Contribution à la caractérisation des performances des
problèmes conjoints de détection et d’estimation
Eric Chaumette
To cite this version:
Eric Chaumette. Contribution à la caractérisation des performances des problèmes conjoints de détection et d’estimation. Traitement du signal et de l’image [eess.SP]. École normale supérieure de
Cachan - ENS Cachan, 2004. Français. �tel-00132161�
HAL Id: tel-00132161
https://tel.archives-ouvertes.fr/tel-00132161
Submitted on 22 Feb 2007
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Modern Monopulse Tracking
Eric Chaumette1, Pierre Saulais1, Nathalie Colin1
1: Thales Naval France, 7-9 av. des Mathurins, 92 Bagneux, France
Abstract : Estimation of the direction of arrival (DOA) of
a signal source by means of a monopulse antenna is one
of the oldest and most widely used high resolution
techniques [1]. However, an analysis of the problem from
the point of view of optimal detection applied to a twosensors system allows to design a new (detector / angle
estimator) solution improving the overall accessible
performance. Performance comparison with standard
monopulse (detector / angle estimator) solution is
illustrated by Monte-Carlo runs.
Keywords: monopulse antenna, optimal detection theory
•
α( t ) represents the complex envelope of the source
(including power budget equation, signal processing
gains),
•
rx =
g∆
is called the monopulse ratio.
gΣ
The angular information θ is contained in the monopulse
g (θ)
ratio curve rx (θ ) = ∆
. Between the first sum pattern
g Σ (θ )
nulls, this curve/relation can be inverted, leading to the
deviation angle function θ = daf (rx ) , as shown on Figure
1.
1 Notation
x denotes a – column – vector (complex or real)
P(D) denotes a probability (of event D)
P(A | D) denotes a conditional probability
f(x) denotes a probability density function (pdf)
f(x | D) denotes a conditional pdf
r̂ denotes an estimator of r
E(x) denotes the expectation of random variable x
Var(x) denotes the variance of random variable x
θ
2 Introduction to standard monopulse DOA estimation
A monopulse antenna (radar or telecom) determines the
angular location of a signal source (radar target or telecom
transmitter) by comparing the returns from difference (∆)
and sum (Σ) antenna pattern [1]. Indeed, a common model
for the received signal vector at time t - after Hilbert
Filtering – is:
v( t ) =
Σ( t )
∆(t)
= α( t )
gΣ
g∆
+
n Σ (t)
= β( t ) x + n ( t )
n ∆ (t)
(1)
where:
T
•
g
β( t ) = α( t ) g Σ , x = 1, rx = ∆
gΣ
•
n ( t ) = (n Σ ( t ), n ∆ ( t ) )T represents an additive receiver
noise. In the problem at hand, the receiver noise is a
circular zero mean, white (both spatially and
temporally) complex Gaussian random vector process
,
[
with covariance matrix C n = E n ( t )n ( t ) H
•
]
= σ 2n Id ,
g Σ and g ∆ represents the one way real antenna
voltage pattern for each channel, at angle θ - offboresight angle - where a narrow band point source is
situated,
g Σ (θ)
g ∆ (θ)
g Σ (θ)
g ∆ (θ)
rx (θ)
g (θ)
rx (θ) = ∆
g Σ (θ)
θ
Figure 1: Monopulse Measurement Principle
If we focus on the static situation in which the signal
source does not alter its relative position (angle θ) with
respect to the monopulse antenna during I independent
observations at times (t1, t2, …, tI), Mosca [2] - in the
sixties - derived the exact maximum likelihood estimator
(MLE) of the monopulse ratio and proposed the following
more practical approximated form – for point source close
to boresight -:
r̂x = Re{ r̂} ≈ Re
ΣH∆
Σ HΣ
(2)
where Σ = [Σ( t1 ),..., Σ( t I )]T , ∆ = [∆( t1 ),..., ∆ ( t I )]T .
If a linear relation rx = kθ is assumed – which is true at
the vicinity of boresight, as shown on Figure 1 (see [1]
and [12] for a more detailed analysis) -, then statistical
RADAR 2004 - International Conference on Radar Systems
r̂
prediction of θˆ = x can be easily derived from statistical
k
prediction of r̂x . It's the reason why, in open literature:
•
the deviation angle function is generally reduced to a
linear function characterised by a Monopulse Slope,
•
most of statistical performance analysis are related to
r̂x (monopulse ratio statistical prediction).
