Robust Adaptive Beamforming via Estimating
Steering Vector Based on Semidefinite Relaxation
Arash Khabbazibasmenj, Sergiy A. Vorobyov, and Aboulnasr Hassanien
Dept. of Electrical and Computer Engineering
University of Alberta, Edmonton, AB T6G 2V4 Canada
Email: khabbazi, vorobyov, hassanie@ece.ualberta.ca
Abstract—Most of the known robust adaptive beamforming
techniques can be unified under one framework. This is to use
minimum variance distortionless response principle for beam-
forming vector computation in tandem with sample covariance
matrix estimation and steering vector estimation based on some
information about steering vector prior. Motivated by such
unified framework, we develop a new robust adaptive beam-
forming method based on finding a more accurate estimate of
the actual steering vector than the available prior. The objective
for finding such steering vector estimate is the maximization
of the beamformer output power under the constraints that the
estimate does not converge to an interference steering vector and
does not change the norm of the prior. The resulting optimization
problem is a non-convex quadratically constrained quadratic
programming problem, which is NP hard in general, but can be
efficiently and exactly solved in our specific case. Our simulation
results demonstrate the superiority of the proposed method over
other robust adaptive beamforming methods.
I. INTRODUCTION
Robust adaptive beamforming is one of the classic array
processing problems with ubiquitous applicability in wireless
communications, radar, sonar, microphone array speech pro-
cessing, radio astronomy, medical imaging, etc. Thus, various
robust adaptive beamforming techniques gained a significant
popularity due to their practical importance [1]. Among first
robust adaptive beamforming techniques are the diagonal
loading [2], [3] and the eigenspace-based beamformers [4].
More recent and more rigorous techniques are the worst-
case-based adaptive beamforming [5]- [8], the probabilistically
constrained robust adaptive beamforming [9], [10], doubly
constrained robust Capon beamforming [11], and the method
of [12], [13] based on steering vector estimation.
In general, most of the known robust adaptive beamforming
techniques can be unified under one framework which can be
summarized as follows. Use minimum variance distortionless
response principle for beamforming vector computation in
tandem with sample covariance matrix estimation and steering
vector estimation based on some prior information about steer-
ing vector. For example, in the worst-case-based robust adap-
tive beamformer of [5], a presumed steering vector is assumed
to be known, while the steering vector mismatch is modeled
as an unknown norm bounded deterministic vector. It has been
widely popularized that the method of [5] can be also obtained
This work is supported in parts by the Natural Science and Engineering Re-
search Council (NSERC) of Canada and the Alberta Innovates – Technology
Futures,, Alberta, Canada.
based on the covariance fitting approach [7] that involves a
steering vector estimation based on the corresponding prior
information. The probabilistically constrained robust adaptive
beamformer of [9] uses different prior information, but it
structurally reduces to the the worst-case-based approach and,
thus, can also be understood in terms of the steering vector
estimation-based framework. The doubly-constrained Capon
robust adaptive beamformer of [11] explicitly estimates the
steering vector using the same prior information as the worst-
case-based approach, while the robust beamformer of [12]
estimates the steering vector by maximizing the beamforming
output power and restricting the estimate from convergence
to any interference steering vector. The later is achieved in
[12] by projecting a steering vector estimate to a certain
subspace obtained from a matrix computed over an angular
sector around the presumed steering vector.
Based on such unified framework to robust adaptive beam-
forming, we develop a new beamforming technique in which
the steering vector is estimated by the beamformer output
power maximization under the constraint on the norm of the
steering vector estimate and the requirement that the estimate
does not converge to an interference steering vector. To satisfy
the latter requirement, we develop a new constraint which
is different from the one in [12] and is convex quadratic.
In general, our new robust adaptive beamforming technique
differers from other techniques by the prior information
about steering vector. The corresponding optimization problem
is a non-convex (due to the steering vector normalization
condition) quadratically constrained quadratic programming
(QCQP) problem which satisfies the strong duality property
(see [14] and [15] for similar problems). However, the analysis
in [14] is based on extended version of the so-called S-lemma,
while the analysis in [15] is based on the so-called rank
reduction technique. Both approaches do not ideally fitthe
purposes of our study. Thus, we take a more common for array
processing linear algebra-based approach while solving the
corresponding optimization problem. It allows us to draw new
links to the previously proposed robust adaptive beamforming
techniques. Our simulation results demonstrate the superiority
of the proposed method over other previously developed robust
adaptive beamforming techniques.
