a note on the cramer rao bound for 2 d direction finding based on 2 d array the authors present explicit expressions for the cramer rao bound crb for estimating the two dimensional 2 d direction of a single source based on 2 d arrays of identical omnidirectional sensors two commonly used models random wave and unknown wave are compared it is shown that the crbs for the two models have the same dependency on the array structure a specialization of the crb to two orthogonal uniform linear arrays ulas is discussed it is found that the joint crbs of the direction angles based on the two orthogonal ulas can be as low as one quarter for a random waveform model with a large number of snapshots and low snr or one half for both models with high snr of the crbs based on each ula

on focusing matrices for wide band array processing a general class of transformation matrices for coherent signal subspace processing is presented these signal subspace transformation sst matrices are shown to generate a sufficient statistic for maximum likelihood bearing estimation two general forms for calculating sst matrices are presented and the rotational signal subspace rss focusing matrices proposed by h hung and m kaveh 1988 are shown to be a special case of the sst matrices an efficient procedure for computing a subset of the sst matrices utilizing householder transformations is presented the procedure reduces the computational load by a factor of 10 compared with that for the rss matrices the application of music to the coherently combined covariance matrix is also discussed and monte carlo simulations comparing the performance of householder sst matrices and rss matrices are performed

a maximum likelihood time delay estimator in a multipath environment using importance sampling in this paper we present a new implementation of the maximum likelihood criterion for the estimation of the time delays in a multipath environment and then extend it to the estimation of the time difference of arrival when the transmitted signal is unknown the new technique implements the concept of importance sampling is to find the global maximum of the compressed likelihood function in a modest computational manner it thereby avoids traditional complex multidimensional grid search or initialization dependent iterative methods indeed one of the most interesting features is that it transforms the multi dimensional search inherent to multipath propagation into a much simpler one dimensional optimization problem in the delays dimension moreover it guarantees convergence to the global maximum contrarily to the popular iterative implementation of the maximum likelihood criterion by the well known expectation maximization em algorithm comparisons with some other methods such as the em algorithm music and accelerated random search ars demonstrates the superiority of the proposed is based multipath delay estimator in terms of estimation performance and complexity

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

a fast algorithm for solving a toeplitz system of equations a fast algorithm for the solution of a toeplitz system of equations is presented the algorithm requires order n log n 2 computations where n is the number of equations for banded toeplitz matrices the order of computations is reduced to only n log n m log m 2 where 2m is the maximum number of nonzero principal subdiagonals of the toeplitz matrix the algorithm is in essence a fast implementation of the trench algorithm in reverse thus the algorithm involves imbedding of the given matrix in a cyclic matrix and a fast hd half divisor algorithm to compute the first row of the inverse matrix the desired solution is then obtained directly from the first row by applying fast fourier transform techniques in order n log n computations finally the extension of the algorithm to block toeplitz matrices is also presented

parameter estimation of exponentially damped sinusoids using higher order statistics a new approach for the estimation of the parameters of exponentially damped sinusoids is introduced based on third or fourth order statistics of the observation signal the method may be seen as an extension of the minimum norm principal eigenvectors method to higher order statistics domains the strong points and limitations of the method are discussed as well as sufficient conditions for the existence of the solution the utilization of the method in the case of finite length signals in the presence of additive gaussian noise white or colored is addressed monte carlo simulations demonstrate the effectiveness of the new method when the additive noise is colored gaussian with unknown autocorrelation sequence for different signal to noise ratios and a single data record the case of an ensemble of data records is studied when the exponentially damped sinusoids are assumed to have random phase

model order selection of damped sinusoids in noise by predictive densities we develop a procedure for the order selection of damped sinusoidal models based on the maximum a posteriori map criterion the proposed method merges the concept of predictive densities with bayesian inference to arrive at a complex multidimensional integral whose solution is achieved by way of the efficient monte carlo importance sampling technique the importance function a multivariate cauchy probability density is employed to produce stratified samples over the hypersurfaces support region centrality location parameters for the cauchy are resolved by exploiting the special structure of the compressed likelihood function clf and applying the fast maximum likelihood fml procedure of umesh and tufts simulation results allow for a comparison between our method and the singular value decomposition svd based information theoretic criteria of reddy and biradar see ieee trans signal processing vol 41 no 9 p 2872 81 1993

joint angle and delay estimation using shift invariance techniques in a multipath communication scenario it is often relevant to estimate the directions and relative delays of each multipath ray we derive a closed form subspace based method for the simultaneous estimation of these parameters from an estimated channel impulse response using knowledge of the transmitted pulse shape function the algorithm uses a two dimensional 2 d esprit like shift invariance technique to separate and estimate the phase shifts due to delay and direction of incidence with automatic pairing of the two parameter sets improved resolution is obtained by enlarging the data matrix with shifted and conjugated copies of itself

linear prediction approach for efficient frequency estimation of multiple real sinusoids algorithms and analyses based on the linear prediction property of sinusoidal signals two constrained weighted least squares frequency estimators for multiple real sinusoids embedded in white noise are proposed in order to achieve accurate frequency estimation the first algorithm uses a generalized unit norm constraint while the second method employs a monic constraint the weighting matrices in both methods are a function of the frequency parameters and are obtained in an iterative manner for the case of a single real tone with sufficiently large data samples both estimators provide nearly identical frequency estimates and their performance approaches crame spl acute r rao lower bound crlb for white gaussian noise before the threshold effect occurs algorithms for closed form single tone frequency estimation are also devised computer simulations are included to corroborate the theoretical development and to contrast the estimator performance with the crlb for different frequencies observation lengths and signal to noise ratio snr conditions

a minimal parameter adaptive notch filter with constrained poles and zeros a new algorithm is presented for adaptive notch filtering and parametric spectral estimation of multiple narrow band or sine wave signals in an additive broad band process the algorithm is of recursive prediction error rpe form and uses a special constrained model of infinite impulse response iir with a minimal number of parameters the convergent filter is characterized by highly narrow bandwidth and uniform notches of desired shape for sufficiently large data sets the variances of the sine wave frequency estimates are of the same order of magnitude as the cramer rao bound results from simulations illustrate the performance of the algorithm under a wide range of conditions

