χmin
w = χmaxw and (12)
wH
w = 1, (13)
where χmax = 1/χmin. We derive the following inferences
when w and χmin satisfy (11)-(13):
1) With the weight vector norm constrained to unity by
virtue of (13), (11) minimizes the term Λ
1/2
s UH
s w 2
2
such that the weight vector w resides in the noise
subspace. We therefore obtain significant peaks along
the source directions in P(θ).
2) The eigenvalue problem in (12) maximizes wH
e1
2
2,
or equivalently the first element of w, maintaining
unit weight vector norm that is imposed by (13). This
condition is similar to the conditions enforced by the
MN algorithm on the zeros of the polynomial. While the
MN algorithm forces the first coefficient of the weight
vector to unity and minimizes the norm wH
w, we here
maintain wH
w = 1 and maximize the first coefficient
of the weight vector.
From the above inferences, we observe that the proposed
technique has a similar working principle as that of the MN
algorithm. By minimizing the weight vector norm the MN al-
gorithm is known to force the extraneous zeros within the unit
circle, consequently suppressing any false peaks in its spatial
spectrum due to these zeros. In contrast, the proposed algo-
rithm maximizes the value of the first element of the weight
vector relative to its fixed unit norm, and as a consequence,
the spatial spectrum of the proposed algorithm is less regulated

3
while the estimation accuracy is not compromised. This is of
less concern since the search range can be localized by low-
resolution algorithms such as the conventional beamformer
(CBF) [3] or the Capon’s estimator that are insensitive to the
estimated model order. We therefore refer to this algorithm as
a minimum norm-like (MNL) algorithm.
The influence of β on DOA estimation can be understood by
studying (4) under extreme values of β. While β = 0 simplifies
the optimization problem to that of the ME method by con-
straining the first element of w to unity, a significantly large
β such that β |wH
e1|2
constrains the weight vector norm
alone, yielding the minimum eigenvector of R as the weight
vector solution. Under this condition, the spatial spectrum
of the estimator exhibits false peaks since the weight vector
solution is not constrained to suppress them. An appropriate
choice for β has to be chosen which provides a balance
between these extreme conditions.
Since R is a positive definite matrix, the weight vector
solution in (6) can also be written in terms of the matrix pencil
{ e1eT
1 + βIL , R} as [12]
R−1
e1eT
1 + βIL w =
1
χmin
w (14)
wH R−1e1eT
1 w + βwH R−1w
w 2
=
1
χmin
= χmax.
Denoting the maximum eigenvalues of R−1
e1eT
1 and R−1
as λmax,R−1e1eT
1
and λmax,R−1 , respectively, we have the
inequality
λmax,R−1e1eT
1
+ βλmax,R−1 ≥ χmax. (15)
Since R−1
e1eT
1 is a unit rank matrix, we have
λmax,R−1e1eT
1
= tr{R−1
e1eT
1 } = tr{eT
1 R−1
e1} =
eT
1 R−1
e1. In order to circumvent the two extreme conditions
discussed above we can choose β such that
βλmax,R−1 = eT
1 R−1
e1
β =
eT
1 R−1e1
λmax,R−1
=
eT
1 R−1e1
uH
1 R−1u1
, (16)
where u1 is the eigenvector of R−1
corresponding to
λmax,R−1 .
B. Computational complexity
For the total number of search directions Ntot, the imple-
mentation of the MUSIC-like algorithm was optimized in [9]
by employing the algorithm presented in [13] to evaluate the
GEV along some coarse directions Ncoarse and a fast minor
component analysis algorithm [14] in the remaining Nfine
directions resulting in a computational cost of O(NcoarseL2
+
2NfineNiterL+NtotL), where Niter is the number of iterations
required for the fast minor component analysis algorithm
to converge, excluding the computation of β. In contrast to
the MUSIC-like algorithm, the proposed algorithm requires
to solve the generalized eigenvalue problem only once. The
computational cost is therefore reduced to O(L2
+ NtotL)
with the GEV problem solved using [13].
0 20 40 60 80 100 120 140 160 180
0
5
10
15
20
25
30
35
40
Direction (degree)
SpatialSpectrum(dB)
MNL
algorithm
CBF
True source
directions
MN
algorithm
(with number of
sources known)
(with number of
sources unknown)
Fig. 1. Spatial spectrum of the CBF, MN and the proposed MNL algorithm
with 100 snapshots simulated at SNR = 5 dB.
IV. SIMULATION RESULTS
We first illustrate the DOA estimation capability of the pro-
posed MNL algorithm in comparison with the MN algorithm
for a scenario with L = 8-sensor uniform linear array (ULA)
and three sources situated at an angle 70◦
, 78◦
and 110◦
from the array axis. The 100 array snapshots employed for
DOA estimation are synthesized such that the signal from each
source direction has a signal-to-noise ratio (SNR) of 5 dB.
We also plot the low-resolution power spectrum of the CBF
which can at best identify regions consisting of the sources.
While a confined search over these regions is sufficient for
DOA estimation, the spatial spectrum of the MNL algorithm
is plotted against the entire azimuth range in Fig. 1.
As expected, the power spectrum of the CBF identifies the
azimuth regions which consists of source directions, although
it fails to uniquely resolve the sources from 70◦
and 78◦
.
Within these azimuth regions, the MNL algorithm provides
distinct peaks along the source directions. Besides the source
direction peaks, the spatial spectra of MN and MNL algo-
rithms exhibit two damped peaks at approximately 40◦
and
130◦
. Interpreting the weight vector solution for the ULA as
a polynomial [5], these peaks are due to the proximity of its
extraneous zeros to the unit circle.
We next consider four sources positioned at 40◦
, 70◦
,
78◦
and 110◦
with respect to the array axis in the current
simulation while using the array setup employed for the
previous simulation. The performance of the MNL algorithm
is evaluated by first studying the probability of resolving all
the source directions against SNR estimated over 200 Monte-
Carlo trials in comparison with the MN algorithm and the
MUSIC-like algorithm in Fig. 2. All the source directions
are considered to be resolved when the algorithm provides
a significant peak for each source θi within the spatial region
[θi − ∆θ, θi + ∆θ], where ∆θ = min{θi − θj}/2, ∀i, j and
i = j ∈ M. In order to evaluate the advantage of the
MNL algorithm we compare the resolution capability of the
MUSIC-like algorithm with unknown number of sources, the