II. S
YSTEM MODEL AND PROBLEM FORMULATION
The narrowband signal received by a linear antenna array
with M omni-directional antenna elements at the time instant
1102978-1-4244-9720-1/10/$26.00 ©2010 IEEE Asilomar 2010
k is
x(k)=s(k)+i(k)+n(k) (1)
where s(k), i(k),andn(k) are the M × 1 vectors of the de-
sired signal, interference, and noise, respectively. The desired
signal, interference, and noise are assumed to be statistically
independent to each other, and the desired signal can be written
as s(k)=s(k)a, where s(k) is the desired signal waveform
and a is the desired signal steering vector.
The beamformer output at the time instant k is
y(k)=w
H
x(k) (2)
where w is the M×1 complex weight (beamforming) vector of
the antenna array and (·)
H
stands for the Hermitian transpose.
Using the available data sample covariance matrix
ˆ
R =
1
K
K
i=1
x(i)x
H
(i) (3)
instead of the unavailable interference-plus-noise covariance
matrix R
i+n
E{(i(k)+n(k))(i(k)+n(k))
H
}, the beam-
forming problem can be formulated as the problem of finding
such vector w that maximizes the beamformer output signal-
to-noise-plus-interference ratio (SINR)
SINR =
σ
2
s
|w
H
a|
2
w
H
ˆ
Rw
. (4)
Here K is the number of training data samples which also
include the desired signal component, E{·} stands for the
statistical expectation, and σ
2
s
is the desired signal power. The
solution to this problem is given by the well-known minimum
variance (MV) sample matrix inversion (SMI) beamforming,
that is,
w
MV−SMI
= α
ˆ
R
−1
a (5)
where α =1/a
H
ˆ
R
−1
a.
Robust adaptive beamforming techniques address the situa-
tions when the steering vector a is not known precisely as well
as when the sample estimate of the data covariance matrix (3)
is inaccurate (for example, because of small sample size). It
is usually assumed that some prior information about steering
vector is available such as the presumed steering vector p as
well as some information about the steering vector mismatch
δ. The actual steering vector differs from the presumed one
and can be expressed as a = p + δ where δ is unknown.
Most of the known robust adaptive beamforming methods
can be represented in the form of (5) with an estimate of the
steering vector
ˆ
a. The estimate
ˆ
a is found using the prior given
by the presumed steering vector p. For example, in [12], [13],
the estimate
ˆ
a is found so that the beamformer output power
is maximized while the convergence of
ˆ
a to any interference
steering vector is prohibited. Specifically, (5) can be written
as a function of unknown δ,thatis,w(δ)=α
ˆ
R
−1
(p + δ).
Using the latter expression, the beamformer output power can
be also expressed as
P (δ)=
1
(p + δ)
H
ˆ
R
−1
(p + δ)
. (6)
Thus, such estimate of δ or, equivalently, a that maximizes
(6) is the best estimate of the actual steering vector a under
the constraints that the norm of
ˆ
a equals
√
M and
ˆ
a does
not converge to any interference steering vectors. The latter is
guaranteed in [12], [13] by requiring that
P
⊥
(p +
ˆ
δ)=P
⊥
ˆ
a =0 (7)
where P
⊥
= I − UU
H
, U =[u
1
, u
2
,...,u
L
], u
l
,
l =1,...,L are the L dominant eigenvectors of the matrix
C =
Θ
d(θ)d
H
(θ) dθ (8)
with d(θ) being the steering vector associated with direction
θ andhavingthestructuredefined by the antenna geometry,
Θ being the angular sector in which the desired signal is
located,
ˆ
δ and
ˆ
a standing for the estimate of the steering
vector mismatch and the estimate of the actual steering vector,
respectively, and I being the identity matrix.
III. N
EW ROBUST BEAMFORMING PROBLEM
FORMULATION
The output power (6) is maximized if the denominator of (6)
is minimized. While maximizing (6) one needs to guarantee
that the norm of the estimate
ˆ
a equals
√
M, i.e.,
ˆ
a =
√
M
and
ˆ
a does not converge to any interference steering vector.
We suggest to achieve the latter purpose by means of the
following new constraint. Let us assume that the desired source
is located in the angular sector Θ=[θ
min
,θ
max
] which can be
obtained using low resolution direction finding methods and
it is distinguishable from general locations of all interfering
signals. Let us define the M ×M matrix
˜
C as
˜
C =
˜
Θ
d(θ)d
H
(θ) dθ (9)
where
˜
Θ denotes the complement of the sector Θ. Then the
new constraint can be expressed as
ˆ
a
H
˜
C
ˆ
a ≤ Δ
0
(10)
where Δ
0
is the maximum value(s) of d
H
(θ)
˜
Cd(θ) within the
sector Θ. Thus, as soon as the sector Θ is known, the value
of Δ
0
is unique.
