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296
IEEE
TRANSACTIONS ON SIGNAL PROCESSING, VOL.
44,
NO.
2,
FEBRUARY 1996
Kjell
of Wing Flexure Induce
S
-Finding A
Gustafsson, Member, IEEE, Frank McCarthy, Member, IEEE, and Arogyaswami Paulraj, Fellow, IEEE
Abstract-Errors in array calibration are. the dominant error
source for direction finding DF) in airborne platforms. This
problem arises since wings in large surveillance aircraft exhibit
significant flexure, and their actual instantaneous positions during
array calibration and operational flight
is
likely to be quite
different. Scattering from time-varying wing structures onto the
belly mounted antennas therefore causes the array responses to
deviate from the array calibration and
gives
rise
to
DF errors.
We present a simple model for array manifold perturbations due
to wing flexure that nicely captures their effect. The model
is
physically motivated and has been validated using experiments
on
a scale-modelaircraft in an anechoic chamber. Our model
can
be exploited to derive new versions of the classical
DF
estimation
schemes including weighted subspace fitting WSF).
I. INTRODUCTION
IRECTION finding (DF) for multiple, co-channel
signals
has received considerable attention over the last 15years.
Many techniques have been proposed to determine the angle-
of-arrival (AOA) of each signal using measurements from an
array of antennas; these include [1]-[8]. For DF, the response
of the antenna array (the array manifold) must be known over
the range of AOA’s and frequency bands of operation. Errors
in the array manifold are an important error source for DF.
Errors in the array manifold arise from changes in the
antenna gain andor phase, changes in the positions of the
antennas, or variations in local scatterers that couple energy
into the antenna. One approach to improving DF accuracy in
the presence of such errors is to model the array manifold in
terms of a few unknown parameters and then estimate these
parameters along with the AOA’s. This results in the so-called
“autocalibration” approach; see
[9]-[ 121.
Another approach to improved DF accuracy is to con-
centrate on the statistical properties of the array manifold
error. An example is presented in [13], where the statistics
of an array manifold perturbation are used to derive an
optimal weighting for MUSIC. Although this leads to a more
robust estimation procedure, it does not fully exploit available
parametric structure of
the
manifold error.
Manuscript received May 29, 1994; revised
June
19, 1995.
This
work was
supported by the Advanced Research Projects Agency of the Department of
Defense and was monitored by the Air Force Office of Scientific Research
under Contract F49620-91-C-0086, The associate editor coordinating the
review of .this paper and approving it for publication was Dr. R. D. Preuss.
K. Gustafsson is with the Ericsson Mobile Communications AB, Lund,
Sweden (email: [email protected]).
F. McCarthy is with the ARGOSystems, Inc., Sunnyvale, CA 94088 USA
(email: mccarthy
@
rascals. stanfordzdu)
A.
Paulraj is with the Information Systems Laboratory, Stanford University,
Stanford, CA 94305 USA (email: paulraj @rascals.stanford.edu).
Publisher Item Identifier S 1053-587X(96)01655-6.
1053-587X/96$05
DF from surveillance aircrafts is one application area where
the problem with array manifold errors has to be addressed to
improve performance. Antenna arrays for communications re-
connaissance are typically mounted on the belly of the aircraft.
The antenna arrays are designed for signals in the VHF band
and lower part of the UHF band. In order to achieve accurate
AOA estimates, one measures or calibrates the response of the
antenna array in flight over the full operating azimuth angle.
This calibration data captures the array response including the
scattering from the aircraft structure but only at the current
position of the aircraft structure. Most of the fuselage is fairly
rigid, but the position of the wing tips
of
an aircraft may move
several meters vertically relative to the fuselage during flight.
Most of the variation in wing flexure is due to changes in
fuel loading, but turbulence and varying flight conditions also
cause wing movements.
The wing flexure makes antenna-array calibration difficult
for airborne platforms. The array manifold is typically col-
lected without regard for wing position. Wing movements,
however, can cause significant changes in the near-field mul-
tipaths and, hence, perturb the array manifold. The deviation
in the array manifold from the calibrated value leads to DF
errors. In order to achieve highly accurate DF, wing flexure
must be addressed.An important first step is to obtain a model
that captures the properties of the array manifold error. In this
paper, we present a simple model for the wing flexure-induced
array manifold errors. The model is physically motivated and
can be used to improve the accuracy in several classical DF
algorithms.
The array manifold can be thought of as consisting of two
parts:
where
~ 6 )
s the manifold collected during calibration,
and ii B,v) is the perturbation caused by the wing positions
deviating
from
the position
Q,,
during calibration. The pertur-
bation depends both on the signal direction 8 and the wing
positions
77.’
The perturbation is zero when
pl
=
vo.
The
wing-tip position, of course, varied as the calibration data
were collected; therefore, qo is a function of 8. The array
manifold also depends on the frequency of the transmitted
signal. The variation over the processing bandwidth
is
small,
and therefore, the dependency is dropped.
