Some Comparisons of Cramer-Rao Bounds for Vector Sensors and Scalar

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Some Comparisons of Cramer-Rao Bounds for Vector Sensors and Scalar

Sensor Arrays for Array Processing

¹

Soren Anderson

2

and Arye Nehorai

Department of Electrical Engineering Yale University

New Haven, CT 06520

Abstract

The eect from

polarization

of emitted wave fronts on the parameter estimation ac- curacy for an array composed only of sensors sensitive to just one polarization direction has not been addressed in the literature this far. Antennas with such characteristics are,

^e.g.

, dipole (or scalar) antennas. A

^vector

sensor, on the other hand, is a sen- sor whose output data consists of, for the electromagnetic case, the complete electric and magnetic elds at the sensor. This paper examines some of the eects on the Cram er-Rao Bound for the elevation and/or azimuth angles to a single source emit- ting a polarized (electromagnetic) waveform. Since only

^one

vector sensor is needed for estimation of both azimuth and elevation, it would be of interest to compare the lower parameter estimation error bound resulting from the vector sensor data model to the \ordinary" one,

^i.e.

the data model used for scalar arrays. Such comparisons, both analytically and numerically, are herein made for an acoustic data model, as well as for an electromagnetic measurement model, for some simple scenarios and array congurations.

1

The work of A. Nehorai was supported by the Air Force O ce of Scientic Research under Grant no. F49620-93-1-0096, the O ce of Naval Research under Grant no. N00014-91-J-1298, and the National Science Foundation under Grant no. MIP-9122753.

2

On leave from the Department of Electrical Engineering, Linkoping University, S-581 83 Linkoping, Sweden.

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1 Introduction

During the last decade, much work in the area of sensor array processing has been devoted to the problem of making performance predictions for estimators of physical parameters as, e.g., direction-of-arrivals and emitted source and measurement noise powers, 1, 2, 3, 4, 5, 6]. Also the possibility of using polarization sensitive sensors for the reception has been analyzed, 7, 8, 9, 10]. This should be useful when the emitted waveforms are exhibiting dierent polarizations, thus improving the reception of signal power at the array.

However, an issue that not has been addressed in the literature is that of examining what eect polarization of the emitted wave fronts has on the estimation accuracy for an array composed only of sensors sensitive to just one polarization direction.

Such characteristics are displayed by, e.g., dipole antennas. Any such sensor will herein be referred to as a scalar sensor, and we assume also that the sensor is small compared to the wavelength of the received wavefronts. This paper examines some of the eects on the Cramer-Rao Bound (CRB, the lower bound for any unbiased estimator) for the elevation and/or azimuth angles to a single source emitting a polarized (electromagnetic) waveform.

In 11, 12], the concept of \vector sensors" is introduced for source localization, see also 13]. A vector sensor is a sensor whose output data consists of, for the electromagnetic case, the complete electric and magnetic elds at the sensor, see e.g.

14]. These sensors can consist, e.g., of two lumped orthogonal triads of scalar sensors that measure the electric and magnetic elds, see e.g. 15]. A possible advantage, to be examined herein, of the vector sensor over the traditional scalar sensors is their greater observability of direction-of-arrival (DOA) estimation, resulting in a smaller array aperture while maintaining performance. Since only one vector sensor is needed for estimation of both azimuth and elevation, it would be of interest to compare the lower bound (CRB) resulting from the vector sensor data model to the \ordinary"

one, i.e. the data model used for scalar arrays. Such comparisons are herein made for an acoustic data model, as well as for the electromagnetic measurement model, for some simple scenarios.

By studying some simple scalar array congurations, explicit bounds on the scalar sensor array sizes are found that unveil when one single vector sensor performs better (w.r.t optimal estimation accuracy) than a specied scalar array. Although the results obtained are specialized to specic array congurations, the observations made give some insight into the limitations inherent in scalar arrays when compared to a single vector sensor.

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2 Data Models

In this section the two dierent models employed are briey reviewed. Moreover, a more general model, used for stating the general Cramer-Rao bound expression, is also given. See 11, 12] for further details.