The constraint (10) forces the estimate
ˆ
a not to converge
to any interference steering vector with the directions within
the angular sector
˜
Θ. Indeed, let us consider as an example
a uniform linear array (ULA) of 10 omni-directional antenna
elements spaced half wavelength apart from each other. Let the
range of the desired signal angular locations be Θ=[0
◦
, 10
◦
].
Fig. 1 shows the values of the quadratic term d
H
(θ)
˜
Cd(θ)
for different angles. The rectangular bar in the figure marks
the directions within the angular sector Θ. It can be observed
from this figure that the term d
H
(θ)
˜
Cd(θ) takes the smallest
values within the angular sector Θ where the desired signal is
located, and increases outside of this sector. Therefore, if Δ
0
equals to the maximum value of the term d
H
(θ)
˜
Cd(θ) within
Θ, the constraint (10) guarantees that the estimate
ˆ
a does not
converge to any interference steering vectors. The constraint
(10) is an alternative to the constraint (7). This can be further
explained later.
1103
−80 −60 −40 −20 0 20 40 60 80
0
10
20
30
40
50
60
θ
d
H
(θ) C d(θ)
Fig. 1. The term d
H
(θ)
˜
Cd(θ) in the constraint (10) versus different angles.
The problem of finding the estimate
ˆ
a can be then formu-
lated as the following optimization problem
min
ˆ
a
ˆ
a
H
ˆ
R
−1
ˆ
a (11)
subject to
ˆ
a =
√
M (12)
ˆ
a
H
˜
C
ˆ
a ≤ Δ
0
. (13)
where the prior p is used only for selecting the sector Θ.
Because of the non-convex equality constraint (12) the QCQP
problems of type (11)–(13) are non-convex and an NP-hard in
general. However, an exact and simple solution for the problem
(11)–(13) can be found.
IV. S
OLUTION VIA SEMIDEFINITE PROGRAMMING
RELAXATION
First, it can be verified that the problem (11)–(13) is feasible
if and only if Δ
0
/M is greater than or equal to the smallest
eigenvalue of the matrix
˜
C. Indeed, if the smallest eigenvalue
of
˜
C is larger than Δ
0
/M , then the constraint (13) can not
be satisfied for any estimate
ˆ
a.However,Δ
0
selected so (see
above) that the aforementioned feasibility condition is always
satisfied and, thus the problem (11)–(13) is feasible.
Second, if the problem (11)–(13) is feasible, the equalities
ˆ
a
H
ˆ
R
−1
ˆ
a = Tr(
ˆ
R
−1
ˆ
a
ˆ
a
H
) and
ˆ
a
H
˜
C
ˆ
a = Tr(
˜
C
ˆ
a
ˆ
a
H
), where
Tr(·) denotes the trace of a matrix, can be used to rewrite it
as the following optimization problem
min
ˆ
a
Tr(
ˆ
R
−1
ˆ
a
ˆ
a
H
) (14)
subject to Tr(
ˆ
a
ˆ
a
H
)=M (15)
Tr(
˜
C
ˆ
a
ˆ
a
H
) ≤ Δ
0
. (16)
Introducing the new variable A =
ˆ
a
ˆ
a
H
, the problem (14)–(16)
can be further rewritten as
min
A
Tr(
ˆ
R
−1
A) (17)
subject to Tr(A)=M (18)
Tr(
˜
CA) ≤ Δ
0
(19)
rank(A)=1 (20)
where rank(·) stands for the rank of a matrix.
The only non-convex constraint in (17)–(20) is the rank-
one constraint (20). Using the SDP relaxation technique,
the relaxed problem can be obtained by dropping the non-
convex rank-one constraint (20) and replacing it by the semi-
definiteness constraint A 0. Thus, the problem (17)–(20) is
replaced by the following relaxed convex problem
min
A
Tr(
ˆ
R
−1
A) (21)
subject to Tr(A)=M (22)
Tr(
˜
CA) ≤ Δ
0
(23)
A 0. (24)
In general, only approximate solution of the original prob-
lem can be found from the solution of the relaxed problem.
However, it has been shown in [14] and [15] that the strong
duality between the primal and dual problems holds for the
considered type of optimization problems and, thus, the exact
solution can be found. It is worth noting that the robust
adaptive beamforming problem considered in [14] is just a
replica of the beamforming problem developed in [11] that
differs from the one in [11] only by adopting an ellipsoidal
uncertainty region instead of spherical uncertainty region.
Thus, the worst-case approach is taken in both aforementioned
works and the choice of the ellipsoid in [14] as well as any
physical meaning of the problem are not given. Moreover, to
establish strong duality, [14] considered an extended version
of S-lemma, while [15] uses the rank reduction technique.
