Joint Detection Estimation Problem of Monopulse Angle Measurement-UI0.pdf

I. INTRODUCTION

Joint Detection Estimation

Problem of Monopulse Angle

Measurement

A multi-mode radar must be able to operate

in a tracking mode. The tracking algorithm needs

to locate the signal source (radar target) direction

more precisely than the detection process performed

with the sum beam alone which gives only a rough

estimate, and this has traditionally been achieved

by performing a monopulse estimation [11, 21].

Monopulse estimation has evolved from an intuitive

engineering technique based on the principle of

interferometry for angle estimation [11, 21] to a very

general principle for precisely estimating multiple

parameters by maximizing a generalized (sum) beam

output [16]. This maximization principle derives

from the maximum likelihood (ML) procedure and

estimates the unknown parameters Μ =( Μ 1 , Μ 2 , ::: )

of a single deterministic target (Swerling 0 radar

target [23]) from an array antenna output snapshot

[7, 13, 14, 16]

JÉROME GALY

LIRMM

France

ERIC CHAUMETTE

French Aerospace Lab

PASCAL LARZABAL, Member, IEEE

SATIE/CNRS

France

n a

Estimation of the direction of arrival (DOA) of a signal

source by means of a monopulse estimation scheme is one of

the oldest and most widely used high-precision techniques in

operational tracking systems. Although the statistical performance

of this estimation technique has been extensively investigated,

it has never been analyzed from the viewpoint of the joint

detection-estimation problem. As a consequence the historical

form of the (detector-angle estimator) solution has become the

usual form implemented in operational tracking systems and may

restrict their accessible performance. Indeed, the application of

the optimal detection theory reveals alternative (detector-angle

estimator) solutions providing other relevant trade-offs in

detection-estimation performances that are worth considering

to optimize the performance of the tracking system according

to the scenario. Both stochastic signal model (Rayleigh-type

signal source) and deterministic signal model (signal source

of unknown amplitude) have been investigated, leading to a

novel but equally simple (detector-angle estimator) solution

through which analytical statistical prediction has been derived.

Comparison of detection-estimation performances of new and

usual solutions is presented.

~z = ®~a ( Μ )+

¡!

n a represents the Gaussian receiver noise plus

external interference contribution, a ( Μ )represents

the one-way complex array antenna voltage pattern

depending on Μ ,and ® represents the complex

amplitude of the signal source (including power

budget equation, signal processing gains). The

maximization of the generalized beam output

can be efficiently solved [13, 14, 16] by a linear

approximation of the type

Μ =

¡!

Μ 0 + S (Re f~rg¡¹ )

@ ¡!

Μ 0 ) H ~z

d 2 (

Μ 0 ) H ~z

(1)

d 1 (

~r =

, :::

¡!

Μ 0 ) H ~z

w (

where

Manuscript received June 30, 2007; revised February 27, 2008,

August 21, 2008; released for publication November 15, 2008.

¡!

Μ 0 is an initial estimate of the desired

parameters,

2) S , ¹ are real correction quantities,

IEEE Log No. T-AES/46/1/935950.

¡!

Μ 0 ) H ~z ,

¡!

Μ 0 ) H ~z , :::g are,

Refereeing of this contribution was handled by M. Rangaswamy.

3) w (

Μ 0 ) H ~z and f !

d 1 (

d 2 (

This work was partially funded by the European Network of

excellence NEWCOM++ under the number 216715.

This was presented in part at the IEEE International Conference on

Acoustics, Speech, and Signal Processing, Montreal, Canada, April

2004 and at the European Signal Processing Conference, Vienna,

Austria, Sept. 2004.

Μ 0 ,and

respectively, the (generalized) sum beam at

a set of estimates of its derivatives

Μ 0 )

@Μ 1

~z , @w (

¡!