2.1 A General Measurement Model

The basic data model we will be working with here is of the following form, where we assume that the number of vector sensors are m and the number of sources to be n .

y

( t ) =

^A

(

)

^x

( t ) +

^e

( t ) : (1)

y

( t )

²

C

^j ¹

is the observed array output vector at time instant t , whereas

^e

( t )

²

C

^j ¹

is an additive measurement noise vector. Assuming that the n emitters generate unknown vectors of wave fronts

^x

k ( t )

²

C

^j

^k¹

, we have

^x

( t )

²

C

^j

¹

where =

P

nk

⁼¹

k . The transfer matrix

^A

(

)

²

C

^j

and the parameter vector

²

I R ^q

¹

are given by

A

(

) =

^h^A¹⁽¹⁾

:::

^A

n

(

n

⁾ⁱ

(2)

=

^h⁽¹⁾

:::

⁽

ⁿ

⁾ⁱ

(3)

with

^A

k

(

k

⁾

2

C

^j

^k

and the parameter vector for the k ^th source

⁽

^k

⁾ ²

I R ^q

^k¹

(thus q =

^P

_nk

⁼¹

q k ). We have also dened

y

( t ) =

^h^y⁽¹⁾

( t ) ^T :::

^y⁽

^m

⁾

( t ) ^T

ⁱ

^T (4)

x

( t ) =

^h^x⁽¹⁾

( t ) ^T :::

^x⁽

ⁿ

⁾

( t ) ^T

ⁱ

^T (5) where

^y⁽

^j

⁾

( t )

²

C

^j ^j¹

is the vector measurement of the j ^th sensor (hence =

P

mj

⁼¹

j ), and

^x⁽

^k

⁾

( t )

²

C

^j

^k¹

is the vector signal of the k ^th source.

2.2 The Acoustic Case

For the acoustic case, we will study the optimal performance for estimation of the azimuth angle only. Let

^u

k be the unit vector pointing towards source number k . Then we have

u

k =

"

cos k

sin k

#

(6)

where k is the azimuth angle of

^u

k . If we denote by

^P

k ( t ) the phasor representation (complex envelope) of the acoustic pressure (at some reference point), resulting from source k we, can write the measurement model as follows (see, e.g., 12]).

y

( t ) =

"

y p ( t )

y

v ( t )

#

=

^X

ⁿ

k

⁼¹

^e

k

^I³

]

"

1

u

k

#

P

( t ) +

^e

pv ( t ) (7)

3

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The pressure measurement is here denoted y p ( t ) while

^y

v ( t ) denotes the measurement of the acoustic particle velocity, which is here assumed to be measured in two dimen- sions (the x - and y -directions of the employed coordinate system). In (7),

is the Kronecker matrix product,

^e

k denotes the k ^th column of the matrix

^E ²

C

^j

^m

ⁿ whose ( jk ) entry is

^E

jk = e

^;

^i!

^c

^jk

. jk is the dierential time delay of the k ^th source signal between the j ^th sensor and the origin of some xed reference coordinate system. The measurement noise vector

^e

pv ( t )

²

C

^j ³¹

is composed of the noise resulting from the pressure measurement as its rst component, and the velocity measurement noise in the last two positions. For further details on this model, see 12].

This is then a model to be compared with the more general one, (1), in order to identify the parametrization that results for the acoustic case. Such a comparison shows, see (1), that it is necessary to multiply the pressure measurements y p ( t ) by r =

v = p (the quotient between the velocity and pressure measurement noise standard deviations) and also assume that r is known (which is the case, e.g., if both the noise variances are unknown but equal). Now, this implies that we have (7) as a special case of (1) with

a

k =

^e

k

^I³

]

"

r

u

k

#

^x

( t ) =

^P¹

( t ) :::

^P

n ( t )] ^T

²

= v

²

=

¹

::: n ] ^T : (8)

Here,

^a

k is the k ^th column of

^A

(

).

2.3 The Electromagnetic Case

By \examination" (see 11]) of the Maxwell equations for planar electromagnetic waveforms, the electric phasor (or complex envelope of the electric eld) can be written as

^E

( t ) =

^V

( t ), where

V

=

2

6

4

;

sin

¹ ^;

cos

¹

sin

²

cos

¹ ^;

sin

¹

sin

²

0 cos

²

3

7

5

(9)

and where the vector

( t ) has the representation 11] (in this case, for single signal, s ( t ), transmission only)

( t ) =

^Qw

s ( t ). Here,

Q

=

"

cos

³

sin

³

;

sin

³

cos

³

#

(10)

w

=

"

cos

⁴

i sin

⁴

#

(11) The orientation and ellipticity angles,

³ ²

(

^;

= 2 = 2 ] and

⁴ ²

^;

= 4 = 4], respectively, describe the polarization of the complex envelope of the emitted signal, see further 11].