‘To fully descnbe the array manifold perturbation, the model must com-
pletely capture the
wing
flexure.
We use 9
to denote whatever parameters it
takes to exactly describe the position of the wings.
~
.OO 0
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et
al.: MITIGATION
OF WING
FLEXURE INDUCED ERRORS
FOR
AIRBORNE DIRECTION-FINDING
APPLICATIONS
29
The relation between the wing flexure and the perturbation
(e, )
s very complicated. The perturbation, however, has a
special structure that can be exploited.As will be demonstrated
later, the dominating part of
ii(0,q)
spans a low-dimensional
subspace. The magnitude of the perturbation in relation to
the nominal manifold vector depends on the wing flexure. Its
direction, however, is only weakly dependent on both
0
and q.
It is well known that sharp corners (like wing tips) are a
major source of electro-magnetic scattering. The wing tips
are also the part of the aircraft structure that move the most,
and it is reasonable to assume that a dominating part of the
perturbation originates from changes in the scattering around
the wing tips. The magnitude and phase of the scattered signal
varies with the wing flexure, but since it originates from within
a reasonably small area near the wing tip, the corresponding
subspace in which i i O , q ) ies will be reasonably invariant.
The subspace invariance of the perturbation can be exploited
to improve the DF accuracy. One straightforward approach is
to reduce the effect of the perturbation by projecting the data
on the space orthogonal to the subspace spanned by
&(O,q).
In some DF methods, e.g., weighted subspace fitting (WSF)
[14], the projection need not be performed explicitly but can
be introduced implicitly in the calculations.
In Section 11, we motivate a simple model for i i O , q),which
captures the gross behavior of the perturbation. In Section
111, it is demonstrated how the structure of the perturbation
model can be exploited to improve the accuracy of the DF.
Simulations indicate that the DF performance gains are good
even with approximate wing tip scattering models. The validity
of the perturbation model was evaluated using experiments
in an anechoic chamber. These experiments are described
in Section IV. Finally, Section V contains some concluding
remarks.
A. Notations
voltages obtained from the antennas, can be modeled as
It is well known that
~ ( t ) ,
hich is the m x 1 vector of
~ ( t )
A(O)s(t)+ n(t)
(2)
where
s ( t )
d x
1 vector containing the
d
transmitted signals,
A(O) m
x
d matrix where column j contains . e,), which
is the sensor response caused by a unit wavefront
impinging from direction O
n(t) m
x 1
vector of additive noise.
Under reasonable assumptions [14], the array output is
a complex Gaussian vector with zero mean and covariance
matrix
R = E
[ ~ ( t ) ~ ~ ( t ) ]A S A ~
& I
3)
where
S
= E [s ( t ) sH( t ) ] . (4)
The rank of S is
d’.
If
d’
< d , here is a linear dependence
among the transmitted signals.
The covariance matrix
R
can be decomposed as
rn
R
= X e e H =
E,A,E:
+ E,A,E: ( 5 )
where A1 > > Ad’ >
=
-
A
=
o2
’
a = 1
The matrix
E,
=
[el,
.
,
d /
]
contains the
d’
eigenvectors
of R corresponding to the d’ largest eigenvalues. These
(signal) eigenvalues are assumed to be distinct. The range
space of E, is called the signal subspace. Its orthogonal
complement-the noise subspace-is spanned by the columns
of E , = [ed/+l,*..,e,].
In WSF (see [14]), the direction of arrival is determined
by estimating
E ,
from the sample covariance matrix and then
minimizing
where P i ( , ) s a projector onto the null space of
AH(B),
nd
W
is a
d’
x
d’ weighting matrix.
11.
A MODEL
OF THE ARRAY
MANIFOLD
PERTURBATIONAUSED
Y THE
W-ING FLEXURE
The nominal array manifold ao(0)contains components that
include scattering at the wing tips of the impinging signal.
However, when the wing tip’s position changes from its
nominal position during array calibration, the new array vector
will be different. In particular, it includes a perturbation on
the form
2
( d , q )
x
CP(Q va)as (va)
(7)
a = 1
where Q, vz)s the array response to a signal originating from
a point source at the ith wing tip, and P ( 0 , q ) is a complex
coefficient representing the difference between the magnitude
and phase of the scattered signal during calibration and at
the current position of the wing tip. The parameter vector
vi
encodes the part of Q that specifies the position of the ith wing
tip. The more the wing deviates from its calibration position,
the larger the magnitude of
p ( 0 , q )
gets. The array response
u,(v;) s fairly insensitive to the vertical movements of the
wing, whereas P ( O , q ) varies quickly with both O and
q.
If the antenna array is mounted symmetrically on the aircraft
(which we will assume in the sequel), the two wing tip sources
cause structurally similar (symmetric) perturbations. In the
unsymmetric case, the model
(7)
contains two factors
a,
(
v a )
with different v,.