Lastly, a monopulse tracking system is generally
completed – after various analogue and/or digital
processing dedicated to maximize the Signal-to-Noise
Ratio (SNR) – by a detection step. On this subject, the
open literature shows an “historical” separate analysis of
detection and estimation. Although, as early as the sixties,
monopulse ratio estimation was covered by theoretical
work on the formulation of its MLE (2), then of its
statistical performance [2][3][4], questioning on detection
step fell somewhat into oblivion. Indeed, the contribution
of the difference channel has always been limited to the
estimation part of the problem, since it was originally
introduced to overcome the Rayleigh resolution limit – the
well known beamwidth at (–3dB) - inherent to a single
sum antenna pattern. As a consequence, monopulse
tracking systems has kept since the same detection
scheme as surveillance systems: a threshold detection
applied to the receiver sum-channel,
Σ HΣ ≥ T
(3)
As the monopulse measurement is performed only on
detected samples, if we denote the event of a threshold
detection by D = Σ / Σ H Σ ≥ T , probability density
function, mean and variance must be computed taking
into account this observation selection criterion requesting
use of conditional statistics: f r̂x D , E r̂x D ,
{
(
}
(
)
)
(
)
Var r̂x D . First statistical predictions including the
detection test appeared as late as the nineties [5][6] and
have been completed lately [7]-[9].
Unfortunately, all these statistical predictions do not
exactly provide the relevant information, which is the
statistical predictions of DOA θ. Indeed as previously
mentioned, actual angle inference characteristics
(deviation angle function) are not linear functions but
a rx
rather closed to θ =
[12]. Therefore, exact
b 2 + rx2
( )
computations of E θˆ D
(
( )
and Var θˆ D
)
3 Improved monopulse processing
3.1) Theoretical Background
As far as we know, the “historical” monopulse processing
was never questioned until recent works [10][11] that
have shown the interest of investigating monopulse
processing from the point of view of optimal detection
theory [13] applied to a two-sensors system. In the
particular case of the monopulse measurement, the
detection problem is to decide, based on I independent
snapshots v( t1 ) , …, v( t I ) , whether to accept the null
hypothesis (noise only) H0, or to accept the alternative
hypothesis (signal plus noise) H1 when the observation
model is described by (1):
H 0 : v( t ) = n ( t )
H 1 : v ( t ) = β( t ) x + n ( t )
If the pdf of the measurement is known under both
hypothesis, the optimal detector – in the Neyman-Pearson
sense– is the Likelihood Ratio Test (LRT)
[
f [v( t 1 ),
f v( t1 ),
( [] ) ( [ ] )
(
[ ](
)
r̂ =
(4)
)
f r̂x D has only been derived so far for:
•
•
mixture of unknown amplitude signals and Rayleigh
type signals [4] and unconditional statistics (T = 0),
Rayleigh type signal source [5] and conditional
statistics.
H0
PFA = P(D H 0 ) , if D denotes the event of a threshold
detection. Nevertheless, the LRT is not directly usable as
some parameters are unknown (at least rx ). They must be
replaced by estimators and the detection problem becomes
a composite hypotheses testing problem (CHTP).
Although not necessarily optimal, the GLRT method
(Generalized LRT) is widely used in such problem. It
consists in replacing the unknown parameters by their
Maximum Likelihood Estimates (MLE). Application of
the theory to the monopulse observation model has been
detailed in the case of a point source with a Rayleigh [10]
or unknown [11] amplitude fluctuation law. One of the
meaningful results - under the hypothesis of a spatially
and temporally white noise of power σ 2n - is the
demonstration of the following common GLRT and
monopulse ratio MLE expressions for both amplitude
fluctuation laws:
λˆ max H1
≥T
(5)
σˆ 2n
request the
h θˆ (r̂x ) f r̂x D dr̂x
H1
that maximises the probability of Detection
PD = P(D H1 ) for a given probability of False Alarm
knowledge of f r̂x D , since for any function h (θˆ ) :
E h θˆ D = E h θˆ (r̂x ) D =
]
]
, v( t I ) H1 >
T
, v( t I ) H 0 <
()
( )2 − 4 R̂ − 2 Σ 2
Tr R̂ + Tr R̂
2∆H Σ
(6)
1 I
v( t i ) v( t i ) H ,
I i=1
•
R̂ =
•
σˆ 2n = λˆ min or σˆ 2n = σ 2n depending on whether the
noise power σ 2n is an unknown parameter or not,
•
λ̂ max and λ̂ min are the 2 eighenvalues of R̂ .
