Many signal processing algorithms utilize singular vectors of the data received by a sensor array or eigenvectors of the sample covariance matrix. The performance of such subspace-based algorithms can be substantially improved by using the optimal signal or noise subspace estimate instead of using the subspaces spanned by the eigenvectors or the singular vectors. The optimal signal and noise subspaces are estimated by exploiting the shift-invariant structure of the sensor array. We develop a methodology to find the optimal subspaces in sparse arrays that possess shift-invariant structure and interpolate the optimal subspaces to match the subspaces of the fully populated arrays with an equal aperture. These shift-invariant sparse arrays’ direction of arrival estimation performance is superior compared to the methods that use eigenvectors or singular vectors directly. Our method also encompasses any symmetric array that is not shift-invariant, thus broadening the class of sparse arrays where our method can be applied.

This paper introduces matrix filters as a tool for localization and detection problems in passive sonar. The outputs of an array of sensors, at some given frequency, can be represented by a vector of complex numbers. A linear filtering operation on the sensor outputs can be expressed as the multiplication of a matrix (called a matrix filter) times this vector. The purpose of a matrix filter is to attenuate unwanted components in the measured sensor data while passing desired components with minimal distortion. Matrix filters are designed by defining an appropriate pass band and stop band and solving a convex optimization problem. This paper formulates the design of matrix filters for passive sonar and gives two examples.

This paper presents a new approach to deriving statistically optimal weights for weighted subspace fitting (WSF) algorithms. The approach uses a formula called a "subspace perturbation expansion," which shows how the subspaces of a matrix change when the matrix elements are perturbed. The perturbation expansion is used to derive an optimal WSF algorithm for estimating directions of arrival in array signal processing.

This paper considers a method for estimating time delays, amplitudes, and Doppler scales of a multipath signal. The method is an extension of work previously reported by Manickam and Vaccaro (1993) which dealt solely with time delays and amplitudes, and extended by Habboosh and Vaccaro (see Proceedings CISS '96, p.633-8, 1996) to include Doppler scale. An algorithm is presented for determining the size of the indicator set to reduce ill-conditioning of the signal subspace matrix. Simulation results are shown and comparisons to the Cramer-Rao lower bound provided; these results show that significant reduction in estimate variances can be achieved using the deconvolution approach with a properly selected indicator set.

In many estimation problems, the set of unknown parameters can be divided into a subset of desired parameters and a subset of nuisance parameters. Using a maximum a posteriori (MAP) approach to parameter estimation, these nuisance parameters are integrated out in the estimation process. This can result in an extremely computationally-intensive estimator. This paper proposes a method by which computationally-intensive integrations over the nuisance parameters required in Bayesian estimation may be avoided under certain conditions. The proposed method is an approximate MAP estimator which is much more computationally efficient than direct, or even Monte Carlo, integration of the joint posteriori distribution of the desired and nuisance parameters. As an example of its efficiency, we apply the fast algorithm to matched-field source localization in an uncertain environment.

We are concerned with signal processing of acoustic signals resulting from active transmissions by high frequency sonar systems. These signals consist of structured interference related to propagation effects in the media, reflections from targets, and measurement noise. The methods herein model these signals as replicas of the transmitted signal, scaled in amplitude and time, and delayed. Furthermore, we are interested in signals with 'simple' time frequency profiles, such as linear frequency modulated (LFM) or hyperbolic frequency modulated (HFM) signals. These signals have the underlying property that the principle ridge of the autoambiguity function crosses the mid-point of the time-frequency plane in a smooth manner, with a simple relationship between time delay and time scaling (frequency shifting). This paper describes a method for estimating the delay and time scale of the signal components using fast maximum likelihood, while preserving the high resolution property of related time delay estimation techniques.