A. An Example
To demonstrate the effects of a perturbation on the form (7),
we will consider a simple example. Assume that the nominal
array manifold
a
0) is given by an m-element uniform linear
array (ULA) mounted along the belly of the aircraft; cf. Fig.
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TRANSACTIONS ON SIGNAL PROCESSING,
VOL.
44,
NO.
2, FEBRUARY 1996
_ - - _
-.
, _--.
I
\ I
;
-->, \$
'&
:,
',
$-?
--------.
,,.---><
I ..
.
-
---,Y -..
{
r
,,
Fig. 1. Uniform linear array with a planar wave impinging from direction 0
and a scatter source emitting a spherical wave from a position
v
= [r 4 ith
respect
to
the array center.
The
array has
m
elements spaced
A
wavelengths
apart. The angles 0 and q5 are regarded positive in counter clockwise direction
and are measured with respect to the axis perpendicular to the array.
1 , i.e.,
-1
()(e) = [. .
e - 3 2 ~ k -
-)A
sin
0
k
=
1, . .
m
8)
where the element spacing
A
is in terms of wavelengths. The
response is normalized with respect to the center
of
the array.
A source (the wing tip scatterer) located at distance r (in
wavelengths) and angle 4 with respect to the array center gives
rise to the array response
where
Y
=
[r
41.
There are two wing tips, and the
pertur-
bation to the array manifold will consist of two factors of the
form (9). The planar wavefront reaches the two wing tips at
different times and gets scattered differently. Hence, the two
factors will have different amplitude and different phase with
respect to the nominal response.
From Fig.
1
=
rcos( f
- e
and by using complex scalars
a1
and
a 2
to denote the
amplitude and change of phase of the two scattered signals,
the
perturbation
to the array manifold
is
given by
q o
l )
=
Q l e 3 2 m 4 4 1 - 0 )
a s
Y1)
Os V2).
(10)
a2e32.rrr~ ~ 0 4 4 2 - 0 )
Assuming a symmetrically mounted antenna array, i.e.,
r1 =
r2
and 4 1 =
?r
- 4 2 ,
(10)
simplifies to (7) with
. (11)
(Q,s>
=
ale32.rrrcos
(4-6 )
+ a2e-32""+" cos (d+0)
Fig. 2 depicts the estimation error that results when using
weighted subspace fitting for DF. The data are generated
assuming a uniform linear array with six elements spaced half
r = i 0
0 =Odeg r=10, 0 =22.5deg
r
I
n l
- 1
0
-1 -1
- 1
0
U
b
Fig. 2. Error in DF due to a perturbation of the type (7) to the array manifold.
The thin curves correspond to cy1 = a: = 0.1 in (11), whereas the thick
curve corresponds
to
a
= a2 = 0.05.
a wavelength apart and an array perturbation
of
the form
(7)
with
u, Y)
and
P(0 ,q )
given by
(9)
and ( l l ), respectively.
The subspace fitting is done using the nominal array manifold
corresponding
to
the uniform linear array. There was no noise,
i.e.,
gE =
0,
in
the data. The four plots in Fig. 2 depict
different scenarios regarding the position of the wing tips. The
thin curves correspond to a1 = a 2 = 0.1, whereas the thick
curves correspond to
a
=
a2
=
0.05.
As is evident from Fig.
2,
the magnitude of the estimation
error is related to the magnitude of
a =
[ a 1 a 2 ] .When
a
grows, so does the estimation error
e
= - 9. n
1151,
the root
mean square (rms) value of e is quantified in terms of a.The
approach is to assume that
a
can be modeled as a circularly
symmetric Gaussian random variable with zero mean, i.e.
E a = 0, E a &
= ~ ~ 2 1 ,
aaT = 0. (12)
The circular symmetry in
a
corresponds to the assumption
of l l phase changes equally likely in the scattering process.
The value
of
2 determines the power of the scattered signal.
Using a Taylor expansion
of
the cost function that WSF
izes, the
rms
value
of
8 can be related to
02 .
In
[16], the
DF
errors due to wing flexure is estimated to about
51 .
This corresponds fairly well to the errors resulting from
g:
= 0.01, i.e., having the power of the scattered signal
at about one hundredth of the direct signal, which seems
physically reasonable. It is of course not meaningful
to
directly
compare the estimation errors in the example with real life
DF errors, but the figures still indicate that a reasonably sized
perturbation on the form
(7)
will cause the type of DF errors
observed in practice.
From Fig. 2, we also note that a wing tip positioned
symmetrically with respect to the array
(4
=
0) creates a
situation with 8-antisymmetric estimation errors. Moreover, a
large distance
T
from the array center to the wing tip causes
the estimation error
to
vary considerably with respect to small
changes in 8. The estimation error is related to the distance
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(in m space) between the nominal manifold ao(6') and the
perturbed manifold
a(0 ,
).