RADAR 2004 - International Conference on Radar Systems
Form (6) of r̂ is a generalization of Mosca result [2] for
complex monopulse ratio (see [3] and [5] for applications
involving the imaginary part). Form (5) of GLRT is a
constant false alarm rate (CFAR) detector which assesses
the noise power σ 2n using the smallest eighenvalue of R̂ .
As most CFAR process, its performance (PD vs. PFA) is
poor for small number of snapshots. This is the reason
why σ 2n estimation is always performed at a different
stage of the processing, generally at the output of the
Range-Doppler Matched Filter, where a large amount of
samples is available.
It is quite obvious that the exact solution of the CHTP forms (5) of the GLRT and (6) of the MLE - seems
unpractical for establishing analytical results. However,
two approximations of the (detector, monopulse ratio
estimator) pair can be derived from GLRT:
Σ
2
≥ T , r̂ =
σˆ 2n
Σ
H1
2
+ ∆
σˆ 2n
2
Σ ∆
H
Σ
(7)
2
ΣH∆
H1
≥ T ' , r̂ =
Σ
2
(8)
based on the standard "historical" approach (7) - point
source close to boresight - or on the correlation between
the two receiving channels (8) [10][11]. In the case where
σ 2n can be estimated precisely enough to be a known
parameter of observation model, (7) and (8) have been
characterized analytically [5]-[11] in terms of conditional
mean and variance. These analytical statistical predictions
show that expression (8) proposes an appreciable
improvement of the performances of the CHTP. While
retaining a comparable estimation Root Mean Square
Error (RMSE), it helps to improve overall detection
performance over the main lobe of the sum channel. A
restriction on the interest of these results is the underlying
assumption of a linear deviation angle function in order to
be extended to angle statistical prediction (see §1).
Unfortunately, this is a quite general restriction, as actual
forms of deviation angle functions combined with the
difficulty of evaluating f r̂x D prevent from performing
(
)
analytical computation.
Therefore, a step forward consists in not focusing on
statistical predictions but in investigating the DOA
estimation performance of exact (5)-(6) or approximate
(7)(8) solutions of the GLRT scheme through a MonteCarlo type simulation with a large number of draws, as it
is possible nowadays with the computing power available.
()
r̂ =
According to section 3.1, the recommended monopulse
processing is the following (detector, estimator) pair:
()
Tr R̂ +
( )2 − 4 R̂
Tr R̂
σˆ 2n
H1
≥T
(9)
(10)
2∆H Σ
θˆ = daf (r̂x ),
r̂x = Re{ r̂}
(11)
where σ̂ 2n is a noise power estimator.
This processing can be easily implemented in actual
single floating point DSP, what could have been a major
issue a few decades ago.
We shall designate hereinafter the various solutions (910), (7) and (8) of the CHTP as “glrt”, “mosca sum” and
“mosca power” respectively.
3.3) Generalisation
The problem is to find out if expressions (9) and (10) can
be the solution of the GLRT method for any kind of
amplitude fluctuation law. Unfortunately, some basic
computations leads rapidly to a negative answer. Let’s
consider the simplest case where the noise power σ 2n is a
known parameter ( σ 2n = 1 ). In that case, all unknown
parameters of the LRT belongs to hypothesis H1 and the
GLRT method now only consists in assessing them in the
ML sense. Assume that β( t1 ) , …, β( t I ) are I
independent identical random variables with a known pdf
f (β ) , independent from the noise. Then the pdf of
observation model (1) is:
[
[
]
] ()
, v( t I ) H1 = f v( t1 ),
f v( t1 ),
, v ( t I ) r , β f β dβ
and the MLE of r – the only unknown parameter left - is
solution of:
[
∂ f v( t 1 ),
, v ( t I ) H1
∂r
H
()
]=0
(12)
I
where: f β = f [β( t1 ),
, β( t I )] = ∏ f (βi )
[
−
f v( t1 ),
i =1
]
, v( t I ) r , β =
e
I
i =1
v ( t i ) −β( t i ) x
2
π 2I
After some straightforward calculi, (12) can be rewritten
as:
[
I
r=
β( t i ) H ∆( t i )f v( t1 ),
i =1
I
i =1
3.2) Improved Monopulse Processing
( )2 − 4 R̂ − 2 Σ 2
Tr R̂ + Tr R̂
2
[
β( t i ) f v( t1 ),
] ()
, v ( t I ) r , β f β dβ
] ()
, v ( t I ) r , β f β dβ
which is an implicit equation in r whose solution has
obviously no reason to be the same whatever the form of
f (β ) . Therefore (9) and (10) can not be the general
solution of the GLRT method applied to monopulse
measurement.