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Now, let

^y

EH ( t ) and

^e

EH ( t ) be the 6 m

1-dimensional electromagnetic sensor phasor measurement and noise vectors,

y

EH ( t ) =

^y⁽¹⁾

_E ( t )

^T

^y⁽¹⁾

_H ( t )

^T :::

^y⁽

_E ^m

⁾

( t )

^T

^y⁽

_H ^m

⁾

( t )

^T

^T (12)

e

EH ( t ) =

^e⁽¹⁾

E ( t )

^T

^e⁽¹⁾

H ( t )

^T :::

^e⁽

E ^m

⁾

( t )

^T

^e⁽

H ^m

⁾

( t )

^T

^T (13) where

^y⁽

E ^j

⁾

( t ) and

^y⁽

H ^j

⁾

( t ) are, respectively, the measured phasor electric and magnetic vector elds at the j ^th sensor. (And similarly for

^e⁽

E ^j

⁾

( t ) and

^e⁽

H ^j

⁾

( t ).) The array output measurement vector can then be modeled by,

y

EH ( t ) =

^X

ⁿ

k

⁼¹

^e

k

^I⁶

]

"

I

(

^u

k

3

)

#

V

k

k ( t ) +

^e

EH ( t ) (14) where the operator (

^u

) is dened by the matrix

(

^u

) =

2

6

4

0

^;

u z u y

u z 0

^;

u x

;

u y u x 0

3

7

5

(15)

where u x u y u z are the x y z components of the vector

u

= cos

¹

cos

²

sin

¹

cos

²

sin

²

] ^T .

Similarly to the acoustic case, we must also here dene a scaled measurement of the array output vector in order to relate (14) to the model (1). In this case we take

y

( t ) =

^h

r

^y

_TE ( t )

^y

_TH ( t )

ⁱ

^T , with r = H = E being the quotient between the magnetic and the electric measurement noise standard deviations. Also here we assume that r is known. This leads to the parametrization (c.f. (1))

a

k =

^e

k

^I⁶

]

"

r

^I³

(

^u

k

)

#

V

k

^Q

k

^w

k

^x

( t ) = s

¹

( t ) ::: s n ( t )] ^T

²

= _H

²

=

^h

⁽¹⁾¹

¹⁽²⁾

:::

⁽¹⁾

_n _n

⁽²⁾ⁱ

^T (16) where

is the Kronecker matrix product,

^e

k denotes the k ^th column of the matrix

E 2

C

^j

^m

ⁿ whose ( jk ) entry is

^E

jk = e

^;

^i!

^c

^jk

. jk is the dierential time delay of the k ^th source signal between the j ^th sensor and the origin of some xed reference coordinate system. Finally,

^a

k is the k ^th column of

^A

(

).

5

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3 Cramer-Rao Bound Expressions

The lower bounds for parameter estimation, using the models described in Section 2, are stated in this section. To be able to make analytical or numerical comparisons, the array geometry must be specied for the scalar-sensor cases. Herein, we study some simple array congurations for the scalar sensor case. Only one vector sensor is used in the subsequent comparisons of the dierent measurement models. The corre- sponding lower bounds are computed using the symbolic software package MAPLE,

16].

Since we are mainly concerned with small arrays, the particular congurations chosen are not believed to be of that much importance for the comparisons to be more generally valid. However, care should denitely be taken when trying to generalize the observations made.

3.1 The General Case

Consider the model (1) and the problem of estimating the parameter vector

of that model, with

,

^P

= E

^h^x

( t )

^x

( t ) ^T

ⁱ

and

²

= E

^h^e

( t )

^e

( t ) ^T

ⁱ

unknown. From 11], we have the following theorem (where N is the number of array output measurements or snapshots)

Theorem 1 The Cram er-Rao bound on the estimation error covariance matrix of any (locally) unbiased estimator of the vector

in the model (1) (under assumptions A5 to A7, see 11]) with

,

^P

,

²

unknown and k = for all k is

CRB(

) =

²

2 N

n

Re

^h

btr

( 1

²^U

)

²

(

^D

c

^D

) ^bT

^io^;1

(17) where

U

=

^P ^A^AP

+

²Î^;1ÂÂP

(18)

c =

^I ^;

(19)

=

^A

(

^A^A

)

^;1^A

(20)

D

=

^h^D⁽¹⁾¹

:::

^D⁽¹⁾

_q

¹

:::

^D⁽¹

ⁿ

⁾

:::

^D⁽

_q ⁿ

ⁿ⁾ⁱ

(21)

D (

k

⁾

l = @

^A

k

@

⁽

_l ^k

⁾

⁽²²⁾

and where 1 ^{denotes a} q

q matrix with all entries equal to one, and the block trace operator btr(

), the block Kronecker matrix product

²

, the block Schur-Hadamard product

²

and the block transpose operator bT are dened with blocks of dimensions

, except for the matrix 1 that has blocks of dimensions q i

q j . Furthermore, the CRB remains the same independently of whether

²

is known or unknown.

