Contains research objectives, summary of research and reports on one research project.Joint Services Electronics Programs (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DAAB07-71-C-0300National Science Foundation (Grant GX-36331

Baggeroer, Arthur B.

Moura, J. M. F.

Van Trees, Harry L., Jr.

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XVIII. DETECTION AND ESTIMATION THEORYAcademic Research StaffProf. H. L. Van TreesProf. A. B. BaggeroerGraduate StudentsJ. P. Albuquerque J. M. F. Moura L. C. PuseyL. S. Metzger S. Orphanoudakis K. B. TheriaultS. A. ParlRESEARCH OBJECTIVES AND SUMMARY OF RESEARCHThe research of this group is focused on the theory of detection and estimationand its application to development of effective signal-processing methods. Specificareas of current interest include nonlinear estimation theory, array processing, detec-tion theory for random processes, and signal processing for oceanographic research.1. Nonlinear Estimation Theory and Array ProcessingArray receiver structures for tracking the motion of targets using only passivesignals have been developed. The "bearings only" problem has been investigated ina state-variable context. The effects of target motion on the temporal structure ofsignals, e. g., Doppler effects, have been incorporated, and lead to a receiver struc-ture that is similar to a phase-locked loop coupled to a split-beam tracker. Spatialeffects of angle of arrival and wavefront curvature have been investigated. A sum-mary of a Master's thesis by J. M. F. Moura, to be submitted to the Department ofElectrical Engineering in February 1973, appears in Section XVIII-A. L. S. Metzger andS. Orphanoudakis are also involved in this research.H. L. Van Trees, A. B. Baggeroer2. Theory of Random-Process Detection and ModelingIn many applications of random-process detection, a stochastic model constructedfrom an observation of a segment of noisy sample function is wanted. Many methodshave been developed to meet this problem. One issue of particular importance isthe sensitivity of these methods and the degree to which different stochastic models canbe separated or detected. Work directed at quantifying this sensitivity is beingdeveloped by J. P. Albuquerque.The theory of estimating temporal random processes has been studied extensively.In contrast, study of the structure of spatial random process estimators is just starting.S. A. Parl is formulating the problem of estimating random processes generated by apropagating dynamical structure. This work involves aspects of distributed random pro-cesses, and he is applying it to the example of estimating oceanic internal waves.A. B. BaggeroerThis work is supported by the Joint Services Electronics Programs (U. S. Army,U. S. Navy, U. S. Air Force) under Contract DAAB07-71-C-0300, and the NationalScience Foundation (Grant GX-36331).QPR No. 108 231(XVIII. DETECTION AND ESTIMATION THEORY)3. Application of Detection and Estimation Methods to the East AtlanticContinental Margin ProgramThis work is being done in cooperation with scientists from the Geology and Geo-physics Department at Woods Hole Oceanographic Institution. It is directed towarddeveloping models and algorithms for shipboard processing for the signals that areemployed in continuous seismic profiling of the sea floor. During a cruise to theWest Coast of Africa last year, a parameter -estimation model for in situ removal ofwater column multiples was developed and reported in Quarterly Progress ReportNo. 