Stephen W. Lang

Array design for MEM and MLM array processing

Much work has been done on designing arrays for use with conventional beamforming techniques. However, high resolution array processing techniques such as the maximum entropy (MEM) and maximum likelihood (MLM) methods impose different constraints on the array. MEM and MLM operate on measured samples of the correlation function and so are expected to perform better when the correlation function is measured at more and more evenly distributed lags. A uniform array gives redundant lags; the proposed array design technique places the sensors to maximize the number of the resulting lags and to distribute them uniformly. The resulting nonuniform arrays have the property that no nonzero lags are repeated. Simulations demonstrate improved resolution and spectral matching properties when the optimized nonuniform arrays are used with MEM and MLM array processing.

The extension of Pisarenko's method to multiple dimensions

Pisarenkos method of spectral estimation, which models the spectrum as a sum of impulses plus a white noise component, was originally formulated for the time series case. The extension of this method to multiple dimensions and non-uniformly spaced correlation samples involves several fascinating problems. Pisarenkos estimate, which in the time series case involves the solution of an eigenvalue problem, is shown more generally to involve the solution of a linear optimization problem. The computation of Pisarenkos estimate by the application of the simplex method to the linear programming problem is considered. The possibility of a faster multiple exchange algorithm is discussed.

A simple proof of stability for all-pole linear prediction models

A simple method of proof is presented for the minimum-phase property of the all-pole model obtained in the autocorrelation method of linear prediction. The proof does not require knowledge of Levinsons recursion and extends easily to some special cases of the covariance method of linear prediction.

Power spectral density bounds (Corresp.)

The determination of a power density spectrum from a finite set of correlation samples is an ill-posed problem. Furthermore. it is not possible even to bound the values that consistent power density spectra can take on at a particular point. A more reasonable problem is to try to determine the total spectral power in some frequency interval. Although this power cannot be determined exactly, upper and lower bounds on its possible values can be determined. This observation leads to a unified treatment of certain classical and modern spectral estimation techniques and to new interpretations for two data adaptive spectral estimators. maximum likelihood method (MLM) and data adaptive spectral estimator (DASE). According to these new interpretations. MLM and DASE provide upper bounds on spectral power in a specified frequency region subject to the assumption that the spectral density is constant in that region. These methods make no use of an extendibility constraint that can be used to obtain tight upper bounds, as well as nontrivial lower bounds on power. Cybenko has studied a related problem of bounding windowed power, for an arbitrary window, with no assumptions about the form of the spectral density. A new type of classical resolution limit for these bounds is derived and a numerical example is presented.

Efficient phase-only frequency estimation

Techniques that estimate frequency and chirp rate as weighted sums of phase differences have received attention because they can be simple to implement. The novel contributions in this paper include the efficient implementation of these estimates through recursive structuring of the computation and trapezoidal approximation of the weighting functions. A multirate cascade structure for the trapezoidal weighting is given that allows frequency and chirp rate estimates to be formed for multiple analysis window lengths with little extra computation. The techniques described are particularly well suited to hardware implementation because they virtually eliminate multiplications while having little impact on performance.

Bounds from noisy linear measurements

An estimation problem in which a finite number of linear measurements of an unknown function is available, and in which the only prior information available concerning the unknown function consists of inequality constraints on its magnitude, is ill-posed in that insufficient information is available from which point estimates of the unknown function can be made with any reliability, even with exact measurements. An alternative to point estimation involves the computation of bounds on linear functionals of the unknown function in terms of the measurements. A generalization is described of the bounding technique to problems in which the measurements are inexact. The bounds are defined in terms of a primal optimization problem. A deterministic interpretation of the bounds is given, as well as a probabilistic one for the case of additive Gaussian measurement noise. An unconstrained dual optimization problem is derived that has an interesting data-adaptive filtering interpretation and provides an attractive basis for computation. Several aspects of the primal and dual optimization problems are investigated that have important implications for the reliable computation of the bounds.

A linear programming approach to bounding spectral power

The mapping from a finite set of correlation samples to a power density spectrum is not unique. Furthermore, power density spectra exist that take on arbitrary values at a particular frequency and yet are consistent with the correlation samples. Thus values of the spectral density function at a particular frequency cannot be determined without further prior information. In a recent paper, Cybenko comments that tight upper and lower bounds on linear functionals of the spectral density can be obtained as solutions of semi-infinite linear programming problems. In this paper, the primal linear programming problem is interpreted as a search for extremal spectra and the dual linear programming problem is interpreted as a data-adaptive window design procedure. The effect of discretization on both the primal and dual problems is noted. Finally, it is shown how the dual window design problem can be used to design fixed classical type windows for the computation of suboptimal bounds.

New interpretations for the MLM and DASE spectral estimators

This paper provides new interpretations for two modern spectral estimators, the Data Adaptive Spectral Estimator (DASE) of Davis and Regier, and the earlier Maximum Likelihood Method (MLM) of Capon, a special case of DASE. These methods provide estimates for spectral power in some region of frequency space, in terms of an estimate for a correlation matrix. They are conventionally interpreted as window-type spectral estimates, where the window is a function of the estimated correlation matrix. Assuming that the estimated correlation matrix is correct, it is shown that the problem of determining the spectral power is ill-posed. Specifically it is shown that DASE and MLM provide upper bounds on spectral power in some region of frequency space where the spectral density is assumed constant. Furthermore, it is shown that the assumptions and constraints that determine these upper bounds yield trivial lower bounds of zero.

Near optimal frequency/Angle of arrival estimates based on maximum entropy spectral techniques

The problem of estimating the parameters of sinusoids corrupted by additive noise arises both in time series analysis and array processing. The performance of a certain type of maximum entropy spectral estimator based on a modified covariance method of linear prediction has been studied analytically in the high signal-to-noise region. The peak positions in the spectral estimate have been shown to provide low bias estimates which approach the Cramer-Rao bound on variance. The frequency/ angle of arrival estimates obtained in such a manner can be used to form nearly efficient estimates of the component amplitudes by a linear least squares fit.

Confidence regions for spectral bounds

Existing variance calculations for spectral estimates are unsatisfactory in that they depend upon information that is usually unavailable in practice. Some recent work in spectral estimation has involved the computation of bounds on the average spectral density in some region from a true correlation matrix. The computation of these bounds involves optimization over a set of spectra that are consistent with the correlation matrix. The specific new work to be reported on involves the construction of confidence regions for the true correlation matrix, based on a Wishart distributed sample correlation matrix. Bounds computed over spectra that are consistent with the true correlation matrix being in this set are valid with a certain minimum a priori probability which does not depend upon unavailable information about the spectrum. The result is a performance characterization for the bounding method which is different and, in some ways, more satisfactory than the existing variance analyses for other spectral estimation methods.