In many estimation problems, the set of unknown parameters can be divided into a subset of desired parameters and a subset of nuisance parameters. Using a maximum a posteriori (MAP) approach to parameter estimation, these nuisance parameters are integrated out in the estimation process. This can result in an extremely computationally intensive estimator. This letter proposes a method by which computationally intensive integration over the nuisance parameters required in Bayesian estimation can be avoided under certain conditions. The proposed method is an approximate MAP estimator, which is much more computationally efficient than direct, or even Monte Carlo, integration of the joint posteriori distribution of the desired and nuisance parameters. As an example, we apply the fast algorithm to matched-field source localization in an uncertain environment.

This paper provides an analytical solution to the problem of determining the data matrix dimensions when a subspace-based algorithm is used to estimate parameters from a noisy signal. A unified expression of mean squared errors (MSEs) of signal parameter estimates is given first in terms of singular values and singular vectors. This expression is then generalized so that the MSEs are a function of physical parameters such as the data length, the signal frequency spacing, and the number of rows of the data matrix formed for the use of a subspace-based algorithm. When all the physical parameters except the data matrix dimensions are fixed, the mean squared errors of the signal parameter estimate are a function of the data matrix dimension. The optimal matrix dimensions, which correspond to the minima of the error function, can then be determined. Examples using synthetic data are included to demonstrate the significance of the results.

Matched-field source localization methods can be sensitive to environmental parameter mismatch. In this paper, a source localization method which is robust to environmental parameter uncertainty is presented. This method allows for a family of propagation models based on the uncertainty intervals of the environmental-parameters. It is a computationally efficient approximation to a full maximum-likelihood search which can be followed by a local exact maximum-likelihood search. The effectiveness of this method is demonstrated using experimental data from the Mediterranean Sea.

A new method for solving DOA tracking problems is presented. The complexity of the method is second order in L, where L is the number of array sensors. One advantage gained by using the invariant subspace updating method is the ability to track rapidly changing DOAs. A second technique allows the immediate detection of appearing and disappearing sources.

This paper studies the Cramer-Rao (CR) bound for the problem of estimating the directions of arrival of narrow-band plane waves impinging on an ESPRIT array with multiple displacement invariances. We call this bound the MI-CR bound. The CR bound for ESPRIT arrays with a single invariance was derived by Ottersten, Viberg and Kailath (see IEEE Trans. on Signal Processing, vol.39, p.1122-35, May 1991), and called the ESPRIT-CR bound. The difference between the ESPRIT-CR bound and the MI-CR bound can be large, especially for highly correlated sources. We give examples to show that an algorithm called WSE achieves the MI-CR bound derived in this paper.

Matched-field source localization methods attempt to estimate the range and depth of a source in an acoustic waveguide. These methods give good results when the waveguide parameters are known precisely; however, matched-field methods have been shown to be very sensitive to model mismatch resulting from errors in the assumed environmental parameters. In this paper, we describe an approach which minimizes model mismatch by estimating the environmental parameters and source location together. We propose an efficient way to initialize the maximum-likelihood search by projecting the received data onto subspaces corresponding to regions in parameter space.

We address the problem of multipath time-delay estimation. When the received data is very long compared to the transmitted signal, the data is expected to consist of a large number of paths. Modeling the entire data becomes computationally expensive. We propose a technique to break the data into short segments and model each segment individually without misfitting or truncating any paths at the ends of any segment. By effectively using overlapping segments, the estimates of time-delays are combined to model the entire data record. The method is extended to the case where only basebanded data are available. The proposed technique is demonstrated on an experimental sea-test data.

Subspace-based direction-of-arrival (DOA) estimation has attracted many excellent performance studies, but limitations such as the analysis of individual algorithms generally exist in these performance studies. The authors previously proposed a unified performance analysis based on a finite amount of data, and achieved a tractable expression for the mean-squared DOA estimation error for the MUSIC, Min-Norm, ESPRIT, and state-space realization algorithms. However, this expression uses the singular values and vectors of a data matrix. Thus, the effects of the original data parameters such as numbers of sensors and snapshots, source coherence, and separations were not explicitly analyzed. In this work, the authors have further unified and simplified their previous results and derived a unified expression based on the original data parameters.>