a parameter estimation scheme for damped sinusoidal signals based on low rank hankel approximation most of the existing algorithms for parameter estimation of damped sinusoidal signals are based only on the low rank approximation of the prediction matrix and ignore the hankel property of the prediction matrix we propose a modified kumaresan tufts mkt algorithm exploiting both rank deficient and hankel properties of the prediction matrix computer simulation results demonstrate that compared with the original kumaresan tufts 1982 algorithm and the matrix pencil algorithm the mkt algorithm has a lower noise threshold and can estimate the parameters of signal with larger damping factors

maximum likelihood direction of arrival estimation in unknown noise fields using sparse sensor arrays we address the problem of maximum likelihood ml direction of arrival doa estimation in unknown spatially correlated noise fields using sparse sensor arrays composed of multiple widely separated subarrays in such arrays intersubarray spacings are substantially larger than the signal wavelength and therefore sensor noises can be assumed to be uncorrelated between different subarrays this leads to a block diagonal structure of the noise covariance matrix which enables a substantial reduction of the number of nuisance noise parameters and ensures the identifiability of the underlying doa estimation problem a new deterministic ml doa estimator is derived for this class of sparse sensor arrays the proposed approach concentrates the ml estimation problem with respect to all nuisance parameters in contrast to the analytic concentration used in conventional ml techniques the implementation of the proposed estimator is based on an iterative procedure which includes a stepwise concentration of the log likelihood ll function the proposed algorithm is shown to have a straightforward extension to the case of uncalibrated arrays with unknown sensor gains and phases it is free of any further structural constraints or parametric model restrictions that are usually imposed on the noise covariance matrix and received signals in most existing ml based approaches to doa estimation in spatially correlated noise

newton algorithms for conditional and unconditional maximum likelihood estimation of the parameters of exponential signals in noise the authors present polynomial based newton algorithms for maximum likelihood estimation mle of the parameters of multiple exponential signals in noise this formulation can be used in the estimation for example of the directions of arrival of multiple noise corrupted narrowband plane waves using uniform linear arrays and the frequencies of multiple noise corrupted complex sine waves the algorithms offer rapid convergence and exhibit the computation efficiency associated with the polynomial approach compact closed form expressions are presented for the gradients and hessians various model assumptions concerning the statistics of the underlying signals are considered numerical simulations are presented to demonstrate the algorithms performance

maximum likelihood estimation of 2 d superimposed exponential signals the problem of maximum likelihood estimation of the parameters i e frequencies amplitudes and noise variance of 2 d superimposed exponential signals is considered in this paper a polynomial rooting approach is proposed to obtain the estimates of the 2 d frequencies strong consistency and limiting distribution are established for the estimates of the parameters furthermore the covariance matrix of the limiting distribution attains the cramer rao lower bound

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

fast maximum likelihood for blind identification of multiple fir channels this paper develops a fast maximum likelihood method for estimating the impulse responses of multiple fir channels driven by an arbitrary unknown input the resulting method consists of two iterative steps where each step minimizes a quadratic function the two step maximum likelihood tsml method is shown to be high snr efficient i e attaining the cramer rao lower bound crb at high snr the tsml method exploits a novel orthogonal complement matrix of the generalized sylvester matrix simulations show that the tsml method significantly outperforms the cross relation cr method and the subspace ss method and attains the crb over a wide range of snr this paper also studies a fisher information fi matrix to reveal the identifiability of the m channel system a strong connection between the fi based identifiability and the cr based identifiability is established

a generalized least squares method for frequency estimation the application of the generalized least squares gls method to the estimation of the frequencies of sinusoids in additive colored noise is discussed an iterative algorithm is proposed starting from the assumption that the parameter vector in the model of the sinusoids is symmetric the algorithm utilizes an adaptive strategy for contracting the poles of the estimated signal model expressions for the probability limit and the asymptotic variance of the estimates are derived for the single sinusoid case possible convergence points and the asymptotic behavior of the algorithm in the case of low snrs signal to noise ratios are analyzed extensive simulation results show that the algorithm represents a simple and reliable tool for practical applications

efficient wideband source localization using beamforming invariance technique a novel scheme for wideband direction of arrival doa estimation is proposed the technique performs coherent signal subspace transformation by a set of judiciously constructed beamforming matrices the beamformers are chosen to transform each of the narrowband array manifold vectors into the one corresponding to the reference frequency regardless of the actual spatial distribution of the sources the focused data correlation matrix can thus be obtained without any preliminary doa estimation or iteration a simplified version of the beamspace root music algorithm is developed and used in conjunction with the proposed method to efficiently localize multiple wideband sources with a linear equally spaced array numerical simulations are conducted to demonstrate the efficacy of the new scheme

computationally efficient angle estimation for signals with known waveforms this paper presents a large sample decoupled maximum likelihood deml angle estimator for uncorrelated narrowband plane waves with known waveforms and unknown amplitudes arriving at a sensor array in the presence of unknown and arbitrary spatially colored noise the deml estimator decouples the multidimensional problem of the exact ml estimator to a set of 1 d problems and hence is computationally efficient we shall derive the asymptotic statistical performance of the deml estimator and compare the performance with its cramer rao bound crb i e the best possible performance for the class of asymptotically unbiased estimators we will show that the deml estimator is asymptotically statistically efficient for uncorrelated signals with known waveforms we will also show that for moderately correlated signals with known waveforms the deml estimator is no longer a large sample maximum likelihood ml estimator but the deml estimator may still be used for angle estimation and the performance degradation relative to the crb is small we shall show that the deml estimator can also be used to estimate the arrival angles of desired signals with known waveforms in the presence of interfering or jamming signals by modeling the interfering or jamming signals as random processes with an unknown spatial covariance matrix finally several numerical examples showing the performance of the deml estimator are presented in this paper

application of maximum likelihood estimation to radar imaging an efficient maximum likelihood ml estimator to obtain the scattering center locations of a target and the relative scattering level of these scattering centers from the scattered field data is described in the proposed method ml estimation is carried out in the image domain rather than in the frequency aspect domain a two dimensional 2 d inverse fourier transform is used to transfer the scattered field data from frequency aspect domain to the image domain down range cross range as expected the scattered field data in the image domain has some regions with high energy the samples in the high energy regions are used to obtain the initial guess for the ml estimator as well as for ml estimation the ml estimator in the image domain is applied to both simulated and experimental scattered fields of some targets