107 (pp. 95-101). We are now developing these methods further using the modelsof spread channel from communication theory. We are also studying state-variablemodels for horizon tracking, parameter estimation for matched filtering of seismicsource signals, and velocity estimation. Experiments resulting from these studies willbe conducted at sea during the forthcoming National Science Foundation InternationalDecade of Ocean Exploration cruise to the East Atlantic Continental Margin offNorthern Africa.A. B. BaggeroerA. AN INTEGRATED APPROACH TO THE ESTIMATION OF THEDYNAMICS OF A MOVING SOURCE BY A PASSIVE OBSERVERJoint Services Electronics Programs (Contract DAAB07 -71 -C -0300)J. M. F. Moura1. IntroductionWe shall consider an integrated model 1 for the estimation of the dynamics of amoving source (MS) from observed acoustical data by a passive linear array.The changes induced by the source dynamics in the spatial and temporal structureof the emitted narrow-band signal are processed by a spatially and temporally coupledreceiver that simultaneously estimates the range and bearing and their time deriva-tives modeled as finite-state dimensional stochastic processes.The MS dynamics is studied in two frameworks, rectangular and polar, both ofwhich lead to a nonlinear estimation problem. After linearizing the problem, theextended Kalman filter and the maximum a posteriori filter (discrete version) areapplied. Monte Carlo simulation studies have shown that regions of convergence forthe geometry, the signal-to-noise ratio, and the driving noise power level can be foundfor all processors. In order to prevent numerical divergence from the polar framethat would result from an ill-conditioned error -covariance matrix, a square-root filterwas implemented. The necessary a priori information to start the filters was obtainedby means of a maximum-likelihood algorithm followed by a triangularization procedure.2. ModelWe describe the stochastic dynamical system modeling the MS and the model for thereceived signal. For simplicity, we assume a stationary observer (discrete linearQPR No. 108 232(XVIII. DETECTION AND ESTIMATION THEORY)array), planar geometry, and a far-field hypothesis.We shall consider two coordinate systems, the polarFig. XVIII-1,shown in Fig.and rectangularPlanar geometry.XVIII-1, with the usual relationsR(t) = x2(t) + y(t)0(t) = tan (x(t)/y(t)).The linear observer has N sensors at locations pl, ...a. Received SignalThe MS generates a narrow-band signal modeled ash(t) = N2p sin w t,where P is the transmitted power.At the reference sensor a delayed version of h(t) is received,noisePN' along the x axis.corrupted by additivero(t) = ho (t) + w(t)= h(t-T (t)) + w (t), (3)where T (t) is the travel time of the wavefront from the MS to the reference element,0which is given byR(t-T (t))T(t) = (4)O Cand c is the medium propagation velocity.thAt the i sensorr.(t) = hi(t) + wi.(t)= h (t+Ti(t)) + w.i(t), (5)0 1 1QPR No. 108frames233(XVIII. DETECTION AND ESTIMATION THEORY)where the relative delay T.i(t) to the reference element issin 6(t- To(t+Ti(t))+Ti(t))Ti(t) = Pi. (6)Expressions 4 and 6 are memory functions of the dynamics, which can be approxi-mated by exploiting the narrow-band assumption on the signal by means of truncatedTaylor series expansions,2 by no-memory functions leading to( R(t) p .r.(t) = 2P i sin w(t) + w.(t) (7)for i = 1, ... , N.The additive noises w.(t) are assumed to be sample functions of uncorrelated zero-1mean white Gaussian noises with spectral heights N o/2.b. Dynamical SystemNewton's law gives the dynamical system equation. Modeling the MS accelerationcomponents by sample functions of a (mathematical) white-noise process yields inrectangular coordinatesx(t) =F x(t) + G u(t)= 0 1 0 0 x(t) + 0 0 ux(t)0 0 0 x(t) 1 0 u y M(t)(8)0 o 0 1 y(t) o 00000 y(t) 0 1with x(0) = x .-0By a simple transformation of coordinates, from (8), we obtain in polar coordinatesx(t) = f(x(t)) + g(x(t)) u(t)R(t) = R(t) + 0 0 ux(t)R(t) R(t) 0(t) 2  sin 0(t) cos (t) u (t)0(t) e(t) 0 00(t) -2R(t) 6(t)/R(t) cos 0(t)/R(t) -sin 0(t)/R(t) (9)with x(0) = x .