The authors consider the problem of estimating the arrival times of overlapping ocean-acoustic signals from a noisy received waveform which consists of scaled and delayed replicas of a deterministic transient signal. It is assumed that the transmitted signal and the number of paths in the multipath environment are known. Previously developed tools for finding least-squares estimates of the amplitude and time delay of each path are combined to form a complete algorithm. The algorithm is a function of several data-dependent parameters, and rules for choosing these parameters are given. The algorithm is demonstrated on a synthetic four-path example in which the paths cannot be resolved by standard cross-correlation techniques.>

A statistical performance analysis of subspace-based direction-of-arrival (DOA) estimation algorithms in the presence of correlated observation noise with unknown covariance is presented. This analysis of five different estimation algorithms is unified by a single expression for the mean-squared DOA estimation error which is derived using a subspace perturbation expansion. The analysis assumes that only finite amount of array data is available.>

A unified performance analysis of subspace-based algorithms for direction-of-arrival (DOA) estimation involving multiple signal arrivals in array signal processing is presented. Following previous analyses of MUSIC and Min-Norm, an analytical expression of the variance of the DOA estimation-error is developed for state-space realization and ESPRIT in a greatly simplified fashion. The tractable formulas provide insight for design. Simulation results verify the analysis.>

Consideration is given to the problem of estimating the arrival times of overlapping or nonoverlapping ocean acoustic signals from a received signal which consists of an unknown deterministic signal along with scaled and delayed versions due to multipath propagation plus additive noise. The objective is to simultaneously determine the transmitted waveform and the arrival times. The proposed algorithm obtains approximately maximum likelihood estimates of the arrival times and the parameters which characterize the unknown signal, with much fewer computations that the direct implementation of the MLE. It is assumed that the number of the different paths is known and that the signal must belong to a parametric class of signals. The algorithm is demonstrated on a class of signals consisting of gated sinusoids.>

The authors present a min-norm algorithm for estimating the directions of arrival of signals impinging on a sensor array of arbitrary geometry. The algorithm is an extension of the min-norm algorithm derived by R. Kumaresan and D.W. Tufts (1983) for the case of a uniform line array. A perturbation analysis for this arbitrary array min-norm method is also presented, and analytical formulas that predict the variance of the estimated directions are given. Simulations with a circular array demonstrate the algorithm as well as verify the performance analysis.>

The problem of analyzing the threshold effect of signal processing algorithms which use the singular-value decomposition (SVD) is addressed. The probability of obtaining an outlier is calculated and used to determine the threshold SNR at which the variance of parameter estimation errors depart from Cramer-Rao bound behavior. Simulation results using low rank approximation and linear prediction for frequency estimation verify the analysis. The same method of analysis can be applied to a broad class of parameter-estimation methods in which the principal-component technique or low rank approximations to matrices are used.>

A discrete-time command generator that accepts position and orientation trajectories in Cartesian space and produces position, velocity, and acceleration trajectories in joint space is presented. Discrete-time command generators can be obtained from their continuous-time counterparts by using any standard numerical procedure to solve the nonlinear differential equation at a discrete set of points. However, existing numerical procedures require multiple function evaluations per input point to achieve an accurate solution. A novel technique is presented that achieves an accurate solution with only one function evaluation per input point. The application of the scheme to six degree-of-freedom manipulators is demonstrated through simulation results for a spherical-wrist robot.>

Many recently developed control schemes for robotic manipulators require as inputs the desired position, velocity, and in some cases, acceleration of each joint of the manipulator. However, it is most natural to specify the desired trajectory of the end effector in Cartesian coordinates. Thus it is desirable to have a command generator which has as input a desired Cartesian trajectory, and as output a vector of joint positions, velocities, and accelerations that correspond to the demanded trajectory. Such a command generator is presented in the form of a nonlinear feedback system that has the advantage of being related to a linear system. The linear system can be used to compute precise bounds on the performance of the nonlinear system. Simulation results for a nonspherical wrist manipulator are given.>

