Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

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Differential Equations and Dynamical Systems

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

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Linear systems

System Identification I

There has been a great deal of material in the area of discrete-time transforms that has been published in recent years. This book does an excellent job of presenting important aspects of such material in a clear manner. The book has 11 chapters and a very useful appendix. Seven of these chapters are essentially devoted to the Fourier series/transform, discrete Fourier transform, fast Fourier transform (FFT), and applications of the FFT in the area of spectral estimation. Chapters 8 through 10 deal with many other discrete-time transforms and algorithms to compute them. Of these transforms, the KarhunenLoeve, the discrete cosine, and the Walsh-Hadamard transform are perhaps the most well-known. A lucid discussion of number theoretic transforms i5 presented in Chapter 11. This reviewer feels that the authors have done a fine job of compiling the pertinent material and presenting it in a concise and clear manner. There are a number of problems at the end of each chapter, an appreciable number of which are challenging. The authors have included a comprehensive set of references at the end of the book. In brief, this book is a valuable contribution to the general areas of signal processing and communications. It can be used for a graduate level course in perhaps two ways. One would be to cover the first seven chapters in great detail. The other would be to cover the whole book by focussing on different topics in a selective manner. This book  by  Elliott  and Rao  is extremely  useful to researchers/engineers who are working  in the areas of signal processing and communications. It i s also an excellent reference book, and hence a valuable addition to one’s library

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Multirate digital signal processing

To prevent unstable behavior or saturation, a frequency response function (FRF) measurement is often performed under closed-loop conditions (e.g., open-loop gain measurements of an operational amplifier). The difficulty of such FRF measurements is that the nonlinear (NL) distortions also perturb the input via the feedback loop. The latter introduces a bias in the estimate of the best linear approximation (BLA) and jeopardizes the interpretation of the output NL distortions. In this paper, we solve these problems via a generalized definition of the BLA that is valid for NL systems operating in feedback. The classical definition for open-loop systems follows as a special case.

FRF Measurement of Nonlinear Systems Operating in Closed Loop

A Frequency Domain Approach

The logarithmic distribution of spectral lines in excitation signals is widely used to measure the transfer functions of dynamic systems over a wide frequency band, covering several decades. Periodic signals are very popular in many advanced dynamic signal analyzers. Generating multitone periodic signals requires an equidistant frequency grid which conflicts with the logarithmic distribution, particularly at the low frequencies. In this paper, we offer an elegant solution to get around this problem using an improved choice for the amplitude spectrum of the multitone. We also offer a simple way to tune the frequency spacing of such a signal to allow for robust identification of the system under test.

Design of Quasi-Logarithmic Multisine Excitations for Robust Broad Frequency Band Measurements

The goal of this article is twofold. Firstly, nonlinear system identification is introduced to a wide audience, guiding practicing engineers and newcomers in the field to a sound solution of their data driven modeling problems for nonlinear dynamic systems. In addition, the article also provides a broad perspective on the topic to researchers that are already familiar with the linear system identification theory, showing the similarities and differences between the linear and nonlinear problem. The reader will be referred to the existing literature for detailed mathematical explanations and formal proofs. Here the focus is on the basic philosophy, giving an intuitive understanding of the problems and the solutions, by making a guided tour along the wide range of user choices in nonlinear system identification. Guidelines will be given in addition to many examples, to reach that goal.

Johan Schoukens

Papers

FRF Measurement of Nonlinear Systems Operating in Closed Loop

A Frequency Domain Approach

Design of Quasi-Logarithmic Multisine Excitations for Robust Broad Frequency Band Measurements

Nonlinear System Identification: A User-Oriented Roadmap

Brief paper: Two nonlinear optimization methods for black box identification compared