Μ 0 )

@Μ 2

@w (

~z , :::

;

Authors’ addresses: J. Galy, LIRMM (laboratoire d’Informatique,

de Robotique et de Micro-Electronique de Montpellier), 161 rue

ada 34392 Montpellier Cedex 5, France; E. Chaumette, ONERA,

DEMR, Chemin de la Huniere, Palaiseau Cedex, FR-91761, France,

E-mail: (eric.chaumette@orange.fr); P. Larzabal, SATIE/CNRS,

Ecole Normale Superieure de Cachan, 61 avenue du President

Wilson—94235 Cachan Cedex, France.

called (generalized) difference beams,

4) ~r is an estimator of the vector of the monopulse

ratios

@ ¡!

Μ 0 ) H a ( Μ )

¡!

d 2 (

Μ 0 ) H a ( Μ )

d 1 (

~r =

, :::

¡!

Μ 0 ) H a ( Μ )

0018-9251/10/$26.00 ° 2010 IEEE

w (

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where

In tracking mode, the unknown parameters are

descriptors of the target direction, e.g., the azimuth

¡!

Μ 0 =( Μ 1 , Μ 2 ) T

is the antenna (sum) beam looked-direction. The

importance of formula (1) is not so much the linearity

but the explicit relationship derived in [14, 16]

between the correction values S , ¹ and arbitrary

sum and difference beams, particularly adaptive

beams synthesized to cancel the contribution of the

interferences while maintaining low sidelobe patterns

[16]. The monopulse technique can produce accurate

parameter estimates for a single target, but it does not

improve the resolution of multiple targets [16]. For

any estimator of type (1), the mean and variance are

given by

( Μ 1 ) and elevation ( Μ 2 ) angles, and

Μ 0 + S ( m Re f~rg ¡¹ )

C Μ = SC Re f~rg S T

(2)

Fig. 1. Beams pattern of monopulse rectangular reflector antenna

(collocated sensors paired in phase with uniform sum excitation

for sum beam and linear odd difference excitation for difference

beam, beamwidth at ¡ 3dB=1deg).

which means that, given the sum and difference beams

(hence S and ¹ , see [16]), the performance of the

generalized monopulse estimates only depends on the

performance of the estimation of ~r .

As a consequence, in a multi-mode radar using

monopulse calculation in its tracking mode, to cope

with time budget constraints (surveillance mode),

each tracking loop (one per track), once initialized,

is maintained by processing the minimum necessary

information, i.e., the monopulse sum and difference

beams only. At each loop the processing principle

consists of a detection step followed by a monopulse

angle estimation (1) that feeds the tracking filter

(Kalman filter) which performs the position prediction

for the next measurement [11]. So far, the detection

step has always been performed with the sum beam

only, since this usual processing is the generalized

likelihood ratio test (GLRT) for an array antenna,

provided the looked-direction is the true direction of

the signal source [10]. However, within a tracking

loop, the looked-direction is only expected to be close

enough to the true signal source direction in order

to ensure the detection and maintain the loop, which

generally means it is within the beamwidth at ¡ 3dB

of the sum beam. Moreover, in a receiver noise only

scenario, the usual shapes of a sum and a difference

beam look like the ones displayed in Fig. 1.

These usual shapes clearly suggest that, as the

predicted looked-direction moves away from the true

direction of the signal source, the usual detection

processing based on the sum beam only may not lead

to the optimal detection performance accessible with

the given set of tracking beams.

Therefore, we propose in this paper to revisit the

detection and estimation steps of a tracking system

using monopulse calculation from the viewpoint of

the joint detection estimation problem (also called

the composite hypothesis testing problem [10]) and to

explore the various trade-offs in detection-estimation

performance available.

Let us recall that the joint detection estimation

problem stems from the optimal detection theory,

which shows that optimal detection rules are a

function of the exact statistics of the observations

[10]. Therefore optimal detection rules are generally

not realizable since they almost always depend

at least on one of the unknown parameters. A

common approach to designing realizable tests is to

replace the unknown parameters with estimates, the

detection problem becoming a composite hypothesis

testing problem (CHTP)/joint detection estimation

problem [10]. Each pair (detector-estimator)

defines a particular trade-off in detection-estimation

performance. Although not necessarily optimal for

detection performance, the estimates are generally

chosen in the maximum likelihood (ML) sense,

thereby obtaining the GLRT (see Section III).

