PublishedPaper

Optimum array design to maximize Fisher information
for bearing estimation
Saurav R. Tuladhara)
and John R. Buck
Department of Electrical and Computer Engineering, University of Massachusetts Dartmouth,
285 Old Westport Road, North Dartmouth, Massachusetts 02747-2300
(Received 1 April 2011; revised 6 September 2011; accepted 10 September 2011)
Source bearing estimation is a common application of linear sensor arrays. The Cramer–Rao bound
(CRB) sets a lower bound on the achievable mean square error (MSE) of any unbiased bearing
estimate. In the spatially white noise case, the CRB is minimized by placing half of the sensors at
each end of the array. However, many realistic ocean environments have a mixture of both white
noise and spatially correlated noise. In shallow water environments, the correlated ambient noise can
be modeled as cylindrically isotropic. This research designs a fixed aperture linear array to maximize
the bearing Fisher information (FI) under these noise conditions. The FI is the inverse of the CRB, so
maximizing the FI minimizes the CRB. The elements of the optimum array are located closer to the
array ends than uniform spacing, but are not as extreme as in the white noise case. The optimum array
results from a trade off between maximizing the array bearing sensitivity and minimizing output noise
power variation over the bearing. Depending on the source bearing, the resulting improvement in
MSE performance of the optimized array over a uniform array is equivalent to a gain of 2–5 dB in
input signal-to-noise ratio. VC 2011 Acoustical Society of America. [DOI: 10.1121/1.3644914]
PACS number(s): 43.60.Fg [EJS] Pages: 2797–2806
I. INTRODUCTION
An array is an arrangement of multiple sensor elements
in a particular geometric configuration used for the space-
time processing of a wave field. As the wave field from a
source propagates across the array aperture, the wave front
reaches the sensor elements at different times. The relative
time delay between the arrival of the same wave front at dif-
ferent sensors is related to the source bearing as described in
Van Trees (2002, Sec. 2.2). In the narrowband case, the rela-
tive time delay is represented as a phase shift in the signals
observed at the sensor. Thus, the relative phase shifts in the
array measurements can be used to estimate the source
bearing.
Source bearing estimation is a common application of a
linear sensor array. The mean square error (MSE) between
the true bearing (h) and the estimated bearing (^h) is a com-
mon performance measure. The Cramer–Rao bound (CRB)
establishes a lower bound on the achievable MSE for any
unbiased bearing estimate (Van Trees, 2001, Sec. 2.4.2). The
CRB is the inverse of the Fisher information (FI). The FI for
bearing estimation depends on the element positions of the
array, the target bearing direction and the signal-to-noise ra-
tio (SNR). For a linear array with fixed aperture, the bearing
FI can be increased by optimizing the sensor element posi-
tions. This lowers the CRB and hence the achievable MSE
on the bearing estimate. MacDonald and Schultheiss (1969)
discuss the optimum element position to minimize the CRB
on bearing estimation for the spatially white noise case.
Their optimum array has half of the elements placed at each
end of the array. In practice however, clustering the sensor
elements at the array end is not feasible because of the finite
size of the sensor elements. Moreover the assumption of
uncorrelated noise from sensor to sensor no longer holds as
the sensors are packed together unless the noise is predomi-
nantly self-generated at each sensor.
Several other approaches have been proposed for opti-
mizing the arrangement of sensors in a linear array. Brown
and Rowlands (1959) approached the design from an infor-
mation theory point of view. They designed the array to use
the least number of sensors required for bearing estimation
of a narrowband source. The sensors are positioned such that
the information available from sensor pairs is maximized.
The Brown and Rowlands design does not constrain the
array aperture and it also does not consider the impact of
noise. The bearing estimation problem can also be consid-
ered as a quantization problem to assign the source to a bear-
ing partition (Zhu and Buck, 2010). Zhu and Buck proposed
the design of a fixed aperture linear array to maximize the
mutual information from the sensors while nulling the for-
ward endfire direction. This design assumes that the array
operates in a spatially white noise environment with a far-
field narrowband source. Murino et al. (1996) used simulated
annealing to design arrays that reduced the peak sidelobe in
the beam pattern by optimizing element positions and weigh-
ing coefficients. They also assume a fixed number of sensor
elements and aperture size. Similarly, Lang et al. (1981)
designed linear arrays maximizing the number of distinct
lags in the co-arrays for use with maximum entropy methods
and maximum likelihood method array processing. This
algorithm also assumed a fixed array aperture and a fixed
number of sensors, but no consideration was made for the
impact of noise. Pearson et al. (2002) proposed a number
theory algorithm for the optimum placement of sensors in a
sparse array to improve the detection and resolution of
a)
Author to whom correspondence should be addressed. Electronic mail:
stuladhar@umassd.edu
J. Acoust. Soc. Am. 130 (5), November 2011 VC 2011 Acoustical Society of America 27970001-4966/2011/130(5)/2797/10/$30.00
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multiple sources in a spatially white noise case. The algo-
rithm generates minimum redundancy sensor locations. All
of those methods design arrays optimizing different array
performance metrics. However, most of these methods make
no consideration for the noise field and the few that do con-
sider the noise are limited to the spatially white noise. More-
over, none of the papers evaluated the bearing MSE
performance of the designed array.
The assumption of noise independence from sensor to
sensor may not always be true in practice. Especially in an
underwater environment, it is likely that both correlated and
white noise components are present in the measurement,
although in different proportions. The spatially correlated
noise can be attributed to the background noise field from
undesirable acoustic sources, such as distant ships and the
breaking of wind-driven surface waves. In the shallow under-
water acoustic channel, the correlated noise is generally mod-
eled as cylindrically (two dimensionally) isotropic. Kuperman
and Ingenito (1980) proposed a model for the noise field in a
stratified shallow ocean due to breaking surface waves. For an
array at a constant depth, the Kuperman and Ingenito model
simplifies to the cylindrically isotropic model of Cox (1973)
and Jacobson (1962). On the other hand, the spatially white
noise component predominantly models the self generated
sensor circuitry noise and the quantization noise due to sam-
pling, together called the instrumentation noise.
This paper presents the design of a fixed aperture horizon-
tal linear array optimized to maximize the FI for the unbiased
bearing estimation of a narrowband source in a predominantly
cylindrically isotropic spatially correlated noise field.
II. MEASUREMENT MODEL
In underwater bearing estimation, a hydrophone array is
used to measure the acoustic pressure field from a source.
The measurement of a narrowband signal at a single sensor
element can be represented as a complex number (Johnson
and Dudgeon, 1992, Sec. 2.2.1). When the array is suffi-
ciently distant from the source, the acoustic field wave front
reaching the array can be approximated by a plane wave.
The direction of propagation of the plane wave is approxi-
mately same for all sensors. A single snapshot measurement
from an N-element linear array is then expressed as an N Â 1
complex vector y, which combines the plane wave signal
and the noise as
y ¼ xðhÞ þ n ¼ svðhÞ þ n; (1)
where s is the signal amplitude, vðhÞ is the N Â 1 array steer-
ing vector, h is the source bearing and n is the N Â 1 com-
plex noise vector. For an N-element nonuniform linear array
geometry with zero phase at the center of the array, the steer-
ing vector vðhÞ is of the form
vðhÞ ¼ ½ejkdÀðNÀ1Þ=2 sin h
; …; ejkdÀ1 sin h
; 1; ejkd1 sin h
; …;
ejkdðNÀ1Þ=2 sin h
ŠT
; (2)
where k is the wave number and di is the distance of the ith ele-
ment from the center of the array (see Fig. 1). For convenience,
the N array elements are indexed symmetrically from
ÀðN À 1Þ=2 to ðN À 1Þ=2. The source power and noise power
are generally unknown and modeled as zero mean complex
Gaussian processes with covariances EðssÃ
Þ ¼ r2
s and
EðnnH
Þ ¼ Knn. The covariance for the measurement vector in
Eq. (1) is
Kyy ¼ Kxx þ Knn ¼ r2
s v hð ÞvH
hð Þ þ Knn: (3)
The structure of the noise covariance matrix Knn is deter-
mined by the noise model. This work models the noise as a
combination of the cylindrically isotropic correlated noise
and white instrumentation noise, i.e., Knn ¼ Kiso þ Kw
where Kiso is the isotropic noise spatial covariance and Kw
is the white noise spatial covariance. The isotropic noise is a
special case of the Kuperman and Ingenito model for a hori-
zontal array at a constant depth, as discussed in Sec. I. The
spatial covariance matrix for the isotropic noise field takes
the form
Kiso ¼ r2
isoJ: (4)
The entries of the matrix J are J½ Špq¼ J0 k dpq

