delay estimation by expected value the estimation of time delay between narrowband signals is discussed a minimum mean square error mmse estimator is used that utilizes the maximum likelihood ml window derived by knapp and carter in obtaining the required conditional probability density function from which the delay estimate is calculated the results of simulations using a narrowband signal are presented which evaluate and compare the mean square error of the mmse and ml processors the mmse estimator shows improvement over the ml estimator particularly in regions below the threshold signal to noise ratio conditions for application of the mmse processor are discussed as well as some important considerations for its implementation

fundamental limitations in passive time delay estimation part i narrow band systems time delay estimation of a noise like random signal observed at two or more spatially separated receivers is a problem of considerable practical interest in passive radar sonar applications a new method is presented to analyze the mean square error performance of delay estimation schemes based on a modified improved version of the ziv zakai lower bound zzlb this technique is shown to yield the tightest results on the attainable system performance for a wide range of signal to noise ratio snr conditions for delay estimation using narrow band ambiguity prone signals the fundamental result of this study is illustrated in fig 3 the entire domain of snr is divided into several disjoint segments indicating several distinct modes of operation if the available snr does not exceed snr 1 signal observations from the receiver outputs are completely dominated by noise thus essentially useless for the delay estimation as a result the attainable mean square error bar epsilon 2 is bounded only by the a priori parameter domain if snr 1 2 the modified zzlb coincides with the barankin bound in this regime differential delay observations are subject to ambiguities if snr snr 3 the modified zzlb coincides with the cramer rao lower bound indicating that the ambiguity in the differential delay estimation can essentially be resolved the transition from the ambiguity dominated mode of operation to the ambiguity free mode of operation starts at snr 2 and ends at snr 3 this is the threshold phenomenon in time delay estimation the various deflection points snr i and the various segments of the bound fig 3 are given as functions of such important system parameters as time bandwidth product wt signal bandwidth to center frequency ratio w 0 and the number of half wavelengths of the signal center frequency contained in the spacing between receivers with this information the composite bound illustrated in fig 3 provides the most complete characterization of the attainable system performance under any prespecified snr conditions

the shortest path problem with two objective functions abstract we present methods to find the shortest path in a network where each path is associated with two objectives we describe how to obtain the nondominated paths and the extreme nondominated paths and compare the expected complexity of the methods an improvement in efficiency can be obtained when quasiconcave or quasiconvex utility functions are assumed in the first case we describe how to find the optimal extreme nondominated path and bounds for the optimal path value then the optimal path can be located by calculating the k th shortest path in the second case we suggest a branch and bound method to solve the problem

generalized dynamic programming for stochastic combinatorial optimization in stochastic versions of combinatorial optimization problems the objective is to maximize or minimize a function of random variables for many problems of this type conventionally applied dynamic programming dp may fail to generate an optimal solution due to the potential violation of the monotonicity assumption of dp we develop a generalization of dp that guarantees optimality even in the absence of monotonicity we illustrate the methodology on a version of the stochastic traveling salesman problem for which a previously proposed dp algorithm e kao is potentially suboptimal due to the violation of monotonicity m sniedovich using generalized dp we are able to modify the algorithm to guarantee optimality

fully nonlinear fourth order equations with functional boundary conditions abstract the aim of this paper consists in to give sufficient conditions to ensure the existence and location of the solutions of a nonlinear fully fourth order equation with functional boundary conditions the arguments make use of the upper and lower solutions method a laplacian operator and a fixed point theorem an application of the beam theory to a nonlinear continuous model of the human spine allows to estimate its deformation under some loading forces

existence of solutions for weighted p r laplacian system boundary value problems this paper investigates the existence of solutions for weighted p r laplacian ordinary system boundary value problems via leray schauder degree the sufficient conditions for the existence of solutions be given as an application we obtain the existence of the radial solutions for the p x laplacian partial differential system dirichlet problems

on the solvability of some discontinuous third order nonlinear differential equations with two point boundary conditions abstract we prove the existence of extremal solutions for the third order discontinuous functional nonlinear problem u t f t u u t u t for a e t a b u a a a r l 1 u u u a u b u a 0 l 2 u u a u b u b 0 by using a fixed point theorem after having established some existence results for some auxiliary second order nonlinear problems we observe that together with the discontinuities allowed on the spacial variable u with adequate modifications of technical type analogous results can be obtained when the equation has a second member not necessarily continuous in u