4
-30 -20 -10 0 10 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
ProbabiltyofResolution
MNL algorithm
(with number of
sources unknown)
MN algorithm
(with number of sources
estimated from AIC)
(with number of
sources unknown)
(with number of
sources known)
MN algorithm
MUSIC-like
algorithm
Fig. 2. Probability of resolving two closely-situated sources plotted against
SNR for the MUSIC-like algorithm, the MN algorithm with the number
of sources estimated using AIC, the MN algorithm with known number of
sources and the proposed MNL algorithm.
MN algorithm with the number of sources estimated using
the AIC algorithm [15] and the MN algorithm with the true
number of sources known. We first note that the proposed
MNL algorithm outperforms the MUSIC-like algorithm be-
cause, under low SNR conditions, the MUSIC-like algorithm
struggles to contain the weight vector solution within the noise
subspace due to the direction-dependent constraint, while the
constraint of the MNL algorithm only regulates the positions
of the extraneous zeros within the unit circle. The proposed
MNL algorithm is observed to have higher probability of
resolving source directions at lower SNRs compared to the
MN algorithm when the model order is estimated using AIC
while the MN algorithm with known model order has the best
probability of resolution of all the algorithms under study.
For the above simulation setup, we study the estimation
accuracy with the average root-mean-squared error (RMSE)
of all sources against SNR, evaluated as
RMSE =
1
M
M
i=1
1
K
K
k=1
(θi − θi)2, (17)
where θi is the estimated source direction and K is the number
of trials. The RMSE is plotted for the MN, MUSIC-like and
the MNL algorithms in Fig. 3. The MN algorithm presumes
the true number of sources present in the scenario. From
Figures 2 and 3, we infer that the MNL algorithm has superior
probability of resolution and better estimation accuracy over
the MUSIC-like algorithm. This observation is inline with the
superior performance of the MN algorithm over the MUSIC
algorithm in its estimation accuracy and resolution capability,
because the MUSIC-like and the proposed MNL algorithms
have their working principles similar to that of the MUSIC
and the MN algorithms, respectively.
For the same simulation setupas above, we study the in-
fluence of β on the probability of resolution of the MNL
algorithm. Since the value of β is dependent on the data in
each trial, we first estimate β using (16) and perform DOA
estimation for various values of β = αβ, where α is varied
from 0.01 to 5 in order to study the resolving capability of the
-10 -5 0 5 10 15 20 25
10
-2
10
-1
10
0
SNR (dB)
RMSE(degree)
MNL algorithm
(with number of
sources unknown)
MN algorithm
(with number of
sources known)
MUSIC-like
algorithm
(with number of
sources unknown)
Fig. 3. Average RMSE plotted against SNR for MN algorithm with known
number of sources, the MUSIC-like algorithm and the proposed MNL
algorithm.
0 1 2 3 4 5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Probabilityofresolution
α, a scaling factor for β
SNR = −5 dB
SNR = −2 dB
SNR = 2 dB
SNR = 0 dB
Fig. 4. Probability of resolution plotted against the scaling factor α for various
SNR values.
MNL algorithm with β. We plot the probability of resolution
against α in Fig. 4 for various values of SNR. We note that the
probability of resolution for the MNL algorithm is maximum
when α = 1 and the resolution capability decreases when
α = 1 suggesting that the derived expression for β in (16) is
suitable for various SNR conditions.
V. CONCLUSION
In this work, we have presented a novel technique for
high resolution DOA estimation over a confined spatial region
with unknown number of sources. The proposed algorithm
is shown to have its working principle similar to that of the
minimum-norm algorithm. With enhanced resolution for DOA
estimation over the MUSIC-like algorithm and the minimum-
norm algorithm supported by AIC for model order estimation,
the proposed minimum-norm-like algorithm is shown with
simulation studies to have superior estimation accuracy over
the MUSIC-like algorithm.
REFERENCES
[1] H. Krim and M. Viberg, “Two decades of array signal processing
research: the parametric approach,” IEEE Signal Process. Magazine,
vol. 13, no. 4, pp. 67–94, Jul 1996.

