Sample covariance based estimation of Capon algorithm error probabilities The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Richmond, Christ D. et al. “Sample Covariance Based Estimation of Capon Algorithm Error Probabilities.” IEEE, 2010. 1842–1845. Web. 11 Apr. 2012. © 2011 Institute of Electrical and Electronics Engineers As Published http://dx.doi.org/10.1109/ACSSC.2010.5757895 Publisher Institute of Electrical and Electronics Engineers (IEEE) Version Final published version Citable link http://hdl.handle.net/1721.1/69987 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Sample Covariance Based Estimation of Capon Algorithm Error Probabilities Christ D. Richmond∗,1 and Robert L. Geddes∗ Ramis Movassagh† and Alan Edelman† 1Senior Member, IEEE Dept. Mathematics, MIT {christ,geddes}@ll.mit.edu Cambridge, MA 02139 {ramis,edelman}@math.mit.edu Abstract— The method of interval estimation (MIE) provides a has MSE approximated by strategy for mean squared error (MSE) prediction of algorithm performance at low signal-to-noise ratios (SNR) below estimation E (θ− θ )2 1 threshold where asymptotic predictions fail. MIE interval error M probabilities for the Capon algorithm are known and depend on 2 1 − Pe(θ = θm|θ1) σ (θ1) the true data covariance and assumed signal array response. ASY MP (1) Herein estimation of these error probabilities is considered m=2 M to improve representative measurement errors for parameter + P (θ = θ |θ )(θ − θ )2 estimates obtained in low SNR scenarios, as this may improve e m 1 m 1 overall target tracking performance. A statistical analysis of m=2 Capon error probability estimation based on the data sample 2 where σASY MP (θ1) is the asymptotic MSE (e.g. based on covariance matrix is explored herein. the CRB or Taylor’s theorem), and Pe(θ = θm|θ1) is the error probability, providing a measure of the likelihood of the parameter estimation algorithm choosing an estimate due to a I. INTRODUCTION peak at θm when the true parameter value is θ1. Ultimately it is desired to explore the improvement in tracking performance Recent analysis of a maximum a posteriori penalty function MIE might provide for low SNR cases. The important enabler (MAP-PF) based tracker demonstrates the benefit of using for MIE is the interval error probability Pe(θ = θm|θ1) that the Cramer-Rao´ bound (CRB) to represent the measurement identifies the boundary between the high SNR / asymptotic error in bearing estimates [1]. Low SNR, however, renders region of performance and the low SNR region. the CRB an inadequate representation of measurement error In practice one must rely on some data based estimate resulting in poorer performance of the tracking algorithm of the error probability, as well as a data based estimate [2]. As an example, Figure 1 shows the root mean squared of the asymptotic MSE. The goal of this paper is to assess error (RMSE) performance of a bearing only Kalman filter the statistical properties of a sample covariance based (SCB) (KF) that received bearing measurements [from maximum- estimate of the Capon interval error probabilities. This will likelihood (ML)] under low SNR conditions, but used the CRB allow quantification of the reliability of such an approach, to represent their MSE. The KF predicts an RMSE that is more including e.g. how the number data samples affects the fidelity optimistic than the actual RMSE realized by the tracker. The of these probability estimates and what strategies are best for goal herein is to use an improved measure of MSE for low such practical estimation. SNR scenarios as a means to (i) reduce the realized RMSE of the KF, (ii) improve the tightness of KF prediction of RMSE, II. STATISTICS FOR SCB ESTIMATE OF CAPON and (iii) to make KF prediction more conservative in the low ALGORITHM ERROR PROBABILITIES SNR regime. Reference [4] provides the details of applying MIE to the The method of interval estimation (MIE) is a technique Capon algorithm [5]. Let the signal at angle θ have array re- for extending asymptotic/high SNR predictions like the CRB sponse v(θ) for an N sensor array; let the true data covariance be R and sample covariance matrix (SCM) obtained from L to lower SNR scenarios [3], [4]. MIE discretizes the integral for mean squared error (MSE) based on the structure of the data observations be denoted R. Consider the Capon algorithm