In this paper, different from [14] and [15], we aim at
straightforwardly studying the s tructure of the primal problem
(11)–(13) that gives us the possibility to obtain and explain the
solution in more traditional for array processing linear algebra
terms. It also proves a necessary intuition for understanding the
differences of proposed robust adaptive beamforming method
over the other methods. Toward this end the following results
are of importance.
First, we consider the result that connects the feasibility of
the relaxed problem (21)–(24) to the feasibility of the original
problem (11)-(13). This result states that problem (21)–(24) is
feasible if and only if the problem (11)–(13) is feasible. Due
to space limitations, the complete proof is not given here, but
will be available in [16].
Second, we observe that if the relaxed problem (21)–(24)
has a rank-one solution, then the principal eigenvector of the
solution of (21)–(24) is exactly the solution to the original
problem (11)–(13). However, if the relaxed problem (21)–(24)
has a solution A
∗
= YY
H
of a rank higher than one, i.e., Y is
an N × r matrix with r>1, the exact solution of the original
problem (11)–(13) can be extracted from the solution of the
relaxed problem (21)–(24) by means of algebraic operations
given in the following main constructive result.
Result: If A
∗
= YY
H
is the rank r optimal minimizer of
the relaxed problem (21)–(24), the estimate
ˆ
a can be found as
ˆ
a = Yv where v is an r×1 vector such that Yv =
√
M and
v
H
Y
H
˜
CYv = Tr(Y
H
˜
CY). Note that one possible choice
for the vector v is proportional to the sum of the eigenvectors
1104
of the following r × r matrix
D =
1
M
Y
H
Y −
Y
H
˜
CY
Tr(Y
H
˜
CY )
. (25)
The proof of this result is lengthy and is given in [16].
The last result aims at showing when the solution of the
relaxed problem (21)–(24) has rank one. It is worth noting
that any phase rotation of
ˆ
a does not change the SINR at
the output of the corresponding robust adaptive beamformer.
Therefore, we say that the solution
ˆ
a is unique regardless of
any phase rotation if the value of the output SINR (output
power) is the same for any
ˆ
a
=
ˆ
ae
jφ
. Then our last result
stands that under the condition that the solution of the original
problem (11)–(13) is unique in the aforementioned sense, the
solution of the relaxed problem (21)–(24) always has rank one.
The proof of this result is also given in [16].
In summary, under the condition of the last result, the
solution of the relaxed problem (21)–(24) is rank-one and the
solution of the original problem (11)–(13) can be found as a
dominant eigenvector of the optimal solution of the relaxed
problem (21)–(24). However, when the uniqueness condition
of the last result is not satisfied for the problem (11)–(13),
we resort to the constructive second result, which shows how
to find the rank-one solution of (11)–(13) algebraically. As
compared to the problem in [12], in which the constraint (7)
enforces the estimated steering vector to be a linear combi-
nation of L dominant eigenvectors U =[u
1
,...,u
L
],the
estimated steering vector by the new method is not restricted
by such linear combination constraint due to the new quadratic
constraint (10) and as a result has more degrees of freedom.
It is thus expected that the new robust adaptive beamforming
method will outperform the one of [12] that can indeed be
seen from the following simulations.
V. S
IMULATION RESULTS
AULAof10 omni-directional antenna elements with
the inter-element spacing of half wavelength is considered.
Additive noise in antenna elements is modeled as spatially
and temporally independent complex Gaussian noise with zero
mean and unit variance. Two interfering sources are impinging
on the antenna array from the directions 30
◦
and 50
◦
, while
the presumed direction towards the desired signal is 3
◦
.The
interference-to-noise ratio (INR) equals 30 dB and the desired
signal is always present in the training data. For obtaining each
point in the examples, 100 independent runs are used.
The proposed robust adaptive beamforming method is com-
pared with three other methods in terms of the output SINR.
These robust adaptive beamformers are (i) the worst-case-
based robust adaptive beamformer of [5], (ii) the robust
adaptive beamformer of [12], and (iii) the eigenspace-based
beamformer of [4]. For the proposed robust adaptive beam-
former and the beamformer of [12], the angular sector of
interest Θ is assumed to be Θ=[θ
p
− 5
◦
,θ
p
+5
◦
] where θ
p
is the presumed direction of arrival of the desired signal. The
value δ =0.1 and 6 dominant eigenvectors of the matrix C are
used in the robust adaptive beamformer of [12] and the value
ε =0.3M is used for the worst-case-based beamformer as it
10 20 30 40 50 60 70 80 90 100
0
5
10
15
20
25
30
NUMBER OF SNAPSHOTS
OUTPUT SINR (DB)
OPTIMAL SINR
WORST−CASE−BASED BEAMFORMER
EIGENSPACE−BASED BEAMFORMER
PROPOSED BEAMFORMER
SQP−BASED BEAMFORMER
Fig. 2. Output SINR versus training sample size K for fixed SNR = 20 dB
and INR = 30 dB. The case of exactly known signal steering vector.