The application of the optimal detection theory

(see Section II) [10] to a sensor array consisting of the

monopulse beams reveals, at least in a receiver noise

only scenario, alternative (detector-estimator) solutions

(see Sections III—IV) providing other relevant

trade-offs in detection-estimation performances that

are worth considering to optimize the performance

of the tracking system according to the scenario.

The proposed approach is novel, since the open

literature on this subject reveals a separate analysis

of detection and estimation [1—3, 7—9, 11, 12, 15—16,

20—22, 24, 26]. The use of the difference beams

has always been limited to the angle estimation part

of the problem, which has been first addressed for

decoupled parameter estimates (azimuth or elevation)

and covered by theoretical work on the formulation of

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m Μ = E [ Μ ]=

estimators of a single real monopulse ratio [12], then

of its statistical performance [8, 9]. The first analyses,

which included the usual sum beam detection test,

appeared as late as the 1990s [20, 24], as did their

extension to coupled parameter estimates (azimuth

and elevation) with correlated beams leading to

the generalized monopulse correction formula (1)

[13—16]. Even lately, authors [2, 3, 26] involved in

radar target tracking have introduced and characterized

an alternative statistical description of monopulse

parameters conditioned on the signal-to-noise ratio

(SNR) on the sum beam at each observation.

Due to the complexity of the analytical

computations (see Appendix, Subsection C), we have

restricted our analysis to the monopulse measurement

of a single angle by means of a two-beams array

(sum ( § ) and difference ( ¢ )), in the static situation

in which the signal source does not alter its relative

position with respect to the monopulse antenna during

I independent measurements at time t i , i2 [1, I ].

Therefore, in the following, the beams’ signal vector

consists of

Throughout the paper, for the sake of simplicity in

expressions,

1) most references to observation time t i , i2 [1, I ]

will be expressed implicitly by use of subscript i ,and

explicit dependence on Μ of g § , g ¢ , r , g , ::: will be

omitted;

2) § =( § 1 = § ( t 1 ), ::: , § I = § ( t I )) T , ¢ =( ¢ 1 =

¢ ( t 1 ), ::: , ¢ I = ¢ ( t I )) T , V =( § T , ¢ T ) T ;

3) R =1 =I

i =1 v ( t i ) v ( t i ) H =1 =I

i =1 v i v i ;

L ( m x , C x ) denotes a complex circular

Gaussian random vector of dimension L with mean

m x ,covariancematrix C x , and probability density

function (pdf)

p CN L ( x ; m x , C x )= e ¡ ( x¡m x ) H C ¡ 1

x ( x¡m x )

¼ L j C x

j :

II. OPTIMAL DETECTOR FOR MONOPULSE

ANTENNAS

Let us consider the detection problem related to

the observation model (3). Based on I independent

array snapshots v ( t 1 ), ::: , v ( t I ), we want to decide

whether to accept the null hypothesis (noise only)

H 0 , or to accept the alternate hypothesis (signal plus

noise) H 1 :

H 0 : v ( t )= n ( t ), H 1 : v ( t )= ® ( t ) g + n ( t ) : (4)

If no a priori probability ( P ( H 0 ), P ( H 1 )) of hypotheses

is available, then the Neyman-Pearson criterion is

often used for binary hypothesis testing [10, §3].

This process consists of searching for the subset of

observations D (detection event) which maximizes

the probability of detection P ( DjH 1 )( P D )foragiven

probability of false alarm P ( DjH 0 )( P FA ). The optimal

detector in the Neyman-Pearson sense is the likelihood

ratio test (LRT):

§ ( t )

¢ ( t )

g § ( Μ )

g ¢ ( Μ )

n § ( t )

n ¢ ( t )

v ( t )=

= ® ( t )

= ® ( t ) g ( Μ )+ n ( t )

( 3 )

n a ( t ) are the noise plus

external interference contribution at the beams’ output.