-oQPR No. 108 234(XVIII. DETECTION AND ESTIMATION THEORY)The stochastic interpretation of Eq. 8 does not offer any ambiguity. With the polarsystem it can be shown2 that either the Ito or the Stratonovitch integral leads to thesame model, since the Wong-Zakai correction terms 3 are zero for the particular g( )matrix in Eq. 9.We observe that in rectangular coordinates the system is linear. The MS parameterrange, bearing, and respective rates are given by a nonlinear transformation of thestate variables. With the polar frame we choose as state variables the MS parameters,but the dynamical system becomes nonlinear.The driving processes ux(t) and u y(t) are assumed to be zero-mean white Gaussiannoise with spectral heights Q, independent of the wi.(t) noise and of the initial con-Adition x which is modeled as a random variable of mean x and covariance matrix-o -oP.-o3. Nonlinear Estimation ProblemFor simplicity, we worked with the sampled version of this continuous time estima-tion problem. Only filters with a first-order approximation are considered.2 In par-ticular, we implemented the extended Kalman filter and the maximum a posteriori filter(MAP). 4With the discrete extended Kalman filter we used the following format.(i) Measurement Update Equations (Correction)The estimator equation isA - T -1xk = x -k + PH Rk (r k-h(x k)) (10)Ax = Ex-0 -oThe covariance propagation equation isS =  +H Rk H (11)Pk -k k k -klwith P = covariant (x , x ).-0 O -0(ii) Time Update Equations (Prediction)The estimator equation isXk+1 = f(xk) ( k) k' (12)The covariance propagation equation isQPR No. 108 235(XVIII. DETECTION AND ESTIMATION THEORY)- f T A TAk+l P k A k) Q g (xk). (13)The MAP filter is similar to the extended Kalman filter. The principal differenceis in the measurement update equation which, instead of (11), becomesa -T -1P= k R (r -h(x , k)) . (14)axkWith_ HR (r k-h(k, k)) > 0, (15)-kAa matrix H k can be found such that^ -1A I T -1HR k k H- k ax R k k (r k- h(x k)) , (16)8xkand so we can formally reduce Eq. 14 to Eq. 11.Given the observations at time k,the error -covariance matrix Pk as computed byusing the IVLAP has a random component that becomes more important as the signal-to-noise ratio decreases and may cause P k to become indefinite and the filter to diverge.In the polar framework, because of the presence in the state vector of the range andthe bearing rate, the filtering problem is computationally ill-conditioned; that is, thecondition numbermaximum eigenvalue of Pminimum eigenvalue of Pof the error covariance matrix is very large. This leads to numerical divergenceproblems. These difficulties were overcome2 by implementing a square-root algorithm(e.g., Kaminski et al. 5). We modified Potter's method 6 by using an argument ofAndrews 7 to get a computable solution for the case wherein the number of observationsis much larger than the dimension of the state vector.4. Mathematical Analysis of the ReceiversWe shall now discuss the filter for the polar frame. Other configurations, andadditional details have been given elsewhere. 2QPR No. 108 236(XVIII. DETECTION AND ESTIMATION THEORY)a. Estimator EquationsFigure XVIII-2 is a block diagram for the estimator equations. Essentially, thefilter combines the innovation processesr(t) - h(x(t))to form a conventional delay and sum array beam Zi and a "difference" beam Z3. Thefarthest right side of Fig. XVIII-2 shows the combining matrix operation P followedby the conventional receiver's copy of the dynamical system.Fig. XVIII-2. Estimator processor: polar system.GAIN MODULATED BYTHE DIFFERENCE (sin x3-sin x3)RANGE CHANNELcosAb3MODULATION BYTHE DIFFERENCE (xl- i )x3 PN pNco s X3BEARING CHANNELFig. XVIII-3. Range and bearing channel in polar coordinates.QPR No. 108I  