M ODELING AND M ISMODELING IN Centrale Supélec – June 2015 R ADAR A PPLICATIONS : P ARAMETER E STIMATION AND B OUNDS Maria S. Greco Dipartimento di Ingegneria dell ’ Informazione University of Pisa (Italy)
Outline of the talk The radar scenario: covariance matrix estimation in non-Gaussian clutter The Misspecified Cramér-Rao Bound (MCRB) and the Huber Bound Matrix estimation for CES distributed disturbance Examples and conclusions 2
Radar scenario • Radar systems detect targets by examining reflected energy, or returns, from objects • Along with target echoes, returns come from the sea surface, land masses, buildings, rainstorms, and other sources. This disturbance is called clutter •Much of this clutter is far stronger than signals received from targets of interest • The main challenge to radar systems is discriminating these weaker target echoes from clutter 3 • Coherent signal processing techniques are used to this end
Radar scenario: the clutter Radar clutter is defined as unwanted echoes, typically from the ground, sea, rain or other atmospheric phenomena. These unwanted returns may affect the radar performance and can even obscure the target of interest. Hence clutter returns must be taken into account in designing a radar system. The function of the clutter model is to define a consistent theory whereby a physical model results in an analytical model which can be used for radar design and performance analysis . 4
Radar scenario: the clutter • In early studies, the resolution capabilities of radar systems were relatively low, and the scattered return from clutter was thought to comprise a large number of scatterers • From the Central Limit Theorem (CLT) , researchers in the field were led to conclude that the appropriate statistical model for clutter was the Gaussian model (for the I & Q components), i.e., the amplitude R is Rayleigh distributed) • In the quest for better performance, the resolution capabilities of radar systems have been improved • For detection performance, the belief originally was that a higher resolution radar system would intercept less clutter than a lower resolution system, thereby increasing detection performance • However, as resolution has increased, the clutter statistics have no longer been observed to be Gaussian, and the detection performance has not improved directly. The radar system is now plagued by target-like “spikes” that give rise to non-Gaussian observations 5
Radar scenario: the clutter • These spikes are passed by the detector as targets at a much higher false alarm rate (FAR) than the system is designed to tolerate The reason for the poor performance can be traced to the fact that the • traditional radar detector is designed to operate against Gaussian noise • New clutter models and new detection strategies are required to reduce the effects of the spikes and to improve detection performance 6 Spikes in the horizontally polarized Spikes in the vertically polarized sea sea clutter data clutter data
Radar scenario: the clutter Empirical studies have produced several candidate models for spiky non- Gaussian clutter, the most popular being the Weibull distribution, the K distribution, the log-normal, the generalized K, the Student-t (or Pareto), etc. 10 2 Weibull Lognormal Measured sea clutter data K 10 1 GK (IPIX database) LNt IG 10 0 Histogram pdf the Weibull , K, 10 -1 log-normal etc. have heavier tails than the 10 -2 Rayleigh 10 -3 0 0.2 0.4 0.6 0.8 1 7 Amplitude (V)
Radar scenario: compound-Gaussian model In general, taking into account the variability of the local power , that can be modeled itself as a random variable, we obtain the so-called compound-Gaussian model, very popular in clutter modeling, where 2 2 r r = p r ( | ) exp u r ( ) 8 = p r ( ) p r ( | ) p ( ) d ; 0 r 0 According to the CG model: = z n ( ) ( ) ( ) n x n = x n ( ) x n ( ) jx n ( ) I Q Texture: non negative random Speckle: complex Gaussian process, takes into account the 8 process, takes into account the local mean power local backscattering
Compound-Gaussian model and CES Particular cases of CG model (amplitude PDF): 4 4 4 K (Gamma texture) = p ( ) r r K r u r R 1 1 2 b b 2 GK (Generalized 2 br r = b 2 p ( ) r exp d Gamma texture) R 0 l ( 1) h t-Student = h 2 p ( ) r 2 r 1 r u r ( ) or Pareto R l c 1 c r W, Weibull = c p ( ) r exp ( r b ) u r R b b 9 The CG model belongs to the family of Complex Elliptically Symmetric (CES) distributions