À Á
where
J0 (ÁÁÁ) is the zeroth order Bessel function and dpq ¼ dp À dq
is the spacing between the pth and qth elements. Here the
rows and columns p and q are also indexed symmetrically
from ÀðN À 1Þ=2 to ðN À 1Þ=2 as are the sensor locations dp
and dq. Using this model of the covariance matrix, the matrix
J is known a priori for given array element positions.
In addition, the instrumentation noise also contributes to
the observations at the sensor elements. The instrumentation
noise is uncorrelated from sensor to sensor and in modern
sensor arrays it is generally very weak compared to the am-
bient noise in the underwater environment. The spatial co-
variance matrix for the instrumentation noise is Kw ¼ r2
wI.
Thus, the combined noise covariance can be expressed as
Knn ¼ r2
isoJ þ r2
wI ¼ r2
nQn; (5)
where the combined noise power r2
n ¼ r2
iso þ r2
w
¼ Tr Knnð Þ=N, where Tr ðÁ Á ÁÞ denotes the trace operator. The
matrix Qn defines the noise’s spatial structure. The matrix
Qn is known a priori for a given array and it can be used to
FIG. 1. Linear array geometry for N-element array with aperture
L ¼ (N À 1)k/2.
2798 J. Acoust. Soc. Am., Vol. 130, No. 5, November 2011 S. R. Tuladhar and J. R. Buck: Optimum array to maximize Fisher information

whiten the measured noise. Thus, in this measurement model
the unknown parameters are the signal power (r2
s ), the com-
bined noise power (r2
n) and the source bearing (h). The three
unknown parameters can be combined into a parameter vec-
tor w ¼ ½r2
s r2
nhŠT
.
III. DESIGN APPROACH
The design of the optimum array starts from the bearing
FI. The FI is a function of the bearing angle (h) and the array
element positions (di). The optimum element positions are
determined by optimizing the FI in the maximum–minimum
sense. The resulting array will be referred to as the isotropic
noise optimum array (INOA). The performance of the INOA
is compared to the uniform array (UA) with k=2 spacing
between elements and the white noise optimum array
(WNOA) discussed in MacDonald and Schultheiss (1969).
A. Bearing Fisher information
The FI for the measurement model in Eq. (1) is a 3 Â 3
matrix. The general expression for the entries of the FI ma-
trix is given by (Van Trees, 2002, Sec. 8.2.3)
Imn ¼ Tr KÀ1
yy
@Kyy
@wm
KÀ1
yy
@Kyy
@wn

