to Steering Vector Mismatch

clanmurderUrban and Civil

Nov 15, 2013 (3 years and 9 months ago)

94 views

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But the linear constraint is no longer valid.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