A remarkable feature of monopulse antennas

introduced by the present paper is that, in the

presence of a temporally and spatially white noise

n ( t )( C n = ¾ n I ), analytical expressions of the GLRT

and associated estimators, in particular that of the

monopulse ratio, can be derived for both deterministic

(Swerling 0 radar target [23]) and stochastic (Swerling

1 or Swerling 2 radar target [23]) signal models.

Moreover, we also show that both the GLRT and

monopulse ratio ML estimator (MLE) have identical

expressions for both signal models whether the

noise power ( ¾ n ) is an unknown parameter or not.

Unfortunately, the exact solution of the GLRT is

unpractical for establishing analytical results if I ¸

2. As a result, we develop two approximations of

the (detector, monopulse ratio estimator) pair (see

Section IV). The first one, called “Sum-Mosca,”

is the usual approximation. The second one,

called “Power-Mosca,” is a novel but equally

simple approximation that has been characterized

analytically (see Section V). Last, a comparison

of detection-estimation performance trade-offs of

new (“GLRT-MLE,” “Power-Mosca”) and usual

(“Sum-Mosca”) solutions is presented for a receiver

noise only scenario (see Section VI).

H 1

H 0 T:

In the problem at hand, the signal source does not

alter its relative position with respect to the array

during the I snapshots (static situation: g is constant)

and n ( t ) »CN

LRT:

p ( V j H 1 )

p ( VjH 0 )

2 (0, ¾ n I ). Therefore,

p CN 2 ( v i ;0, ¾ n I )= e ¡ ( I=¾ n )Tr( R )

p ( VjH 0 )=

( ¼ ( ¾ n )) 2 I : (5)

i =1

A. Stochastic Signal Model

1 (0, ¾ ® ) independent

identically distributed (IID) and independent from the

noise; therefore,

p CN 2 ( v i ;0, C v i )= e ¡I Tr( C ¡ 1 R )

p ( VjH 1 )=

( ¼ 2 j C j ) I

(6)

i =1

C = C v i = ¾ ® gg H + ¾ n I :

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4) x»CN

n a ( t ), n ¢ ( t )= d ( Μ 0 ) H !

where g § ( Μ )= w ( Μ 0 ) H a ( Μ ), g ¢ ( Μ )= d ( Μ 0 ) H a ( Μ )are

the amplitudes at angle Μ of the sum and difference

beams steered at the looked-direction, Μ 0 ,and n § ( t )=

w ( Μ 0 ) H ¡!

The amplitudes ® i are CN

Under these assumptions the LRT takes the form of

Therefore, a common approach to designing realizable

tests is to replace the unknown parameters by

estimates, the detection problem becoming a CHTP,

also called joint detection estimation problem [10, §6].

Although not necessarily optimal for detection

performance, the estimates are generally chosen in

the ML sense, thereby obtaining the GLRT. Let us

denote by ' j the unknown parameters vector under

hypothesis j . The GLRT for deciding whether to

accept H 0 or to accept H 1 is given equivalently by (12)

or (13):

LRT:

p ( VjH 0 ) = e ¡I Tr([ C ¡ 1 ¡ ( ¾ n I ) ¡ 1 ] R )

H 1

H 0

T 0

j C ( ¾ n I ) ¡ 1 j I

(7)

¯ ¯ ¯ ¯

2 H 1

H 0

g H v i

kgk

i =1

Then P FA and P D are given by (chi-square test):

¾ n

P FA = e ¡T=¾ n e I¡ 1

¾ n + ¾ ®

max ' 1

fp ( Vj ' 0 ) g = p ( V j ' 1 )

H 1

H 0 T (12)

P D = e ¡T= ( ¾ n + ¾ ® kgk 2 ) e I¡ 1

GLRT:

(8)

p ( Vj ' 0 )

kgk 2

max ' 0

x n

n ! :

f ln p ( Vj ' 0 ) g H H 0 ln T

(13)

where ' j stands for the MLE [10, §6] of the unknown

parameters under hypothesis j .

