Richard J. Vaccaro

Performance analysis for DOA estimation algorithms: unification, simplification, and observations

Subspace based direction-of-arrival (DOA) estimation has motivated many performance studies, but limitations such as the assumption of an infinite amount of data and analysis of individual algorithms generally exist in these performance studies. The authors have 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 multiple signal classification (MUSIC). Min-Norm, estimation of signal parameters using rotational invariance techniques (ESPRIT), and state-space realization algorithms. However, this expression uses the singular values and vectors of a data matrix, which are obtained by the highly nonlinear transformation of the singular value decomposition (SVD). Thus the effects of the original data parameters such as numbers of sensors and snapshots, source coherence and separations were not explicitly analyzed. The authors unify and simplify this previous result and derive a unified expression based on the original data parameters. They analytically observe the effects of these parameters on the estimation error. >

Unified analysis for DOA estimation algorithms in array signal processing

Abstract In this paper, a unified statistical performance analysis using perturbation expansions is applied to subspace-based algorithms for direction-of-arrival (DOA) estimation in array signal processing. The analysis assumes that only a finite amount of array data is available at high signal-to-noise ratio. The MUSIC, Min-Norm, State-Space Realization (TAM) and ESPRIT algorithms are analyzed in a common framework. A significant feature of this analysis is that it includes different types of error sources, such as the finite sample effect induced by additive observation noise, the sensor error effect induced by the inaccurate knowledge of sensor response and location, and the effect of a coherent noise field with unknown structure. All of the algorithms considered in this paper are based on a singular value decomposition of a data matrix. A general expression for the perturbation of singular vectors as a function of data matrix perturbations is derived and used to obtain an analytical expression for the mean-squared DOA estimation error in a simple and self-contained fashion. The tractable formulas provide insight into the algorithms. Simulation results verify the analytically predicted performance.

A least-squares algorithm for multipath time-delay estimation

We consider the problem of estimating the arrival times of overlapping ocean-acoustic signals from a noisy received waveform that consists of attenuated and delayed replicas of a known transient signal. We assume that the transmitted signal and the number of paths in the multipath environment are known and develop an algorithm that gives least-squares (LS) estimates of the amplitude and time delay of each path. Direct computation of the LS estimates would involve minimization of a highly oscillatory error function. By allowing the amplitudes to be complex valued, a much smoother error function that is easier to minimize using gradient-based techniques is obtained. Using this property and the knowledge (derived from the data) of the spacing between adjacent minima in the actual LS error function, an efficient algorithm is devised. The algorithm is a function of a data-dependent parameter, and we give rules for choosing this parameter. The algorithm is demonstrated on a broad-band signal, using simulated data. The proposed method is shown to achieve the Cramer-Rao lower bound over a wide range of SNRs. Comparisons are made with alternating projection (AP) and estimate maximize (EM) algorithms. >

Analysis of Min-Norm and MUSIC with arbitrary array geometry

A nonasymptotic performance comparison is presented between the Min-Norm and MUSIC algorithms for estimating the directions of arrival of narrowband plane waves impinging on an array of sensors. The analysis is based on a finite amount of sensor data. The analysis makes the assumption of high signal-to-noise ratio (SNR), and it applies to arrays of arbitrary geometry. It is shown that Min-Norm can be expressed as a certain data-dependent weighted MUSIC algorithm, and that this relationship allows a unified performance comparison. It is also shown that the variances of the estimated directions-of-arrival from the MUSIC algorithm are always smaller than those of the Min-Norm algorithm at high SNR when both algorithms employ a numerical search procedure to obtain the estimates. >

Performance degradation of DOA estimators due to unknown noise fields

A statistical performance analysis of subspace-based directions-of-arrival (DOA) estimation algorithms in the presence of correlated observation noise with unknown covariance is presented. The 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 a finite amount of array data is available. >

Statistical Modeling of Rate Gyros

Gyroscopes are integral components of inertial measurement units, which are used for guidance and stabilization of many platforms. This paper presents an algorithm for estimating the statistical parameters that govern the performance of rate gyros, i.e., the spectral densities R and Q of the angle random walk and rate random walk components, respectively. Previous work on gyro modeling is based on computing the Allan variance of a gyro signal and using a well-known formula for its mean. The algorithm in this paper uses these as well as the following quantities, which are derived in this paper: the theoretical variance of the Allan variance and the covariance between different Allan variance points. The algorithm is developed using the formulation of the best linear unbiased estimator from statistical estimation theory. The performance of the algorithm is demonstrated using simulated and experimental data. A bound on the error in the integral of the gyro output, as a function of Q and R, is also derived.

Performance analysis of the state-space realization (TAM) and ESPRIT algorithms for DOA estimation

The performance of signal-subspace-based algorithms for directions-of-arrival estimation involving multiple signal arrivals in array signal processing is analyzed. An analytical expression of the variance of the DOA estimation error is developed for three signal subspace based algorithms, state-space realization (SSR) (TAM), ESPRIT, and matrix pencil. Simulation results that verify the analysis are reported. >

Optimal matrix-filter design

A matrix filter produces N output values given a block of N input values. Matrix filters are particularly useful for filtering short data records (e.g. N/spl les/20). We introduce a new set of matrix-filter design criteria and show that the design of a matrix filter can be formulated as a convex optimization problem. Several examples are given of lowpass and bandpass designs as well as a Hilbert transformer design.

Least‐squares time‐delay estimation for transient signals in a multipath environment

The problem of estimating the arrival times of overlapping ocean‐acoustic signals from a noisy received waveform that consists of scaled and delayed replicas of a deterministic transient signal is considered. It is assumed that the transmitted signal and the number of paths in the multipath environment are known, and an algorithm is developed that gives least‐squares estimates of the amplitude and time delay of each path. A method is given to ensure that the global minimum of the error surface is found in spite of the existence of numerous local minima. The algorithm is then extended to the case in which the transmitted signal is not known precisely, but is assumed to belong to a parametric class of signals. The extended algorithm additionally obtains the parameters that characterize the transmitted signal. The algorithm is demonstrated on the class of signals consisting of gated sinusoids, using both simulated and experimental data.

A joint-space command generator for Cartesian control of robotic manipulators

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. >