−20 −15 −10 −5 0 5 10 15 20
−20
−15
−10
−5
0
5
10
15
20
25
30
SNR (DB)
OUTPUT SINR (DB)
OPTIMAL SINR
WORST−CASE−BASED BEAMFORMER
EIGENSPACE−BASED BEAMFORMER
PROPOSED BEAMFORMER
SQP−BASED BEAMFORMER
Fig. 3. Output SINR versus SNR for training data size of K =30and
INR = 30 dB. The case of exactly known signal steering vector.
has been recommended in [ 5]. The dimension of the signal-
plus-interference subspace is assumed to be always estimated
correctly for the eigenspace-based beamformer and equals 3.
First, we consider the case when the actual desired signal
steering vector is known exactly. Even in this case, the
presence of the signal of interest in the training data can
substantially reduce the convergence rates of adaptive beam-
forming algorithms as compared to the signal-free training data
case. In Fig. 2, the mean output SINRs for the aforementioned
methods are illustrated versus the number of training snapshots
for the fixed single-sensor SNR = 20 dB. Fig. 3 displays the
mean output SINR of the same methods versus the SNR for
fixed training data s ize of K =30. It can be seen from these
figures that the proposed beamforming technique outperforms
the other techniques even in the case of exactly known signal
steering vector. It is especially true for small sample size.
Second, we consider the situation when the signal steering
vector is distorted by wave propagation effects in an inho-
mogeneous medium. Independent-increment phase distortions
1105
10 20 30 40 50 60 70 80 90 100
5
10
15
20
25
30
NUMBER OF SNAPSHOTS
OUTPUT SINR (DB)
OPTIMAL SINR
WORST−CASE−BASED BEAMFORMER
EIGENSPACE−BASED BEAMFORMER
PROPOSED BEAMFORMER
BEAMFORMER OF [12]
Fig. 4. Output SINR versus training sample size K for fixed SNR = 20 dB
and INR = 30 dB. The case of signal steering vector mismatch due to
wavefront distortion.
−10 0 10 20 30 40 50
−20
−10
0
10
20
30
40
50
60
SNR (DB)
OUTPUT SINR (DB)
OPTIMAL SINR
WORST−CASE−BEAMFORMER
EIGENSPACE−BASED BEAMFORMER
PROPOSED BEAMFORMER
BEAMFORMER OF [12]
Fig. 5. Output SINR versus SNR for training data size of K =30and
INR = 30 dB. The case of signal steering vector mismatch due to wavefront
distortion.
are accumulated by the components of the presumed steering
vector. It is assumed that the phase increments remain fixed
in each simulation run and are independently chosen from
a Guassian random generator with zero mean and variance
0.04. The performance of the methods tested is shown ver-
sus the number of training snapshots for fixed single-sensor
SNR=20dB in Fig. 4 and versus the SNR for fixed training
data size K =30in Fig. 5. It can be seen from these
figures that the proposed beamforming technique outperforms
all other beamforming techniques. Interestingly, it outperforms
the eigenspace-based beamformer even at high SNR. This
performance improvement compared to the eigenspace-based
beamformer can be attributed to the fact that the knowledge of
sector which includes the desired signal steering vector is used
in the proposed beamforming technique. Fig. 5 also illustrates
the case when SNRINR where INR stands for interference-
to-noise ratio. This case aims to illustrate the situation when
SIR→∞. As it can be expected, the proposed and the worst-
case-based methods perform almost equivalently.
VI. C
ONCLUSION
A new approach to robust adaptive beamforming in the
presence of signal steering vector errors has been developed.
According to this approach, the actual steering vector is first
estimated using its presumed value, and then this estimate
is used to find the optimal beamformer weight vector. It
has been shown that the corresponding optimization problem
for estimating the actual steering vector can be solved using
the SDP relaxation technique and the exact solution for the
signal steering vector can be found efficiently. As compared to
the eigespace-based method, the proposed technique does not
suffer from the subspace swap phenomenon since it does not
use eigenvalue decomposition of the sample covariance matrix.
Moreover, our simulation results demonstrate the superior
performance for the proposed method over the existing state
of the art robust adaptive beamforming methods.
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