Radar scenario: adaptive detection Any radar detector should be designed to operate in an unknown interference environment, i.e. it should adaptively estimate the disturbance covariance matrix R = 2 S (i.e. power 2 and scatter matrix S ). Detectors that estimate adaptively (on-line) the disturbance covariance matrix are named “ adaptive detectors ”. When the interference environment is a-priori unknown, several approaches are possible. The most commonly used are: 1) Assume that the disturbance is white, i.e. R = 2 I , and implement the non-adaptive detector which is optimal for that scenario (clearly, it will perform suboptimally in correlated disturbance). 2) Model the clutter as an autoregressive (AR) process and estimate the clutter covariance matrix by estimating the AR parameters [see e.g. S. Haykin and A. Steinhardt. Adaptive Radar Detection and Estimate . John Wiley and Sons, 1992]. 10
Radar scenario: adaptive detection 3) Assume that a set of secondary data x i = d i , i=1,…, M is available, free of target signal components, sharing the same correlation characteristics as the disturbance in the Cell Under Test (CUT), then estimate 2 and S from M > N secondary data vectors. This scenario is usually referred to as homogeneous environment and this is the case we treat here. Secondary data are usually obtained by processing range gates in spatial proximity with the CUT. The data from the CUT are termed primary data. Problem : we want to estimate the scatter matrix ( shape matrix ) S of Complex Elliptically Symmetric (CES) distributed clutter for adaptive radar detection 11
Radar scenario: disturbance matrix estimation A fundamental assumption underlying most estimation problems is that the true data model and the model assumed to derive an estimation algorithm are the same, that is, the model is correctly specified. If the unknown parameters are deterministic, a typical estimator is the Maximum Likelihood one and its performance asymptotically tends to the Cramér-Rao Lower Bound (CRLB). But for some non-perfect knowledge of the true data model or for operative constraints on the estimation algorithm (i.e. easy of implementation, computational cost, very short operational time) there can be a mismatch between assumed and true data model. In this case, which is the bound on the performance of ML and other estimators? Problem : we want to estimate the scatter matrix S of Complex Elliptically Symmetric (CES) distributed clutter in presence of 12 model mismatch
Parameter estimation and bounds Most treatments of parameter bounds assume perfect knowledge of the data distribution. When the assumed distribution for the measured data differs from the true distribution, the model is said to be misspecified . While much attention has been devoted to understanding the performance of Maximum Likelihood (ML) and Least Squares (LS) estimators under model misspecification (see e.g. [Hub67], [Whi81], [Fom03]), little consideration appears to have been given to the concomitant bounds on performance . 3
Parameter bounds under misspecified models Problem : Given a parameter vector , we are interested in finding a lower bound on the mean square error (MSE) of any estimator, based on the assumed probability density function (pdf) f X ( x ; ) in the presence of misspecified data model misspecified bound (MCRB) = = p ( ) x true pdf of , x f ( ; ) x θ assumed pdf, p ( ) x f ( ; ) x θ X X X X Choice of score function The tightest bounds are obtained via a choice of score function with two properties [McW93]: (1) zero mean; (2) dependence on the sufficient statistic T ( x ) for estimating . 14
Parameter bounds under misspecified models First, we define a score function by subtracting to the score function used to derive the classical CRLB its possibly non-zero mean value : s x ( ) ln f ( ; ) x θ E ln f ( ; ) x θ θ θ X p θ X = ln f ( ; ) x θ D p f θ X θ θ where we expressed the 2nd term as a function of the Kullback- Leibler (KL) divergence between the assumed density and the true density : p p ( ) x = D p f E ln ln X p ( ) x x d p X f f ( ; ) x θ X = = D p f 0 f ( ; ) x θ p ( ) x 15 X X
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