This
distance depends on ,l?, which
for large radii T varies more rapidly in 8; cf. 11). This,
together with the general form of 1 1 ) , indicates that a modified
DF
method that relies on the direct estimation of p may be
sensitive to errors in
v.
B. The Structure of the Perturbation
In order to derive a new
DF
method that is able to cope with
the array manifold perturbation
ii(6',
),we note that with (7)
and one impinging signal, the data model
(2)
can be written
as (we assume only one wing tip in the sequel, the model
can easily be extended to two or more dominant scatterer
scenarios)
From
this expression, it is evident that the scenario can be
viewed as having two coherent signals arriving from different
directions. One of these directions is known from the structure
of the array manifold perturbation, e.g., a,
v).
The scenario is
similar for several signals. In that case, there will be d signals
impinging on the array. One of these signals is the scattered
signal, and there will only be d' =
d -
1 independent signals,
i.e.
4 8 ) = b o ( 6 )
* * *
U d ' ( 6 ' ) as(v)
T
44 =
[S l ( t )
* * S d ' ( t > E=
P ( e k ,
) ) S k ( t )
in
(2).
111.
DF
METHODS
XPLOITING
THE STRUCTURED PERTURBATION
There are several ways to exploit the perturbation model
The array response could be parameterized using 8 and
the perturbation model parameters, e.g., v and ,l? or
v and a.
WSF
could then be used to estimate all pa-
rameters simultaneously. This approach would require
an investigation of the identifiability of the different
parameterizations.
If treating the array perturbation as a random variable, the
perturbation model could be used to quantify its mean and
variance. This information could then be used as outlined
in, e.g., [13].
The perturbation lies in a (known) low-dimensional sub-
space. This could be exploited to improve DF accuracy by
projecting
the data onto the null space
of
the perturbation
subspace prior to performing
DF.
Comparing different
DF
algorithms is outside the scope of
this paper. We will, however, use the projection method to
exemplify how the perturbation model can be used to improve
DF
accuracy.
(7)
in
DF
algorithms:
In general, a,
)
s not parallel to a (6 ) for any
6
because
the scattered signal is a near-field source. The nominal array
response uo(8) s related
to
the planar wave front of a far-
field source. Hence, the effect of the scatter source can be
removed by projecting the received data onto the null space
of
a:(.)
before doing
DF.
The null space of
U:(.)
has
dimension
m
-
1,
and the
DF
will qualitatively behave as
if the array consisted of m - 1 sensors. Any
DF
algorithm
can be used, provided that the nominal array manifold
( L O ( @ )
is also projected onto the null space of U: (
v)
before the
DF
is carried out.
A.
A
ModiJied
WSF
Criterion
When using
WSF
for
DF,
the projection of the data using
P ) need not be performed explicitly. The projection can
instead be incorporated into the
WSF
criterion function. The
modified criterion is (cf. 6))
where
71=
[Ao(B)
a,(v) ,
and
Ao(6')
s the m x
d
matrix
of the nominal array response vectors for the d' sources. The
only difference between the criterion function
14)
and the
standard one 6) is that the sensor array response
a, v)
or
the dth signal (the signal from the scatter source) is known.
Let B and C be two arbitrary matrices with the same
number of rows, and let P I
B
c denote the projection matrix
onto the space spanned by the columns of [ B C ] . hen
where BC is the residual of the columns of B when projected
on
C,
i.e.
BC
= ( I
- PC)B.
Using 14) and 15), we have
P
- 1 P [
Ao(q
a , v) = I
- PX - Pa, v)
A -
= Pk - pa@)
with X =
P )Ao(6').
The projection onto Pus(v)does
not depend on 8 and can be excluded in the minimization
of 14). Consequently, 14) can be interpreted as minimizing
the projection of ,WE, onto the null space of the part of
A f ( 8 ) hat is orthogonal to
a , ( v ) .
Consider the performance of the modified
WSF
criterion in
the absence of noise when the position
U
of the scatter source is
known. We will consider the case of one impinging signal; the
case for multiple signals is a straightforward generalization.
The signal
subspace E ,
=
~ o ( 6 ' )+ 4(6' , v)a , ( v)s a he a r
combination of
00
(6 ) and as v) Multiplying E , from the left
by the projection matrix
Pl
esults in a zero, provided the
projection matrix is formed based on the same value of
6
and
v
that defined E,. Consequently, the AOA estimate obtained
using the modified criterion is correct, i.e., 6 = 8
AH
A
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E E E TRANSACTIONS ON SIGNAL PROCESSING, VOL. 44, NO.