on the variance of the height of random binary search trees let h n be the height of a random binary search tree on n nodes we show that there exists a constant alpha 4 31107 ldots such that rm p h n alpha log n beta log log n to 0 where beta 15 alpha ln 2 93 2933 ldots the proof uses the second moment method and does not rely on properties of branching processes we also show that var h n o log log n 2

exact maximum likelihood parameter estimation of superimposed exponential signals in noise a unified framework for the exact maximum likelihood estimation of the parameters of superimposed exponential signals in noise encompassing both the time series and the array problems is presented an exact expression for the ml criterion is derived in terms of the linear prediction polynomial of the signal and an iterative algorithm for the maximization of this criterion is presented the algorithm is equally applicable in the case of signal coherence in the array problem simulation shows the estimator to be capable of providing more accurate frequency estimates than currently existing techniques the algorithm is similar to those independently derived by kumaresan et al in addition to its practical value the present formulation is used to interpret previous methods such as prony s pisarenko s and modifications thereof

error analysis of maximum likelihood estimates of physical parameters from one or more dispersive waves measurement of physical parameters from one or more dispersive waves propagating in a band of frequencies across a linear array of receivers is considered the waves propagate according to a known physical model having real valued parameters and have spatially uncorrelated white gaussian noise added to them the objective of the measurement is to estimate a specified subset of the physical parameters the remaining physical parameters are externally provided and have known error statistics existing maximum likelihood ml estimation and cramer rao cr bounds theory describes the measurement under the assumption of correct values for the externally provided parameters a review of this theory is presented for completeness then a bounding estimate on the total error on the estimated parameters is derived which takes into account both the cr bounds and errors in the externally provided parameters suggestions for the application of the theory to practical problems are provided a criterion is derived for determining whether additional unwanted nuisance parameters should be included in the estimate to improve the total error on the desired parameters

the most efficient implementation of the iqml algorithm the work by clark and scharf 1992 showed a new implementation of the iqml iterative quadratic maximum likelihood algorithm which requires at each iteration computational flops of order n sup 2 where n is the dimension of signal vector or length of data sequence they also indicated that the implementation of other related algorithms such as the steiglitz mcbride 1965 algorithm would also require order n sup 2 computations the present author gives a better way of implementation which requires computational flops of the order n this better way of implementation is shown in detail for the iqml algorithm following the same idea shown in the present paper one can also straightforwardly design the order n implementation of the steiglitz mcbride algorithm the present implementation is also the most efficient in that no implementation can be made less than order n sup 2

estimation of parameters of exponentially damped sinusoids using fast maximum likelihood estimation with application to nmr spectroscopy data we present fast maximum likelihood fml estimation of parameters of multiple exponentially damped sinusoids the fml algorithm was motivated by the desire to analyze data that have many closely spaced components such as the nmr spectroscopy data of human blood plasma the computational efficiency of fml lies in reducing the multidimensional search involved in ml estimation into multiple 1 d searches this is achieved by using our knowledge of the shape of the compressed likelihood function clf in the parameter space the proposed fml algorithm is an iterative method that decomposes the original data into its constituent signal components and estimates the parameters of the individual components efficiently using our knowledge of the shape of the clf the other striking features of the proposed algorithm are that it provides procedures for initialization has a fast converging iteration stage and makes use of the information extracted in preliminary iterations to segment the data suitably to increase the effective signal to noise ratio snr the computational complexity and the performance of the proposed algorithm are compared with other existing methods such as those based on linear prediction kiss iqml alternating projections ap and expectation maximization em

doa estimation and detection in colored noise using additional noise only data in a typical array processing scenario noise acting on the array can not be assumed spatially white it is in many cases necessary to use quiet periods when only noise is received to estimate the noise covariance if estimation of the signal parameters and noise covariance is performed jointly performance can be improved this is especially true when stationarity considerations limit the amount of available valid noise only data an asymptotically valid approximative maximum likelihood method aml for the estimation problem is derived in this work the resulting criterion is when concentrated with respect to the signal parameters relatively simple in numerical experiments aml shows promising small sample performance compared to earlier methods the criterion function is also well suited for numerical optimization the new criterion function allows for the development of a novel mode like non iterative estimation procedure if the array belongs to the important class of uniform linear arrays the resulting procedure retains the asymptotic properties of maximum likelihood and numerical simulations indicate superior threshold performance when compared to an optimally weighted subspace fitting wsf formulation of mode for the detection problem no method has been presented that takes the unknown noise covariance into account here a well known detection scheme for wsf is extended to work in this scenario as well the derivations of this scheme further stress the importance of using the correct weighting in wsf when the noise covariance is unknown it is also shown that the minimum value of the criterion function associated with aml can be used for the detection purpose numerical experiments indicate very promising performance for the aml detection scheme

detection of coherent signals using weighted subspace smoothing a new approach to detecting the number of coherent signals incident upon a uniformly spaced linear array is presented the approach combines a modified spatial smoothing scheme and a modified mdl criterion the modified spatial smoothing scheme referred to as weighted subspace smoothing is actually a generalization of the post smoothing approach proposed by krim and proakis in 1994 it is shown that the noise eigenvalues obtained with weighted subspace smoothing are more accurate than those obtained with the original spatial smoothing we thus attempt to improve the detection performance using these more accurate eigenvalues a novel modification of the minimum description length mdl criterion is proposed to accomplish this computer simulations are presented to study the performance improvement of this new approach

estimation of the directions of arrival of signals in unknown correlated noise i the map approach and its implementation the authors propose a method of direction of arrival doa estimation of signals in the presence of noise whose covariance matrix is unknown and arbitrary other than being positive definite they examine the projection of the data onto the noise subspace the conditional probability density function pdf of the projected data given the signal parameters and the unknown projected noise covariance matrix is first formed the a posteriori pdf of the signal parameters alone is then obtained by assigning a noninformative a priori pdf to the unknown noise covariance matrix and integrating out this quantity a simple criterion for the maximum a posteriori map estimate of the doas of the signals is established some properties of this criterion are discussed and an efficient numerical algorithm for the implementation of this criterion is developed the advantage of this method is that the noise covariance matrix does not have to be known nor must it be estimated

