Martin Morf

Displacement Ranks of Matrices and Linear Equations

Abstract It takes of the order of N3 operations to solve a set of N linear equations in N unknowns or to invert the corresponding coefficient matrix. When the underlying physical problem has some time- or shift-invariance properties, the coefficient matrix is of Toeplitz (or difference or convolution) type and it is known that it can be inverted with O(N2) operations. However non-Toeplitz matrices often arise even in problems with some underlying time-invariance, e.g., as inverses or products or sums of products of possibly rectangular Toeplitz matrices. These non-Toeplitz matrices should be invertible with a complexity between O(N2) and O(N3). In this paper we provide some content for this feeling by introducing the concept of displacement ranks, which serve as a measure of how ‘close’ to Toeplitz a given matrix is.

New results in 2-D systems theory, part II: 2-D state-space models—Realization and the notions of controllability, observability, and minimality

In this part, a comparison between the different state-space models is presented. We discuss proper definitions of state, controllability and observability and their relations to minimality of 2-D systems. We also present new circuit realizations and 2-D digital filter hardware implementation of 2-D transfer functions.

Fast calculation of gain matrices for recursive estimation schemes

A sequence of vectors {x(t)} with dimension N is given, such that x(t+1) is obtained from x(t) by introducing p new elements, deleting p old ones, and shifting the others in some fashion. The sequence of vectors

Inverses of Toeplitz Operators, Innovations, and Orthogonal Polynomials

is sought. This paper presents a method of calculating these vectors with proportional-to-Np operations and memory locations, in contrast to the conventional way which requires proportional-to-N 2 operations and memory locations. A non-symmetric case is also treated.

New inversion formulas for matrices classified in terms of their distance from Toeplitz matrices

We describe several interconnections between the topics mentioned in the title. In particular, we show how some previously known formulas for inverting Toeplitz operators in both discrete- and continuous-time can be interpreted as versions of the Christoffel-Darboux formula for the biorthogonal Szegö and Krein polynomials on the circle and the line, respectively. The discrete-time inversion result is often known as Trenchs formula, while the continuous-time result was apparently first deduced (in radiative transfer theory) by Sobolev. The concept of innovations is used to motivate the definitions of the Szegö and especially the Krein orthogonal functionals, and connections to work on the fitting of autoregressive models and inversion of the associated covariance matrices are also noted.

Recursive Multichannel Maximum Entropy Spectral Estimation

The problem of solving linear equations, or equivalently of inverting matrices, arises in many fields. Efficient recursive algorithms for finding the inverses of Toeplitz or displacement-type matrices have been known for some time. By introducting a way of characterizing matrices in terms of their “distance” from being Toeplitz, a natural extension of these algorithms is obtained. Several new inversion formulas for the representation of the inverse of non-Toeplitz matrices are also presented.

Square-root algorithms for least-squares estimation

We present a generalization to the multichannel case of the well-known Burg maximum entropy technique for spectral estimation. The extension is obtained by first obtaining the proper generalization of the scalar reflection coefficients to the multichannel (or matrix) case.

Some new algorithms for recursive estimation in constant, linear, discrete-time systems

We present several new algorithms, and more generally a new approach, to recursive estimation algorithms for linear dynamical systems. Earlier results in this area have been obtained by several others, especially Potter, Golub, Dyer and McReynolds, Kaminski, Schmidt, Bryson, and Bierman on what are known as square-root algorithms. Our results are more comprehensive. They also show bow constancy of parameters can be exploited to reduce the number of computations and to obtain new forms of the Chandrasekhar-type equations for computing the filter gain. Our approach is essentially based on certain simple geometric interpretations of the overall estimation problem. One of our goals is to attract attention to non-Riccati-based studies of estimation problems.

The signal subspace approach for multiple wide-band emitter location

Certain recently developed fast algorithms for recursive estimation in constant continuous-time linear systems are extended to discrete-time systems. The main feature is the replacement of the Riccati-type difference equation that is generally used for such problems by another set of difference equations that we call of Chandrasekhar-type. The total number of operations in the new algorithm is in general less than with the Riccati-equation based Kalman filter, with significant reductions being obtained in several important special cases. The algorithms are derived via a factorization of increments of the Riccati equation variable, a method that can be extended to nonsymmetric Riccati equations as well.

Square root covariance ladder algorithms

The rational vector space generalization of the signal subspace approach is presented and applied to the estimation of multiple wide-band emitter locations from the signals received at multiple sensors. The signal subspace and array manifold concepts first introduced by Schmidt are generalized to rational vector space. These concepts are used to develop the rational signal subspace theory and prove the signal subspace theorem, on which signal subspace algorithms are based. The theory is applied in discrete time to derive a class of rational signal subspace algorithms for source location and spectral estimation using unit circle eigendecomposition of multivariate rational models of sensor outputs. Simulation results are presented for an algorithm in this class, including sample statistics from Monte Carlo trials and comparisons with the Cramer-Rao bound.