underlying ambiguity function for parameter estimation. The evaluated at point θ: parameter estimate θ of the true parameter θ1, for example, 1 1 PCapon(θ)= · . L − N +1 vH (θ)R −1v(θ) (2) ∗ The authors currently work for MIT Lincoln Laboratory. Opinions, The two-point error probability facilitating application of MIE interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the United States Government. is given by † This work was sponsored by the National Science Foundation under grants Capon CCF-0829421 and DMS-1035400. Pe (θa|θb) = Pr [PCapon(θa) >PCapon(θb)] . (3) 978-1-4244-9721-8/10/$26.00 ©2010 IEEE 1842 Asilomar 2010 Define parameters P+(F ),P−(F ), and Iab(F ) such that: be obtained via numerical integration: H −1 H −1 v (θb)R v(θb) ± F v (θa)R v(θa) P (F ) = M ± 2 Capon (4) E Pe˜ (θa|θb,FΔ) = H −1 2 and Iab(F ) = F v (θa)R v(θb) ; ∞ M (11) P (F ) PCapon(θ |θ ,F) dF. FΔ e˜ a b and also define λ±(F ), lλ(F ), and Sλ(F ): 0 2 Capon λ±(F )=P−(F ) ± P+(F ) − Iab(F ), The exact probability density function for Pe˜ (θa|θb,FΔ) λ+(F ) (5) can likewise be deduced with some additional effort. lλ(F ) = − , Sλ(F ) = sign [λ−(F )] . λ−(F ) Regarding (6), define the N × 2 matrix V =[v(θa)|v(θb)]. One can estimate λ±(F ) and lλ(F ) via the eigenvalues of the It can be shown that the exact pairwise error probability for 2 × 2 matrix the Capon algorithm is given by [4] λ±( F )= P Capon(θ |θ )= e a b −F 0 H −1 −1 (12) (6) λ1,2 (V R V) . 0.5 ·{1+Sλ(1)}−Sλ(1) ·F[lλ(1),L− N +2] 01 l (F ) where F(x, J) is the cumulative distribution function (cdf) for Since results apply regardless of how λ is formed, the a special case of the complex central F statistic: condition number of (12) is chosen for convenience: arg max |λ(F )| J J−1 x 2J − 1 λ−,λ+ F(x, J) = · xk. l (F ) = − 2J−1 (7) λ (13) (1 + x) k + J arg min |λ(F )| k=0 λ−,λ+ While eq(6) is exact, it is somewhat challenging to analyze Let K = L − N +2; It can be shown that its pdf is given a data based estimate of this probability. Ignoring the apparent exactly by statistical dependence between the two points PCapon(θ), θ = (l +1)lK−2 a − la 2K−1 θa,θb, however, yields a simple approximation of the error l ∼ c · 1+ 1 2 λ 2K−1 (14) probability that is surprisingly accurate in many scenarios: (a1 − la2) la1 − a2 [P (θ ) >P (θ )] l ≥ 0 where the constants a1 and a2 are determined from R, V Pr Capon a Capon b χ2 vH (θ )R−1v(θ ) and F ; also where sign(a2)=−sign(a1) and the normalizing a L−N+1 > a a Pr χ2 vH (θ )R−1v(θ ) constant is given by b L−N+1 b b vH (θ )R−1v(θ ) (8) 22(K−1)(−a a )K Γ(K − 1 )Γ(2K − 1) = F a a ,L− N +1 1 2 2 H −1 c = √ . (15) v (θb)R v(θb) πΓ(K − 1)(a1 − a2) = P Capon(θ |θ ). e˜ a b This paper will focus on the error probability estimator in eq(10) given its relative simplicity. Analysis of a SCB estimate The assumed independence of points θa,θb allows use of the of eq(6) will be the subject of future analysis. Capon-Goodman result [6], leading to a ratio of indepen- dent chi-squared random variables that is known to be F - III. NUMERICAL EXAMPLES distributed. Note that (8) is parameterized by a ratio of Capon A. Statistics for Estimated Capon Error Probabilities spectral points θa,θb. Define the ratio of SCB Capon spectral Using eq(11) one can explore the fidelity of using a SCB estimates as the following random variable: approach to estimating error probabilities for the Capon al- H −1 v (θa)R v(θa) gorithm. Consider a bearing estimation scenario involving a FΔ = . (9) H −1 single planewave source and a set of signal bearing snapshots v (θb)R v(θb) 2 H x(l) ∼CN[0, I + σSv(θT )v (θT )], l =1, 2,...,L, for an N =18 The goal of this paper is first to assess the statistical properties element uniform linear array (ULA) with slightly less λ/2 of the following SCB estimator of the Capon error probability: than element spacing. The array has a 3dB beamwidth of 7.2 degrees and the desired target signal is arbitrarily placed Capon θ =90 Pe˜ (θa|θb,FΔ)=F [FΔ,L− N +1] (10) at T degrees (array broadside). The signal parameter search space of interest is defined to be θ ∈ [60◦, 120◦]. Figure and second to begin discussion of initial results for the 2 illustrates the Capon error probability for the peak sidelobe statistical analysis of a SCB estimate of the exact two-point as a function of element level SNR for L =1.5N,3N, probability found in [4].