cc/ sin ct -v/P cos aN sin 5 N237(XVIII. DETECTION AND ESTIMATION THEORY)Figure XVIII-2 can be rearranged in the form of Fig. XVIII-3 which shows thedecoupled range and bearing channels.Figure XVIII-4 illustrates the estimator channel for the four-state variable. Thecoupling effect between the channels is introduced mainly by the crosscovariances.IfP11 >> P13(17)P33 >> P13'the range channel and the bearing channel perform as phase-locked loops to track thewaveformswcS (t) c R(t) (18)and_2 (t) = sin 0(t). (19)The filter can estimate the range and bearing rates through the modulation of thereceived waveforms induced by the MS. At each rate channel both beams Z1 and Z31,2 3,.4 Fig. XVIII-4. Estimator channel.DELAYZ3 Pzl.are present; the weights are the respective error crosscovariances Pli and P3j' j = 2, 4.Because the filter behaves like a phase-locked loop, it exhibits a primary lock-inregion of an interval of one wavelength centered on the true range.We observe that the dependence of the MAP covariance equation on the actual errorsof the estimates shows that the propagated covariances, and consequently the Kalmangain, will tend to follow the actual errors. Physically, the filter bandwidth is directlymodeled by the errors in the estimates.b. Covariance EquationSince the term B = HTR-1H is the Fisher information matrix, the error covari-ances propagated by the Kalman filter are bounded above by the Cramer-Rao boundsQPR No. 108 238(XVIII. DETECTION AND ESTIMATION THEORY)for the nonrandom unknown parameter case. This reflects the incorporation of thea priori information when modeling the processes 6(t) and R(t) generated by a finite-dimensional dynamical system. For this problem we have1P < B - (20)RR <  11 2 (20)2PT co01P0 < B33 2 (21)2PT wc 2 2NPT } cos 0 T p5. Monte Carlo SimulationThe receiver structures were simulated for several geometric configurations anddifferent sets of parameters and initial conditions on a digital computer.1 s (1) 10.5t .- b(1)soA (2) P 221.00.5 b(2)-0.510-3P44 (4)100TIME (s)2PT/N = 12.5 R(0) 104ftQ/T = 2 R(0) =30 ft/sN = 20Fig. XVIII-5. Typical statistical history for 4 linearized filters.QPR No. 108 239A0.2 -E0.1 -50 100 150TIME (s)Fig. XVIII-6. Bearing-angle history for two different initial bearing rates(SQRT, Polar, EKF).S1040 60TIME (s)40TIME (s)Fig. XVIII-7. Study of an initial error of range and range-rate estimateson the standard deviations (Polar, EKF, SQRT).QPR No. 1080.020.010-0.01-0.02-0.03240(XVIII. DETECTION AND ESTIMATION THEORY)The performance was measured by the bias(22)(i) = (ib(i) 4 (i)and the standard deviations(i) = . .4 i)We fixed the wavelength at X = 50 ft. In Figs. XVIII-5 and XVIII-6 the typical behaviorof the 4 filters is summarized.s(3)00 50 100 150(a)50 100 150TIME (s)2PT/N = 12.5 N = 20Q/T = 2 R(0) = 5000 ft(b)Fig. XVIII-8. History of (a) Y-channel and (b) X-channel for EKFrectangular- coordinate receiver.QPR No. 108(23)241(XVIII. DETECTION AND ESTIMATION THEORY)The 4 filters work in a linear region in the sense thats. - P.. j = 1, ..... 4 (24)and they give unbiased estimates.Figure XVIII-7 considers a nonzero initial error on the range and range rate forthe polar configuration. As long as the range is inside the primary lock-in region,the filters converge to the right range value. The filters are not very sensitive to theinitial error in the range rate.Figure XVIII-8 gives a comparison of the X- and Y-channels in rectangular coor-dinates for a broadside geometry. We observe that the Y-channel has a behaviorsimilar to the range channel of the polar frame.If for the rectangular system we derive bounds similar to Eqs. 17 and 18 (or equiv-alently compute the Cramer-Rao bounds for the X and Y coordinates which are nowconsidered nonrandom unknown parameters), we would