(6)
where the subscript mn is the mth row and nth row element
of FI matrix and m, n ¼ 1, 2, 3. The matrix entry correspond-
ing to bearing FI is I33 and is a function of both the bearing
angle and the array element positions. For the discussion in
this section the bearing FI I33 is represented as IðhÞ and the
dependence on the element positions are suppressed in the
sequel except where necessary for clarity.
Ye and DeGroat (1995) derived that the signal and noise
power can be estimated separately from the bearing parameter
for the model in Eq. (1). Thus, for bearing parameter,
CRB(h)¼ IðhÞÀ1
. They derive the expression for the bearing FI
assuming an unknown noise covariance. However, in this work,
the structure of the noise covariance is known [Eq. (5)]. Appen-
dix A shows how the result from Ye and DeGroat (1995) is
adapted to obtain the FI expression for the known noise covari-
ance structure. The final expression for the bearing FI is
IðhÞ ¼ 2
r4
s
r2
n
d~vH
ðhÞ
dh
P?
~vðhÞ
d~vðhÞ
dh

vH
ðhÞKÀ1
yy vðhÞ

; (7)
where ~vðhÞ ¼ QÀ1=2
n vðhÞ is the whitened steering vector and
P?
~vðhÞ ¼ I À ~vðhÞð~vH
ðhÞ~vðhÞÞÀ1
~vH
ðhÞ is the projection ma-
trix for the subspace orthogonal to the whitened signal sub-
space. The first term in Eq. (7) equals zero in the endfire
direction, hence no information on source bearing is avail-
able in the endfire direction. As derived in Appendix A, Eq.
(7) can further be expressed in terms of the input
SNR ¼ r2
s =r2
n as
IðhÞ ¼2 SNRð Þ P?
~vðhÞ
d~vðhÞ
dh

2
!
SNR ~vðhÞk k2
1 þ SNR ~vðhÞk k2
!
:
(8)
The expression for the bearing FI [Eq. (8)] is a product of three
terms. The first term is the input SNR. This expression makes it
clear that the bearing FI is directly proportional to the input
SNR. The second term is the projection of the derivative of the
whitened steering vector onto the subspace orthogonal to the
whitened signal subspace. The third term is a scaling term
related to the input SNR and the norm of whitened steering
vector. The third term can be identified as the common Wiener
filter gain once the term SNRjj~vðhÞjj2
is recognized as the
equivalent input SNR for the whitened signal model.
The derivative of a steering vector is the measure of the
sensitivity of an array to a bearing direction and it will be
referred to as the sensitivity vector. The CRB and the FI for
bearing depends on the sensitivity vector as seen in the deri-
vations discussed in Section 8.4 of Van Trees (2002). The
sensitivity vector (dvðhÞ=dh) is tangential to the manifold
traced by the steering vector (vðhÞ) in N-dimensional com-
plex space. Thus, the steering vector and the sensitivity vec-
tor are orthogonal along the manifold. In the whitened space
this orthogonality may not always hold. Only the component
of the whitened sensitivity vector orthogonal to the whitened
steering vector provides bearing information. The compo-
nent collinear to the steering vector itself only adds to infor-
mation about signal power (r2
s ). As a result the FI in Eq. (8)
depends on the length of the projection of the whitened sen-
sitivity vector onto the subspace orthogonal to the whitened
signal subspace.
B. Maximum–minimum (max-min) optimization
The max-min optimization approach chooses the opti-
mum element positions such that the minimum FI over the
operating bearing range is maximized. The array element
positions can be expressed as a vector
d ¼ ½dÀðNÀ1Þ=2; …; dÀ1; 0; d1; …; dðNÀ1Þ=2ŠT
;
where di is the distance of the ith element from the center of
the aperture. The bearing FI is now represented as Iðd; hÞ
with the element position vector as an explicit argument
along with the bearing. The minimum FI over the bearing
range is a function of the position vector only,
IminðdÞ ¼ min
0 h h0
Iðd; hÞ; h0 p=2; (9)
where h0 is the upper limit for the bearing range. As men-
tioned in Sec. III A, the bearing FI in the endfire direction
(h ¼ 0:5p) is identically zero for any choice of d. This is the
minimum FI over the bearing range (0 h 0:5p). Thus, it
is not possible to increase the FI at the endfire direction by
changing the element positions. Rather the minimum FI is
computed over a reduced bearing range with an upper limit
set at h0. Evaluating the FI over a reduced bearing range is
also consistent with the concept of a maximum scan angle
described in Van Trees (2002, Sec. 2.5). The solution to the
minimization problem in Eq. (9) was approximated by an ex-
haustive search over the bearing range discretized at 100
points. Although the optimization evaluated I(d; h) at 100
discrete points for the interval 0 h h0, the angle result-
ing in Imin was consistently h ¼ h0. The overall optimization
problem can now be expressed as
J. Acoust. Soc. Am., Vol. 130, No. 5, November 2011 S. R. Tuladhar and J. R. Buck: Optimum array to maximize Fisher information 2799