2 ,
FEBRUARY
1996
5
5
-
P
E O 3 0
5 5
-5 -5
-50
0 50
-50 0 50
-50 0 50
-50
0 50
e [degl e Idegl
Fig 3 Level curves for the rms value of the DF error
6
=
6
-
'
as a
function of AOA 6 and wing tip position error The top four plots depict the
scenano when the radius r used in the DF differs from the true radius T O ,
whereas the lower-four plots correspond to error in the angle 4 errors The
contours for
rms(0
= 0 1 , 0 2 , , 1 0 are plotted The error is zero when
r = TO and g5 = 40
B. Sensitivity to Errors in the Scatter Source Position
The result for the WSF criterion in absence of noise relies
on perfect knowledge of the position of the scatter source.
If the actual position v differs from the value YO used when
forming the projection matrix, the result will be an error in
the direction estimate. If
v
is sufficiently close to vo, he end
result is still a reduction in error compared with DF based on
the nominal array manifold uo(I9).
To investigate how sensitive the projection method is to
errors in vo, we return to the previous ULA e_xample. Fig. 3
depict the rms value of the estimation error I9
=
I9 -
9
that
results when using WSF and the criterion function (14) for
DF. The DF was done based on wing tip positions defined
by v = [TO
401,whereas the
true
position was Y =
[T
41.
The data in the figure were obtained by averaging over
100
simulations with a (cf. 1 1) and (12)) drawn from a circularly
symmetric Gaussian distribution with variance 02 =
0.01.
There was no noise present, i.e., 02 =
0.
As
is
seen from Fig. 3, the new estimation scheme is fairly
insensitive to errors in the scatter source position. Using the
standard criterion function
6),
the DF leads to an average
rms estimation error of about
0.5'.
The error in the scatter
source position has to be substantial for the new DF method
to perfom worse than that.
The estimation scheme is very robust to errors in r . The
radius TO used when forming the projection matrix can be
about f 20% in error and still not cause estimation errors
larger than 0.2' rms (cf. Fig. 3). The sensitivity to an error in
T grows as T becomes smaller. The larger the distance from
the antenna array to the scatter source, the closer the spherical
wave
from
the scatter source resembles a planar wave at the
antenna array.
A
change in position of a distant source will
cause a small change in curvature of the spherical wave. A
similar change in position for a source close to the array causes
a larger change in the curvature of the spherical wave. This
makes the scenario with ro
= 2
more sensitive to an error in
radius than the case with ro =
10.
From the simulation, we see that the new estimation scheme
improves the direction of arrival-estimate even when q3 differs
from 40 by several degrees. Within a sector of about f O the
rms value of the estimation error stays below 0.2'. This sector
more or less covers the width of
a
wing at the wing tip. From
Fig. 3, it is clear that errors in 4 are more severe than errors
in
r
Intuitively, this also makes sense because the direction of
arrival of the scattered signal should be more important than
the exact curvature of the spherical wave it produces.
We use a point source at the wing tip to approximate the
complex diffuse scattering that takes place in reality. The
robustness to errors in the position of this source demonstrated
by the new estimation method further motivates the choice of
such a simple perturbation model. The new estimation method
would perform satisfactorily for airborne DF as long as the
changes in the scattering due to wing flexure predominantly
originate from a region near the position defined by ro and
q50.
C . Potential Problems with the Perturbation Model
The perturbation model
(7)
models the array manifold error
as lying in the subspace
U,
Y). potential problem is if this
subspace
is
close to the planar wave array response
ao 19).
n
such a case, the DF algorithm may interpret a valid signal as
a perturbation to the array manifold. The problem is inherent
in the model formulation, although different algorithms would
handle the situation more or less gracefully.
In an attempt to quantify the problem, we will return
to
the example in Section
II
Assume that a signal is impinging
from a location given by
U,
.e., r (measured in wavelengths
A)
and q3 , in relation to the uniform linear array. By letting
r
-+
and having
q5
= 8 , this corresponds to a planar wave
from direction
19.
The wing-tip scatter source is located at
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= i o ,
=Odeg
-i----?
a
ot
1
1
20
100
i Y
1
20
4 ~
&
22.5dej
20
5
TJ
r =2, =22.5deg
4 1
.\
I
Fig. 4. Level curves
(-20,
-15 , -10, and -5 dB) for the gain g
(17).
Signals originating from a neighborhood of the position ('x') defined by T O
and 40 are substantially attenuated. The projection matrix corresponding to
TO = 10 also causes a fairly strong attenuation for signals coming from
relative far field (large
T ) .
Note the logarithmic scale in
T .
v o =
[TO
$01. The quantity
measures the closeness of a, v) nd
us vo).
f g
= I,
the two
vectors are perpendicular, whereas g
= 0
corresponds to two
parallel vectors. Fig.
4
depicts how
g
varies with v.The further
the wing-tip source is located from the array, the more closely
waves from the wing-tip source resemble a planar wave. This
is demonstrated in Fig. 4 as small g values for large
T
and
$
$0 in the case
TO =
10.
A consequence of this behavior
is that the perturbation model will be less useful for scenarios
were the distance from the center of the antenna array to the
wing tip is much larger than the array aperture. In such cases,
the wave from the point source will be very close to a planar
wave when reaching the array.