In the problem at hand, we not only search for

a realizable detection test, but we also search for

an estimator of the monopulse ratio r in order to

implement (1). Therefore, recasting (3) as

f ln p ( Vj ' 1 ) g¡ max

' 0

GLRT: max

' 1

e N ( x )=

n =0

B. Deterministic Signal Model

The amplitudes ® i are deterministic values which

a priori fluctuation law is unknown. Let

¾ ® = ® H ®

= 1

j® i

j 2

i =1

v i = ¯ i x + n i , ¯ i = ® i g §

x =(1, r ) T , r = g ¢

g §

Then,

(14)

p CN 2 ( v i ; ® i g , ¾ n I )= e ¡ ( I=¾ n )Tr( C )

p ( VjH 1 )=

( ¼ ( ¾ n )) 2 I

i =1

(9)

allows us to derive a GLRT based on the MLE of r ,

the possible unknown parameters becoming fr , ¾ n , ¾ ¯

C = 1

[ v i

¡® i g ][ v i

¡® i g ] H :

for the stochastic signal model, and fr , ¾ n , ¯ =

( ¯ 1 , ::: , ¯ I ) T g for the deterministic signal model.

In practice the reception beams § and ¢ of a

monopulse antenna are paired in phase so as to obtain

a real monopulse ratio. Thus, initially, Mosca [12]

has investigated the formulation of estimators of a

real monopulse ratio. Nevertheless, Asseo in [1] has

shown that it was worth considering the monopulse

ratio as a complex value where the real part supplies

the angular measurement, and the imaginary part

provides a simple detection test for the simultaneous

presence of several signal sources, i.e., for the validity

of the angular measurement [1, 9, 11, 20, 24]. In this

case, Im frg6 =0 although Im frg =0 in the case of a

single signal source. We follow Asseo’s approach

hereinafter and consider a complex monopulse ratio.

A remarkable feature of monopulse antennas is

that, in the presence of a temporally and spatially

white noise n ( t ), analytical expressions of the GLRT

and associated estimators, in particular that of the

monopulse ratio, can be derived for both deterministic

and stochastic signal models. Indeed, almost always,

one must resort to numerical maximization methods to

assess some of the unknown parameter estimators [25,

§8], of which the simplest implementation is a discrete

search on a fixed set of parameters possible values

i =1

Under these assumptions the LRT takes the form of

LRT:

p ( VjH 0 ) = e ¡ ( I=¾ n )Tr( C ¡ R ) H H 0

T 0

h P

(10)

g H

i =1 ® ¤

i v i

H 1

H 0

¾ n

2 [ I¾ ®

kgk 2 ]

Then P FA and P D are given by (real Gaussian law)

P FA = 1 ¡ erf( T )

1 ¡ erf

T¡

2 I¾ ®

kgk 2

(11)

P D =

erf( x )= p

¡t 2

dt:

III. GLRT FOR MONOPULSE ANTENNAS

As shown in the previous section, LRTs are

generally not realizable since they almost always

depend at least on one of the unknown parameters.

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p ( V j H 1 )

fp ( Vj ' 1 ) g

p ( V j H 1 )

(joint detection and estimation algorithm). Moreover,

GLRT derivations (see Appendix, Subsections A

and B for details) show that both the GLRT and

monopulse ratio MLE have identical expressions for

both signal models. The final form of GLRT (13)

only depends on whether the noise power ( ¾ n )isan

unknown parameter (15) or not (16), whereas the

MLE of r remains unchanged (17):

Tr( R ) 2 ¡ 4 j R j¼k§k 2 ¡k¢k 2 +2 j§ H ¢j 2 =k§k 2 and

CHTP:

k§k 2 H H 0 T , ˆ r = § H ¢

k§k 2 : (19)

The real part (Re f ˆ rg ) of this approximated form of

ˆ r ML was introduced by Mosca [12] as the solution

of “the problem of estimation of angle of arrival in

amplitude comparison monopulse radars,” but with

no reference to the associated detection test. A more

global approach is the theoretical approach disclosed

above. It leads to a symmetrical form (relative to

¢ and § ) of the CHTP and therefore suggests an

approximation based on a symmetrical criterion, such

as the correlation of the two channels under H 1 with

¾ ® large. In this case j§ H ¢j 2 =k§k 2 k¢k 2 ¼ 1. Then,

Tr( R )+

Tr( R ) 2 ¡ 4 j R j

2 ¾ n

GLRT:

