1
A Novel Beamformer Robust
to Steering Vector Mismatch
P. P. Vaidyanathan and Chun

Yang Chen
California Institute of Technology
2
Adaptive Beamforming
•
Adaptive beamforming:
–
Linear combination
–
Extract the signal of
interest (SOI).
–
Suppress interferences
and noise.
Interferences
Antenna array
Signal of interest (SOI)
…
w
0
*
w
1
*
w
2
*
w
N

1
*
+
Beamforming
3
Outline
•
Optimal beamformer
•
Optimal beamformer with mismatch
•
Other robust beamformers
•
New robust beamformer
4
Beamforming
•
By
linearly combining
the
array output signals,
the
signal of interest (SOI) can
be extracted
while the
interferences and noise is
suppressed.
Interferences
Antenna array
Signal of interest (SOI)
…
w
0
*
w
1
*
w
2
*
w
N

1
*
+
Linear
combination
•
Beamforming has long been
used in many areas, such as
radar, sonar, seismology,
medical imaging, speech
processing and wireless
communications.
5
Beamforming (2)
…
d
d
q
x(t)e
j(
2
p
ft

kx
)
dsin
q
(N

1)dsin
q
2dsin
q
y
1
y
2
y
3
y
N
Steering
vector
6
The beamformer output
…
d
d
q
x(t)e
j(
2
p
ft

kx
)
w
1
*
+
w
2
*
w
3
*
w
N
*
Beamformer
response of the
signal from angle
q
.
7
Optimal Beamforming
•
The goal is to extract the signal of interest (SOI)
while suppressing the interferences and noise. In
other word,
maximize SINR
.
•
Instead of
R
v
, only
R
y
can be estimated in
practice.
Therefore we obtain the optimal
beamformer by minimizing the total variance
while constraining the signal response to be unity.
8
Optimal Beamforming (2)
•
The solution to the above problem can be
expressed as
•
The solution is well know as the Capon
beamformer [Capon 1969].
9
Example of an optimal beamformer
0
10
20
30
40
45
50
60
70
80
90
80
60
40
20
0
20
40
Angle (degree)
Beam pattern (dB)
SOI
SINR =
19.70
dB
10
Steering vector mismatch
…
d
d
q
x(t)e
j(
2
p
ft

kx
)
q
m
•
Now the beamformer becomes the solution to the
following optimization problem:
11
Steering vector mismatch (2)
•
Note that the objective function (total variance)
contains the magnitude response of the signal
Magnitude
response of
the signal
•
But the linear constraint is no longer valid.
•
The signal is therefore interpreted as an interference
and is attenuated.
12
An example of steering vector
mismatch
The signal is interpreted
as interference and is
seriously attenuated.
SINR =
19.70
dB

5.10
dB
13
Robust beamforming methods
Modify the objective function
so that the SINR is not that
sensitive to the mismatch
Modify the constraint
so that
it covers more steering
vectors.
•
To overcome this problem, many methods have been
proposed during the past three decades.
•
The goal is to
maintain a high SINR even when there
exists mismatch
.
14
Diagonal loading
Modify the objective
function as [Calson 1988]
•
It increases the variance of the artificial white noise by
g
.
•
It forces the beamformer
put more effort on suppressing
white noise rather than interferences
(The mismatched
signal is interpreted as an interference).
•
However,
it changes the objective function
. That
decreases the SINR.
15
Uncertainty set based methods
Modify the constraint as
•
Modified the constraint so that the magnitude response of
an uncertainty set
of steering vectors exceed unity.
•
If the mismatched steering vector is in this set, It will not
be attenuated.
•
In order to fit the standard optimization tools, the set is
often selected as ellipsoid [Lorenz 2005], sphere, or
polyhedron [Wu 1999].
16
Optimization robust beamformer
•
Minimize the total variance subject to the constraint that
the magnitude response in a range of angle exceed unity.
•
This problem, however, dose not fit any standard
optimization tool. Therefore
it is not clear how to find the
optimal solution.
17
Optimal robust beamformer versus
uncertainty set based methods
•
The set we need is actually a curve. But it is not clear how to
solve the problem with this constraint.
•
In order to solve the problem, the uncertainty set often
selected as ellipsoid, sphere, or polyhedron.
•
However,
this includes too many constraints to minimize the
objective function properly
.
18
New robust beamformer
•
Our approach does this in an opposite way. We start looking
for the solution by loosening the constraint instead of adding
extra constraints.
•
The minimum is a
lower bound
of the original one.
•
The original robust constraint
might not be satisfied
.
19
The two

point quadratic constraint
problem
•
Recast the problem into the following equivalent form
•
Define the following function with the Lagrange multiplier
b
.
20
The two

point quadratic constraint
problem (2)
•
Taking the gradient and equating it to zero and applying the
constraint, we can obtain
•
Substituting
w
0
into the objective function, we obtain
where
21
The two