dopt ¼ arg max
d
minðdÞf g; jdij L=2; (10)
where L is the array aperture which is constrained to be
L ¼ ðN À 1Þk=2: This is the aperture of a uniform array with
N elements at k=2 spacing. The fixed aperture constraint
requires that one element is placed at each end of the aper-
ture leaving N À 2 elements to be optimized.
The optimization problem in Eq. (10) can be solved by
either an exhaustive search or a gradient search method as
described in Zhu and Buck (2010, Sec. II A). The exhaustive
search scans through a grid of array configurations to find
the optimum array that maximizes IminðdÞ. The grid of array
configurations is generated by shifting the array elements in
steps of Dd. If Dd is small enough the exhaustive search will
find an array close to the exact optimum solution although
the exact optimum array may not be in the grid. Further, the
number of possible array configurations in the search grid is
of the order OðCn
Þ where C ¼ L=Dd is a constant and n is
the number of elements free to be shifted in discrete steps to
generate the different array configurations. This is a fairly
loose bound on the growth of the search grid. The exhaustive
search method is suitable for arrays with smaller numbers of
sensors so that the number of possible configurations is trac-
table for search.
The gradient search method is suitable for arrays with a
larger number of elements. The gradient search starts from
an initial value of the element position d0 and converges to
the optimum result dopt by iteratively updating the position
vector as
dkþ1 ¼ dk þ ardIminðdÞ; (11)
where a is the update step size. The gradient of minimum
FI rdIminðdÞ was approximated numerically using central dif-
ference method. The iteration stops when dkþ1 À dkk k ,
where is the tolerance. The minimum FI IminðdÞ is not guar-
anteed to have a single maximum point within the search
region. Thus, the search may converge to a local maximum.
In order to improve the chance of finding the global maximum
of IminðdÞ, the gradient search is executed multiple times with
a random starting value (d0) each time. The element position
vector, which gives the highest maxima of IminðdÞ from all
the searches is the estimate of the global optimum array. Also,
in order to ensure that the search satisfies the fixed aperture
constraint, the entries of the updated position vector dkþ1 are
checked after each iteration. Any jdij L=2 is reset to
di ¼ L=2 before the next iteration.
The noise model in Sec. II accounts for the cylindrically
isotropic ambient noise and the instrumentation noise from
the sensors, both of which exist in practice. In an ideal case
without the instrumentation noise, considering only the iso-
tropic ambient noise would suffice. The isotropic noise co-
variance (Kiso), however is a function of the inter-element
spacing (dpq). As the search for the optimum array goes
through different configurations of element positions, the
noise covariance (Kiso) also changes due to the different ele-
ment spacing. The noise covariance matrix (Kiso) can
become ill-conditioned for some configuration of the sensor
positions. The ill-conditioned covariance matrix will skew
the FI (Eq. (7)) during the optimization process and lead to a
false optimum. Although the white noise term in the covari-
ance of Eq. (5) models the real phenomena of instrumenta-
tion noise, this term has the additional benefit of effectively
performing diagonal loading on the isotropic noise covari-
ance, thus solving the conditioning issue.
IV. RESULTS
This section presents array designs for N¼ 7 and N ¼ 21
elements with their apertures fixed at L ¼ 3k and L ¼ 10k,
respectively, as described in Sec. III B. The minimum FI is
computed over a bearing range with the upper limit set to
h0 ¼ 0:40p. This choice for h0 falls in the range defined by the
maximum scan angles for N¼ 7 and N¼ 21 element uniform
linear arrays (Van Trees, 2002, Fig. 2.24). The noise covariance
(Knn) is a combination of isotropic noise and instrumentation
noise such that r2
w=r2
iso ¼ À30 dB. All computations were per-
formed using Matlab (v7.7.0.0471) on a standard system with
Core 2 Duo (2.2 GHz) processor running Linux (v2.6.38).
A. Optimum arrays
The optimum array for N ¼ 7 can be easily found with
the exhaustive search approach assuming a symmetric place-
ment of sensor elements. The symmetry and fixed aperture
constraints forces one element to be placed at the array cen-
ter and one element at each of the array ends. Among the
remaining four elements, two are positioned at distances d1
and d2 to the right of the center and the other two are placed
at distances Àd1 and Àd2 to the left side of the center. This
reduces the optimization problem to just two degrees of
FIG. 2. Optimum element positions
(diamonds) and uniform array ele-
ment positions (circles) for N ¼ 7
elements (top panel) and N ¼ 21 ele-
ments (bottom panel).