3.2 The Acoustic Case

This section gives the explicit expressions for the Cramer-Rao bounds resulting from the acoustic data model using a single vector sensor, as well as the bounds for two particular array congurations used for the scalar sensor case.

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3.2.1 CRB for One Vector Sensor

If we assume that we only have one vector sensor and one source, we nd by a direct calculation that the Cramer-Rao bound for estimation of the azimuth of the source is

CRB

^ACO

( ) = 1 + %

2 N%% v (23)

where we have dened

% = _s

²

^k²

= _p

²

_v

²

_p + _v

²

% v = _s

²

_v

²

: (24)

Note that CRB

^ACO

( ) is independent of the azimuth angle. This is due to the sym- metry resulting when only one sensor is employed.

3.2.2 CRB for Some Scalar Sensor Arrays { Acoustic Case

The arrays under consideration here are assumed to be located in the x

^;

y -plane, with the reference point for the time delays jk being the origin of the coordinate system. The azimuth angle is dened from the x -axis towards the y -axis. For the scalar arrays, only the acoustic pressure is measured. Therefore, in order to make a more \fair" comparison, we assume that the number of scalar sensors is equal to the number of measurements obtained from the vector sensor which, in this particular study, is three.

A Uniform Linear Array (ULA)

The array response vector for the scenario of one wave front impinging on the array from the azimuth direction and three sensors, located at the ( xy ) positions

(0 0) (0

^;

d ) (0

^;

2 d )], is given by

a

( ) =

2

6

4

e

^;

^ik 1

^sin

e

^;2

^ik

^sin

3

7

5

(25)

where k = 2 d= and is the wavelength of the emitted (pressure) wave form. The measurement model is thus

^y

( t ) =

^a

( )

^P

( t ) +

^e

p ( t ), where

^e

p ( t ) is the measurement noise in the measurements of the acoustic pressure. By making use of Theorem 1 and MAPLE, we obtain the following lower azimuth angle estimation error bound for the

ULA CRB

^ULA

( ) = 1

2 N cos

²

^d

²

1

5 SNR

^p

+ 1 20

SNR

²^p

!

(26)

where the signal-to-noise ratio is dened by SNR

^p

=

²

_s = _p

²

with E

^h^e

p ( t )

^e

p ( t ) ^T

ⁱ

=

_p

²^I³

. Observe that the expression (26) has singularities for =

= 2 rad. This is due to the fact that the derivative of the array response vector (corresponding to

^D

in (17)) is the zero vector for that particular azimuth angle. This can also be viewed as a loss of sensitivity for small variations of around

= 2 radians.

7

(8)

By comparing the bounds (23) and (26), we nd that the scalar array will produce a higher bound than the vector sensor, provided that

d <

0

@ 1

5SNRp

+

^20SNR¹ ²p

(2 )

²

cos

²

_s

⁴

_v

²

_s +

^k²

1

A

: (27)

Hence, for the values of d , a measure of the array aperture, satisfying (27) the vector sensor will perform better than a ULA composed of three scalar sensors.

A \Triangular" Array

Let the three sensors be positioned at ( d 0) (0 d ) (

^;

d 0)]. Since we have as- sumed that the reference point for the time delays is the origin, we have for this conguration the array response vector

a

( ) =

2

6

4

e ^ik

^cos

e ^ik

^sin

e

^;

^ik

^sin

3

7

5

: (28)

The Cramer-Rao bond for this array can now be calculated to be CRB

^TRI

( ) = 1

2 N (1 + 2cos

²

)

²

^d

²

1

2

3

SNR

^p

+ 1 2

SNR

²^p

!

: (29)

As for to the ULA, by comparing the bounds (23) and (29), we nd that the scalar array will produce a higher bound than the vector sensor, provided that

< d

0

B

@

3

^SNR¹ p

+

^3SNR¹ ²p

2(2 )

²

(1 + 2cos

²

)

_s

⁴

_v

²

_s

²

+

^k²

1

C

A

: (30)

Thus, for values of d satisfying (30), the vector sensor will perform better than the triangular array composed of three scalar sensors. Note also the strong similarity between the two bounds (27) and (30).

The eect of dierent values for the parameters on the bounds above will be examined numerically in Section 4. The scalar array that will be compared to the vector sensor is the triangular array, since that bound does not have any singularities for any value of .

3.3 The Electromagnetic Case

Expressions for the Cramer-Rao bounds resulting from the electromagnetic data model, employing one vector sensor, and bounds for some array congurations used for the scalar sensor case are stated here. The parameters of interest are in this case both the azimuth (denoted by

¹

) and elevation (

²

) angles as well as the polarization angles,

³

and

⁴

. The azimuth angle is dened from the x -axis, increasing towards the y -axis, and the elevation angle is dened from the xy -plane, increasing towards the z -axis.

