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

The large number of carriers and the huge number of possible transmitted symbols in an Orthogonal Frequency Domain Multiplexing (OFDM) modulation makes it a challenge to estimate the Bit-Error-Rate (BER) in the presence of nonlinear distortion. This paper proposes a quasi analytical method for estimating the BER of such OFDM modulation schemes using a relative small number of measurements on the nonlinear circuit. The presented theory is compared with Monte Carlo simulations and is demonstrated on RF measurements. The latter are performed on an RF amplifier using a vectorial nonlinear network analyser.

Bit-Error-Rate Estimation for OFDM Based Telecommunication Systems Schemes in the Presence of Nonlinear Distortions

This paper studies the identification of a block-structured nonlinear Wiener-Hammerstein system that is captured in the feedforward or the feedback path of a closed-loop system. First, nonparametric initial estimates are generated for the three dynamic blocks, which are modeled by their frequency response function, and the static nonlinear system, which is described using an input-output table. Next, a parametric model is identified. Random or periodic excitations can be used in the experiments. The method is illustrated on the identification of an Agilent HP420C crystal detector.

Identification of a Block-Structured Nonlinear Feedback System, Applied to a

A completely automatic identification system is described which uses the latest results of the frequency domain identification approach to provide an easy-to-use and reliable tool to the inexperienced users. The proposed system accepts periodic measurement data and performs the whole identification procedure automatically from the data-preprocessing step to model and parameter selection. The validation phase provides information on the results at the level of the nonexpert user. The performance of the system is illustrated by results obtained from real measurement data.

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Automatic Model Selection for Linear Time-Invariant Systems – Practical Issues

This paper investigates the measurement errors involved in estimating frequency response functions from weighted overlapped segment averaging, a technique that has become a standard in modern data analysers due to its computational advantages. A particular attention is paid to leakage errors, for which this technique has been frequently criticised. Our main result is that a Half-sine or Diff window with 67% overlap achieves the best compromise to reduce leakage errors, and this is independently of the system frequency response function. This conclusion is to be contrasted with the customary habit of using a Hanning window with 50% overlap.

FRF Measurements with the Weighted Overlapped Segment Averaging Technique

This paper considers an estimator of the parameters in a linear-in-the-parameter signal, based on its binary quantization. It shows an analysis of the systematic errors for growing lengths of the data set. It also shows how to express the cost function, when the minimum is reached. Zero-mean additive noise is assumed in the model, having independent outcomes. Furthermore, a method is introduced to validate the estimation process. Monte Carlo results validate the proposed analyses.

Johan Schoukens

Papers

Bit-Error-Rate Estimation for OFDM Based Telecommunication Systems Schemes in the Presence of Nonlinear Distortions

Identification of a Block-Structured Nonlinear Feedback System, Applied to a

Automatic Model Selection for Linear Time-Invariant Systems – Practical Issues

FRF Measurements with the Weighted Overlapped Segment Averaging Technique

Asymptotic Properties of a One-bit Estimator of Parametric Signals