freedom. An exhaustive search is performed over a grid of
almost 104
array configurations generated by incrementing
d1 and d2 at steps Dd ¼ 0:01k starting from the array center.
The search was completed in 70.9 s.
The elements of the resulting INOA are located closer
to the array ends than the elements of the UA, but are not as
extreme as the elements of the WNOA. The INOA element
positions expressed as a factor of k are listed in Table I and
Fig. 2 shows these positions in reference to the UA element
positions. Figure 3(a) shows how the FI from the INOA
compares to the UA and the WNOA as a function of the
bearing. The INOA improves the FI over the UA FI at most
but not all bearings. The FI increases by the greatest factor at
the bearings closer to the endfire, while at broadside the rela-
tive improvement is only a small amount.
The same grid of Dd ¼ 0:01k produces almost 1030
sym-
metric arrays for the N ¼ 21 element case, making exhaustive
search impractical for optimizing arrays for the size. Instead
the gradient search method was used to find the N ¼ 21 ele-
ment INOA. Because the gradient method does not enforce a
symmetry constraint, the fixed aperture constraint on the array
leaves 19 elements free for optimization. The search step size
was set at a ¼ 10À5
. The gradient search was executed 20
times, each time initialized with random symmetric starting
positions to improve the likelihood that the global optimum is
found. On average the search converged to the optimum result
in 320 s. The INOA element positions for N ¼ 21 are listed in
Table I. Figure 2(b) shows the INOA element positions com-
pared to the UA element positions. As in the case of the N ¼ 7
array, the elements of the INOA are again pushed toward the
array ends but not to the extreme extent as in the WNOA.
Although no symmetry constraints were enforced during the
gradient search iterations, the resulting INOA still has a sym-
metric arrangement of elements. This result agrees with the
implicit symmetry constraint on the optimal sensor locations
(see Appendix B for the outline of a proof of this constraint).
A comparison of the FI in Fig. 3(b) shows that the improve-
ment in the FI of the INOA over the UA across the bearing
(h) is more consistent in contrast to the case of N ¼ 7.
B. Bearing MSE performance
The INOA bearing estimation performance is quantified
by estimating the MSE of the maximum likelihood (ML)
bearing estimator in Monte Carlo trials. The MSEs for the
INOA and the UA are compared against each other and their
respective CRBs over a range of input SNR values for several
bearings. In the results presented in the following, the MSE
value for each SNR at each bearing is obtained from 1000
Monte Carlo trials. The signal power and the noise power are
assumed to be known for the bearing MLE as both parameters
can be estimated separately from the bearing as mentioned in
Sec. III A. The MSE performance is evaluated for both N ¼ 7
and N ¼ 21 arrays when steered to bearings 0p and 0:4p.
The main lobe of the array beam pattern widens as it is
steered from broadside to the endfire (Fig. 4). As the main
lobe widens, the beam pattern rolls off from the peak at a
slower rate than when the main lobe is narrower. This results
in a greater probability for additive noise of a fixed power to
shift the peak of the likelihood function when the main lobe
is wider. Hence, the bearing MSE and CRB at 0p should be
lower compared to the bearing MSE at 0:4p, which is closer
to endfire. Similarly, the array with N ¼ 21 elements and the
aperture L ¼ 10k has a narrower main lobe compared to
the array with N ¼ 7 elements and aperture L ¼ 3k. Thus, the
N ¼ 21 elements array should have a lower bearing MSE and
CRB compared to the N ¼ 7 element array.
When the arrays are steered to broadside (h ¼ 0), the
CRBs of the INOA and the UA are so close as to be indistin-
guishable (overlapping solid and dashed curves) for both the
N ¼ 7 and N ¼ 21 element arrays as seen in the left-hand pan-
els of Fig. 5. Further, the CRB for the N ¼ 21 array is lower
TABLE I. Optimum element positions.
N Optimum positions as factor of k (dopt)
7 À1.5 À1.34 À1.17 0 1.17 1.34 1.5
21 À5.00 À4.82 À4.49 À4.12 À3.74 À3.34 À2.96 À2.61 À2.36 À0.20
0 0.20 2.36 2.61 2.96 3.34 3.74 4.12 4.49 4.82 5.00
FIG. 3. Comparison of FI for the isotropic noise optimum array (INOA,
solid), the uniform array (UA, dashed-dotted) and the white noise optimum
array (WNOA, dashed) for (a) N ¼ 7 and (b) N ¼ 21. The input SNR is 10
dB and the noise field is combination of cylindrically isotropic spatially cor-
related noise and white sensor noise with the white noise 30 dB below the
isotropic noise component.

than the CRB for N ¼ 7 array as discussed previously. The
MSE curve for both N ¼ 7 and N ¼ 21 exhibits three distinct
regions, namely, the asymptotic region for high SNR, the
threshold region in the neighborhood just below the threshold
SNR and the no information region for low SNR as described
in Fig. 1 of Richmond (2006). This type of behavior is charac-
teristic of ML bearing estimators (Van Trees, 2002, Sec. 8.5).
In the threshold region, for both N ¼ 7 and 21 the MSE is sig-
nificantly larger than the CRB. Also the INOA has a larger
MSE (diamonds) than the UA (circles). As the MSE curves
converge to the CRB, for N ¼ 7 the INOA has a higher thresh-
old SNR than the UA, but for N ¼ 21 the threshold SNR is
essentially the same for both arrays. In the asymptotic region
the MSE follows the CRB for both arrays.
FIG. 4. Beam patterns for the array
with N ¼ 7 elements and N ¼ 21 ele-
ments steered to 0p and 0:4p. When
steered toward endfire at h ¼ 0:4p
the UA beam pattern develops a
lobe at À0:5p, which is higher than
the sidelobes of the INOA. This
sidelobes behavior results in
improved input SNR performance of
the INOA compared to the UA at
endfire direction.
FIG. 5. ML bearing estimator MSE
and CRB at bearings h ¼ 0p and
0:4p for arrays with N ¼ 7 and 21
elements. The MSE value for each
SNR was generated from 1000
Monte Carlo trials. At broadside, the
INOA (solid) and the UA (dashed)
both have essentially the same CRB,
but at bearing 0:4p the INOA has a
lower (improved) CRB than the UA
for both N ¼ 7 and N ¼ 21. The MSE
curves exhibits the classic behavior
with the no information region, the
threshold region and the asymptotic
region. The MSE follows the CRB
in the asymptotic region but it devi-
ates away from the CRB in the
threshold region for both the INOA
and the UA with N ¼ 7 and 21.

