PublishedPaper

Optimum array design to maximize Fisher informationfor bearing estimation

Saurav R. Tuladhara) and John R. BuckDepartment 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:

[email protected]

J. Acoust. Soc. Am. 130 (5), November 2011 VC 2011 Acoustical Society of America 27970001-4966/2011/130(5)/2797/10/$30.00

Downloaded 17 Nov 2011 to 134.88.13.40. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp

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� 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� 1 array steer-

ing vector, h is the source bearing and n is the N� 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 ½ejkd�ðN�1Þ=2 sin h;…; ejkd�1 sin h; 1; ejkd1 sin h;…;

ejkdðN�1Þ=2 sin h�T ; (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

�ðN � 1Þ=2 to ðN � 1Þ=2. The source power and noise power

are generally unknown and modeled as zero mean complex

Gaussian processes with covariances Eðss�Þ ¼ r2s and

EðnnHÞ ¼ Knn. The covariance for the measurement vector in

Eq. (1) is

Kyy ¼ Kxx þKnn ¼ r2s v hð ÞvH hð Þ þKnn: (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 þKw

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 r2isoJ: (4)

The entries of the matrix J are J½ �pq¼ J0 k dpq

�� where

J0 (��) is the zeroth order Bessel function and dpq ¼ dp � dq

is the spacing between the pth and qth elements. Here the

rows and columns p and q are also indexed symmetrically

from �ðN � 1Þ=2 to ðN � 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 r2wI.

Thus, the combined noise covariance can be expressed as

Knn ¼ r2isoJþ r2

wI ¼ r2nQn; (5)

where the combined noise power r2n ¼ r2

iso þ r2w

¼ Tr Knnð Þ=N, where Tr ð� � �Þ 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

L¼ (N� 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 (r2s ), the com-

bined noise power (r2n) and the source bearing (h). The three

unknown parameters can be combined into a parameter vec-

tor w ¼ ½r2s r

2nh�

T.

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� 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 K�1yy

@Kyy

@wm

K�1yy

@Kyy

@wn

� �(6)

where the subscript mn is the mth row and nth row element

of FI matrix and m, n¼ 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 IðhÞ�1. 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 2r4

s

r2n

d~vHðhÞdh

P?~vðhÞd~vðhÞ

dh

� �vHðhÞK�1

yy vðhÞ� �

; (7)

where ~vðhÞ ¼ Q�1=2n vðhÞ is the whitened steering vector and

P?~vðhÞ ¼ I� ~vðhÞð~vHðhÞ~vðhÞÞ�1~vHðhÞ 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 r2s=r

2n as

IðhÞ ¼2 SNRð Þ P?~vðhÞd~vðhÞ

dh

2 !

SNR ~vðhÞk k2

1þ 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 (r2s ). 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 ½d�ðN�1Þ=2;…; d�1; 0; d1;…; dðN�1Þ=2�T ;

where di is the distance of the ith element from the center of

the aperture. The bearing FI is now represented as Iðd; hÞwith 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 min0�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 maxd

minðdÞf g; jdij � L=2; (10)

where L is the array aperture which is constrained to be

L ¼ ðN � 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� 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 OðCnÞ 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

FIrdIminðdÞ was approximated numerically using central dif-

ference method. The iteration stops when dkþ1 � 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 N¼ 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 r2w=r

2iso ¼ �30 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 N¼ 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 �d1 and �d2 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 N¼ 7

elements (top panel) and N¼ 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 N¼ 21 element case, making exhaustive

search impractical for optimizing arrays for the size. Instead

the gradient search method was used to find the N¼ 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 10�5. 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 N¼ 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 N¼ 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 N¼ 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 N¼ 7

and N¼ 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 N¼ 21 elements and the

aperture L¼ 10k has a narrower main lobe compared to

the array with N¼ 7 elements and aperture L¼ 3k. Thus, the

N¼ 21 elements array should have a lower bearing MSE and

CRB compared to the N¼ 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

N¼ 7 and N¼ 21 element arrays as seen in the left-hand pan-

els of Fig. 5. Further, the CRB for the N¼ 21 array is lower

TABLE I. Optimum element positions.