existence of solutions of boundary value problem for 3rd order nonlinear system by appropriate bounding function pair and modified functions using the theory of differential inequality and the theory of topological degree this paper presents the existence criterions of solutions for the system of 3rd order differential equations under nonlinear boundary conditions and extends many known results

bounding functions methods for fully nonlinear boundary value problems abstract using the bounding functions method and the theory of topological degree this paper presents the existence criterion of solution for third order bvp with nonlinear boundary conditions and extends the existing results

lower bounds on worst case probability of large error for two channel time delay estimation a lower bound on the worst case probability of large error for two channel time delay estimation using random signals is developed the bound is based on the minimum probability of error m ary hypothesis test using a maximum likelihood estimator mle the bound can only be evaluated approximately however it can be determined arbitrarily well via simulation if the mle can be instrumented the bound is evaluated in detail for uniform lowpass signal and noise spectra and large bandwidth time product it is then compared with another lower bound based on a binary hypothesis test with computer simulation results for an approximate mle instrumentation and with an upper bound established via computer simulation for a correlator for a uniform lowpass spectrum the m ary bound is shown to be very nearly a greatest lower bound over the full range of input signal to noise ratio further the correlator is shown to be very nearly an optimal instrumentation in the sense of reaching the minimum probability of large error

delay estimation using narrow band processes array processing of narrow band gaussian signals is studied with emphasis on delay estimation the barankin bound is used to examine the effect of ambiguity on mean square measurement error when the bound is plotted as a function of signal to noise ratio one observes a distinct threshold above the critical signal to noise ratio the lower bound on mean square error is given by the cramer rao inequality which is approached by the barankin inequality under these conditions below the threshold the barankin bound can exceed the cramer rao bound by large factors the relative magnitude of the bounds in that region depends critically on the ratio of signal center frequency to signal bandwidth

statistics of the mle and approximate upper and lower bounds part i application to toa estimation in nonlinear deterministic parameter estimation the maximum likelihood estimator mle is unable to attain the cramer rao lower bound at low and medium signal to noise ratios snrs due the threshold and ambiguity phenomena in order to evaluate the achieved mean squared error mse at those snr levels we propose new mse approximations msea and an approximate upper bound by using the method of interval estimation mie the mean and the distribution of the mle are approximated as well the mie consists in splitting the a priori domain of the unknown parameter into intervals and computing the statistics of the estimator in each interval also we derive an approximate lower bound alb based on the taylor series expansion of noise and an alb family by employing the binary detection principle the accuracy of the proposed mseas and the tightness of the derived approximate bounds are validated by considering the example of time of arrival estimation

limitations on snr estimator accuracy we consider the samples of a pure tone in additive white gaussian noise awgn for which we wish to determine the signal to noise ratio snr defined here to be spl alpha a sup 2 2 spl sigma sup 2 where a is the tone amplitude and spl sigma sup 2 is the noise variance a and spl sigma sup 2 are assumed to be deterministic but unknown a priori if the variance of an unbiased estimator of spl alpha is spl sigma sub spl alpha spl circ sup 2 we show that at high snr the normalized standard a deviation satisfies the cramer rao lower bound crlb according to spl sigma sub spl alpha spl circ spl alpha spl ges spl radic 2 n where n is the number of independent observables used to obtain the snr estimate spl sigma spl circ

composite bound on the attainable mean square error in passive time delay estimation from ambiguity prone signals corresp the location of a radiating source can be determined by measuring the relative time delay of its signal wavefront to several spatially separated receivers a new technique is presented to investigate the performance of time delay measurement schemes based on a modified version of the ziv zakai lower bound this technique is shown to yield a tight bound on the attainable mean square measurement error for any prespecified signal to noise ratio conditions