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3.3.1 Mean Square Angular Error Lower Bound for One Vector Sensor

For one vector sensor and one source, expressions for the Cramer-Rao bound for estimation of azimuth and elevation angles can be found in 11]. These expressions are rather complicated functions of

¹

,

²

as well as of the polarization parameters,

³

and

⁴

. However, by instead employing another lower bound, the Mean Square Angular Error (MSAE

^CR

) 11], dened by

MSAE

^CR

= N

^h

cos

²

CRB

^EM

(

¹

) + CRB

^EM

(

²

)

ⁱ

: (31) This bound has the property of being invariant to the choice of reference coordinate frame, see further 11]. For the special case discussed here, we obtain 11]

MSAE

^EM

(

¹

) = (1 + % )(1 + r

²

)

²

2 %

²^h

r

²

+ (1

^;

r

²

)

²

sin

²

⁴

cos

²

⁴ⁱ

⁽³²⁾ where we have dened r = H = E and

% = _s

²

^k²

= _E

²

_H

_E

²

+ _H

²

: (33)

This bound is appropriate for evaluation of estimators of both azimuth and elevation angles, since it is a scalar measure of the optimal estimation accuracy of these angles.

3.3.2 MSAE for Some Scalar Sensor Arrays { Electromagnetic Case

We will here take a closer look at the MSAE expressions for two dierent ar- ray congurations one planar array and one \three-dimensional", where the sensors are distributed on a sphere. Since we assume the sensors to be \scalar", only one component of the electric eld will be measured for each sensor.

A Planar Array

We let the number of sensors be ve, located in the yz -plane at the ( xyz ) po- sitions (0 0 0) (0 d 0) (0 0 d ) (0

^;

d 0) (0 0

^;

d )]. Only the z -component of the electric eld is measured by each sensor. Assuming that one wave front parametrized as in (10) impinge on the array, the array response vector is found to be

A

(

) = z (

)

2

6

4

e ^ik

^sin

1

¹^cos

²

e ^ik

^sin

²

e

^;

^ik

^sin

¹^cos

²

e

^;

^ik

^sin

²

3

7

5

(34)

where k = 2 d= and the factor

z (

) = cos

²

(

^;

sin

³

cos

⁴

+ i cos

³

sin

⁴

) (35) describes the inuence of the polarization and the fact that only the z -component of the electric eld is measured.

9

(10)

By computation of the Cramer-Rao bound, we nd the MSAE lower bound for this array conguration to be

MSAE

^PLA^EM

= 1 2

²

^d

²

1

2 SNR + 1 10

SNR

²

1 + 1

cos

²

¹

cos

²

(36) where the signal-to-noise ratio has been dened as

SNR = (

³

⁴

)

²

_s

_E

²

^cos

²

(37)

(

³

⁴

) = sin

²

³

cos

²

⁴

+ cos

²

³

sin

²

⁴

(38) Observe here that

(

= 4

⁴

)

1 = 2

(

³

= 4) (39) (from 11] we have that

^j

⁴^j

= = 4 corresponds to circular polarization and that

⁴

= 0 implies linear polarization).

Note also that the SNR is zero for wave fronts (with arbitrary polarization param- eters) impinging from the

z -directions (

²

=

= 2). This is, however, expected by physical reasons since the electric eld then is orthogonal to the direction in which the wave traverses. Hence, in the present case, no component of the electric eld will exist in the z -direction and no signal power will therefore be received. This is a singularity of the MSAE expression. Other singularities can be found for

¹

=

= 2.

As in the acoustic case, this corresponds to a loss of \sensitivity" of the array, that is, the derivative of the array response vector is zero for these particular

¹

values.

Remark: Due to the similarity, for dierent array congurations, of the bounds describing for which array aperture ( d ) the vector sensor outperforms the scalar sensor array, the corresponding bound for this particular planar array is not stated here. See instead the bound for the \spherical" array, (42).

A \Spherical" Array

We let the number of sensors be six, symmetrically located in the xyz -coordinate system at ( xyz )-positions ( d 0 0) (0 d 0) (

^;

d 0 0) (0

^;

d 0) (0 0

^;

d ) (0 0 d )].