pseudo randomly generated estimator banks a new tool for improving the threshold performance of direction finding a new powerful tool for improving the threshold performance of direction finding is considered the main idea of our approach is to reduce the number of outliers in the doa estimates using a previously proposed joint estimation strategy jes for this purpose multiple different doa estimators are calculated in a parallel manner for the same batch of data i e for a single data record employing these estimators simultaneously the jes improves the threshold performance because it removes outliers and exploits only successful estimators that are sorted out using a hypothesis testing procedure we consider an efficient modification of the jes with application to the pseudo randomly generated eigenstructure estimator banks based on secondand higher order statistics weighted music estimators based on the covariance and contracted quadricovariance matrices are chosen as appropriate underlying techniques for the second and fourth order estimator banks respectively computer simulations with uncorrelated sources verify the dramatic improvements of threshold performance as compared with the conventional second and fourth order music algorithms simulations also show that in the second order case the threshold performance of our technique is close to that of the wsf method and stochastic deterministic ml methods which are known today as the most powerful in the sense of estimation performance and at the same time as the most computationally expensive doa estimation techniques the computational cost of our algorithm is much lower than that of the wsf and ml techniques because no multidimensional optimization is required

asymptotic behavior of maximum likelihood estimates of superimposed exponential signals strong consistency and asymptotic normality are derived for the maximum likelihood estimates mles of the unknown parameters omega sub 1 omega sub p alpha sub 1 alpha sub p and sigma sup 2 in the superimposed exponential model for signals y sub t sigma alpha exp it omega sub k e sub t where the summation is from k 1 to p t 0 1 n 1 and sigma sup 2 is the variance of the complex normal distribution of e sub t as a by product it is found that the mles of the parameters attain the cramer rao lower bound for the asymptotic covariance matrix

estimating the parameters of exponentially damped sinusoids and pole zero modeling in noise we have presented techniques 1 6 based on linear prediction lp and singular value decomposition svd for accurate estimation of closely spaced frequencies of sinusoidal signals in noise in this note we extend these techniques to estimate the parameters of exponentially damped sinusoidal signals in noise the estimation procedure presented here makes use of backward prediction in addition to svd first the method is applied to data consisting of one and two exponentially damped sinusoids the choice of one and two signal components facilitates the comparison of estimation error in pole damping factors and pole frequencies to the appropriate cramer rao cr bounds and to traditional methods of linear prediction second our method is applied to an example due to steiglitz 8 in which the data consists of noisy values of the impulse response samples composed of many exponentially damped sinusoids of a linear system having both poles and zeros the poles of the system are accurately determined by our method and the zeros are obtained subsequently using shanks method

improving threshold performance of the iqml algorithm in this article improvements to the iterative quadratic maximum likelihood iqml algorithm are presented the modifications are twofold modification of the nontriviality constraint on the polynomial coefficient vector and the addition of a noise preprocessing step based on the eigendecomposition of the data covariance matrix simulation results are provided

the risk unbiased cramer rao bound for non bayesian multivariate parameter estimation how accurately can one estimate a deterministic parameter subject to other unknown deterministic model nuisance parameters the most popular answer to this question is given by the cramer rao bound crb the main assumption behind the derivation of the crb is local unbiased estimation of all model parameters the foundations of this paper rely on doubting this assumption generally in multivariate parameter estimation each parameter in its turn can be treated as a single parameter of interest whereas the other model parameters are treated as nuisance as their misknowledge interferes with the estimation of the parameter of interest this approach is utilized in this paper to provide a fresh look at deterministic parameter estimation a new cramer rao cr type bound is derived without assuming unbiased estimation of the nuisance parameters rather than that we apply lehmann s concept of unbiasedness for a risk that measures the distance between the estimator and the locally best unbiased estimator which assumes perfect knowledge of the nuisance parameters the proposed risk unbiased crb rucrb is proven to be asymptotically attainable by the maximum likelihood estimator while being tighter than the conventional crb furthermore simulations verify the asymptotic achievability of the rucrb by the maximum likelihood estimator for an array processing problem

an algorithm for pole zero modeling and spectral analysis an explicit connection between fitting exponential models and pole zero models to observed data is made the fitting problem is formulated as a constrained nonlinear minimization problem this problem is then solved using a simplified iterative algorithm the algorithm is applied to simulated data and the performance of the algorithm is compared to previous results

maximum likelihood array processing in spatially correlated noise fields using parameterized signals this paper deals with the problem of estimating signal parameters using an array of sensors this problem is of interest in a variety of applications such as radar and sonar source localization a vast number of estimation techniques have been proposed in the literature during the past two decades most of these can deliver consistent estimates only if the covariance matrix of the background noise is known in many applications the aforementioned assumption is unrealistic recently a number of contributions have addressed the problem of signal parameter estimation in unknown noise environments based on various assumptions on the noise herein a different approach is taken we assume instead that the signals are partially known the received signals are modeled as linear combinations of certain known basis functions the exact maximum likelihood ml estimator for the problem at hand is derived as well as computationally more attractive approximation the cramer rao lower bound crb on the estimation error variance is also derived and found to coincide with the crb assuming an arbitrary deterministic model and known noise covariance

bayesian detection and estimation of cisoids in colored noise the problem of estimating the number of cisoids in colored noise is addressed it is assumed that the noise can be modeled by an autoregression whose order has also to be estimated a new criterion is proposed for estimating the number of cisoids and the autoregressive model order as well as a new algorithm for estimating the cisoidal frequencies in the derivation a bayesian methodology and subspace decomposition are employed the proposed criterion significantly outperforms the popular mdl and aic as applied in a paper by nagesha and kay in addition an algorithm that reduces the computational complexity of the solution is developed computer simulations that demonstrate the performance of the criterion are included

