Generalizations of the sampling theorem: Seven decades after Nyquist

The sampling theorem is one of the most basic and fascinating topics in engineering sciences. The most well-known form is Shannon's uniform-sampling theorem for bandlimited signals. Extensions of this to bandpass signals and multiband signals, and to nonuniform sampling are also well-known. The connection between such extensions and the theory of filter banks in DSP has been well established. This paper presents some of the less known aspects of sampling, with special emphasis on non bandlimited signals, pointwise stability of reconstruction, and reconstruction from nonuniform samples. Applications in multiresolution computation and in digital spline interpolation are also reviewed

A kernel method for non-linear systems identification – infinite degree volterra series estimation

Volterra series expansions are widely used in analyzing and solving the problems of non-linear dynamical systems. However, the problem that the number of terms to be determined increases exponentially with the order of the expansion restricts its practical application. In practice, Volterra series expansions are truncated severely so that they may not give accurate representations of the original system. To address this problem, kernel methods are shown to be deserving of exploration. In this report, we make use of an existing result from the theory of approximation in reproducing kernel Hilbert space (RKHS) that has not yet been exploited in the systems identification field. An exponential kernel method, based on an RKHS called a generalized Fock space, is introduced, to model non-linear dynamical systems and to specify the corresponding Volterra series expansion. In this way a non-linear dynamical system can be modelled using a finite memory length, infinite degree Volterra series expansion, thus reducing the source of approximation error solely to truncation in time. We can also, in principle, recover any coefficient in the Volterra series

Review of Unbiased FIR Filters, Smoothers, and Predictors for Polynomial Signals

Extracting an estimate of a slowly varying signal corrupted by noise is a common task. Examples can be found in industrial, scientific and biomedical instrumentation. Depending on the nature of the application the signal estimate is allowed to be a delayed estimate of the original signal or, in the other extreme, no delay is tolerated. These cases are commonly referred to as filtering, prediction, and smoothing depending on the amount of advance or lag between the input data set and the output data set. In this review paper we provide a comprehensive set of design and analysis tools for designing unbiased FIR filters, predictors, and smoothers for slowly varying signals, i.e. signals that can be modeled by low order polynomials. Explicit expressions of parameters needed in practical implementations are given. Real life examples are provided including cases where the method is extended to signals that are piecewise slowly varying. A critical view on recursive implementations of the algorithms is provided

On designing H∞ filters with circular pole and error variance constraints

Copyright [2003] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.In this paper, we deal with the problem of designing a H∞ filter for discrete-time systems subject to error variance and circular pole constraints. Specifically, we aim to design a filter such that the H∞ norm of the filtering error-transfer function is not less than a given upper bound, while the poles of the filtering matrix are assigned within a prespecified circular region, and the steady-state error variance for each state is not more than the individual prespecified value. The filter design problem is formulated as an auxiliary matrix assignment problem. Both the existence condition and the explicit expression of the desired filters are then derived by using an algebraic matrix inequality approach. The proposed design algorithm is illustrated by a numerical example

A kernel method for non-linear systems indentification - infinite degree volerra series estimation

Volterra series expansions are widely used in analyzing and solving the problems of non-linear dynamical systems. However, the problem that the number of terms to be determined increases exponentially with the order of the expansion restricts its practical application. In practice, Volterra series expansions are truncated severely so that they may not give accurate representations of the original system. To address this problem, kernel methods are shown to be deserving of exploration. In this report, we make use of an existing result from the theory of approximation in reproducing kernel Hilbert space (RKHS) that has not yet been exploited in the systems identification field. An exponential kernel method, based on an RKHS called a generalized Fock space, is introduced, to model non-linear dynamical systems and to specify the corresponding Volterra series expansion. In this way a non-linear dynamical system can be modelled using a finite memory length, infinite degree Volterra series expansion, thus reducing the source of approximation error solely to truncation in time. We can also, in principle, recover any coefficient in the Volterra series

Signal Reconstruction via H-infinity Sampled-Data Control Theory: Beyond the Shannon Paradigm

This paper presents a new method for signal reconstruction by leveraging sampled-data control theory. We formulate the signal reconstruction problem in terms of an analog performance optimization problem using a stable discrete-time filter. The proposed H-infinity performance criterion naturally takes intersample behavior into account, reflecting the energy distributions of the signal. We present methods for computing optimal solutions which are guaranteed to be stable and causal. Detailed comparisons to alternative methods are provided. We discuss some applications in sound and image reconstruction

Biorthogonal partners and applications

Two digital filters H(z) and F(z) are said to be biorthogonal partners of each other if their cascade H(z)F(z) satisfies the Nyquist or zero-crossing property. Biorthogonal partners arise in many different contexts such as filterbank theory, exact and least squares digital interpolation, and multiresolution theory. They also play a central role in the theory of equalization, especially, fractionally spaced equalizers in digital communications. We first develop several theoretical properties of biorthogonal partners. We also develop conditions for the existence of biorthogonal partners and FIR biorthogonal pairs and establish the connections to the Riesz basis property. We then explain how these results play a role in many of the above-mentioned applications

Estimation of Overspread Scattering Functions

In many radar scenarios, the radar target or the medium is assumed to possess randomly varying parts. The properties of a target are described by a random process known as the spreading function. Its second order statistics under the WSSUS assumption are given by the scattering function. Recent developments in operator sampling theory suggest novel channel sounding procedures that allow for the determination of the spreading function given complete statistical knowledge of the operator echo from a single sounding by a weighted pulse train. We construct and analyze a novel estimator for the scattering function based on these findings. Our results apply whenever the scattering function is supported on a compact subset of the time-frequency plane. We do not make any restrictions either on the geometry of this support set, or on its area. Our estimator can be seen as a generalization of an averaged periodogram estimator for the case of a non-rectangular geometry of the support set of the scattering function