Sample Covariance Based Estimation of Capon Algorithm Error Probabilities

Sample covariance based estimation of Capon algorithm error probabilities

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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)

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]. The pdf of FΔ is derived in [4] and and 6N. Note that the approximation in eq(8) is illustrated includes the affects of statistical dependence between points by the blue circle curves and is very accurate as it falls Capon θa,θb. Thus, the desired moments of Pe˜ (θa|θb,FΔ) can nearly right on top of the exact error probability in eq(6)

1843 illustrated by the solid blue curve. The mean of the SCB asymptotic value based on MIE and defined by equation (1) Capon estimate Pe˜ (θa|θb,FΔ) is illustrated by the red curve and (accounting for both local and global errors). its standard deviation by the black dashed curves. Clearly, a The Kalman filter is run 5000 times, each time producing a small bias persists even with increasing sample support L. filtered target bearing and bearing rate which are compared to Analysis shows that this bias is the result of a SNR loss due the true trajectory to obtain average errors. The sample error to reliance on an estimated covariance matrix. Surprisingly the covariance from the 5000 Monte Carlo trials is compared to value of this bias is very weakly dependent on array size N the Kalman filters own predicted error covariance. and sample support L. It is approximately ∼2dB for many cases. When one corrects for this bias, a much better fit of the Capon C. Kalman Filter RMSE Performance statistics for Pe˜ (θa|θb,FΔ) to the true value is obtained. This improved fit is illustrated in the lower right image of Herein a simple N =18element horizontal line array that Figure 2 for the L =3N case. The KF generates an estimate of measures data from a distant target near array broadside is 2 the signal power level σS [2] that can be used to help apply this considered. The target bearing trajectory used for simulation correction in practice. It is also noteworthy that the variance is shown in Figure 3. The bearing estimates are obtained of the SCB estimates of probability decrease with increasing via the Capon algorithm where each estimate uses L =2N SNR and increasing sample support L. Consequently, one can training samples in the covariance estimate. Figure 4 illustrates expect increasing accuracy as the SNR increases toward the the resulting RMSE performance of the KF for varied SNR threshold SNR. levels. The threshold SNR (element level) for this simulation is ∼−8dB. The first plot in Figure 4 shows the KF RMSE B. Framework for Kalman Filter Sensitivity Analysis when the signal level is above threshold SNR. While only How does the KF perform if the error bars (MSE estimate) visible in the other images there are four curves shown. The for the bearing measurements are optimistically small but the solid curves show the true bearing RMSE realized by the KF bearing measurement is rather erroneous? One can explore and the dashed lines show the KF’s prediction of the bearing the sensitivity of the KF for varied levels of accuracy for RMSE. The red lines are obtained when the MSE estimates the MSE estimates it’s given for the bearing measurements accompanying the Capon bearing estimates derive from Taylor via simulation. Figure 3 illustrates via a block diagram the series [7] (asymptotic), and the blue lines are obtained when framework for the KF analysis via simulation. First a target the MSE estimates derive from MIE (non-asymptotic). The bearing trajectory is generated based on a Markov model for second and third images in Figure 4 shows the performance the bearing rate. Simulated array data can then be generated when the signal level falls below threshold. It is noteworthy based on this trajectory using one’s choice of target fluctuation that for these low SNR signals the true KF bearing RMSE models. The Capon algorithm (or other, e.g. ML) can then has been reduced by using MIE and that the tightness of fit be applied to generate bearing estimates over time that can between the true RMSE and the KF predicted RMSE improves be subsequently fed into the KF. One can then examine the significantly. Lastly, use of MIE results in a KF predicted error resulting RMSE of the KF as one varies the accuracy of the that is more conservative at low SNRs as desired. MSE estimates that accompany the bearing measurements fed to the KF. IV. CONCLUSIONS The state vector within the Kalman filter is a three state system composed of: target bearing, target bearing rate and This paper explores the notion of using MIE to improve the target received power. The target bearing rate is assumed to accuracy of MSE estimates attached to bearing measurements have the form of a first order Markov random variable, while fed to a Kalman filter (KF) tracker. MIE requires calculation of the target received power is modeled as a random bias. The error probabilities; therefore, a simple SCB estimate of these target bearing is just the integral of the bearing rate. These probabilities was proposed and analyzed. It was found that states are first represented in the form of a set of continuous an SNR loss of ∼2dB was present and nearly invariant to stochastic differential equations that are easily transformed array size N and sample support L. Correcting for this bias into a discrete set of difference equations [8]. was shown to significantly improve the match between the The actual true target trajectory is generated as a single statistics of these estimated error probabilities and the true realization of this same model for target bearing and bearing values. It was also shown that the accuracy of these SCB rate. An example is shown in Figure 3. The two parameters error probability estimate improves with SNR and sample for the first order Markov for target bearing rate are the support, getting better as one approaches the threshold SNR time constant defined to be 1200 seconds, and the standard from below. A simulation framework was exploited to assess deviation of the target bearing rate which is 3.5e-3 (deg/sec). the benefit of improved MSE estimates to the KF overall The time step is one second and the duration of the simulation performance. It is shown that for low SNR signals the true is 500 time steps. Measurements are processed at each time KF bearing RMSE is reduced by using MIE and that the step. The variance assigned to each measured Capon bearing tightness of fit between the true RMSE and the KF predicted estimate is assigned either an asymptotic value based on RMSE improves significantly. Use of MIE also results in a the Taylor series [7] (accounting for local errors) or a non- more conservative prediction of error variance by the KF.