Also for this array conguration, only the z -component of the electric eld is measured by each sensor. Assuming that one wave front, parametrized as in (10), impinge on the array, the array response vector is found to be

A

(

) = z (

)

2

6

4

e ^ik

^cos

¹^cos

²

e ^ik

^sin

¹^cos

²

e

^;

^ik

^cos

¹^cos

²

e

^;

^ik

^sin

¹^cos

²

e ^ik

^sin

²

e

^;

^ik

^sin

²

3

7

5

(40)

where, as for the planar array, k = 2 d= , and z (

) is dened by (35).

Computing the Cramer-Rao bound also for this array conguration, we nd the MSAE lower bound to be

MSAE

^SPH^EM

= 1 2

²

^d

²

1

2 SNR + 1 12

SNR

²

(41)

10

(11)

where the signal-to-noise ratio has been dened as in (37). The same observations as those following (37) is of course valid also for this case, with the exception that the only singularities here are for

²

=

= 2 (then implying SNR = 0).

As for the acoustic data model, we may also here determine the lower bound for d , at which the scalar array performs as good as the single vector sensor

d < ² %

²

r

²

+ 1

^;

r

²

]

²

sin

²

⁴

cos

²

⁴

(2 )

²

(1 + % )(1 + r

²

)

²

1

2 SNR + 1 12

SNR

²

(42) with SNR dened by (37) and % as in (33). Hence, for the values of d satisfying (42), the vector sensor will perform better than the spherical array consisting of the six scalar sensors. Very similar bounds are obtained also for planar arrays with four and

ve scalar sensors, which is why those bounds are not stated herein.

Measuring Dierent Polarization Directions

We shall also study the MSAE for a spherical array for which the electric eld component is measured by each sensor along the coordinate axis at which the sen- sor under consideration is located. The two sensors on the x -axis thus measure the x -component of the eld (mutatis mutandis for sensors at the y - and z -axis, respec- tively).

Using the parametrization (10) of the emitted wave front, we obtain the array response

A

(

) =

2

6

4

x (

) e ^ik

^cos

¹^cos

²

y (

) e ^ik

^sin

¹^cos

²

x (

) e

^;

^ik

^cos

¹^cos

²

y (

) e

^;

^ik

^sin

¹^cos

²

z (

) e ^ik

^sin

²

z (

) e

^;

^ik

^sin

²

3

7

5

(43)

where k = 2 d= and

x (

) = cos

⁴

(cos

¹

sin

²

sin

³^;

sin

¹

cos

³

) +

+ i sin

⁴

(

^;

sin

¹

sin

³^;

cos

¹

sin

²

cos

³

) (44)

y (

) = cos

⁴

(cos

¹

cos

³

+ sin

¹

sin

²

sin

³

) + + i sin

⁴

(cos

¹

sin

³^;

sin

¹

sin

²

cos

³

)

z (

) = cos

²

(

^;

sin

³

cos

⁴

+ i cos

³

sin

⁴

)

describe the eect from the measurements of the dierent polarization directions for dierent sensors. This array is then a \mixture" of the more common \uniformly sensitive" scalar array (as those described by (25) and (28)) and an array where each sensor is sensitive to more than one polarization direction, see e.g. 8, 9]. Due to the complex expressions in (44), a closed-form MSAE expression could not be calculated using the symbolic software package MAPLE. Instead, this case will be examined by means of numerical evaluation of (17), see Section 4.

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4 Numerical Illustrations

By numerical evaluation of the Cramer-Rao bounds and the Mean Square An- gular Error lower bounds for the acoustic and electromagnetic data modeling cases, respectively, this section compares one vector sensor with some dierent scalar array congurations. The aim is at examining what kind of inuence the parameter sets for the dierent modeling assumptions have on the aforementioned bounds. In all exam- ples involving one emitter, the signal power is assumed to be

^P

= _s

²

= 1 and we let N = 1 in all comparisons. We also normalize the wavelength of the waveforms to be = 1 . Furthermore, we will consider only the most \favorable" (w.r.t lowest possible CRB's/MSAE's) azimuth and/or elevation angles for the scalar arrays (which turns out to be

¹

= 0 =

²

for almost all cases examined here, since this implies largest SNR). All angles are stated in radians. Observe also that, for the scalar arrays, only the pressure or the electric measurements are assumed to be used.

4.1 The Acoustic Data Modeling Case

The scalar array under consideration here will be the \triangular" one, as dened in equation (28).

Case AC1: (Equal velocity and pressure measurement noise power, ( r = v = p = 1).) According to the bounds (23) and (29), the parameters we may vary are d , the distance from each scalar sensor to the origin of the coordinate system, and the measurement noise variances, _v

²

and _p

²

(of which the last two ones shall be the same in this case). In Figure 1, we let the noise variances vary, and as can be seen, the vector sensor will perform better than the scalar array provided that d= is less than about 0 : 1. Thus, if it is considered to be of importance to have a physically small sensor device, the vector sensor is an interesting alternative.