N Optimum positions as factor of k (dopt)

7 �1.5 �1.34 �1.17 0 1.17 1.34 1.5

21 �5.00 �4.82 �4.49 �4.12 �3.74 �3.34 �2.96 �2.61 �2.36 �0.20

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) N¼ 7 and (b) N¼ 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 N¼ 7 array as discussed previously. The

MSE curve for both N¼ 7 and N¼ 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 N¼ 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 N¼ 7 the INOA has a higher thresh-

old SNR than the UA, but for N¼ 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 N¼ 7 elements and N¼ 21 ele-

ments steered to 0p and 0:4p. When

steered toward endfire at h ¼ 0:4pthe UA beam pattern develops a

lobe at �0:5p, 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 N¼ 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 N¼ 7 and N¼ 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 N¼ 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 N¼ 7, the INOA’s

CRB is �8 dB lower than the UA’s CRB for the SNR range

shown in Fig. 5. For N¼ 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 N¼ 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 N¼ 7 and N¼ 21. In the asymptotic region

for N¼ 7, the INOA requires �5 dB less SNR to achieve the

same MSE as the UA. For N¼ 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 N¼ 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 N¼ 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 N¼ 7 and only by factor of 2 for N¼ 21. As a result,

for N¼ 7 the INOA has a greater threshold SNR than the UA,

whereas for N¼ 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 N¼ 7 and 21 the UA’s beam pattern has a

lobe at �0:5p 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 �0:5p bearing being

erroneously chosen as the peak instead of the mainlobe peak.

Also, the error introduced due to selecting the lobe at �0:5pbearing 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 �0:5p bearing, for N¼ 7 the

UA has a 5 dB greater threshold SNR than the INOA,

whereas for N¼ 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 (vðhÞHKnnvðhÞ). 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 N¼ 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 N¼ 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 (vðhÞHKnnvðhÞ) 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 N¼ 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 N¼ 7 and the gradient search method was used for

N¼ 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 N¼ 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

(vðhÞHKnnvðhÞ) 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 FISHERINFORMATION

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

r2n

RedvHðhÞ

dhQ�1

n~P?vðhÞ

dvðhÞdh

� �

� r2s vHðhÞK�1

yy vðhÞr2s

� �o; (A1)

where, ~P?vðhÞ ¼ I� vðhÞðvHðhÞQ�1n vðhÞÞ�1

vHðhÞQ�1n . The

bearing FI expression in Eq. (A1) is a product of two quadratic

terms. Consider the inner matrices in the first quadratic terms,

Q�1n

~P?vðhÞ ¼ Q�1n I � vðhÞvHðhÞ

ðvHðhÞQ�1n vðhÞÞ

Q�1n

!

¼ Q�1=2n I �Q�1=2

n vðhÞvHðhÞQ�1=2n

ðvHðhÞQ�1n vðhÞÞ

!Q�1=2

n

¼ Q�1=2n P?~vðhÞQ

�1=2n :

Using this result in Eq. (A1)

IðhÞ ¼ 2

r2n

R edvHðhÞ

dhQ�1=2

n P?~vðhÞQ�1=2n

dvðhÞdh

� �

� r2s vHðhÞK�1

yy vðhÞr2s

� ��

¼ 2

r2n

d~vHðhÞdh

P?~vðhÞd~vðhÞ

dh

� �r2

s vHðhÞK�1yy 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 � � �ð 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

vHðhÞK�1yy vðhÞ ¼ vHðhÞðr2

s vðhÞvHðhÞ þKnnÞ�1vðhÞ:

Using the matrix inversion lemma (Van Trees, 2002, Eq.

A.50),

vHðhÞK�1yy vðhÞ ¼ vHðhÞ K�1

nn � r2s

K�1nn vðhÞvHðhÞK�1

nn

1þ r2s vHðhÞK�1

nn vðhÞ

!

� vðhÞ

¼ vHðhÞK�1nn vðhÞ

1þ r2s vHðhÞK�1

nn vðhÞ

¼ 1

r2s

r2s

r2nvHðhÞQ�1

n vðhÞ

1þ r2s

r2nvHðhÞQ�1

n vðhÞ

0@

1A

¼ 1

r2s

SNR vHðhÞQ�1n vðhÞ

1þ SNR vHðhÞQ�1n vðhÞ

!(A3)

where SNR¼ (r2s=r

2n) is the input SNR. Substituting Eq.

(A3) in Eq. (A2),

IðhÞ¼2r2s

r2n

d~vHðhÞdh

P?~vðhÞd~vðhÞ

dh

� �SNRvHðhÞQ�1

n vðhÞ1þSNRvHðhÞQ�1

n vðhÞ

!