maximum likelihood estimation of multiple frequencies with constraints to guarantee unit circle roots an approximate maximum likelihood estimator mle of multiple exponentials converts the frequency estimation problem into a problem of estimating the coefficients of a z polynomial with roots at the desired frequencies theoretically the roots of the estimated polynomial should fall on the unit circle but mle as originally proposed does not guarantee unit circle roots this drawback sometimes causes merged frequency estimates especially at low snr if all the sufficient conditions for the z polynomial to have unit circle roots are incorporated the optimization problem becomes too nonlinear and it loses the desirable weighted quadratic structure of mle in the present paper the exact constraints are imposed on each of the first order factors corresponding to individual frequencies for ensuring unit circle roots the constraints are applied during optimization alternately for each frequency in the absence of any merged frequency estimates the rms values more closely approach the theoretical cramer rao cr bound at low snr levels

frequency domain prony method for autoregressive model identification and sinusoidal parameter estimation in this study the weighted integral method for identifying differential equation models is extended to a discrete time system with a difference equation de model and a finite length sampled data sequence and obtain a frequency domain algorithm for short time signal analysis and frequency estimation the derivation consists of three steps 1 provide the de autoregressive model with unknown coefficients which is satisfied in a finite observation interval 2 discrete fourier transform dft the de to obtain algebraic equations aes among the fourier coefficients two mathematical techniques are introduced to maintain the circulant nature of time shifts 3 simultaneously solve a sufficient number of aes with least squares criterion to obtain unknowns exactly when the driving term is absent or to obtain unknowns that minimize the driving power when it is present the methods developed enable a decomposed processing of identification and estimation in the frequency domain thus they will be suitable for maximizing statistical efficiency smallness of estimation error variance reducing the computational cost and use in a resolution enhanced time frequency analysis of real world signals the performance of the proposed methods are compared with those of several dft based methods and cramer rao lower bound also the interference effect and its reduction in frequency decomposed processing are examined

an efficient approach for two dimensional parameter estimation of a single tone in this paper parameter estimation of a two dimensional 2 d single damped real complex tone in the presence of additive white gaussian noise is addressed by utilizing the rank one property of the 2 d noise free data matrix the damping factor and frequency for each dimension are estimated in a separable manner from the principal left and right singular vectors according to an iterative weighted least squares procedure the remaining parameters are then obtained straightforwardly using standard least squares the biases as well as variances of the damping factor and frequency estimates are also derived which show that they are approximately unbiased and their performance achieves crame r rao lower bound crlb at sufficiently large signal to noise ratio snr and or data size conditions we refer the proposed approach to as principal singular vector utilization for modal analysis puma which performs estimation in a fast and accurate manner the development and analysis can easily be adapted for a tone which is undamped in at least one dimension furthermore comparative simulation results with several conventional 2 d estimators and crlb are included to corroborate the theoretical development of the puma approach as well as to demonstrate its superiority

exploiting arrays with multiple invariances using music and mode this paper describes several new techniques for direction of arrival doa estimation using arrays composed of multiple translated and uncalibrated subarrays the new algorithms can be thought of as generalizations of the music root music and mode techniques originally developed for fully calibrated arrays the advantage of these new approaches is that the doas can be estimated using either a simple one dimensional 1 d search or by rooting a polynomial as opposed to the multidimensional search required by multiple invariance mi esprit when it can be applied the proposed mi mode algorithm shares the statistical optimality of mi esprit while mi music and root mi music are only optimal for uncorrelated sources they perform better than a single invariance implementation of esprit and are thus better suited for finding the initial conditions required by the mi esprit search

full newton and constraint methods for semilinear signal modeling the authors derive an expression for the hessian matrix of the variable projection functional vpf and implement the hessian using qr factorization this is incorporated into a full newton variable projection fnvp algorithm for estimating parameters in semilinear signals they introduce a deflation technique for constraining the vpf to contain known basis vectors for modeling exponential signals the computational cost of the fnvp algorithm is shown to vary linearly with n the size of the data vector while other algorithms vary as n sup 2 or n log sub 2 n

estimating frequencies of exponentials in noise using joint diagonalization the problem of estimating frequencies of superimposed exponentials in white noise collected by a single sensor is considered we discuss a solution based on a modification of an analytic algorithm that was proposed for separation of constant modulus signals received by an antenna array by van der veen and paulraj see ibid vol vol 44 p 1136 55 1996 we show that with an appropriate implementation of certain constraints the algorithm produces excellent super resolution results an advantage of the algorithm is that it does not require multidimensional search and therefore it is free from problems associated with search algorithms like convergence to local extremum or proper initialization another advantage is its applicability in nonuniform sampling conditions where methods based on linear prediction cannot be applied we carry out a detailed performance analysis under the assumption of long data record and derive conditions for estimating all the frequencies the performance of the proposed algorithm compares favorably with the performance of the tufts kumaresan method 1982

music maximum likelihood and cramer rao bound further results and comparisons the problem of determining the direction of arrival of narrowband plane waves using sensor arrays and the related problem of estimating the parameters of superimposed signals from noisy measurements are studied a number of results have been recently presented by the authors on the statistical performance of the multiple signal characterization music and the maximum likelihood ml estimators for the above problems this work extends those results in several directions first it establishes that in the class of weighted music estimators the unweighted music achieves the best performance i e the minimum variance of estimation errors in large samples next it derives the covariance matrix of the ml estimator and presents detailed analytic studies of the statistical efficiency of music and ml estimators these studies include performance comparisons of music and mle with each other as well as with the ultimate performance corresponding to the cramer rao bound finally some numerical examples are given which provide a more quantitative study of performance for the problem of finding two directions with uniform linear sensor arrays

robust estimation of structured covariance matrices in the context of the narrowband array processing problem robust methods for accurately estimating the spatial correlation matrix using a priori information about the matrix structure are developed by minimizing the worse case asymptotic variance robust structured maximum likelihood type estimates of the spatial correlation matrix in the presence of noises with probability density functions in the in contamination and kolmogorov classes are obtained these estimates are robust against variations in the noise s amplitude distribution the kolmogorov class is demonstrated to be the natural class to use for array processing applications and a technique is developed to determine exactly the size of this class performance of bearing estimation algorithms improves substantially when the robust estimates are used especially when nongaussian noise is present a parametric structured estimate of the spatial correlation matrix that allows direct estimation of the arrival angles is also demonstrated