However, the GLRT method originates from asymptotic
ML Estimates properties [13] which are - under
reasonably general conditions - unbiasness and efficiency
(estimator variance reaches the Cramer Rao lower bound),
RADAR 2004 - International Conference on Radar Systems
when the number of independent observations tends to
infinity. A thorough implementation of the GLRT method
consists in finding ML estimates not only of the
monopulse ratio r but also of the parameters ω of the
performance of “glrt” and “mosca power” solutions,
which makes statistical prediction of “mosca power”
solutions a very attractive problem in future.
amplitude fluctuation law f β ω . Actually the use of (10)
5 Conclusion
( )
as MLE of r for any given value of β = [β( t1 ),
, β( t I )]T is
equivalent to look for estimators of r (and β ) that are
unbiased for each value of the nuisance parameter β ,
whereas the true GLRT method looks for estimator of r
(and ω ) that are unbiased only over the set of possible
values of the nuisance parameter β . Such estimation
scheme can be related to Miller and Chang lower bound
[14] (a modified Cramer-Rao lower bound) as optimality
criterion. Thus, the use of (9) and (10) whatever the
amplitude fluctuation law can be regarded as a Modified
GLRT method and therefore as a reasonable general
(detector, estimator) solution pair of the CHTP as shown
in next section.
This paper, generalizing results established lately in [10]
and [11], emphasizes the existence of a better (detector,
estimator) solution pair than the “historical” one for the
monopulse CHTP. It provides a significant improvement
of overall performance and proves to be robust to
amplitude fluctuation law type. In addition to the expected
impact on the future implementation of monopulse
antennas and tracking performance, it contributes to
illustrate the often unacknowledged or underestimated
interaction between the components of the (detector,
estimator) solution pairs of the CHTP.
6 References
[1]
4 Performance Comparison
[2]
As an example of performance comparison, we consider
the multifunction Radar case tracking a signal source
which amplitude fluctuation law is either of Swerling 0,
Swerling 2 or Swerling 4 type. Due to time budget
constraint, the maximum number of observations
available per target is generally 2 (I = 2). The noise power
is assumed to be known. A likely probability of false
alarm for such radar mission is PFA = 10-4. Whatever the
Swerling case, the Signal-to-Noise-Ratio (SNR) is
matched to obtain PD = 0.9 when signal source is on
boresight and detected on sum (Σ) channel only. The
monopulse antenna model corresponds to a rectangular
surface sum antenna (Σ, 1° beamwidth) with a plane
surface uniform current distribution associated with an
appropriate difference beam (∆, linear odd current
distribution).
All simulation results have been gathered on a single page
to offer an overview. They are divided in 2 figures (Figure
2 and 3) consisting of 3 plots, one per Swerling case, from
SW0 to SW4 (top to bottom). On each plot, the
performance of the 3 solutions “glrt”, “mosca sum” and
“mosca power” of the CHTP are displayed.
Figure 2 and Figure 3 depicts respectively the variation of
PD and normalized RMSE of the angle estimator (11)
within the main sum beam [− θ3dB , θ3dB ] . In Figure 2
“Theo” stands for Theoretical, i.e. assessed using
analytical formulas.
All PFA measurements have been performed on 109
independent trials. All PD and RMSE measurements have
been performed on 106 independent trials.
The two figures illustrates the on average superiority of
GLRT solution (9)-(10) over Standard solution (7): almost
equal RMSE and improved on average PD. This result is
also obtained for non Gaussian pdf of observations (SW4
mixture) that confirms the robustness of the Modified
GLRT method applied to monopulse measurement.
Another valuable result is the almost equivalent
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
S.M. Sherman: "Monopulse Principles and Techniques",
Artech House, 1984
E. Mosca: "Angle estimation in amplitude comparison
monopulse", IEEE Trans. AES. Vol. 17, March 1969.