¼2SNR jjP?~vðhÞd~vðhÞ

dhjj2

� �SNR~vHðhÞ~vðhÞ

1þSNR~vHðhÞ~vðhÞ

� �

¼2SNR jjP?~vðhÞd~vðhÞ

dhjj2

� �SNRjj~vðhÞjj2

1þSNRjj~vðhÞjj2

!:

(A4)

This is the final expression for bearing FI.

APPENDIX B: PROOF OF THE SYMMETRYCONSTRAINT ON THE OPTIMAL SENSORLOCATIONS

The optimal array element locations must be symmetric

about the array center, i.e., dopt ¼ �dopt. This can be demon-

strated using proof by contradiction. Assume the optimal sen-

sor locations are not symmetric, i.e., dopt 6¼ �dopt. This

implies

Imin dopt

� �> Imin �dopt

� �;

I dopt;hmin

� �> I �dopt; hmin

� �; (B1)

where hmin is the bearing corresponding to minimum FI. But

from Eq. (2) it is easy to see that v(�d, h)¼ v(d,�h) which

leads to

I �dopt;hmin

� �¼ I dopt;�hmin

� �: (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; � hmin

� �¼ I dopt; hmin

� �: (B3)

Combining Eqs. (B1)–(B3) yields the contradiction

I dopt;hmin

� �> I dopt; hmin

� �. Thus dopt must be symmetric.

Brown, J. L., and Rowlands, R. O. (1959). “Design of directional arrays,”

J. Acoust. Soc. Am. 31, 1638–1643.

Cox, H. (1973). “Spatial correlation in arbitrary noise fields with application

to ambient sea noise,” J. Acoust. Soc. Am. 54, 1289–1301.

Jacobson, M. J. (1962). “Space time correlation in spherical and circular

noise fields,” J. Acoust. Soc. Am. 34, 971–978.

Johnson, D., and Dudgeon, D. (1992). Array Signal Processing: Conceptsand Techniques (Prentice-Hall, Englewood Cliffs, NJ), Sec. 2.2.1.

Kuperman, W. A., and Ingenito, F. (1980) “Spatial correlation of surface

generated noise in a stratified ocean,” J. Acoust. Soc. Am. 67,

1988–1996.

Lai, H., and Bell, K. (2007). “Cramer–Rao lower bound for DOA estima-

tion using vector and higher-order sensor arrays,” Proceedings of theAsilomar Conference on Signals, Systems, and Computers (ACSSC), pp.

1262–1266.

Lang, S., Duckworth, G., and McClellan, J. (1981). “Array design for MEM

and MLM array processing,” Proceedings of the International Conferenceon Acoustics, Speech and Signal Processing (ICASSP), Vol. 6, pp.

145–148.

MacDonald, V. H., and Schultheiss, P. M. (1969). “Optimum passive bear-

ing estimation in a spatially incoherent noise environment,” J. Acoust.

Soc. Am. 46, 37–43.

Murino, V., Trucco, A., and Regazzoni, C. S. (1996). “Synthesis of

unequally spaced arrays by simulated annealing,” IEEE Trans. Signal Pro-

cess. 44, 119–122.

Pearson, D., Pillai, S. U., and Lee. Y. (2002). “An algorithm for near-

optimal placement of sensor elements,” IEEE Trans. Inf. Theory 36,

1280–1284.

Richmond, C. D. (2006). “Mean-squared error and threshold SNR prediction

of maximum-likelihood signal parameter estimation with estimated col-

ored noise covariances,” IEEE Trans. Inf. Theory 52, 2146–2164.

Van Trees, H. L. (2001). Detection, Estimation, and Modulation Theory,Part I (Wiley-Interscience, New York), Sec. 2.2.1.

Van Trees, H. L. (2002). Optimum Array Processing: Part IV of Detection,Estimation and Modulation Theory (Wiley-Interscience, New York), Secs.

2.2, 2.5, 2.24, 8.2.3, and 8.4.

Ye, H., and DeGroat, D. (1995). “Maximum likelihood DOA estimation and

asymptotic Cramer-Rao Bounds for additive unknown colored noise,”

IEEE Trans. Signal Process. 43, 938–949.

Zhu, X., and Buck, J. R. (2010). “Designing nonuniform linear arrays to

maximize mutual information for bearing estimation,” J. Acoust. Soc.

Am. 128, 2926–2939.



Date post:	15-Aug-2015
Category:	Documents
Upload:	saurav-tuladhar
View:	32 times
Download:	1 times

PublishedPaper

Documents