the cramer rao bound on frequency estimates of signals closely spaced in frequency this paper examines the cramer rao cr lower bound on the variance of frequency estimates for the problem of n signals closely spaced in frequency the main results presented are simple analytic expressions for the cr bound in terms of the maximum frequency separation delta omega snr and the number of data vectors n that are valid for small delta omega the results are applicable to the conditional deterministic signal model the results show that the cr bound on frequency estimates is proportional to delta omega sup 2 n 1 n snr therefore the bound increases rapidly as the signal separation is reduced examples indicate that the expressions closely approximate the exact cr bounds whenever the signal separation is smaller than one resolution cell based upon the results it is argued that the threshold snr at which an unbiased estimator can resolve n closely spaced signals is at least proportional to delta omega sup 2n n the results are quite general and apply to many different types of temporal and spatial sampling grids

a weighted linear prediction method for near field source localization this paper proposes a low computational cost method for the near field narrowband source localization problem which does not require multidimensional search or high order statistics the proposed method is based on the second order statistics sos of the outputs of a uniform linear array ula more precisely the range and angle parameters are estimated through a weighted linear prediction lp algorithm applied to a properly chosen array output correlation sequence detailed performance analysis and derivation of the optimal weightings are provided simulation results are finally presented to validate the theoretical analysis results and to assess the performance of the proposed method

total least squares for affinely structured matrices and the noisy realization problem structured rank deficient matrices arise in many applications in signal processing system identification and control theory the author discusses the structured total least squares stls problem which is the problem of approximating affinely structured matrices i e matrices affine in the parameters by similarly structured rank deficient ones while minimizing an l sub 2 error criterion it is shown that the optimality conditions lead to a nonlinear generalized singular value decomposition which can be solved via an algorithm that is inspired by inverse iteration next the author concentrates on the so called l sub 2 optimal noisy realization problem which is equivalent with approximating a given data sequence by the impulse response of a finite dimensional time invariant linear system of a given order this can be solved as a structured total least squares problem it is shown with some simple counter examples that classical algorithms such as the steiglitz mcbride 1965 iterative quadratic maximum likelihood and cadzow s 1988 iteration do not converge to the optimal l sub 2 solution despite misleading claims in the literature

a combined order selection and parameter estimation algorithm for undamped exponentials we propose an approximate maximum likelihood parameter estimation algorithm combined with a model order estimator for superimposed undamped exponentials in noise the algorithm combines the robustness of fourier based estimators and the high resolution capabilities of parametric methods we use a combination of a wald 1945 statistic and a map test for order selection and initialize an iterative maximum likelihood descent algorithm recursively based on estimates at higher candidate model orders experiments using simulated data and synthetic radar data demonstrate improved performance over mdl map and aic in places of practical interest

employing symmetry constraints for improved frequency estimation by eigenanalysis methods the problem of extracting sinusoid signals from noisy observations made at equally spaced times is considered eigenanalysis methods such as pisarenko s method and the extended prony method find the eigenvector with minimum eigenvalue of a suitably chosen matrix and then obtain the complex sinusoids as the roots of the polynomial that has the components of the eigenvector as coefficients for the sinusoids to be undamped it is necessary that the roots lie on the unit circle and hence that the eigenvector be conjugate symmetric it is shown how this symmetry constraint can be incorporated into eigenanalysis estimation methods in a routine way the practical importance of the constraint is investigated by cramer rao variance bound calculations and by simulation the following conclusions are made 1 the symmetry constraint is straightforward to implement and reduces the amount of computation required 2 the relative reduction in variance of the constrained over the unconstrained frequency estimators

on the complexity of iqml algorithms the authors study the computational complexity of two methods for solving least squares and maximum likelihood modal analysis problems in particular they consider the steiglitz mcbride and iterative quadratic maximum likelihood iqml algorithms j h mcclellan and d lee ibid vol 39 no 2 p 509 12 1991 have shown the iterations of the two methods to be equivalent however they suggest that the steiglitz mcbride algorithm may be computationally preferable a method for reducing the dimension of the matrix inversion required at each iteration of iqml is provided the resulting reduction in the computation makes the computational complexity of iqml commensurate with that of the steiglitz mcbride algorithm

perfect blind restoration of images blurred by multiple filters theory and efficient algorithms we address the problem of restoring an image from its noisy convolutions with two or more unknown finite impulse response fir filters we develop theoretical results about the existence and uniqueness of solutions and show that under some generically true assumptions both the filters and the image can be determined exactly in the absence of noise and stably estimated in its presence we present efficient algorithms to estimate the blur functions and their sizes these algorithms are of two types subspace based and likelihood based and are extensions of techniques proposed for the solution of the multichannel blind deconvolution problem in one dimension we present memory and computation efficient techniques to handle the very large matrices arising in the two dimensional 2 d case once the blur functions are determined they are used in a multichannel deconvolution step to reconstruct the unknown image the theoretical and practical implications of edge effects and weakly exciting images are examined finally the algorithms are demonstrated on synthetic and real data

parametric identification using weighted null space fitting in identification of dynamical systems the prediction error method with a quadratic cost function provides asymptotically efficient estimates under gaussian noise but in general it requires solving a nonconvex optimization problem which may imply convergence to nonglobal minima an alternative class of methods uses a nonparametric model as intermediate step to obtain the model of interest weighted null space fitting wnsf belongs to this class starting with the estimate of a nonparametric arx model with least squares then the reduction to a parametric model is a multistep procedure where each step consists of the solution of a quadratic optimization problem which can be obtained with weighted least squares the method is suitable for both open and closed loop data and can be applied to many common parametric model structures including output error armax and box jenkins the price to pay is the increase of dimensionality in the nonparametric model which needs to tend to infinity as function of the sample size for certain asymptotic statistical properties to hold in this paper we conduct a rigorous analysis of these properties namely consistency and asymptotic efficiency also we perform a simulation study illustrating the performance of wnsf and identify scenarios where it can be particularly advantageous compared with state of the art methods

a modified prony algorithm for exponential function fitting a modification of the classical technique of prony for fitting sums of exponential functions to data is considered the method maximizes the likelihood for the problem unlike the usual implementation of prony s method which is not even consistent for transient signals proves to be remarkably effective in practice and is supported by an asymptotic stability result novel features include a discussion of the problem parametrization and its implications for consistency the asymptotic convergence proofs are made possible by an expression for the algorithm in terms of circulant divided difference operators