IV. EXPERIMENTALALIDATION
An important question is how well the simple perturbation
model (7) represents the actual manifold perturbations due to
aircraft wing flexure. To evaluate our model, we performed
experiments in an anechoic chamber using a model (1/20th
scale) of the RC-135 aircraft (cf. Fig.
5).
A wing extension was fabricated and attached using a hinge
on one of the wings of the aircraft model. The angle between
the extension and the wing can be varied to simulate variations
in wing flexure. The angle between the extension and the wing
can be varied between 0 and 90 ,with the extension pointing
downwards at the 90 position. Care was taken to make the
extension a smooth continuation of the original wing. Copper
tape was applied around the hinge in order to remove sharp
comers and bends at the interface between the extension and
the wing.
We used a single unmodulated, sinusoidal signal to illumi-
nate the aircraft. For each case, we collected a 2000-sample
/ =
60 deg
Fig. 5. RC-135 ircraft model used in the anechoic radar chamber experi-
ments. An extension was fabricated and attached using a hinge on one of the
wings. The locations of the antennas are marked with black squares.
data set from five antennas mounted along the belly of the
aircraft. We also recorded the transmitted signal in the sixth
channel. Two carrier frequencies were used: 1.0 and 1.5 GHz.
The distance from the wing tip to the center of the antenna
array was approximately 3 and 5 wavelengths, respectively.
The array response was calculated by solving (in the least
squares sense) for a(0) in
[ Z ( t l > * * *
z(tiv) U ( O ) [ S ( t l )
. . S ( t N )
using the collected data. Simultaneous samples
of z ( t )
and
s ( t ) are highly correlated because the distance between the
transmitter and the aircraft model is small. Signal powers were
adjusted to provide a high signal-to-noise ratio (SNR) (> 40
dB)
.
A. Repeatability
The perturbation effects of the wing tips are a subtle
phenomena. We carefully designed our experimental setup
so
that we could detect and measure the perturbation effects.
We verified the sensitivity of our measurements by explicitly
testing repeatability of the setup. The repeatability of the setup
sets the sensitivity of our measurement setup. The measured
perturbation effects were 25 to 40 times larger than the errors
in due to repeatability.
The first set of experiments tested the repeatability of the
setup due to measurement noise. Repeated data measurement
and corresponding array response calculationsresulted in array
manifold vectors separated by less than 0.05' in m space.
The aircraft model was located on a turntable. By rotating
the turntable, the orientation of the aircraft can be varied,
thereby varying the AOA of the impinging signal. The next
setup of experiments tested the repeatability of the setup
due
to
turntable repositioning. Repositioning the turntable to
a previous measurement point, and recalculating the array
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response, resulted in array manifold vectors separated by less
than 0.1 in m space. Consequently, the manifold errors due to
inaccuracy in the turntable positioning have approximately the
same magnitude as errors due to noise. For comparison, array
manifold vectors separated by 0.3 in AOA are typically one
degree apart in m space.
Four array response vectors were obtained for each AOA by
moving the turntable, repositioning it, and computing another
array-response vector. These four vectors were then stacked
into a matrix. The singular value decomposition
(SVD)
of the
stacked matrix was calculated. Typical resulting (normalized)
singular values were
Crepeat = [1.0
0.0005
0.0003
0.0001].
(17)
Ideally, the four vectors should have been identical, leading to
zero second, third, and fourth singular values. The nonzero val-
ues represent the spread in array vectors due to measurement
noise and inaccuracy in the positioning of the turntable.
When trying to quantify the perturbation
i i (O ,q) ,
the re-
peatability of the array vector estimation constitutes a limit
on the detectability of array manifold perturbations. This
threshold can be expressed either as the values of the singular
values in (18) or as the corresponding deviation in m-space
between the array vectors.
B. The Dimension of the Perturbation
The model (7) assumes that the perturbation
i i ( 0 , q )
due to
wing flexure lies in a fixed l-D subspace. Variations in the
wing flexure scale the perturbation (through p 0,q))but do
not change the subspace itself. The second set of experiments
aimed at checking the assumption of a l-D perturbation
subspace.
Three different main AOA's were chosen: one
in
front of
the wing (8 = -15 ), one over the wing 0 =
25
O), and
one behind the wing 0 =
60 );
cf. Fig.
5.
The different
angles corresponded o turntable readings of
255,295,
and 330,
respectively. The wing extension was held at zero degrees, and
the nominal array manifold vector a0 ( e )was determined for 0,
+0.3, and 2~0.6 round the main AOA's. The experimental
conditions were the same as in the repeatability test.
After obtaining the nominal response, the angle of the wing
extension was varied, and the corresponding perturbed array
response
a(0, )
was determined. Fig.6 depicts how the angle
in m space between
ao(0)
and a(0,q)varies with the angle
of the wing extension. The more the extension is tilted, the
stronger the perturbation gets. Clearly, the magnitude of the
perturbation is proportional to the deviation in wing flexure
from the nominal wing position.