Case AC2: (Dierent velocity and pressure measurement noise power ( r

⁶

= 1).) Figure 2 depicts the case where the velocity measurement noise power has been

xed to _v

²

= 1, while the pressure measurement noise power is varied. We can, as expected, see that the vector sensor manages ne since the velocity measurement noise is small while, in comparison, the scalar array performs badly for small apertures, d , and a wide range of pressure noise variances.

In Figure 3, the velocity noise power varies and the pressure measurement noise power is _p

²

= 1. Here, we nd that the performance of the vector sensor is degrading with increasing velocity noise variance. This is also expected, since the only reason- ably \accurate" measurement is the pressure and only one \good" measurement is not enough for accurate azimuth angle estimation. Once again, the scalar array needs a fairly large aperture for good performance.

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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

0.5 1 1.5 2 2.5 3 3.5 4

Pressure and Velocity Noise Variances

Azimuth Cramer-Rao Bounds ^d=0.125

d=0.1 d=0.075

d=0.15 Solid Line (-): One Vector Sensor

Dashed Lines (-.): Scalar Array

Figure 1: (Case AC1) The inuence on the CRB's from varying the array aperture, d , and the pressure and velocity measurement noise powers for the acoustic data modeling case (with r = p = v = 1).

4.2 The Electromagnetic Modeling Case

We will consider only the \spherical" array (40) for the electromagnetic data model, but we will include the case when dierent scalar sensors are sensitive to dif- ferent polarization directions, see (43) (this array is here referred to as the \polarized"

array).

Case EM1: (Equal electric and magnetic measurement noise power ( r = 1).) We rst consider what inuence dierent polarization angles have on the MSAE.

This is depicted in Figures 4a) and b), where r = H = E = 1. Note, in Figure 4a), the dramatic increase of the MSAE for the scalar array for polarization angles

³

and

⁴

both close to zero. This corresponds to having the SNR = 0, cf. (37). For small SNR's, the MSAE is proportional to 1 = SNR

²

, see (41). Similar eects can be found also in Figure 4b) for the\polarized" array, but here for other combinations of polarization angles. It is important, however, to observe that the performance of the two versions of scalar arrays examined here is quite unsatisfactory for many combinations of polarization angles. These shortcoming for scalar arrays is an issue that not really has been \examined" before.

Case EM2: (Dierent electric and magnetic measurement noise power ( r

⁶

= 1).) Figures 6 and 7 show the MSAE's for dierent noise powers (

²

_H and

²

_E ). We nd

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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

0.5 1 1.5 2 2.5 3 3.5 4

Pressure Noise Variance

Azimuth Cramer-Rao Bounds

d=0.125 d=0.1

d=0.15 d=0.075

Solid Line (-): One Vector Sensor Dashed Lines (-.): Scalar Array

Figure 2: (Case AC2) The inuence on the CRB's from variation of the pressure measurement noise power for the acoustic data modeling case, _p

²

, and the array aperture, d , while _v

²

= 1.

here eects similar to the corresponding acoustic data model example (Case AC2).

Notice, however, that the \polarized" array requires a smaller aperture than the scalar array sensitive to polarization only in the z -direction.

Case EM3: (Two emitters with decreasing azimuth angular separation.)

We end this series of examples by considering the MSAE's for one vector sensor and the \sperical", six-element scalar array with all sensors sensitive to polarization in the z -direction only. The scenario-of-interest consists in this example of two uncorrelated (i.e.,

^P

= _s

²^I²

), closely spaced emitters. Also here we take

²

_s = 1.

Figure 8 shows the MSAE's for dierent azimuth angles,

¹

, ellipticity polarization angles (

⁴

) and array sizes, d . The scenario is the following. Both emitters are located at the elevation angles

⁽¹⁾²

= 0 =

²⁽²⁾

. The rst one has a varying azimuth angle

⁽¹⁾¹

, while the azimuth for the second source is constant,

¹⁽²⁾

= 0. The orientation angles are identical

⁽¹⁾³

= = 4 =

⁽²⁾³

for both sources, while the ellipticity angles are as:

⁴⁽¹⁾

= = 4 is constant for the rst emitter, and varies for the second one according to

⁽²⁾⁴

= = 12 = 6 = 4.