parametric modeling for damped sinusoids from multiple channels the problem of parametric modeling for noisy damped sinusoidal signals from multiple channels is addressed utilizing the shift invariance property of the signal subspace the number of distinct sinusoidal poles in the multiple channels is first determined with the estimated number the distinct frequencies and damping factors are then computed with the multi channel weighted linear prediction method the estimated sinusoidal poles are then matched to each channel according to the extreme value theory of distribution of random fields simulations are performed to show the performance advantages of the proposed multi channel sinusoidal modeling methodology compared with existing methods

an exact direct method of sinusoidal parameter estimation derived from finite fourier integral of differential equation in this paper we propose a novel method for estimating the parameters frequency amplitude and phase of real sinusoids to derive the estimator we start from the characteristic differential equation of a sinusoid to remove differentials and obtain an algebraic relation for frequency we introduce finite period weighted integrals of the differential equation which become equivalent to the differential equation when a sufficient number of weight functions are applied as weight functions we show that fourier kernels have excellent properties terms related to integral boundaries are readily eliminated observations are provided by fourier coefficients and the relation becomes independently accurate for multiple sinusoids if they are sufficiently spaced we solve the obtained equations in two ways one is for approaching to the cramer rao lower bound crlb and the other is for enhancing the interference rejection capability also methods are proposed to calculate the weighted integrals from sampled signals with an improved accuracy proposed algorithms are examined under noise and sinusoidal interference error variances are compared with the crlb and other fast fourier transform fft based methods

time delay and spatial signature estimation using known asynchronous signals this paper considers the problem of estimating the parameters of known signals received asynchronously by an array of antennas the parameters of primary interest are the time delays of the signals and their spatial signatures at the array estimates of the signal directions of arrival are also considered but are of secondary importance in this work maximum likelihood algorithms and more computationally efficient approximations are developed for both the case where all received signals are identical the channel estimation overlapping echo problem and where they are all distinct conditions are also derived under which the standard matched filter approach yields consistent and statistically efficient parameter estimates the issue of solution uniqueness is addressed and in particular an upper bound on the number of signals whose parameters may be uniquely identified is derived for a number of different cases typically the bound is far greater than the number of sensors and is limited only by the number of data samples collected some representative simulation examples are also included to illustrate the algorithms performance relative to the cramer rao bound

worst case cramer rao bounds for parametric estimation of superimposed signals with applications the problem of parameter estimation of superimposed signals in white gaussian noise is considered the effect of the correlation structure of the signals on the cramer rao bounds is studied for both the single and multiple experiment cases the best and worst conditions are found using various criteria the results are applied to the example of parameter estimation of superimposed sinusoids or plane wave direction finding in white gaussian noise and best and worst conditions on the correlation structure and relative phase of the sinusoids are found this provides useful information on the limits of the resolvability of sinusoid signals in time series analysis or of plane waves in array processing the conditions are also useful for designing worst case simulation studies of estimation algorithms and for the design of minimax signal acquisition and estimation procedures as demonstrated by an example

matrix pencil method for estimating parameters of exponentially damped undamped sinusoids in noise a study of a matrix pencil method for estimating frequencies and damping factors of exponentially damped and or undamped sinusoids in noise is presented comparison of this method to a polynomial method svd prony method shows that the matrix pencil method and the polynomial method are two special cases of a matrix prediction approach and that the pencil method is more efficient in computation and less restrictive about signal probes it is found through perturbation analysis and simulation that for signals with unknown damping factors the pencil method is less sensitive to noise than the polynomial method an expression of the cramer rao bound for the exponential signals is presented

general smoothing techniques for estimating deterministic undamped sinusoidal frequencies from noisy data general high resolution smoothing techniques for estimating deterministic sinusoidal frequencies from short record noisy data are presented symmetric smoothings are proven to provide conjugate reciprocal polynomial roots also in the noisy case and thereby eliminate spurious frequency estimates in numerical simulations they provided the lowest threshold of any currently existing method

second order hamilton jacobi equations in infinite dimensions some second order hamilton jacobi equations connected to stochastic optimal control problems for infinite dimensional systems driven by a white noise are studied a direct method to prove existence and uniqueness of mild solutions is developed then this solution is identified as the value function of the related stochastic control problem and a feedback formula for optimal controls is derived

detection and estimation in sensor arrays using weighted subspace fitting the problem of signal parameter estimation of narrowband emitter signals impinging on an array of sensors is addressed a multidimensional estimation procedure that applies to arbitrary array structures and signal correlation is proposed the method is based on the recently introduced weighted subspace fitting wsf criterion and includes schemes for both detecting the number of sources and estimating the signal parameters a gauss newton type method is presented for solving the multidimensional wsf and maximum likelihood optimization problems the global and local properties of the search procedure are investigated through computer simulations most methods require knowledge of the number of coherent noncoherent signals present a scheme for consistently estimating this is proposed based on an asymptotic analysis of the wsf cost function the performance of the detection scheme is also investigated through simulations

exact equivalence of the steiglitz mcbride iteration and iqml the exact equivalence of the steiglitz mcbride mode 1 algorithm and the iterative quadratic maximum likelihood algorithm iqml is demonstrated other methods published earlier by l l scharf et al 1986 and by a g evans and r fischl 1973 are also known to give the same equations and answer as iqml hence all these methods are equivalent to the steiglitz mcbride iteration since the iqml method was originally formulated to solve a multiple measurement case an extended version of the steiglitz mcbride mode 1 algorithm applicable to the multiple measurement case is also stated and shown to be equivalent to the general form of iqml

towards generalized fri sampling with an application to source resolution in radioastronomy it is a classic problem to estimate continuous time sparse signals like point sources in a direction of arrival problem or pulses in a time of flight measurement the earliest occurrence is the estimation of sinusoids in time series using prony s method this is at the root of a substantial line of work on high resolution spectral estimation the estimation of continuous time sparse signals from discrete time samples is the goal of the sampling theory for finite rate of innovation fri signals both spectral estimation and fri sampling usually assume uniform sampling but not all measurements are obtained uniformly as exemplified by a concrete radioastronomy problem we set out to solve thus we develop the theory and algorithm to reconstruct sparse signals typically sum of sinusoids from nonuniform samples we achieve this by identifying a linear transformation that relates the unknown uniform samples of sinusoids to the given measurements these uniform samples are known to satisfy the annihilation equations a valid solution is then obtained by solving a constrained minimization such that the reconstructed signal is consistent with the given measurements and satisfies the annihilation constraint thanks to this new approach we unify a variety of fri based methods we demonstrate the versatility and robustness of the proposed approach with five fri reconstruction problems namely dirac reconstructions with irregular time or fourier domain samples fri curve reconstructions dirac reconstructions on the sphere and point source reconstructions in radioastronomy the proposed algorithm improves substantially over state of the art methods and is able to reconstruct point sources accurately from irregularly sampled fourier measurements under severe noise conditions