The dimensionality can be tested by stacking the nominal
array response vector together with the vectors obtained for
different tilt angles in a matrix and then calculating the
singular values. If the perturbation is l-D, the second singular
value should increase compared with its value in (18). The
other singular values should stay virtually unchanged. If the
perturbation has higher dimensionality, one would see an
increase in magnitude for several of the other singular values.
..O
Wing extension ilt angle [deg]
Fig. 6 .
Perturbation
ii(6',r))
grows stronger
as
the wing position deviates
more from its nominal value. This can be seen from the increase in angle in
m space between ao 8) nd 4 6 , ) ) as a function
of
the wing extension tilt
angle. The full lines correspond to the 1-GHz data set and
the
dashed lines to
the 1.5-GHz set. Moreover, the crosses correspond to
6
=
1 5 O the circles
to 6
=
2 5 O , and the plus signs to 6 = 60
'.
Stacking the vectors for tilt angle 0, 12, 22, 38,49, 60, and,
70 ' and calculating the singular values, leads in the 1-GHz
case to the following (normalized) result:
C-15
= [1.0 0.021 0.0016
0.000
62
0.000
291
C 5
= [1.0 0.013 0.0010
0.00063
0.00047]
=
[1.0
0.019
0.0012 0.000 69 0.000 281.
The second singular value has increased almost two orders of
magnitude compared with (18), whereas the increase in the
other singular values are much less pronounced. This clearly
indicates that the
gross
behavior of the perturbation
zi 0,
q)
is
indeed confined to a l-D subspace. Fig.
6
indicates that the
perturbation for the 1-GHz case is strongest for 0 =
-15
and
weakest for 6
= 25. This
agrees well with the relative size of
the the second singular value of the three data sets above.
The 1.5-GHzdata set demonstrates the same type of proper-
ties as the 1-GHz set. The singular values again indicate a 1-D
perturbation in m space. In this case, though, the perturbation
is
strongest for 0 = 25 and weakest for 0 = 60 .
B.
The Direction
of
the Perturbation
The previous calculations demonstrated that the manifold
perturbation is confined to a 1-D subspace. One of the as-
sumptions underlying
(7)
is that this subspace
is
independent
of 0, i.e., not only is the perturbation subspace l-D, but it also
points in the same direction irrespective of the value of
0.
One way to test this assumption
is
to measure the perturba-
tion ii 0, q) or different 0 and see if the different perturbation
vectors span the same space. This can be achieved using the
data sets collected for the dimensionality test in Section IV-B.
Our data sets include both the antenna signals z( t )as well
as the transmitted signal s ( t ) .When solving for
u O,q),
we
will obtain a scaled version of the true manifold vector. The
scaling factor depends on the gain and phase of the amplifiers
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used in the data processing channels. The signals are sampled
synchronously, using exactly the same setup for each data set.
The scaling factor will be the same (but unknown) for each
data set. A vector proportional to the perturbation
i i 0 , ~ )
an
be obtained by forming the difference between array manifold
vectors calculated for different wing extension angles.
It is important to note that it is not possible to obtain
the perturbation
ii 0,
) based on array manifold vectors cal-
culated from only the antenna signals. The array manifold
vector is normally obtained as the left singular vector of
X = [ ~ ( t l ) (t2) . . . z ( t ~ ) ]orresponding o the largest
singular value. This vector is related to the true array manifold
through an unknown scaling factor. If the scaling factor differs
among data sets, then
i i (0 ,
q) cannot be calculated by forming
differences between array manifold vectors calculated for
different wing extension angles.
The actual calculation of the perturbation vector was per-
formed as follows. For a fixed
0,
the wing extension was
held at eight different tilt angles
0,
12, 22, 30, 38, 49,
60,
and 70”) and the corresponding array manifold vector was
calculated from measured data.
A
set
of
perturbation vectors
were computed from differences between the eight manifold
vectors. Differences between array vectors closer than 1.5’
in m space were avoided in order to reduce the effect of
noise. The perturbation vectors were stacked in a matrix,
and the SVD was used to calculate the dominating direction
(the first left singular vector). The one dimensionality of the
perturbation was again evident from the obtained singular
values. The ratio between the first and the second singular
value was typically around 10. The above measurements were
repeated for the same AOA’s and frequencies used in Section
IV-B. For each of the two frequencies, we obtained 15 different
perturbation vectors: one for each AOA.
We have assumed that the perturbation vectors correspond-
ing to the different AOA’s
all
span the same subspace. This
can be checked by stacking the different perturbation vectors
in a matrix and calculating the singular values. If they span the
same space, one of the singular values should be significantly
larger than the others. We obtained the following values:
C1 = [1.0 0.0505
0.0137
0.0044 0.00341
C15 = [1.0 0.0491 0.0152 0.0045 0.00341
which clearly demonstrates the similarity among the different
perturbation vectors. They all point in more or less the
same direction, independently of the AOA. Another way to
measure this is to calculate the angle in
m
space between
the perturbation vectors for different AOA’s. The angular
separation was typically a few degrees with maximum values
around 7” for both frequencies.
