Philip J. Parker

Adaptive frequency response identification

Given a stable, discrete time, single input single output system G(z), but with only the input signal and the noise corrupted output signal available for measurement, we seek to find an approximation G(z) - a finite impulse response (FIR) filter - with ||G - ¿¿|| = sup |G(ej¿) - ¿(ej¿)| ¿¿(-¿,¿] bounded and small. The infinity norm in (1) has application in control theory and signal processing; furthermore, it is a measure of the deviation in frequency response between G and ¿. Several previous papers, attempt to identify G(z) in the frequency domain; these papers fail to bound G-¿ in any norm. Central to our method of identification is interpolation. First, one estimates accurately G(z) at n equally spaced frequencies. Here, n is a design parameter one may freely choose. This estimation relies on filtering the input and output signals appropriately. Then estimates of G(eJ2¿k/n) come from a bank of n/2 decoupled least mean squares algorithms, each of two parameters; ¿(z) is then the unique FIR filter of degree n-1 with transfer function interpolating to these estimates. ¿(z) is computationally easy to evaluate. The resulting error bound has the form ||G - ¿||¿ ¿ MRn + K(1 + log2n) Here M and R are constants, dependent on G(z), with R<1; the accuracy of estimating G(z) at the interpolation points determines K.

A Kalman filtering approach to short-time Fourier analysis

The problem of estimating time-varying harmonic components of a signal measured in noise is considered. The approach used is via state estimation. Two methods are proposed, one involving pole-placement of a state observer, the other using quadratic optimization techniques. The result is the development of a new class of filters, akin to recursive frequency-sampling filters, for inclusion in a parallel bank to produce sliding harmonic estimates. Kalman filtering theory is applied to effect the good performance in noise, and the class of filters is parameterized by the design tradeoff between noise rejection and convergence rate. These filters can be seen as generalizing the DFT.

Frequency tracking of nonsinusoidal periodic signals in noise

Abstract For a periodic signal measured in noise, this paper applies extended Kalman filtering to the problem of estimating the signals frequency and the amplitudes and phases of the signals first m harmonic components. The resultant estimator will also track the signals frequency and its amplitudes and phases should these change over time. In this respect, it is unique among approaches to this problem. A partial theoretical analysis of the estimator appears in the paper. This analysis shows that there is some measure of decoupling in the estimator: the amplitudes are estimated as if the phase and frequency estimates are correct; the phases and frequency are estimated as if the amplitude estimates are correct. For the special case that the signal is a sinusoid and has known amplitude, the estimator becomes the well-known phase-locked loop. The paper also contains extensive simulations demonstrating both the tracking and the asymptotic behaviour of the estimator. The asymptotic behaviour is compared with the results for another known estimator, and the relative strengths of each method are examined.

Frequency domain conditions for the robust stability of linear and nonlinear dynamical systems

The authors establish a generalized frequency-domain criterion for checking families of polynomials for root confinement in open subsets of the complex plane. The authors show how this criterion reduces to checking certain curves in the complex plane for zero confinement. Moreover, in some special cases, it further reduces to some complex functions with pointwise phase differences that are always less than pi in magnitude. Most of the currently available results on the robust stability of linear systems with parametric uncertainties can be viewed within the unifying frequency-domain framework presented. The framework encapsulates not just finite-dimensional systems, but any linear-time-invariant (LTI) system that can be characterized by transfer functions of a single variable. It also covers robust stability of LTI systems under passive feedback. >

Unstable rational function approximation

The problem is considered of approximating a transfer function with stable and unstable poles by a lower-order transfer function with the same number of unstable poles. A method is suggested and compared with alternative known methods based on Hankel-norm approximation. Examples suggest that no one method always outperforms other methods in terms of minimizing the approximation error.

Hilbert transforms from interpolation data

This paper studies the generation of stable transfer functions for which the real or imaginary part takes prescribed values at discrete uniformly spaced points on the unit circle. Formulas bounding the error between a particular interpolating function and any function consistent with the data are presented, which have the desirable property that the error goes to zero exponentially fast with the number of interpolating points. The paper also examines generation of stable minimum phase transfer functions for which the magnitude takes prescribed values at uniformly spaced points on the unit circle, and presents error bounds for this problem. Connection with the discrete Hilbert transform is made. The effect of uncertainty in the original data is also examined.