I. Kanter: "Multiple Gaussian targets, the track-on –jam
problem", IEEE Trans. AES, vol. AES-13, pp 620-623,
1977
I. Kanter: "The probability density function of the
monopulse ratio for N looks at a combination of constant
and Rayleigh targets”, IEEE Trans., IT-23, pp 643-648,
1977
B-E Tullsson: "Monopulse tracking of Rayleigh targets,
a simple approach", IEEE Trans., AES-27, May 1991
A-D. Seifer: "Monopulse-radar angle tracking in noise or
noise jamming", IEEE Trans. AES, vol. AES-28, pp
622-637, 1992
E. Chaumette and P. Larzabal: "Monopulse Tracking of
Signal Source of Unknown Amplitude using Multiple
Observations", submitted to IEEE Trans. AES, sept 2003
E. Chaumette and P. Larzabal: "Monopulse-Radar
Tracking of Swerling 3-4 Targets using Multiple
Observations.", submitted to IEEE Trans. AES, Nov
2003
E. Chaumette and P. Larzabal: "Statistical Prediction of
Monopulse Tracking: General Expressions", submitted
to IEEE Trans. AES, Nov 2003
E. Chaumette and P. Larzabal: "Optimal Detection
theory applied to monopulse antennas", ICASSP 2004,
2004
E. Chaumette and P. Larzabal: "Optimal Monopulse
Tracking of Signal Source Of Unknown Amplitude",
submitted to EUSIPCO 2004
T. E. Connoly: "Statistical Prediction of Monopulse
errors for fluctuating targets", IEEE Radar conference
1980
H-L Van Trees : “Detection, estimation and modulation
theory, Part 1”, New York Wiley, 1968
R.W. Miller, C.B. Chang: "A modified Cramer-Rao
Bound and its Apllication", IEEE Trans., IT. vol 24,
p398-400, May 1978
RADAR 2004 - International Conference on Radar Systems
SW0, PFA = 1e-4, I = 2, NumSamp = 1e6
SW0, PFA = 1e-4, I = 2, NumSamp = 1e6
1
0.55
RMSE (Linear, unit = Sum Beamwidth)
0.9
Probability of Detection
0.8
0.7
0.6
0.5
0.4
0.3
Neyman-Pearson Theo
Mosca Sum Theo
Mosca Sum
Mosca Power Theo
Mosca Power
GLRT
0.2
0.1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.5
0.45
0.4
Mosca Sum
Mosca Power
GLRT
0.35
0.3
0.25
0.2
0.15
0.1
0.6
0.8
0.05
-1
1
-0.8
0.9
0.5
RMSE (Linear, unit = Sum Beamwidth)
0.55
Probability of Detection
0.8
0.7
0.6
0.5
0.4
Neyman-Pearson Theo
Mosca Sum Theo
Mosca Sum
Mosca Power Theo
Mosca Power
GLRT
0.2
0.1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0.15
0.05
-1
1
-0.8
RMSE (Linear, unit = Sum Beamwidth)
Probability of Detection
0.5
0.4
Neyman-Pearson Theo
Mosca Sum Theo
Mosca Sum
Mosca Power Theo
Mosca Power
GLRT
0.2
0.4
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
SW4, PFA = 1e-4, I = 2, NumSamp = 1e6
0.6
0
-0.6
Normalised Angle Deviation (unit = Sum Beamwidth)
0.7
-0.2
1
0.1
0.8
-0.4
0.8
0.2
0.5
-0.6
0.6
0.25
0.9
-0.8
0.4
Mosca Sum
Mosca Power
GLRT
0.3
0.55
-1
0.2
0.35
SW4, PFA = 1e-4, I = 2, NumSamp = 1e6
0.1
0
0.4
1
0.2
-0.2
0.45
Normalised Angle Deviation (unit = Sum Beamwidth)
0.3
-0.4
SW2, PFA = 1e-4, I = 2, NumSamp = 1e6
SW2, PFA = 1e-4, I = 2, NumSamp = 1e6
1
0.3
-0.6
Normalised Angle Deviation (unit = Sum Beamwidth)
Normalised Angle Deviation (unit = Sum Beamwidth)
0.45
0.4
Mosca Sum
Mosca Power
GLRT
0.35
0.3
0.25
0.2
0.15
0.1
0.6
0.8
Normalised Angle Deviation (unit = Sum Beamwidth)
Figure 2: Probability of Detection
1
0.05
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Normalised Angle Deviation (unit = Sum Beamwidth)
Figure 3: Normalized RMSE of angle estimator
RADAR 2004 - International Conference on Radar Systems
1