OPTIMAL MONOPULSE TRACKING OF SIGNAL SOURCE OF UNKNOWN

AMPLITUDE

(1)(2)Eric Chaumette and (2)Pascal Larzabal

(1) Thales Naval France, Bagneux, France.(2) SATIE/CNRS, Ecole Normale Superieure de Cachan, Cachan, France

ABSTRACT

Estimation of the direction of arrival of a signal source bymeans of a monopulse antenna is one of the oldest and mostwidely used high resolution techniques [1]. Although the sta-tistical performance of this estimation technique has beenextensively investigated for decades, recent work [2] basedon an analysis of the problem from the point of view of op-timal detection applied to a two-sensors system, has shownthat the common solution (detector/estimator) restricts theaccessible performance. Indeed, changing the detector is nec-essary to optimize the overall performance. First derived inthe particular case of Rayleigh-type signal source, this ap-proach can be extended to the case of a signal source of un-known amplitude (including the non fluctuating case). Thepresent paper establishes analytical performance of both newand common solutions in that case.

1. NOTATION

X,x denote vectors (complex or real)

C,R,m denote matrices (complex or real)i, i, g, ni , r, . . . denote scalar values (complex or real)P () denotes a probabilityf() denotes a probability density function (pdf)F () denotes a cumulative distribution function (cdf)r denotes an estimator of r

CX

denotes the covariance matrix of random vector

X

IdI denotes Identity matrix with dimensions (I,I)

fI2(t, 2, 2) =

e

(t+2)2

2II1

(2t

2

)(t

)(I1)

eN (T ) =Nn=0

Tn

n!, II (t) =

1

2

20

et cos() cos (I)d

erf(x) =2

x0

et

2

dt

2. INTRODUCTION

A monopulse antenna (radar or telecom) determines the an-gular location of a signal source (radar target or telecomtransmitter) by comparing the returns from difference ()and sum () antenna pattern [1][3]. It is a particular solu-tion of the more general problem of finding the directionof arrival (DOA) of a signal source. As mentioned latelyin [2], the usual approach related to this particular solu-tion [3][4][5](ref.[3],[12],[20],[27]) limits the contribution ofthe difference channel to the estimation part of the problem,and therefore reveals an "historical" separate analysis of de-tection and estimation. A improved (detector, estimator)couple has been derived in [2] in the case of Rayleigh-typesignal source by applying optimal detection theory to themonopulse antenna. Nevertheless, if the Rayleigh case is of

fundamental interest for some applications (mainly radar ap-plications), most of signal sources (telecom for example) havedifferent amplitude fluctuation laws. If it seems unrealisticto try to solve the problem of optimal detection/estimationusing the statistics averaged over all possible observations,for every amplitude fluctuation law of interest, a suboptimalbut very general approach consists in solving the problemfor each observation, regarding the amplitude fluctuation lawas unknown (the most likely hypothesis when measuring anunknown actual signal source). Additionally, this character-ization may also be useful to support tracking performanceanalysis where monopulse measurement is a preliminary stepof nonstationary process as the Kalman Filter.

In order to allow straightforward comparison with resultsderived in [2], the present paper keeps the same analysisbreakdown of the optimal detection problem [6], includingequations numbering. Thus, we first formulate the optimaldetector the Neyman-Pearson criterion applied to themonopulse antenna, and its associated composite hypothe-ses testing problem, as certain parameters are unknown. Tosolve the composite hypotheses testing problem, we applythe GLRT method and establish the analytical expressionsof the detector and associated estimators, in particular thatof the monopulse ratio. Lastly, we develop two approxima-tions of the (detector, monopulse ratio estimator) pair. Thefirst is based on the common "historical" approach [4][7].The second, which we characterized analytically, proposes anappreciable improvement of the performances of the compos-ite hypotheses testing problem. While retaining a compara-ble estimation Root Mean Square Error (RMSE), it helpsachieve better on average detection performance character-istics, an improvement which is illustrated by an example.

3. PROBLEM FORMULATION

A common model for the receiver signal vector is:

v (t) =

((t) (t)

)= (t)g (0) +

n (t) (1)

where g () = (g(), g())T is the array response vector

(steering vector). It represents the array complex responseto a narrowband point source situated at an angle . Thecomplex envelope of the source is denoted by (t), and n (t)is an 2x1 additive noise vector. Consider the following de-tection problem:

H0 :v (t) = n (t)

H1 :v (t) = (t)g (0) +

n (t)(2)

Based on an observation consisting of I independent arraysnapshots v (t1),...,

v (tI), we want to decide whether toaccept the null hypothesis (noise only) H0, or to accept thealternate hypothesis (signal plus noise) H1.