In the case of the arrays steered to the bearing h ¼ 0:4p
(right-hand panels of Fig. 5), the INOA has a lower CRB
than the UA over the SNR range. For N ¼ 7, the INOA’s
CRB is $8 dB lower than the UA’s CRB for the SNR range
shown in Fig. 5. For N ¼ 21 the INOA’s CRB is $5 dB
lower. The MSE curves exhibit similar three-region behavior
as discussed earlier. In the threshold region, the INOA has a
lower MSE than the UA for both N ¼ 7 and 21. This is in
contrast to the threshold region MSE behavior in the broad-
side case. Further, the INOA has a lower threshold SNR than
the UA for both N ¼ 7 and N ¼ 21. In the asymptotic region
for N ¼ 7, the INOA requires $5 dB less SNR to achieve the
same MSE as the UA. For N ¼ 21, the INOA requires $2 dB
less SNR for same performance.
As discussed in Richmond (2006), the behavior of the
bearing MSE over a range of SNR values is related to the array
beam pattern. In the asymptotic range the MSE is largely
driven by the main lobe width and it follows the CRB closely.
As seen in the left-hand panel of Fig. 4, when the beam pat-
terns are steered to broadside (h ¼ 0), the main lobe of the
INOA is narrower than the main lobe of the UA for both N ¼ 7
and 21 arrays. The effect of the narrower main lobe on MSE is
insignificant as seen by almost overlapping MSE values of the
INOA (diamonds) and the UA (circle) in the asymptotic region
in Fig. 5. In the threshold region, the high MSE values are
driven by the sidelobes. At low SNR the MSE increases due to
noise perturbing the peak of the bearing likelihood function to
a sidelobe location instead of the true bearing. The INOA has
higher sidelobes when steered to broadside than the UA and
for both N ¼ 7 and 21 (left-hand panels of Fig. 4). These
higher sidelobes of the INOA drive its MSE to be larger than
the UA’s MSE in the threshold region. Moreover, the sidelobe
for the INOA is higher than the sidelobe for the UA by a factor
of 4 for N ¼ 7 and only by factor of 2 for N ¼ 21. As a result,
for N ¼ 7 the INOA has a greater threshold SNR than the UA,
whereas for N ¼ 21 the threshold SNRs for the INOA and the
UA are indistinguishable.
In the case of the beam pattern steered to bearing
h ¼ 0:4p, for both N ¼ 7 and 21 the UA’s beam pattern has a
lobe at À0:5p bearing, which is higher than any of the UA’s
other sidelobes, as well as the sidelobes of INOA. This
increases the likelihood of the lobe at À0:5p bearing being
erroneously chosen as the peak instead of the mainlobe peak.
Also, the error introduced due to selecting the lobe at À0:5p
bearing is much greater than due to any other sidelobes. As a
result the MSE for the UA is driven higher than the MSE for
the INOA as shown in the right-hand panel of Fig. 5. Also as
a result of the higher lobe at À0:5p bearing, for N ¼ 7 the
UA has a 5 dB greater threshold SNR than the INOA,
whereas for N ¼ 21 the UA’s threshold SNR is only $2 dB
greater than the INOA.
V. DISCUSSION AND CONCLUSION
A. Discussion
In Sec. IV it was shown that the INOA has elements
placed closer to the ends, but not to the extreme extent of the
WNOA presented in MacDonald and Schultheiss (1969). A
comparison of the FI performance of the INOA (solid) and
the WNOA (dashed) for spatially correlated noise environ-
ment is shown in Fig. 3. It is seen that the INOA at least
matches or exceeds the FI of the UA over the bearing range.
In contrast, the WNOA has a significantly reduced FI in the
broadside region for a nominal gain over the UA’s FI toward
endfire. Hence, for the spatially correlated noise considered
for this work, the INOA performs better than the UA and the
WNOA in terms of bearing FI.
In order to understand the reasons for the particular
element positions for the INOA designed in Sec. IV A,
it is insightful to look at two quantities: first, the norm
of the projected whitened bearing sensitivity term
ðkP?
~vðhÞd~vðhÞ=dhkÞ in the expression for FI [Eq. (8)], and
second the output noise power (vðhÞH
KnnvðhÞ). These two
quantities depend on the element positions as well as the
bearing parameter. Analyzing how these quantities vary with
the bearing and the element positions should give an under-
standing of the design result in Sec. IV. For this discussion,
the two quantities are evaluated over the entire bearing range
for 51 different configurations of the N ¼ 7 element array.
The different configurations are generated by increasing d1
and d2 at discrete steps such that the four free elements
move linearly from the UA element positions to the WNOA
element positions. The INOA is also included as one of the
configurations evaluated. This approach is equivalent to
evaluating the projected sensitivity and the output noise
power along a line from the point (0.5,1,0) to (1.5,1.5) in
Fig. 8 of Zhu and Buck (2010). There are actually many con-
tours that could be chosen to traverse the parameter space
(d1; d2) evolving from the UA to the WNOA, but this choice
is a simple contour and includes all three of the array config-
urations of primary interest (UA, INOA, and WNOA).
In Fig. 6, the right-hand panel shows a color plot of the
variation of the norm of the projected whitened bearing
sensitivity term (kP?
~vðhÞd~vðhÞ=dhk) over the bearing range
(x-axis) for each configuration of the N ¼ 7 element array
FIG. 6. (Color online) Color plot of the variation of the norm of the
projection of the whitened sensitivity vector ððkP?
~v hð Þd~v hð Þ=dhkÞÞ over the
bearing range (x-axis) for different configurations of a seven-element array
(y-axis). A set of 12 array configurations are shown on the left-hand panel
and the INOA configuration is shown with diamond markers. The INOA
configuration maximizes the sensitivity term and this results in maximiza-
tion of the FI according to Eq. (8).