H 1

H 0 T (15)

Tr( R )+

Tr( R ) 2 ¡ 4 j R j

2 ¾ n

GLRT:

H 1

H 0

T (16)

ˆ r ML = Tr( R )+

Tr( R ) 2 ¡ 4 j R j¡ 2 k§k 2

2 ¢ H §

Tr( R ) 2 ¡ 4 j R j¼ Tr( R ) ¡ 2 j R j= Tr( R )and

: (17)

Form (15) of GLRT is a constant false alarm rate

(CFAR) detector [10] which assesses the noise power

( ¾ n ) under H 0 and H 1 u sing the small est eigenvalue

of R : ¾ n = 2 (Tr( R ) ¡

CHTP: Tr( R )= k§k 2 + k¢k 2 H H 0 T

(20)

k¢k 4 + j§ H ¢j 2

¢ H § ( k§k 2 + k¢k 2 ) :

Tr( R ) 2 ¡ 4 j R j ). Whatever

the observation model considered, the performance

( P D versus P FA ) of CFAR detectors is poor for small

number of snapshots [10]. This is the reason why ¾ n

estimation is always performed at a different stage of

the processing, generally at the output of the matched

filter, where a large amount of samples is available.

Therefore, hereinafter, it is assumed that ¾ n can be

estimated precisely enough to be a known parameter

of observation model (14) leading to form (16) of

GLRT.

ˆ r 1 =

Under this form the CHTP becomes a simple

quadratic detector based on the use of the energy

available on the two reception channels. For

the stochastic observation model, the detection

performance ( P D versus P FA ) of this type of detector is

well known (4 I order chi-square tests) and is close to

that of the optimal detector (2 I order chi-square tests)

(8). However, the approximated form ˆ r 1 of ˆ r ML (17)

obtained is not a great deal simpler. It is simplified

when k§k 2 Àk¢k 2 . We then have again the form (19)

of ˆ r ML and (20) becomes

IV. PRACTICAL GLRT APPROXIMATIONS

k§k 2 + k¢k 2 H H 0 T , ˆ r = § H ¢

Except for case I =1,where

CHTP:

k§k 2 : (21)

j¢j 2 + j§j 2 H H 0 T , ˆ r ML = ¢

CHTP:

(18)

We designate hereinafter the various solutions

(16)—(17), (19), (20), and (21) of the CHTP

as GLRT-MLE, Sum-Mosca, Power-E1, and

Power-Mosca, respectively. First, a large number of

Monte-Carlo simulations not fully disclosed in the

present paper (see Figs. 5—9) and 6—11 for examples),

have shown that solution Power-Mosca (21) offers

better performances than solution Power-E1 (20) over

the complete mainlobe of beam § :same P D but lower

root mean square error (RMSE). Second, an analytical

formulation of the performance of the Power-Mosca

(21) solution has been derived and is introduced in the

next section.

the exact solution of the GLRT (16)—(17) is not

practical for establishing analytical results. Although

the computing power of today’s computers allows

a precise study of its performance through a

Monte-Carlo type simulation with a large number

of draws (see Section VI), it is always interesting

to be able to establish analytical results based on

approximated solutions which may be used as

calibration tools for this type of simulation (number

of draws necessary for a reliable estimation).

The usual approximation consists in restricting

the use of the difference beam ¢ to the computation

of ˆ r ML only, where the detection is achieved using

the sum beam § only. Under this assumption, the

samples which pass the detection test and participate

in the estimation process mostly belong to the sum

beam width and verify k§k 2 Àk¢k 2 .Inthiscase,

V. STATISTICAL PREDICTION

Assessing the statistical performances of the

CHTP requires a joint analysis of the performance

of the detector and the estimators of the unknown

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