3.1 Optimal Detector: LRT

If the pdf of the measurement is known under both hypothe-ses, the optimal detector - in the Neyman-Pearson sense [6] -

1445

is the Likelihood Ratio Test (LRT). In the problem at hand,the additive noise n (t) is a circular, zero mean, white (bothtemporally and spatially), complex Gaussian random vec-tor process with variance 2

n. The signal (t) represents

the complex envelope of the source (including power bud-get equation, signal processing gains) at time t and its apriory fluctuation law is unknown. The signal source doesnot alter its relative position with respect to the array dur-ing the I snapshots (static situation: 0 is constant). De-

note by

V =(

T ,

T)T

, where

= (1, ..,I)Tand

= (1, ..,I)T, the 2I dimensional observation vector

related to the I snapshots, then:

f(V | H0

)=

e

I

2n

Tr(CH0)

(22n)I

f

(V | H1

)=

e

I

2n

Tr(CH1)

(22n)I

CH0 =1

I

Ii=1

v (ti)

v (ti)H

CH1 =1

I

I

i=1

[v (ti) (ti)g (0)] [

v (ti) (ti)g (0)]

H

Under these assumptions the LRT takes the form of:

LRT =f(V | H1

)

f(V | H0

) = eI

2n

Tr(CH1CH0 )H1

H0

T (3)

and can be reduced to:

Re

{

g (0)H

[I

i=1

(ti)H

v (ti)

]} 2n

2

[g (0)

2

(Ii=1

| (ti)|2

)] H1H0

T

Denote by D the event of a threshold detection. Then, Prob-ability of False Alarm - PFA = P (D | H0) - and Probabilityof Detection - PD = P (D | H1) - are given by (real Gaussianlaw):

PFA =1 erf (T)

2(4a)

PD =

1 erf

(T

2I2

g (0)

2

)

2(4b)

where: 2=

H

I=

1

I

(I

i=1

|i|2

)

3.2 GLRT

For cases in which some of parameters are unknown, thedetection problem in (2) becomes a composite hypothesestesting problem (CHTP) [6]. Although not necessarily opti-mal, the GLRT (Generalized LRT [6]) is widely used in suchproblem. Lets denote by j the unknown parameters vec-tor under hypothesis j, the GLRT for deciding whether toaccept H0 or to accept H1 is given by:

GLRT =max 1

f(V | 1

)

max 0

f(V | 0

) = f(V |

1

)f(V |

0

) H1H0

T (5)

where j stands for the Maximum Likelihood Estimates(MLE) [6] of the unknown parameters under hypothesis j.In the problem at hand, the observation equation (1) maybe rewritten according to an equivalent form:

v (ti) = (ti)x +n (ti) (6)

where: (ti) = (ti) g(0),x = (1, r(0))

T, r() =

g()

g()

This is the "Monopulse Ratio" reformulation of the observa-tion equation. Under this formulation, the possible unknown

parameters are{2n, r,

}

,

= (1, ..,

I)T , and the final

form of (5) depends on whether the noise power (2n) is an

unknown parameter (7) or not (8) [8]:

GLRT

Tr

(R

)+

Tr

(R

)2 4

R2

2

n

H1

H0

T (7)

GLRT

Tr

(R

)+

Tr

(R

)2

4

R22

n

H1

H0

T (8)

where: R= 1I

I

i=1

v (ti)

v (ti)

H

In both cases:

r =Tr

(R

)+

Tr

(R

)2 4

R 222

H

(9)

Form (7) of GLRT is a constant false alarm rate (CFAR)detector which assesses the noise power (2n) under H0 and

H1 [8] using the smallest eighenvalue of R:

2

n=

Tr

(R

)

Tr

(R

)2

4

R2

As most of CFAR process, its performance (PD vs. PFA) ispoor for small number of snapshots. This is the reason why2n estimation is always performed at a different stage of theprocessing, generally at the output of the Matched Filter,where a large amount of samples is available. Therefore,hereinafter, it is assumed that 2n can be estimated preciselyenough to be a known parameter of observation model (6)leading to form (8) of GLRT. It is worth noticing that in caseof Rayleigh signal source (8) and (9) have the same form [2].

For completeness, lets mention that in the particularcase where

1=

2= .. =

I(non fluctuating case), ex-

pressions (7), (8), (9) and 2n are different [8]. From anoperational point of view, it is preferable not to use expres-sions derived in that case because of their specificity and thedu