doa estimation of wideband sources using a harmonic source model and uniform linear array we consider the problem of estimation of the doas of multiple wideband sources incident on a uniform linear array ula in the presence of spatially and temporally white gaussian noise wgn the approach presented builds up on the iqml algorithm suggested by bresler and macovski 1986 for the case of narrowband doa estimation it is shown that the concept of an arma model for the observed data vector for the narrowband case can be generalized to model an appropriately stacked space time data vector obtained by combining the space time samples the coefficients of the corresponding 2 d predictor polynomial can be used to represent the null subspace of the wideband array steering matrix and rooting of the polynomial at each frequency separately gives the doa estimates these separate estimates at multiple frequencies are combined into a single doa estimate in a least squares sense this leads to the formulation of an iqml like procedure for the spatial parameter estimation of wideband sources like its narrowband counterpart the proposed approach is applicable to both noncoherent and coherent sources the performance of the proposed method is studied via extensive computer simulations and by comparison with the cramer rao bounds

perturbation analysis of subspace based methods in estimating a damped complex exponential we present a study of mode variance statistics for three svd based estimation methods in the case of a single mode damped exponential the methods considered are namely kumaresan tufts matrix pencil and kung s direct data approximation through first order perturbation analysis we derive closed form expressions of the variance of the complex mode frequency and damping factor estimates these expressions are used to compare the different methods and to determine the optimal prediction order for matrix pencil and direct data approximation methods application to the undamped case shows the coherence of the results with those already stated in the literature it is also found that the variances converge linearly towards the cramer rao bound finally the theoretical results are verified using monte carlo simulations

a parameter estimation approach to estimation of frequencies of sinusoids it is shown that with uniform sampling a sample of the sum of m sinusoids at time nt can be uniquely expressed as a linear combination of the 2m samples at times n 1 t n 2m t the 2m coefficients of linear combination are also the coefficients of a 2m order polynomial whose roots are e pmj omega i omega i being the frequencies of the sinusoids given the samples of sinusoids plus noise a consistent estimate of the 2m coefficients are obtained by the instrumental variable method of parameter estimation it is also possible to track time varying frequencies by a recursive algorithm that exponentially weighs out the past data simulation examples of one and two sinusoids are given

iterative inverse filtering approach to the estimation of frequencies of noisy sinusoids a signal modeling approach is proposed to reduce the problem of estimating frequencies of noisy sinusoids to the parameter estimation problem both white and colored noise are considered the maximum likelihood approach leads to an algorithm computationally inefficient for low values of signal to noise ratio based on the equation error formulation an iterative inverse filtering algorithm is derived in the case of colored noise and a generalized least squares algorithm in the case of white noise it is shown on the basis of numerous experimental results that the iterative inverse filtering algorithm provides highly accurate estimates of unknown frequencies for low values of signal to noise ratio even in the case of a small number of sampling points

reduced complexity maximum likelihood direction of arrival estimation based on spatial aliasing spatial aliasing is an undesirable phenomenon that prevents the unique determination of the direction of arrival doa of impinging signals in array signal processing however the characteristics of spatial aliasing that generates ambiguous doas can be also used to reduce the computational complexity in maximum likelihood ml doa estimation this paper proposes a structural method to dramatically reduce the computational complexity of the ml doa estimation using the spatial aliasing generated by a nested array structure with a doubly scaled aper ture an ml full grid search is computationally simplified by the highly compressed searching range and the small number of candidate values to be searched which are derived based on spatial aliasing performance analyses based on the theoretical bounds and computational complexity with computer simulations show that the proposed method requires an extremely reduced computational load compared to the conventional ml doa estimation with a uniform linear array ula while achieving the performance enhancement in the estimation accuracy due to the enlarged array aperture

an l shaped array for estimating 2 d directions of wave arrival a simple structured 2 d array the l shaped array is presented the l shaped array consists of two uniform linear arrays ula connected orthogonally at one end of each ula it is shown that the cramer rao bounds crb of the estimated wave directions based on the l shaped array are about 37 smaller than those for the cross array the crb indicates the accuracy potential because it is the reachable lower bound on the variance of any unbiased estimate an efficient maximum likelihood algorithm is developed utilizing the ula structure inherent in the l shaped array

model selection for sinusoids in noise statistical analysis and a new penalty term detection of the number of sinusoids embedded in noise is a fundamental problem in statistical signal processing most parametric methods minimize the sum of a data fit likelihood term and a complexity penalty term the latter is often derived via information theoretic criteria such as minimum description length mdl or via bayesian approaches including bayesian information criterion bic or maximum a posteriori map while the resulting estimators are asymptotically consistent empirically their finite sample performance is strongly dependent on the specific penalty term chosen in this paper we elucidate the source of this behavior by relating the detection performance to the extreme value distribution of the maximum of the periodogram and of related random fields based on this relation we propose a combined detection estimation algorithm with a new penalty term our proposed penalty term is sharp in the sense that the resulting estimator achieves a nearly constant false alarm rate a series of simulations support our theoretical analysis and show the superior detection performance of the suggested estimator

a newton based ziskind wax alternating projection algorithm applied to spectral estimation a newton based alternating projection ap algorithm for estimating the parameters of exponential signals in noise is presented its performance is compared with that of modified forward backward linear prediction backward linear prediction mfblp blp total least squares tls and iterative quadratic maximum likelihood iqml methods using computer simulations in the case of undamped sinusoids the ap algorithm yielded a lower threshold snr than the others in the case of damped sinusoids its threshold snr was the same as that of tls but was 1 db lower than that of mfblp and iqml