1844 ML MSE Performance: MIE Prediction

Threshold SNR ~0dB True RMSE

L = 3N CRB L = 1.5N

KF Predicted RMSE Probability Probability

Fig. 1. Kalman ﬁlter performance: Below threshold SNR bearing estimates, Element Level SNR (db) Element Level SNR (db) but CRB used to represent MSE

Future work will further explore the practicalities of such L = 6N L = 3N Correcting for Probability an approach, as well as consideration of more dynamic target Probability ~2dB SNR Loss trajectories with larger variations in SNR levels.

ACKNOWLEDGEMENTS Element Level SNR (db) Element Level SNR (db) The authors would like to thank Dr. Kristine Bell for useful PCapon(θ |θ ,F ) θ N =18 discussions on tracking and for providing MATLAB scripts for Fig. 2. Statistics of e˜ a b Δ for a of peak sidelobe: the MAP-PF tracker. We would also like to thank the National element ULA Science Foundation for support through grant CCF-0829421 and for support through grant DMS-1035400. Kalman Filter Sensitivity Analysis

Target ArrayA Signal Bearing Kalman REFERENCES Trajectory Generator Estimator Filter Generator [1] K. L. Bell, R. E. Zarnich, R. Wasyk, “MAP-PF Wideband Multitarget and Colored Noise Tracking,” Proceedings of the ICASSP, Dallas, Texas, • Constant Target • Max-Likelihood • Measurement Errors: • Fluctuating Target • Capon-MVDR -CRB March 2010. -MIE MSE [2] K. L. Bell, H. L. Van Trees, “Posterior Cramer-Rao´ Bound for Tracking Simulated Target Tracking Scenario: Target Bearing,” Proceedings of the Adaptive Sensor and Array Pro- cessing Workshop, MIT Lincoln Laboratory, 2005. V(t) [3] H. L. Van Trees and K. L. Bell, Eds., Bayesian Bounds for Parameter Estimation and Nonlinear Filtering/Tracking. New York:Wiley, 2007. [4] C. D. Richmond, “Capon Algorithm Mean Squared Error Threshold Target t)) SNR Prediction and Probability of Resolution,” IEEE Transactions on Θ( Signal Processing, Vol. 53, No. 8, pp. 2748-2764, Aug. 2005. [5] J. Capon, “High–Resolution Frequency–Wavenumber Spectrum Analy- sis,” Proceedings of the IEEE, Vol. 57, 1408–1418 (1969). ULA [6] J. Capon and N.R. Goodman,“Probability Distributions for Estimators of Frequency Wavenumber Spectrum,” Proceedings of the IEEE, Vol. 58, No. 10, 1785–1786 (1970). [7] C. Vaidyanathan, K. M. Buckley, “Performance Analysis of the MVDR Fig. 3. Kalman ﬁlter sensitivity analysis framework Spatial Spectral Estimator,” IEEE Transactions on Signal Processing, Vol. 43, No. 6, pp. 1427–1437, June 1995. [8] A. Gelb, Editor, Applied Optimal Estimation, MIT Press, 1989.

0dB SNR -10dB SNR

• Above threshold SNR (~ -8dB) no performance difference

-12dB SNR • Below threshold Taylor series underestimates bearing MSE

• Improved MSE for low SNR improves overall KF performance

Fig. 4. Kalman ﬁlter realized MSE Performance: Capon algorithm bearing measurements as input

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