(y-axis). The left-hand panel shows a set of 12 configurations
of the array starting from the UA at the bottom and progress-
ing to the WNOA at the top, with the INOA indicated by dia-
mond markers roughly two-thirds of the way up. In Fig. 6,
the arrays in the left-hand panel are essentially the y-axis for
the color plot on the right. Note that the color axis is shown
on a linear, not log, scale. The norm of the projected whit-
ened bearing sensitivity increases as the array elements
move outward from the uniform position and maximizes for
the INOA configuration. The norm reduces as the elements
move further outward beyond the INOA and is minimized
when elements are at the WNOA configuration. Increasing
the norm of the projected bearing sensitivity increases the FI
[Eq. (8)], and thus should improve the bearing estimation
performance of the array.
Similarly, in Fig. 7 the right-hand panel shows the color
plot of the output noise power (vðhÞH
KnnvðhÞ) variation over
the bearing range for the same set of array configurations
(left panel) as described above for Fig. 6. Again the color
axis is on a linear scale. The UA has a low output noise
power for broadside bearing but it rises sharply toward end-
fire resulting in a variation in the output noise power over
the bearing. On the other hand, the WNOA results in
increased output noise power across all bearings. The INOA
configuration minimizes the variation in output noise power
over the bearing range in comparison to the UA, while keep-
ing the noise power less than the WNOA output noise power.
As a result, the INOA beam former’s output noise power is
more uniform over bearing. Combining the data in Figs. 6
and 7 indicates that moving the array elements outward from
the UA positions provides a dual benefit of increased sensi-
tivity and slightly reduced output noise up to the INOA ele-
ment positions. Progressing beyond the INOA element
positions toward the UA positions reduces sensitivity and
increases output noise both of which reduce bearing estima-
tion performance. Hence the INOA array trades off between
maximizing the norm of the projected whitened bearing sen-
sitivity and minimizing the output noise power and its varia-
tion over the bearing.
Further, the isotropic noise covariance matrix Kiso is
also a function of the element positions. The off-diagonal
entries of the matrix J of Eq. (4) specify the correlation
between the isotropic noise samples measured at the sensor
elements. The inter-element correlation values for the INOA
should differ from the inter-element correlation values for
the UA to improve the estimation process. Reduced correla-
tion between noise samples will result in the noise samples
destructively interfering when the measurements are com-
bined for beam forming, ultimately resulting in lower output
noise power and better bearing estimation performance.
Figure 8 shows a histogram of the entries of the correlation
matrix J for the UA (upper panel) and for the INOA (lower
panel) for the N ¼ 21 arrays. With the UA, only about 25%
of the correlation values are in the range jqj 0:1, but for
the INOA the number of correlation values in the range
jqj 0:1 increases to 40%. Thus, the INOA has a smaller
correlation between noise samples compared to the UA.
B. Conclusion
In conclusion, this work designs an optimum linear hori-
zontal array that maximizes the bearing FI in a predomi-
nantly cylindrically isotropic spatially correlated noise field.
The optimum element positions were determined by opti-
mizing the FI in max-min sense over the bearing range. The
exhaustive search method was used to find the optimum
array for N ¼ 7 and the gradient search method was used for
N ¼ 21. It was found that the elements of the INOA are
pushed toward the array edge from their UA positions, but
not all the way to the ends as in the case of the WNOA. The
optimum element positions result in a trade off between
maximizing the norm of the orthogonal component of the
whitened bearing sensitivity and minimizing the output noise
power and its variation over the bearing. The performance of
FIG. 8. Histogram of entries of the matrix J of Eq. (7) for N ¼ 21 elements.
The INOA element positions result in many more pairs of elements with
lower noise correlation than the UA allowing the noise variance to be
reduced when the elements are coherently combined for beam forming.
FIG. 7. (Color online) Color plot of the variation of the output noise power
(vðhÞH
KnnvðhÞ) over the bearing range (x-axis) for different configurations
of a seven-element array (y-axis). A set of 12 array configurations are shown
on the left-hand panel and the optimum array is shown with diamond
markers. The INOA configuration minimizes the variation in output noise
power over the bearing range in comparison to the UA while keeping the
noise power less than the WNOA output noise power.

the optimum array was verified using Monte Carlo experi-
ments with a ML bearing estimator and depending on the
bearing angle, the optimum array had a improvement in per-
formance equivalent to a gain of 2–5 dB of input SNR.
There are clear opportunities to extend this approach.
These results assume only a narrowband wave field source,
whereas wideband sources are also common in the under-
water environment. Considering a wideband source obvi-
ously complicates the array design and the interpretation of
the results. The structure of the isotropic noise covariance
matrix is also modified for a wideband case. This could be
one future direction of this research.
ACKNOWLEDGMENT
The research was supported by the Office of Naval
Research Code 321US under Grant No. N00014-09-1-0167.
APPENDIX A: DERIVATION OF BEARING FISHER
INFORMATION
This appendix derives the expression for the FI in
Eq. (7) based on the results published by Ye and DeGroat
(1995). In their measurement model, they assume unknown
correlated noise parameterized by some unknown parameter.
However, for this work the structure of the noise covariance
is known from Eq. (4). Hence Eq. (28) of Ye and DeGroat
(1995) reduces to a scalar expression for the bearing FI. Lai
and Bell (2007) used the same result to derive the CRB for
bearing estimation in a correlated noise field for the case of
multiple parameters per source. For this work we start from
Eq. (29) of Ye and DeGroat (1995),
IðhÞ ¼
2
r2
n
Re
dvH
ðhÞ
dh
QÀ1
n
~P?
vðhÞ
dvðhÞ
dh