the method of lower and upper solutions for third order periodic boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem l n u t t u t u i 0 u i 2 i r i 0 n 1 with ln an nth order linear operator and a caratheodory function we obtain sufficient conditions to guarantee the validity of the monotone method for this problem for this we study the sign of the green function of the operator t 1n with tnu lnu mu m 0 furthermore existence results are obtained we show that the results obtained are optimal given a lower solution and an upper solution we obtain the existence of solution between and when either or this general study is applied to the third order problem u t pu t qu t t u t u i 0 u i 2 i r i 0 1 2 with p q r for this problem we obtain the best value on the constant m to guarantee that the green function of the operator t 13 is positive

on the effect of nuisance parameters on the threshold snr value of the barankin bound the performance of nonlinear estimators of signal parameters in noise appears to exhibit a threshold phenomenon below a critical value of signal to noise ratio snr the performance of these estimators deviates significantly from the cramer rao bound in a large error ambiguity prone snr region the barankin bound has been proved to be an advantageous tool to assess the attainable performance and the threshold value obviously when the estimation problem involves additional unknown nuisance parameters the mean square error mse of the estimator does not decrease however the impact of these nuisance parameters on the threshold value is not clear in this correspondence we discuss the influence of unknown nuisance parameters on the threshold value the analysis is done for the common problem concerning estimating parameters of a gaussian process we confine our scope to a simplified problem concerning only two estimated parameters however we explain how this simplified analysis can be used to handle a more complicated problem comprising multiple nuisance parameters we derive a sufficient condition applied to the structure of the data covariance matrix when the condition is satisfied then the need to estimate additional nuisance parameters does not change the threshold value using the proposed condition we prove that the threshold snr in passive source localization does not increase when the spectral parameters of the source are unknown whereas the presence of another source at an unknown bearing may change the threshold snr

on the cramer rao bound for time delay and doppler estimation corresp using a theorem due to whittle simple derivations of the cramer rao lower bound are presented for some delay estimation problems related to a single source multiple sources and multipath the problem of doppler estimation is briefly discussed

statistics of the mle and approximate upper and lower bounds part ii threshold computation and optimal pulse design for toa estimation threshold and ambiguity phenomena are studied in part i of this paper where approximations for the mean squared error mse of the maximum likelihood estimator are proposed using the method of interval estimation mie and where approximate upper and lower bounds are derived in this part we consider time of arrival estimation and we employ the mie to derive closed form expressions of the begin ambiguity end ambiguity and asymptotic signal to noise ratio snr thresholds with respect to some features of the transmitted signal both baseband and passband pulses are considered we prove that the begin ambiguity threshold depends only on the shape of the envelope of the acr whereas the end ambiguity and asymptotic thresholds only on the shape of the acr we exploit the results on the begin ambiguity and asymptotic thresholds to optimize with respect to the available snr the pulse that achieves the minimum attainable mse the results of this paper are valid for various estimation problems

the joint estimation of differential delay doppler and phase corresp in radio and sonar applications it sometimes happens that narrow band signals originated from a remote source and observed at a pair of receivers differ by unknown differential phase and doppler shift in addition to the differential delay corresponding to the range difference the correspondence presents the joint maximum likelihood ml estimate of the differential delay doppler and phase and examines their accuracy by deriving the cramer rao bound it is shown that the joint ml estimators are the values of the delay and doppler that maximize the magnitude of a generalized ambiguity function analogous to the one used in radar it is also shown that for long observation time and high enough signal to noise ratio there is no degradation in the accuracy of the time delay estimator due to the additional phase and doppler uncertainty and that the differential doppler is uncorrelated with the differential delay and phase estimators

positive solutions of nonlinear singular third order two point boundary value problem abstract in this paper we are concerned with the existence of single and multiple positive solutions to the nonlinear singular third order two point boundary value problem u t a t f u t 0 0 t 1 u 0 u 0 u 1 0 where is a positive parameter under various assumptions on a and f we establish intervals of the parameter which yield the existence of at least one at least two and infinitely many positive solutions of the boundary value problem by using krasnoselskii s fixed point theorem of cone expansion compression type

