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:

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

It is the common conviction that frequency domain system identification suffers from the drawback that it cannot handle arbitrary signals without introducing systematic errors. This paper shows that it is possible to deal with nonperiodic signals without any approximation and under the same assumptions as in the time domain, by estimating simultaneously some initial conditions and the system model parameters.

Frequency domain system identification using arbitrary signals

This paper describes a benchmark for nonlinear system identification A Wiener-Hammerstein system is selected as test object In such a structure there is no direct access to the static nonlinearity starting from the measured input/output, because it is sandwiched between two unknown dynamic systems The signal-to-noise ratio of the measurements is quite high, which puts the focus of the benchmark on the ability to identify the nonlinear behaviour, and not so much on the noise rejection properties The benchmark is not intended as a competition, but as a tool to compare the possibilities of different methods to deal with this specific nonlinear structure

/pdf/wiener-hammerstein-benchmark-2m64tfd3du.pdf

Wiener-Hammerstein Benchmark

In nonlinear system identification, the system is often represented as a series of blocks linked together. Such block-oriented models are built with static nonlinear subsystems and linear dynamic systems. This paper deals with the identification of the Hammerstein-Wiener model, which is a block-oriented model where a linear dynamic system is surrounded by two static nonlinearities at its input and output. The proposed identification scheme is iterative and will be demonstrated on measurements. It will be proven that on noiseless data and in absence of modeling errors, the optimization procedure converges to the true system locally.

Hammerstein-Wiener System Estimator Initialization

Electrical impedance spectroscopy (EIS) can be used to characterize biological materials in applications ranging from cell culture to body composition, including tissue and organ state. The emergence of cell therapy and tissue engineering opens up a new and promising field of application. While in most cases classical measurement techniques based on a frequency sweep can be used, EIS based on broadband excitations enables dynamic biological systems to be characterized when the measuring time and injected energy are a constraint. Myocardial regeneration, cell characterization in micro-fluidic systems and dynamic electrical impedance tomography are all examples of such applications. The weakness of such types of fast EIS measuring techniques resides in their intrinsic loss of accuracy. However, since most of the practical applications have no restriction over the excitation used, the input power spectrum can be appropriately designed to maximize the accuracy obtained from the measurements. This paper deals with the problem of designing the optimal multisine excitation for electrical bioimpedance measurements. The optimal multisine is obtained by the minimization of the Cramer–Rao lower bound, or what is the same, by maximizing the accuracy obtained from the measurements. Furthermore, because no analytical solution exists for global optimization involving time and frequency domains jointly, this paper presents the multisine optimization approach partially in both domains and then combines the results. As regards the frequency domain approach, a novel contribution is made for the multisine amplitude power spectrum. In the time domain, multisine is optimized by reducing its crest factor. Moreover, the impact on the information and accuracy of the impedance spectrum obtained from using different multisine amplitude power spectra is discussed, as well as the number of frequencies and frequency distributions. The theory is supported by a set of validation measurements when exciting with the optimal and flat multisine signals and compared to a single frequency ac impedance analyzer when characterizing an RC circuit. In vivo healthy myocardium tissue electrical impedance measurements show that broadband EIS based on multisine excitations enable the characterization of dynamic biological systems.

https://iopscience.iop.org/article/10.1088/0957-0233/22/11/115601/pdf

Johan Schoukens

Papers

Frequency domain system identification using arbitrary signals

Wiener-Hammerstein Benchmark

Hammerstein-Wiener System Estimator Initialization

Optimal multisine excitation design for broadband electrical impedance spectroscopy

Hammerstein-Wiener system estimator initialization