This paper presents a statistical analysis of the state-variable balancing and Prony methods for estimating the parameters of exponential signals in the presence of additive noise. The case of frequency estimation for a single sinusoid is carried out in detail. Analytical expressions for the variances of the frequency estimates at high signal-to-noise ratios are derived. The calculated variances are compared to the Cramer-Rao bound. The results are validated by simulations over a wide range of signal-to-noise ratios. The analysis and simulations show that the state-variable balancing method can provide slightly more accurate frequency estimates while avoiding the problem of selecting the signal zeros of the Prony polynomial.

The problem of statistically analyzing the performance of signal processing algorithms which use the singular value decomposition is adressed in this paper. Such decomposition, which is widely used in system identification and parameter estimation, is a non-linear operation. Consequently, when applied to random data, statistical results are extremely difficult to obtain. The first-order Taylor series expansion is generally used in computing the statistics but the derivative term makes the staistical analysis very difficult. In this paper, a power-like method which results in a simple expression is proposed. The singular vector perturbation using both approaches for the case of low rank approximation is examined.

The problem considered in this paper is the following: given a symmetric sequence{r_{j}}which is not positive, (i.e. its Fourier transform takes on negative values), find a positive sequence{\bar{r}_{j}}which is a good approximation to{r_{j}}. This problem arises, for example, when{r_{j}}is an estimated covariance sequence which is not positive, and a positive covariance sequence is desired. A solution to the positivity problem is given which uses state-space models and a scaled algebraic Riccati equation.

A command generator that accepts position and orientation trajectories in Cartesian space and produces position, velocity, and acceleration trajectories in joint space is presented. A new representation of orientation is introduced in which angular velocities are integrable combinations of finite angular rotations. This is accomplished by an internal overparameterization. The resulting command generator has the following properties: 1) errors are decoupled, and 2) bounds on those errors are easily computed. The application of the scheme to 6 degree of freedom manipulators is demonstrated through simulation results for a spherical-wrist robot.

An algorithm for obtaining a state-space (Markovian) model of a random process from a finite number of estimated covariance lags is presented and compared to other SVD-based methods. The new algorithm performs well for small data sets in which the estimated covariance sequence is perturbed by estimation errors, and may not even be positive definite. Simulation results are presented for a second-order Markov process.

The problem of locating spectral lines (sinusoids) in white noise has been studied intensively in the past few years, especially since the work of Pisarenko who showed that unbiased estimates of the sinusoidal frequencies could be obtained from the λ0eigenvector (i.e. the eigenvector associated with the smallest eigenvalue) of the covariance matrix of the process. Recently, adaptive implementations of Pisarenko's method have been developed in which the estimates can be updated as new data is observed, and so these algorithms have the ability to track slowly time-varying processes. In this paper, a new adaptive algorithm based on the inverse power method is developed and compared by simulation to other methods. Conclusions are drawn which affect any adaptive implementation of Pisarenko's method.

A method for the extraction of features for pattern recognition by system identification is presented. A test waveform is associated with a parameterized process model (PM) which is an inverse filter. The structure of the PM corresponds to the redundant information in a waveform, and the parameter values correspond to the discriminatory information. The PM used in this research is a linear predictive system whose parameters are the linear predictive coefficients (LPC's). This technique is applied to feature extraction of electrocardiograms (ECG's) for differential diagnosis. The LPC's are calculated for each ECG and used as a feature vector in a hypergeometric affine N-space spanned by the LPC's. The efficacy of this feature extraction technique is tested by three different perturbation methods, namely noise, matrix distortion, and a newly developed method called directed distortion. Both the Euclidean and Itakura distances between feature vectors in N-space are shown in increase with increasing perturbation of the template waveform. The monotonic behavior of a distance measure is a necessary attribute of a valid feature space. Thus the perturbation analyses done in this research verify the viability of using the parameters of a process model as a feature vector in a pattern recognition scheme.