explicit ziv zakai lower bound for bearing estimation the extended ziv zakai bound for vector parameters is used to develop a lower bound on the mean square error in estimating the 2 d bearing of a narrowband planewave signal using planar arrays of arbitrary geometry the bound has a simple closed form expression that is a function of the signal wavelength the signal to noise ratio snr the number of data snapshots the number of sensors in the array and the array configuration analysis of the bound suggests that there are several regions of operation and expressions for the thresholds separating the regions are provided in the asymptotic region where the number of snapshots and or snr are large estimation errors are small and the bound approaches the inverse fisher information this is the same as the asymptotic performance predicted by the local cramer rao bound for each value of bearing in the a priori performance region where the number of snapshots or snr is small estimation errors are distributed throughout the a priori parameter space and the bound approaches the a priori covariance in the transition region both small and large errors occur and the bound varies smoothly between the two extremes simulations of the maximum likelihood estimator mle demonstrate that the bound closely predicts the performance of the mle in all regions

optimal interval lengths for nonlocal boundary value problems associated with third order lipschitz equations abstract for the third order differential equation y f x y y y where f x y 1 y 2 y 3 is lipschitz continuous in terms of y i i 1 2 3 we obtain optimal bounds on the length of intervals on which there exist unique solutions of certain nonlocal three and four point boundary value problems these bounds are obtained through an application of the pontryagin maximum principle from the theory of optimal control

optimal source localization and tracking from passive array measurements if the signal of a moving source is observed at several spatially separated locations during a common time interval the received signals exhibit differential delays and doppler shifts which provide information about source location and velocity this paper deals with the optimal in the minimum mean square error sense source localization and tracking by utilizing differential delay and doppler observations at a linear array of receivers we shall be concerned with source ranges large compared with array baseline and with the distance traversed by the source during the observation interval it is shown that the optimal estimation procedure can be carried out in two sequential steps without any loss in performance in the first step source location is estimated from the differential delay group of measurements in the second step source velocity is estimated from the differential doppler group of measurements and the outcome of the first step furthermore by employing an intermediate track dependent parameter set the two step algorithm yields a closed form solution of the estimation problem

multilevel dynamic programming for general multiple linear quadratic control in discrete time systems abstract this paper extends the reach of dynamic programming to general multiple linear quadratic control problems in discrete time systems where the overall objective is a nonlinear function of multiple quadratic performance indices a multilevel dynamic programming solution procedure is proposed to obtain an analytical linear control law for this nonseparable control problem at the lower level the formulated auxiliary lagrangian problem is of a parametric linear quadratic structure and is solved by dynamic programming the weighting vector in the auxiliary lagrangian problem is adjusted by the upper level iteratively this two level process repeats until an optimal condition is satisfied

first and second order discontinuous functional differential equations with impulses at fixed moments abstract we give sufficient conditions for the existence of extremal solutions to discontinuous and functional differential equations with impulses our main results are new even for ordinary differential equations without impulses

time delay estimation via cross correlation in the presence of large estimation errors the estimate of the difference in time of arrival of a common random signal received at two sensors each of which also receives uncorrelated noise is examined for both small and large estimation errors it is shown that as the post integration signal to noise ratio decreases the correlator exhibits a thresholding effect that is the probability of a large error an anomalous estimate increases rapidly approximate theoretical results for the probability of an anomaly are presented and are verified experimentally the variance of the time delay estimate is examined for both a gated mode in which the time delay corresponding to the correlation peak closest to the true time delay is used as the estimate of time delay and an ungated mode in which the time delay corresponding to the largest peak over the full range of the correlator delay times is used as the estimate the observed variance for both modes is compared with the theoretical variance based on a small error analysis for the gated modes the signal to noise ratio below which the observed variance begins to differ significantly from the small error theory can be reliably predicted from a linearity criterion it is shown however that the expected variance for the ungated mode can depart from the small error theory at a higher signal to noise ratio than for the gated modes thus the variance due to anomalies can be the most important factor in determining the region of applicability of the small error analysis

on general multiple linear quadratic control problems general multiple linear quadratic control problems are studied where the overall objective is a nonlinear function of multiple quadratic performance indices a multilevel solution procedure is proposed to obtain an analytical linear control law for this nonseparable control problem the auxiliary lagrangian problem formulated at the lower level is of a parametric linear quadratic structure in which the weighting vector is adjusted by the upper level until a derived optimal condition is met

