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 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. ! "# $% &''( , * * , * -* /0 +3 -- 1 / %$7 3 1 ) " ! 5 -. 7 ; * + . --2 --2 4 4 / " 7:(&<= ,6 8 " 9'&: 7 4 !"#"$% & (')*+' ,-.0/ 13254)6879;:0:<9 ]^45_Q/A4)`;a!45-Ob4GcE4)` =!>)[email protected][email protected]?QP)RSBSTEUV?AW5P)BIX5YGTEKZM[?AP)R;B8\ [email protected][email protected])m)BAMonkBIX)M[BST5p?SRQ>)?AP)TEKNMO?AP)RqB8\ r sZtEuwvxrZy!u zG{}|~o|<~Qw|Q A~<|~QNO0g~ S5N{}~A5{CIq0O! 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JU = [email protected] AcHJ#/G &U) Q,.*J0<''#464!<B%&-64!<HJ,!1,&-B$.0<!1645.7D'GCHUB$"& (4!1#4H+, BH+,"HJ$.5"0 #4'.-)Adgip(/& ,"#cAA}")8:HJ0cQK}Ja|b)$D$cA? .ba ?D| ?U)| JJ = [email protected] Xag c"}U'!<Io'()H+,DH+$"5.01#4'ag(/& 7"&J(K&J,D*+01'64(/&-G/hU!1,.* !1,%,.HU! #4'QHU( ,"H+! #'Z&JBB! ,"*E-)UdgiZ(/&-,.# cJ}~) 8UHU0oc:QK}:a?JU)U$"$:?U?ba .JU)| JU? = [email protected] YZafc!10<0 ' 6)Qc cKA<J ¡Q¢£A-f¤¥U¦-<§J¨R¡S© ª§+44«-< ¬41§+ « ¬o®A««°¯Q§J4<±§+U¬/J "²>³Z«4¬4<´ -Sµ§J.§J¶"·.1¸4«xµ« ¸4·"4«¤ ¨«.¬4¸ ©-¡ ¹gººiº3»/-E¸¼½QºK¥~¡/¾U§Uo¼ ½QºK¥J¤4¿ ÀD¡+¶D¶ ÁU¿+¿b¤Á-Â:á:Ä-Å+ÅU¿ Æ Ç-È ¯X¤gÉi¼"³K-»Z4««¸¡i©/ÊQ«¬4«C´¬41§+~¡«C¸4¬41¨% ¬41§+3-"²¨S§U²D·"< ¬41§+ ¬4¦"«§+fËU¡.ÌiJ4¬QÍ ÎD¡.Ï«®Ð¢i§+4Ñ~¡UÒt <«ËU¡Í À Ç Ã Æ ÓbÈ ºA¼ª¦"J·.¨«¬4¬4« J.²RÌZ¼EÉZJ4± Ô" ¡Õ©µ3§J.§J¶"·.1¸4«m»Z/ ´ÑU<.ÖM§+× ¥D Ö+-E¥§+·D4´ «§+×DØDÑ..§b® ½Q¨¶.<1¬4·.²D«K·"¸4 "ÖQµ·.<¬41¶.1«lÙXÔD¤ ¸4« f¾:-¬4 §U.¸ ©-¡"¸4·.Ô"¨ ¬4¬4«C²R¬4§¹gººiº;»Z/ D¸¼E½QºK¥n¡E¸4«¶D¬Ä ÅJÅU¿ Æ Ã È ºA¼ ª¦" ·"¨«¬4¬4«J¡_© ÙQ¶.¬4 ¨%J Êm«¬4«´¬4 §U>¬4¦.«§+fËÚ-¶.¶" <«²>¬4§ ¨§+"§J¶.·" ¸4«Û-.¬4«."J¸Üݬ4¦D«p·".ÑU.§-®Þ-¨S¶" 1¬4·.².«p´ -¸/«U©-¡ µ« ¨§x»AÏXßiàJKáQ¥.àJºiÌáâ¤jÅU¿JàJÅ+ÂD¡»A¦" 1«¸_Ïm¾:JQß"/ "´«J¡ AJÖ+D«·.ã¡DßE/-"´«+¡"ÄJÅ+ÅJ 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
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