point quadratic constraint
problem (3)
•
To further minimize the objective function, we choose
•
The objective function can be further minimized by
solving the following optimization problem. The
problem can be solved by using the KKT condition.
22
The two

point quadratic constraint
problem (4)
23
Example of the two

point quadratic
constraint beamformer
•
The two

point quadratic
condition is the necessary
condition of the original
problem. Therefore the
minimal of this problem is the
lower bound of the original
problem.
•
If the solution happens to
satisfied the original constraint,
it is also the solution of the
original problem
.
24
Example of the two

point quadratic
constraint beamformer (2)
•
However, in general the
original condition is not
guaranteed to be satisfied.
•
When the signal is strong,
the beamformer tends to
place a zero around the
signal arrival angle
.
•
As before, this seriously
decrease the SINR
performance.
25
Example of the two

point quadratic
constraint beamformer (3)
•
Comparing these two figures, we see that the norm of
w
becomes very large when the constraint is not satisfied.
26
Norm increasing when the
condition is not satisfied
:Quadratic
constraints
:Zero
Im{z}
Re{z}
•
This is because when the constraint is not satisfied, the beamformer
has a zero whose angle is between the constraint angles.
•
When
the zero is close to the constraints
it attenuates their response.
But the magnitude response is constrained to be unity. Thus
the total
energy of the beamformer must adjust to certain high level
to meet the
constraint.
27
Diagonal loading and two

point
quadratic constraint
•
By observing the fact that the norm increases when the
constraint is not satisfied, we can impose some penalty
on the norm to avoid it.
Diagonal
loading
•
The solution can be found by first performing
then applying the previous algorithm.
28
Iterative algorithm for finding the
diagonal loading factor
•
However, introducing diagonal loading changes the objective
function. We should
choose
g
as small as possible
so that the
robust condition
29
Convergence of the algorithm
•
When
g
goes to infinity, the solution converges to
•
The following lemma gives the condition for which w
satisfies the robust condition
•
If the condition is satisfied, there exist a
g
>0 such that the
w
g
satisfied the robust condition.
30
The interactive algorithm
w
0
w
d
w
w
s
w
g
w
10
w
100
Lemma 1 ensures that w
is robust.
Iterative algorithm find this solution
.
31
New robust beamformer
•
Then a diagonal loading method is applied to force the magnitude
responses at a range of arrival angles to exceed unity.
•
Instead of directly impose constraint on all possible steering
vectors, the method constrains the magnitude responses of only
two steering vectors.
32
Numerical examples
•
Parameters
–
Number of antennas = 10
–
Distance between adjacent antennas =
l
/2
–
Variance of noise = 1
–
Interference 1: 30
o
, 40dB above noise
–
Interference 2: 75
o
, 20dB above noise
–
Angle of arrival: 43
o
–
Assumed angle of arrival: 45
o
–
SNR = 10dB
33
SINR versus
g
w
0
w
d
w
w
s
w
g
w
10
w
100
34
SINR versus SNR
One of the
Uncertainty based method
Objective function
based methods
Our method
Optimal beamformer
with mismatch
Optimal beamformer
without mismatch
35
SINR versus Mismatch angle
36
SINR versus number of antennas
Our method
Optimal beamformer
without mismatch
37
Conclusion
•
This approach constrains the magnitude responses
of only
two steering vectors
and then uses a
diagonal loading method
to force the magnitude
response at a range of arrival angles to exceed unity.
•
Unlike the traditional diagonal loading based method,
this method provides
a systematic algorithm for
finding the diagonal loading factor
g.
•
The complexity of the algorithm is compatible to the
traditional Capon beamformer
•
It has a very good SINR performance under a wide
range of conditions.
38
References
[1] J. Capon, “High

resolution frequency

wavenumber spectrum
analysis,” Proc. IEEE, vol. 57, no. 8, pp. 1408

1418, Aug. 1969.
[2] B. D. Carlson, “Covariance matrix estimation errors and diagonal
loading in adaptive arrays,” IEEE Trans. Aerosp. Electron. Syst.,
vol. 24, pp. 397

401, July 1988.
[3] R. G. Lorenz and S. P. Boyd, ``Robust Minimum Variance
Beamforming,'‘IEEE Trans. SP, vol. 53, pp. 1684

1696, May
2005.
[4] J. Li, P. Stoica, and Z. Wang, ``On Robust Capon Beamforming
and Diagonal Loading,'' IEEE Trans. SP, pp. 1702

1714, July
2003.
[5]S. Shahbazpanahi, A. B. Gershman, Z.

Q. Luo, and K. M. Wong,
“Robust adaptive beamforming for general

ranksignal models,
“ IEEE Trans. SP, pp. 2257

2269, Sept. 2003.
39
Thank you
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