Observe that the scalar array cannot separate too closely spaced emitters, as is seen in Figure 8, while the MSAE for the vector sensor remains bounded even for identical azimuth and elevation angles, see further 11]. This ability to estimate the directions to two sources loacated at the same position, but having dierent

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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Velocity Noise Variance

Azimuth Cramer-Rao Bounds

d=0.125 d=0.1

d=0.15 d=0.075

d=0.175 Solid Line (-): One Vector Sensor

Dashed Lines (-.): Scalar Array

Figure 3: (Case AC2) The inuence on the CRB's from varying the array aperture, d , and the velocity measurement noise power, _v

²

, while

²

_p = 1. Acoustic data model.

polarization parameters, is one of the main advantages of the vector sensor over scalar arrays. In order to obtain better performance (for more widely separated emitters) from the scalar array, we may conclude that it is important to increase the array aperture.

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-0.5 0 0.5

-2 -1

0 1

2 0

2 4 6 8 10 12

Ellipticity Orientation

Mean Square Angular Errors (MSAE)

d=0.125

d=0.25

-1 -0.5 0 0.5 1

-1 -1.5 0 -0.5

0 1 2 3 4 5 6 7 8

Ellipticity Orientation

Mean Square Angular Errors (MSAE)

d=0.075

d=0.25

a) b)

Figure 4: (Case EM1) MSAE's for scalar and \polarized" arrays, a) and b) respec- tively, with dierent polarization angles (orientation,

³

, and ellipticity,

⁴

). Electro- magnetic data model. The \planar" surface is the MSAE for the vector sensor.

0 2 4 6 8 10

0 20 40 60 80 100 120

Magnetic and Electric Noise Variances

Mean Square Angular Errors (MSAE)

d=0.15 d=0.125 d=0.1

d=0.125 d=0.075

d=0.25 d=0.025

Solid Line (-): One Vector Sensor Dashed Lines (-.): Scalar Array Dotted Lines (..): "Polarized" Array

Figure 5: (Case EM1) MSAE's for dierent measurement noise powers (with r =

H = E = 1). Electromagnetic data model.

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0 2 4 6 8 10 0

2 4 6 8 10 12

Magnetic Noise Variance

Mean Square angular Errors (MSAE)

d=0.25 d=0.025 d=0.125 d=0.075

d=0.1 Solid Line (-): One Vector Sensor

Dashed Lines (-.): Scalar Array Dotted Lines (..): "Polarized" Array

Figure 6: (Case EM2) MSAE for the electromagnetic data model. The magnetic measurement noise power, _H

²

, is varied and _E

²

= 1.

0 2 4 6 8 10

0 10 20 30 40 50 60 70 80

Electric Noise Variance

Mean Square angular Errors (MSAE) _d=0.25

d=0.125 d=0.125 Solid Line (-): One Vector Sensor

Dashed Lines (-.): Scalar Array Dotted Lines (..): "Polarized" Array

Figure 7: (Case EM2) MSAE for the electromagnetic case. The electric measurement noise power, _E

²

, is varied and _H

²

= 1.

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0 0.5 1 1.5 2 2.5 3 10

⁰

10

¹

10

²

Azimuth Angular Separation

Mean Square Angular Errors (MSAE)

d=0.10

d=0.175 d=0.25

Solid Lines (-): One Vector Sensor 1 - ellipticity = pi/12

2 - ellipticity = pi/6 3 - ellipticity = pi/4

Dashed Lines (-.): Six-Element Scalar Array

1 2

3 Figure 8: (Case EM3) MSAE comparison for one vector sensor and a \spherical"

scalar array with 6 sensors. The eect of azimuth angular separation between two emitters is studied here.

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5 Summary

This paper examines and compares the performance of some specic scalar sensor arrays to a single vector sensor.

By numerical evaluation of the azimuth and elevation angle Cramer-Rao bounds for some simple scenarios and specic array congurations, some of the limitations inherent in the scalar arrays are illuminated. Explicit bounds on the aperture of some scalar arrays are stated that reveals when a single vector sensor performs better with respect to estimation accuracy of azimuth and elevation angles.

For instance, the scalar arrays are fairly sensitive to the direction from which polarized waveforms impinge on the array, and also to the particular values of the polarization parameters themselves. In order to obtain performance similar to a single vector sensor, the array aperture can not be chosen too small. This then implies that the vector sensor is \suitable" in applications where it is necessary/important to have a physically small sensor device, see further 11].

For two closely spaced emitters, a scalar array performs rather porly when trying to resolve the two emitters if the array aperture is not large enough or if the number of sensor elements is too few. The vector sensor, on the other hand, has the ability to resolve sources that has identical azimuth and elevation angles (provided that, at least one, of the polarization parameters dier).

It is, however, important to note that due to the necessity to specify the array conguration in a study like this, care should be exercised when trying to generalize the observations stated herein.

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Some Comparisons of Cramer-Rao Bounds for Vector Sensors and Scalar