find that for a broadside con-figuration the bound for the X-channel is much larger than that for the Y -channel; thisis shown in the experimental results given in Fig. XVIII-8.Figure XVIII-9 shows the typical behavior of the EKF and MAP filters when we varyQ/T and 2PT/N . As we increment the driving noise level, the steady-state error0.8 x 10-20.8 x 10- 3  30.2 x 10-2 0.4 x 10- 3320.70.65 15 10 302PT/No Q/T(a) ( b)Fig. XVIII-9. EKF, Polar, SQRT performance. (a) Effect ofsignal-to-noise ratio. (b) Effect of Q.QPR No. 108 2421.220.62521.054(ft/S)0.749 x 10-333 (rad)0.27 x 10-3i I I I I I I I I I I ,4 8 16 20 4 8 16 20 4 8 16 20NUMBER OF ARRAY ELEMENTSFig. XVIII-10. Effect of array length on filter(Polar, EKF, SQRT).performance2 PT N 50Q/T 0.210 20 30 6 (deg)Fig. XVIII-11.0.1136 x 10-3(rad/s)0.23 x 10- 3(rad)Effect of range on bearing-channel performance(EKF, Polar, SQRT).0. 729 x 10- 42 x 104  5 x 104RANGE (ft)Effect of bearing on bearing-channel(EKF, Polar, SQRT).performanceQPR No. 1080.16 x 10- 30.15 x 10- 35 x 103Fig. XVIII-12.243(XVIII. DETECTION AND ESTIMATION THEORY)covariances increase, and so the Kalman gain increases. This means that a greaterbandwidth is necessary for the filter to follow the larger source dynamics.The filters exhibited a sharp threshold for the signal-to-noise ratio which is in theneighborhood of 1. In several runs the MAP filter diverged even for values of 2PT/Nabove threshold. This divergence was traced to the fact that the error-covariancematrix lost its positive semidefinite character. The square-root algorithm cannotobviate this.Figures XVIII-10 through XVIII-12 show the effects of array length and trackinggeometry on performance.6. ConclusionThese simulation studies have shown that the modulation induced on the temporaland spatial structure of the signal by a moving source (MS) permits simultaneousestimation of MS dynamical parameters.The (computationally) ill-conditioned polar-frame estimation problem can besolved by the square-root algorithm. All filters examined showed a sharp thresh-old on the signal-to-noise ratio and the performance of the filters depended alsoon the driving noise power level, array length, and tracking geometry.Because of the nonlinear character of the estimation problem and the fact that thefilter behaves like a phase -locked loop, some a priori information is needed in order tostart the filters. By remodeling the received signal and considering the bearing as a non-2random unknown quantity, provided we use a sufficiently large array, we are able tosolve for the starting information by means of a maximum-likelihood estimation algo-rithm (composite hypothesis test) followed by a triangularization procedure in whichthe curvature of the incoming wavefronts is considered.References1i. H. L. Van Trees, Memorandum, Research Laboratory of Electronics, M.I.T.,October 1971 (unpublished).2. J. M. F. Moura, S. M. Thesis, Department of Electrical Engineering, M. I. T.(to be submitted February 1973).3. A. H. Jazwinski, Stochastic Processes and Filtering Theory (Academic Press, Inc.,New York, 1970).4. A. P. Sage and J. L. Melsa, Estimation Theory with Applications to Communica-tions and Control (McGraw-Hill Book Company, New York, 1972).5. P. G. Kaminski, A. E. Bryson, and S. F. Schmidt, "Discrete Square Root Filtering:A Survey of Current Techniques," IEEE Trans., Vol. AC-16, No. 6, pp. 727 -736,December 1971.6. R. H. Battin, Astronautical Guidance (McGraw-Hill Book Company, New York, 1964).7. A. Andrews, "A Square Root Formulation of the Kalman Covariance Equation,"AIAA J. 6, 1165-1166 (1968).QPR No. 108 244

Detection and Estimation Theory

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Detection and Estimation Theory

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