James A. Cadzow

High performance spectral estimation--A new ARMA method

In this paper a method for generating an ARMA model spectral estimate of a wide-sense stationary time series from a finite set of observations is presented. The method is based upon a set of error equations which are dependent on the ARMA models parameters. Minimization of a quadratic functional of these error equations with respect to the ARMA models parameters produces the desired spectral estimate. In examples treated to date, this ARMA spectral estimator has provided significantly better performance when compared to such standard procedures as the maximum entropy and Box-Jenkins methods. The computational requirements of this new method basically entail the solving of a system of p linear equations in the autoregressive coefficients where p denotes the order of the ARMA model. Since an ARMA model will typically be of lower order than its autoregressive model counterpart for a specified fidelity of match, the new ARMA procedure is generally more efficient computationally than the maximum entropy method. With this in mind, this ARMA method offers the promise of being a primary tool in many spectral estimation applications.

Two-dimensional spectral estimation

In this paper, effective methods for generating two-dimensional quarter-plane causal autoregressive (AR) and autoregressive moving average (ARMA) spectral estimation models are developed. These procedures are found to provide super resolution capabilities when compared to other more classical methods such as the Fourier transform. The ARMA method involves manipulation of the model equation \sum\min{k = 0}\max{p_{1}} \sum\min{k = 0}\max{p_{2}} a_{km}x(n_{1} - k, n_{2} - m) = \sum\min{k = 0}\max{q_{1}} \sum\min{k = 0}\max{q_{2}} b_{km}\epsilon(n_{1} - k, n_{2} - m) and utilizes the given finite set of observations x(n_{1}, n_{2}) for 1 \leq n_{1} \leq N_{1},1 \leq n_{2} \leq N_{2} . In the above relationship, the random excitation {\epsilon(n_{1}, n_{2})} is taken to be white. This ARMA models autoregressive a km coefficients are selected to minimize a weighted least-squares criterion composed of error elements while the moving average b km coefficients are obtained using an alternative approach. The spectral estimation performance of the AR and ARMA methods will be empirically demonstrated by considering the problem of resolving two sinusoids embedded in noise.

A recursive procedure for ARMA modeling

This paper presents a two-part fast recursive algorithm for ARMA modeling. The algorithm first obtains estimates of the p autoregressive coefficients from a set of p extended Yule-Walker equations. An exact recursive lattice algorithm for this estimator is then derived. The q + 1 numerator spectrum coefficients are then obtained by using one of the output data sequences of this lattice algorithm. The complete recursive algorithm is fast in the sense that O(p + q) computations are required for each update. Moreover, an exponential forgetting factor is incorporated to facilitate tracking of time variations in the time series.

Adaptive ARMA spectral estimation

A novel adaptive method for efficiently obtaining an ARMA model spectral estimate of a wide-sense stationary time series is presented. It is adaptive in the sense that as a new element of the time series is observed, the coefficients of a (p,p)th order ARMA model may be algorithmically updated. This algorithms computational complexity (i.e., the number of multiplications and additions required) is of the order p \log(p) for a particular version of the method. Moreover, the spectral estimation performance of this new method is found typically to be far superior to such contemporary approaches as the Box-Jenkins, maximum entropy, and, Widrows LMS methods. This performance in conjunction with its computational efficiency mark this algorithm as being a primary spectral estimation tool.

An algebraic approach to super-resolution adaptive array processing

In this paper, an algebraic characterization is made of the problem of resolving two or more closely spaced (in frequency wave number) plane waves incident on a linear array. This algebraic characterization in turn suggests a number of adaptive procedures for effecting the desired resolution. One of these procedures is herein empirically shown to provide significantly better performance when compared to other contemporary procedures used in array processing such as the Wiener filter, Pisarenko and MLM algorithms. This includes both a better frequency resolving capability and a faster convergence rate.

Reconstruction of signals from their linear mapping image

Many signal processing problems can be formulated as that of reconstructing a signal given an observed linear mapping (or transformation) of that signal. As an example, the extrapolation of a time-truncated version of a band-limited signal may be so characterized. More specifically, we shall herein consider the task of reconstructing a signal x, which lies in a prespecified closed subspace of a Hilbert space, and, in which the only information available is the linear mapping y = Lx of that signal. This reconstruction will be implemented by means of an algorithmic procedure. Necessary and sufficient conditions which ensure the algorithms convergence are presented.

Signal rank and model order determination

In this paper, a measure of a signals structure as indicated by the effective rank of its associated autocorrelation matrix is given. The measure is conceptually related to the Singular Value Decomposition (SVD) of the autocorrelation matrix but is quite easily computed as it avoids explicit calculation of an SVD. The measure can be directly applied to order determination in ARMA methods based upon the extended-order Yule-Walker equations.

Inversion of signal operations

In a variety of applications, one is given the response signal which results when a known linear system (operator) is excited by an unknown input signal. From this information, it is then desired to ascertain the input signal which gave rise to the observed response behavior. An efficient operational, as well as an algorithmic, procedure shall be presented for achieving the desired signal inversion operation. Conditions under which this inversion is well-posed mathematically shall be detailed. The development is carried out in a general normed space setting thereby making the results applicable to a variety of signal inversion problems.