5
[2] J. Capon, “High-resolution frequency-wavenumber spectrum analysis,”
Proc. of the IEEE, vol. 57, no. 8, pp. 1408–1418, 1969.
[3] L. C. Godara, Smart Antennas. CRC press, 2004.
[4] R. Schmidt, “Multiple emitter location and signal parameter estimation,”
IEEE Trans. Antennas and Propagation, vol. 34, no. 3, pp. 276–280,
1986.
[5] R. Kumaresan and D. Tufts, “Estimating the angles of arrival of multiple
plane waves,” IEEE Trans. Aerospace and Electronic Systems, vol. 19,
no. 1, pp. 134–139, Jan 1983.
[6] R. Roy and T. Kailath, “ESPRIT-estimation of signal parameters via
rotational invariance techniques,” IEEE Trans. Acoust., Speech, and
Signal Process., vol. 37, no. 7, pp. 984–995, 1989.
[7] W.-J. Zeng, X.-L. Li, and X.-D. Zhang, “Direction-of-arrival estimation
based on the joint diagonalization structure of multiple fourth-order
cumulant matrices,” IEEE Signal Process. Lett., vol. 16, no. 3, pp. 164–
167, March 2009.
[8] W.-J. Zeng and X.-L. Li, “High-resolution multiple wideband and
nonstationary source localization with unknown number of sources,”
IEEE Trans. Signal Process., vol. 58, pp. 3125–3136, Jun. 2010.
[9] Z. Ying and B. P. Ng, “Music-like DOA estimation without estimating
the number of sources,” IEEE Trans. Signal Process., vol. 58, no. 3, pp.
1668–1676, 2010.
[10] W.-J. Zeng, X.-L. Li, and H. So, “Direction-of-arrival estimation based
on spatial-temporal statistics without knowing the source number,”
Signal Processing, vol. 93, no. 12, pp. 3479–3486, 2013.
[11] B. N. Parlett, The Symmetric Eigenvalue Problem. Prentice-Hall, 1998.
[12] V. Reddy, B. Ng, and A. Khong, “Insights into MUSIC-like algorithm,”
IEEE Trans. Signal Process., vol. 61, no. 10, pp. 2551–2556, May 2013.
[13] Y.N.Rao and J.C.Principe, “An RLS type algorithm for generalized eigen
decomposition,” in Proc. IEEE Signal Process. Society Workshop on
Neural Network for Signal Process., Sep. 2001, pp. 263–272.
[14] S. Attallah and K. Abed-Meraim, “A fast algorithm for the generalized
symmetric eigenvalue problem,” IEEE Signal Process. Lett., vol. 15, pp.
797–800, 2008.
[15] H. Akaike, “A new look at the statistical model identiﬁcation,” IEEE
Trans. Automatic Control, vol. 19, no. 6, pp. 716–723, Dec. 1974.

2015_Reduced-Complexity Super-Resolution DOA Estimation with Unknown Number of Sources

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Viewers also liked

Viewers also liked (13)

Similar to 2015_Reduced-Complexity Super-Resolution DOA Estimation with Unknown Number of Sources

Similar to 2015_Reduced-Complexity Super-Resolution DOA Estimation with Unknown Number of Sources (20)

2015_Reduced-Complexity Super-Resolution DOA Estimation with Unknown Number of Sources