Christopher J. Zarowski

QR-factorization method for computing the greatest common divisor of polynomials with inexact coefficients

This paper presents a novel means of computing the greatest common divisor (GCD) of two polynomials with real-valued coefficients that have been perturbed by noise. The method involves the QR-factorization of a near-to-Toeplitz matrix derived from the Sylvester matrix of the two polynomials. It turns out that the GCD to within a constant factor is contained in the last nonzero row of the upper triangular matrix R in the QR-factorization of the near-to-Toeplitz matrix. The QR-factorization is efficiently performed by an algorithm due to Chun et al. (1987). A condition number estimator due to Bischof (1990) and an algorithm for rank estimation due to Zarowski (1998) are employed to account for the effects of noise.

The MDL criterion for rank determination via effective singular values

Konstantinides and Yao (1988) have considered the problem of rank determination by use of effective singular values. In this correspondence, we show how to use the minimum description length criterion of Rissanen to provide an alternative means of estimating the index of the smallest nonzero singular value of a matrix when given estimates of the singular values.

Factorable FIR Nyquist filters with least stopband energy under sidelobe level constraints

Spectrally factorable Nyquist filters are used in data communications to avoid intersymbol interference. An approach is developed for obtaining a Nyquist filter that is factorable having the smallest stopband energy for a given sidelobe level. The resulting constrained minimization problem is solved efficiently and reliably using the Goldfarb-Idnani (1983) algorithm. Some examples are presented comparing the present method with a previous approach from the literature.

On lower bounds for the smallest eigenvalue of a Hermitian positive-definite matrix

Presents an improvement to Demhos (1988) lower bound on the smallest eigenvalue of a Hermitian positive-definite matrix. Unlike Dembos bound the improved bound is always positive. >

Schur algorithms for Hermitian Toeplitz, and Hankel matrices with singular leading principal submatrices

It is shown how a simple matrix algebra procedure can be used to induce Schur-type algorithms for the solution of certain Toeplitz and Hankel linear systems of equations when given Levinson-Durbin algorithms for such problems. The algorithm of P. Delsarte et al. (1985) for Hermitian Toeplitz matrices in the singular case is used to induce a Schur algorithm for such matrices. An algorithm due to G. Heinig and K. Rost (1984) for Hankel matrices in the singular case is used to induce a Schur algorithm for such matrices. The Berlekamp-Massey algorithm is viewed as a kind of Levinson-Durbin algorithm and so is used to induce a Schur algorithm for the minimal partial realization problem. The Schur algorithm for Hermitian Toeplitz matrices in the singular case is shown to be amenable to implementation on a linearly connected parallel processor array of the sort considered by Kung and Hu (1983), and in fact generalizes their result to the singular case. >

Limitations on SNR estimator accuracy

We consider the samples of a pure tone in additive white Gaussian noise (AWGN) for which we wish to determine the signal-to-noise ratio (SNR) defined here to be /spl alpha/=(A/sup 2//2/spl sigma//sup 2/), where A is the tone amplitude, and /spl sigma//sup 2/ is the noise variance. A and /spl sigma//sup 2/ are assumed to be deterministic but unknown a priori. If the variance of an unbiased estimator of /spl alpha/ is /spl sigma//sub /spl alpha//spl circ///sup 2/, we show that at high SNR, the normalized standard a deviation satisfies the Cramer-Rao lower bound (CRLB) according to /spl sigma//sub /spl alpha//spl circ////spl alpha//spl ges//spl radic/(2/N), where N is the number of independent observables used to obtain the SNR estimate /spl sigma//spl circ/.

Some algorithms for circadian rhythm identification

We present two approaches to the detrending of physiological data derived from brain-injured human patients. This is important in circadian rhythym characterization. The proper identification of a patients rhythm is known to be important in clinical treatment. One detrending approach is an ad hoc polynomial fitting strategy, while the other is a modification of the so-called cosinor model where subharmonic terms are added. The latter method is based on the fact that many nonlinear dynamic systems when driven with a sinusoidal input produce an output containing subharmonics of the input. The latter method seems to provide consistently superior performance to the polynomial method.

Circadian rhythm of temperature in head injury

It has been shown in a previous study that head injured patients appear to have a circadian rhythm of their body functions. This needed to be confirmed using additional data, better collection methods and analysis. Additional goals were to develop a method of detrending of physiological time series in order to improve rhythm detection when it may be hidden behind a low frequency trend and the creation of a computer system for data acquisition and analysis. The temperature data of 10 head injured patients was studied using the Iterative Cosinor method. In one case, prior to the Cosinor method, detrending of the data was used using a specially designed polynomial fitting technique. The Iterative Cosinor method showed circadian rhythms in nine out of 10 patients. After detrending, a rhythm was found in the data of the 10th patient as well. The periods of the rhythm were around, but were not equal to, 24 hours. The results show that comatose head injured patients have a circadian rhythm of their core temperature. The detection of a circadian rhythm may, in some cases, be improved by using a detrending technique. The deviation of the rhythm period from 24 hours suggests that the rhythms found in these patients are free-running, meaning that head injured patients are not synchronized with their surroundings.

A Schur algorithm and linearly connected processor array for Toeplitz-plus-Hankel matrices

A Levinson-Durbin type algorithm for solving Toeplitz-plus-Hankel (T+H) linear systems of equations is used to induce a Schur-type algorithm for such systems. A Schur-type algorithm is defined as one which efficiently computes the LDU-decomposition of the matrix. On the other hand, Levinson-Durbin type algorithms are defined as those algorithms which efficiently compute the UDL-decomposition of the inverse of a matrix. It is shown that the Schur algorithm so obtained is amenable to efficient implementation on a linearly connected array of processors in a manner which generalizes the results of S.-Y. Kung and Y.H. Ku (1983) for symmetric Toeplitz matrices. Specifically, if T+H is of order n, then the Schur algorithm runs on O(n) processors in O(n) time. >

An approach to initializing the wavelet packet transform

This article presents an approach to the initialization of the wavelet packet transform (WPT), which is a generalization of the discrete wavelet transform (DWT), by an extension of the interpolatory graphical display algorithm (IGDA). The exact computation of the WPT of functions that are piecewise constant on dyadic intervals is demonstrated. The method is for piecewise constant signals, as the details of how to do this do not appear to be readily available in the open literature. Furthermore, the solution is conveniently placed in a multirate signal processing framework.