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

/pdf/differential-equations-and-dynamical-systems-3chsjx2k45.pdf

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:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

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

http://ieeexplore.ieee.org/iel5/5/31347/01457444.pdf

Multirate digital signal processing

Least-squares periodic signal modeling using orbits of nonlinear ODEs and fully automated spectral analysis

This paper deals with the performance of several up-to-date nonparametric frequency estimators. The algorithms of frequency estimation are introduced and their bias, standard deviation and consumption time are compared with regard to the most common signal parameters.

Comparison of nonparametric frequency estimators

Most stochastic processes in engineering applications have an intrinsic continuous-time nature. Think, for example, of the noise generated by resistors and semi-conductor devices. A few methods exist that identify continuous-time noise models from sampled data. In this paper we introduce a new method based on the concept of filtered band limited white noise, make an in depth analysis of the basic assumptions made by the different approaches, and discuss the pros and cons of each method. The new method is particularly highlighted and illustrated on real measurements.

Continuous-Time Noise Modelling from Sampled Data

The forced oscillation technique (FOT) measures the respiratory impedance in a noninvasive way to obtain the mechanical properties of a respiratory system. The respiratory impedance ( $Z_{\text {rs}}$ ) is defined by the frequency-dependent ratio of pressure and flow at the airway opening. A major problem when measuring $Z_{\text {rs}}$ at low frequencies (0.1–2 Hz) on spontaneously breathing patients is the disturbance generated by the patients breathing (0.1–1 Hz). A method is presented that aims at eliminating breathing disturbances from low frequent FOT measurements by applying a combination of amplitude and phase modulated signal models and regularized least squares. The method is validated on simulations using breathing measurements, and estimates of $Z_{\text {rs}}$ with a very low error are obtained. In addition, the method has been applied to measurements performed on a group of healthy subjects, and the results are obtained that can be used for diagnostic purposes or for further studies on parametric modeling of $Z_{\text {rs}}$ at low frequencies.

Estimating Respiratory Impedance at Breathing Frequencies Using Regularized Least Squares on Forced Oscillation Technique Measurements

This paper concerns the identification of nonlinear systems using a variant of the Wiener G-Functionals. The system is modeled by a cascade of a single input multiple output (SIMO) linear dynamic system, followed by a multiple input single output (MISO) static nonlinear system. The dynamic system is described using orthonormal basis functions. The original ideas date back to the Wiener G-functionals of Lee and Schetzen. Whereas the Wiener G-Functionals use Laguerre orthonormal basis functions, in this work Takenaka-Malmquist orthonormal basis functions are used. The poles that these basis functions contain, are estimated using the best linear approximation of the system. The approach is illustrated on the identification of a Wiener system.

http://www.nt.ntnu.no/users/skoge/prost/proceedings/cdc-ecc-2011/data/papers/0122.pdf

Johan Schoukens

Papers

Least-squares periodic signal modeling using orbits of nonlinear ODEs and fully automated spectral analysis

Comparison of nonparametric frequency estimators

Continuous-Time Noise Modelling from Sampled Data

Estimating Respiratory Impedance at Breathing Frequencies Using Regularized Least Squares on Forced Oscillation Technique Measurements

Identifying a Wiener system using a variant of the Wiener G-Functionals