existence of solution for discontinuous third order boundary value problems abstract in this paper we obtain existence results for the problem u q u f t u with boundary conditions u a a u b b u a c and u a u b u a u b u a c we assume f a caratheodory function q l r 0 such that 1 q l loc r 0 and suppose the existence of lower and upper solutions the existence of solution for the first considered conditions is obtained as a consequence of the fixed points theorems we obtain the solution of the second problem as a limit of solutions of the first case for the first problem the monotone method is developed

theory and applications of generalized dynamic programming an overview abstract extensions of dynamic programming dp into generalized preference structures such as exist in multicriteria optimization have invariably assumed sufficient structure to ensure the validity of bellman s principle of optimality or the presence of monotonicity hence ensuring the validity of the functional equations of dp often however the only reasonable formulation of problem in terms of the size of the resulting state space is one for which neither monotonicity nor the strong principle of optimality is satisfied generalized dp based on a weaker principle of optimality is a generalization of dp that produces optimal solutions even in the absence of monotonicity it is shown both to subsume conventional dp and to provide an alternative means of implementing mitten s preference order dp this paper provides an overview of generalized dp and its applications to date summarizing our computational experiences in the process

existence and uniqueness of solutions for third order nonlinear boundary value problems in this paper we present some general results of existence and uniqueness of solutions of nonlinear two point boundary value problems for third order nonlinear differential equations by using the shooting method as applications we give certain concrete sufficient conditions for the existence and uniqueness

realizable lower bounds for time delay estimation the accuracy of time delay estimates obtainable in active localization systems is studied focusing on the effect of ambiguities in the time delay estimates such ambiguities occur when the transmitted signal has small relative bandwidth then for signal to noise ratios below a certain threshold the commonly used cramer rao lower bound is not realizable the study concentrates on the region of intermediate snr values where the cramer rao bound is no longer achievable but useful information on time delays can still be obtained from the measurement realizable bounds for the single and two echo cases are obtained by deriving a new form of the barankin 1949 bound for active time delay estimation this form maintains the realizability property of the most general form but is of reasonable complexity new bounds are derived for the multiple echo case examples are presented to illustrate the dependence of the bound on parameters such as snr relative bandwidth and echo separation

extremality and comparison results for discontinuous third order functional initial boundary value problems extremality and comparison results are derived for third order functional initial boundary value problems differential equations of problems may involve discontinuous nonlinearities

a bound on mean square estimate error a lower bound on mean square estimate error is derived as an instance of the covariance inequality by concatenating the generating matrices for the bhattacharyya and barankin bounds it represents a generalization of the bhattacharyya barankin cramer rao hammersley chapman robbins kiefer and mcaulay hofstetter bounds in that all of these bounds may be derived as special cases the bound is applicable to biased estimates of functions of a multidimensional parameter termed the hybrid bhattacharyya barankin bound it may be written as the sum of the mth order bhattacharyya bound and a nonnegative term similar in form to the rth order hammersley chapman robbins bound it is intended for use when small error bounds such as the cramer rao bound may not be tight unlike many large error bounds it provides a smooth transition between the small error and large error regions as an example application bounds are placed on the variance of unbiased position estimates derived from passive array measurements here the hybrid bhattacharyya barankin bound enters the large error region at a larger snr and in the large error region is noticeably greater than either the hammersley chapman robbins or the bhattacharyya bounds

fourth order problems with fully nonlinear boundary conditions differential inequality method bounding function method and topological degree are applied to obtain the existence criterions of at least one solution for the general fourth order differential equations under nonlinear boundary conditions and many existing results are complemented

extension of dynamic programming to nonseparable dynamic optimization problems abstract the use of dynamic programming is extended to a general nonseparable class where multiobjective optimization is used as a separation strategy the original nonseparable dynamic optimization problem is first embedded into a separable albeit multiobjective optimization problem where multiobjective dynamic programming using the envelope approach is used as a solution scheme under certain conditions the optimal solution of the original nonseparable problem is proven to be attained by a noninferior solution

the dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation the dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation m r grossinho a b 1 st a tersian c 2 a departamento de matem atica iseg universidade t ecnica de lisboa rua do quelhas 6 1200 lisboa portugal b cmaf universidade de lisboa av prof gama pinto 2 1699 lisboa codex portugal c center of applied mathematics and informatics university of rousse 8 studentska str 7017 rousse bulgaria

remarks on the lower and upper solutions method for second and third order periodic boundary value problems we study the solvability and the approximation of the solutions by monotone iteration of the second and third order periodic boundary value problems bvps u cu t u u 0 u email protected u 0 u email protected and email protected au bu t u u 0 u email protected u 0 u email protected u 0 u email protected in the presence of lower and upper solutions which may not satisfy the usual ordering condition to this end some maximum and antimaximum principles are stated and proved for certain linear differential operators naturally associated with the previously stated problems

a barankin type lower bound on the estimation error of a hybrid parameter vector the barankin 1949 bound is a realizable lower bound on the mean square error mse of any unbiased estimator of a nonrandom parameter vector we present a barankin type bound which is useful in problems where there is a prior knowledge on some of the parameters to be estimated that is the parameter vector is a hybrid vector in the sense that some of its entries are deterministic while other are random variables we present a simple expression for a positive definite matrix which provides bounds on the covariance of any unbiased estimator of the nonrandom parameters and an estimator of the random parameters simultaneously we show that the barankin bound for deterministic parameters estimation and the bobrovsky zakai 1976 bound for random parameters estimation are special cases of our proposed bound

hierarchical control for large scale systems with general multiple linear quadratic structure abstract a class of large scale linear control systems where the overall objective function is a nonlinear function of multiple quadratic performance indices is investigated in this paper this type of large scale control problem with a general multiple linear quadratic structure is nonseparable in the sense of conventional hierarchical control hierarchical control is extended in this paper to large scale nonseparable control problems where multiobjective optimization is used as a separation strategy the large scale general multiple linear quadratic control problem is embedded under certain conditions into a family of the weighted lagrangian formulation the weighted lagrangian formulation is separable with respect to subsystems and can be effectively solved using the interaction prediction approach at the two lower levels in the proposed three level solution structure at the third level the weighting vector for the weighted lagrangian formulation is adjusted iteratively to search the optimal weighting vector with which the optimal control of the original large scale nonseparable control problem is attained one feature of this hierarchical control scheme is a derived analytical solution for the large scale general multiple linear quadratic control

uniqueness comparison and existence results for third order initial boundary value problems abstract we derive maximum principles for a nonlinear third order differential operator and prove uniqueness comparison and existence results for different types of third order initial boundary value problems including existence results for functional problems

large and small error performance limits for multipath time delay estimation the cramer rao lower bound crlb and a lower bound on the probability of large errors for multipath time delay estimation for a single resolvable multipath are developed these theoretical results are compared to the performance of the maximum likelihood estimator mle and an autocorrelator via computer simulation for small errors and large snr the mle reaches the crlb the variance of the time delay estimate for the autocorrelator is much larger than the crlb if the reflection coefficient for the delayed path is near unity for small errors and small snr the mle reduces to an autocorrelator hence their performance is identical for large errors and bandwidth time products greater than 500 the error probability for the autocorrelator is within 1 db in input snr of the lower bound on large error probability thus the autocorrelator is nearly an optimal instrumentation from the point of view of large errors at least for large bandwidth time products and the flat low pass spectrum considered here

solvability of some third order boundary value problems with asymmetric unbounded nonlinearities abstract in this paper we present existence and location results for the third order separated boundary value problems u t f t u t u t u t with the boundary conditions u a a u a b u b c or u a a c 1 u a c 2 u a b c 3 u b c 4 u b c with c 1 c 2 c 3 c 4 r and a b c r we assume f a b r 3 r is a continuous function satisfying one sided nagumo type condition which allows an asymmetric unbounded behaviour the arguments used concern leray schauder degree and lower and upper solution techniques

the method of lower and upper solutions for second third fourth and higher order boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u n t t u t u i a u i b i r i 0 n 1 where is a caratheodory function we obtain necessary and sufficient conditions in to guarantee the existence of solutions between a lower solution and an upper solution for n 2 if n 3 either or and n 4 if furthermore we obtain sufficient conditions in for n 2 k 6 when