Â r2
s vH
ðhÞKÀ1
yy vðhÞr2
s
o
; (A1)
where, ~P?
vðhÞ ¼ I À vðhÞðvH
ðhÞQÀ1
n vðhÞÞÀ1
vH
ðhÞQÀ1
n . The
bearing FI expression in Eq. (A1) is a product of two quadratic
terms. Consider the inner matrices in the first quadratic terms,
QÀ1
n
~P?
vðhÞ ¼ QÀ1
n I À
vðhÞvH
ðhÞ
ðvHðhÞQÀ1
n vðhÞÞ
QÀ1
n
!
¼ QÀ1=2
n I À
QÀ1=2
n vðhÞvH
ðhÞQÀ1=2
n
ðvHðhÞQÀ1
n vðhÞÞ
!
QÀ1=2
n
¼ QÀ1=2
n P?
~vðhÞQÀ1=2
n :
Using this result in Eq. (A1)
IðhÞ ¼
2
r2
n
R e
dvH
ðhÞ
dh
QÀ1=2
n P?
~vðhÞQÀ1=2
n
dvðhÞ
dh

Â r2
s vH
ðhÞKÀ1
yy vðhÞr2
s
'
¼
2
r2
n
d~vH
ðhÞ
dh
P?
~vðhÞ
d~vðhÞ
dh

r2
s vH
ðhÞKÀ1
yy vðhÞr2
s

:
(A2)
This is the expression for the bearing FI. From this equation
[Eq. (A2)] it is clear that IðhÞ evaluates to a real value. The
explicit notation for real value Re Á Á Áð Þ is henceforth sup-
pressed in the expression for IðhÞ. The second quadratic
term in Eq. (A1) can further be simplified to express the FI
in terms of input SNR as
vH
ðhÞKÀ1
yy vðhÞ ¼ vH
ðhÞðr2
s vðhÞvH
ðhÞ þ KnnÞÀ1
vðhÞ:
Using the matrix inversion lemma (Van Trees, 2002, Eq.
A.50),
vH
ðhÞKÀ1
yy vðhÞ ¼ vH
ðhÞ KÀ1
nn À r2
s
KÀ1
nn vðhÞvH
ðhÞKÀ1
nn
1 þ r2
s vHðhÞKÀ1
nn vðhÞ
!
Â vðhÞ
¼
vH
ðhÞKÀ1
nn vðhÞ
1 þ r2
s vHðhÞKÀ1
nn vðhÞ
¼
1
r2
s
r2
s
r2
n
vH
ðhÞQÀ1
n vðhÞ
1 þ
r2
s
r2
n
vHðhÞQÀ1
n vðhÞ
0
@
1
A
¼
1
r2
s
SNR vH
ðhÞQÀ1
n vðhÞ
1 þ SNR vHðhÞQÀ1
n vðhÞ
!
(A3)
where SNR¼ (r2
s =r2
n) is the input SNR. Substituting Eq.
(A3) in Eq. (A2),
IðhÞ¼
2r2
s
r2
n
d~vH
ðhÞ
dh
P?
~vðhÞ
d~vðhÞ
dh

SNRvH
ðhÞQÀ1
n vðhÞ
1þSNRvHðhÞQÀ1
n vðhÞ
!
¼2SNR jjP?
~vðhÞ
d~vðhÞ
dh
jj2

SNR~vH
ðhÞ~vðhÞ
1þSNR~vHðhÞ~vðhÞ

¼2SNR jjP?
~vðhÞ
d~vðhÞ
dh
jj2

SNRjj~vðhÞjj2
1þSNRjj~vðhÞjj2
!
:
(A4)
This is the final expression for bearing FI.
APPENDIX B: PROOF OF THE SYMMETRY
CONSTRAINT ON THE OPTIMAL SENSOR
LOCATIONS
The optimal array element locations must be symmetric
about the array center, i.e., dopt ¼ Àdopt. This can be demon-
strated using proof by contradiction. Assume the optimal sen-
sor locations are not symmetric, i.e., dopt 6¼ Àdopt. This
implies
Imin dopt
À Á
Imin Àdopt
À Á
;
I dopt;hmin
À Á
I Àdopt; hmin
À Á
; (B1)
where hmin is the bearing corresponding to minimum FI. But
from Eq. (2) it is easy to see that v(Àd, h) ¼ v(d,Àh) which
leads to
I Àdopt;hmin
À Á
¼ I dopt; Àhmin
À Á
: (B2)

Straightforward algebra demonstrates that the log likelihood
function is even symmetric in bearing (h). The second deriv-
ative of an even function is itself even, so the even symmetry
of the log likelihood implies
I dopt; À hmin
À Á
¼ I dopt; hmin
À Á
: (B3)
Combining Eqs. (B1)–(B3) yields the contradiction
I dopt;hmin
À Á
I dopt; hmin
À Á
. Thus dopt must be symmetric.
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