To quantify the difference between the perturbation vectors
for different AOA’s, it is worthwhile to return to the ULA
example presented earlier. There, we noted that the new
estimation scheme could, very successfully, handle moderate
errors in the position U. An error in the position means
that the perturbation vector a, uo) will differ from the true
vector a, u).The example demonstrated that the scheme
could easily handle an error in 4 of a f 2 ” and a deviation
in T of about f2 0%. The corresponding angular differences
in m-space, i.e., the angle between
a, u)
and a, ug), s
about 5-10’. It is, of course, difficult to directly compare
figures from the experiments and the ULA example. It is,
however, worth noting that the variation in the experimentally
derived perturbation vectors is well within the values that the
estimation scheme could handle in the ULA example.
V. CONCLUSIONS
DF from airborne platforms relies on flight calibration of
the array manifold. Varying wing flexure changes the near-
field scattering and perturbs the manifold from the calibrated
value. This
is
a dominating error source in the DF.
Most of the changes of the scattering originates from the
outer part of the wings. We modeled the perturbation by a
scatter source positioned at each wing tip. The perturbation
model assumes that the perturbation is 1-D in m space and that
its direction is unaffected by the AOA of the impinging signal.
Each of these properties was validated through experiments in
an anechoic chamber. The results of the experiments are quite
encouraging.
The simple structure of the model facilitates its incorpo-
ration into standard DF schemes to reduce their sensitivity
to wing-flexure induced changes to the array response. One
way to reduce the effect of the perturbation is to project the
data onto the null space of the perturbation. The appropriate
projection is given by the perturbation model. This projection
can be implicitly incorporated into DF schemes such as WSF.
The projection operation is fairly insensitive to parameter
variations, and a substantial decrease in estimation error is
achieved even when the parameters of the perturbation model
are slightly inaccurate. The robustness to parameter variations
is sufficient for the scheme to provide improvements even for
diffuse scattering originating from the region around the wing
tips.
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Kjell Gustafsson (M’91) was educated at Lund
Institute of Technology, Lund, Sweden. He received
the M.S. degree in 1985, the Lic. Eng. degree in
1988, and the Ph.D. degree in 1992,
all
in electrical
engineering.
He
joined the Department of Automatic Control,
Lund Institute of Technology, in 1985. From 1992
to 1993, he was a Visiting Assistant Professor in
the Scientific Computing and Computational Math-
ematics group at Stanford University, Stanford, CA,
USA. He worked on different problems related to
sensor-array processing. In 1994, he joined a research group at Ericsson
Mo-
bile Communications, Lund, Sweden.His current work focuses
on
application
of digital signal processing to telecommunication problems.
Frank McCarthy (M’91) received the B.S.E.E.,
M.S.E.E., and
M.S..
degrees in statistics from Stan-
ford University, Stanford, CA, USA, between 1979
and 1985. He is currently working toward the Pb.D.
degree in electrical engineenng at Stanford Univer-
sity.
He has worked at ARGOSystems, Sunnyvale,
CA, USA, from 1980 until the present. Since 1988,
he has worked mainly
on
co-channel interference
reduction and direct ion finding techniques for com-
munications reconnaissance applications. In 1993,
he participated on a team that successfully demonstrated real-nme cochan-
ne1 interference reduction from an airborne reconnaissance platform using
a constant-modulus beamforming algorithm.
His
current research interests
include antenna-amy calibrahon, co-channel interference reduction, direction
findmg, and real-time signal processing.
Arogyaswami Paul raj (F‘91) was educated at the
Naval Engineering College, India, and at the Indmn
Institute of Technology, New Delhi, where he re-
ceived the Ph.D. degree in 1973.
A large part of his career to date has been
spent in research laboratories in India, where he
supervised the development of several electronic
systems.
His
contributions include a sonar receiver
(1973-1974), a surface ship sonar (1976-1983), a
parallel computer (1988-1991), and telecommuni-
cations systems. He has held visiting appointments
at several universit ies, includmg the Indian Institute of Technology, Delhi,
from 1973
to
1974, Loughborough University of Technology,
UK,
f rop 1974
to 1975, and Stanford University, Stanford, CA, USA, from 1983 to 1986.
His
research has spanned several disciplines, emphasizing estimation theory,
sensor signal processing, antenna array processing, parallel computer architec-
tures/algonthms, and commumcation systems. He is currently a Professor of
Electrical Engineering at Stanford University, working in the area of mobile
communications. He is the author of about 90 research papers and holds
several patents. He has won a number of national awards in India for his
contributions to technology development.