There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

Introduction to Electromagnetic Compatibility

dix A. Even if you don’t choose to memorize them this system aids in reference and retreival of important formulas. The book was compiled from notes developed during eight years of teaching a graduate course on the subject and was used as a text. Thus it has been student tested. Appendix F contains a number of problems, grouped to be used on a chapter by chapter basis The problems are designed to illustrate practical applications of the text material. The parts of this book of most interest and value to the EMC engineer will be the chapters on Thermal Noise, Antennas, Propagation and Transmission Lines, and  Reflection and Refraction. This is not  to downpade the chapters on Statistics and Its Applications, Signal Processing and Detection, and Some System Characteristics which also contain much potentially useful materials. Additional plus values for the book include a list of 40 references, a table of symbols used throughout the book, and a subject index. Some readers may find the condensed type and close line spacing hard to read. It was apparently set up by typewriter using an elite type face with single line spacing. When reduced down to a 6 by 9 5 inch size page it is too crowded for easy reading. In spite of this shortcoming your reviewer recommends this book as a worthwhile reference in this field of interest.

http://ieeexplore.ieee.org/iel5/5/31269/01455368.pdf

Signal analysis

This paper provides an effective solution for the simulation of cables and interconnects with the inclusion of the effects of parameter uncertainties. The problem formulation is based on the telegraphers equations with stochastic coefficients, whose solution requires an expansion of the unknown parameters in terms of orthogonal polynomials of random variables. The proposed method offers accuracy and improved efficiency in computing the parameter variability effects on system responses with respect to the conventional Monte Carlo approach. The approach is validated against results available in the literature, and applied to the stochastic analysis of a commercial multiconductor flat cable.

/pdf/stochastic-analysis-of-multiconductor-cables-and-4ke8dugivg.pdf

Stochastic Analysis of Multiconductor Cables and Interconnects

This paper addresses the development of computational models of digital integrated circuit input and output buffers via the identification of nonlinear parametric models. The obtained models run in standard circuit simulation environments, offer improved accuracy and good numerical efficiency, and do not disclose information on the structure of the modeled devices. The paper reviews the basics of the parametric identification approach and illustrates its most recent extensions to handle temperature and supply voltage variations as well as power supply ports and tristate devices.

M/spl pi/log, macromodeling via parametric identification of logic gates

This paper focuses on the derivation of an enhanced transmission-line model allowing the stochastic analysis of a realistic multiconductor interconnect. The proposed model, which is based on the expansion of the well-known telegraph equations in terms of orthogonal polynomials, includes the variability of geometrical or material properties of the interconnect due to uncertainties like fabrication process or temperature. A real application example involving the frequency-domain analysis of a coupled microstrip and the computation of the parameters variability effects on the transmission-line response concludes this paper.

Parameters Variability Effects on Multiconductor Interconnects via Hermite Polynomial Chaos

Addresses the development of macromodels for input and output ports of a digital device. The proposed macromodels consist of parametric representations that can be obtained from port transient waveforms at the device ports via a well established procedure. The models are implementable as SPICE subcircuits and their accuracy and efficiency are verified by applying the approach to the characterization of transistor-level models of commercial devices.

/pdf/parametric-macromodels-of-digital-i-o-ports-3w28sov61c.pdf

Parametric macromodels of digital I/O ports

The aim of this article is to provide an overview of polynomial chaos (PC) based methods for the statistical analysis of transmission lines. The underlying idea of PC is to represent stochastic line voltages and currents as expansions of predefined orthogonal polynomials. The determination of the expansion coefficients allows obtaining pertinent statistical information and is generally much faster than running, e.g., a Monte Carlo (MC) analysis. There exist several approaches to calculate the PC expansion coefficients. The article briefly reviews virtually all existing methods, whilst focusing on the popular and accurate stochastic Galerkin (SG) method as well as on the recent, more efficient and non-intrusive formulation of the so-called stochastic testing (ST) method. These two techniques are introduced by way of a simple illustrative example, i.e., a single-wire line running above a ground plane. Numerical comparisons in terms of accuracy and efficiency are also provided for a four-wire line with nonlinear terminations.

/pdf/stochastic-transmission-line-analysis-via-polynomial-chaos-2b2lv4v6q9.pdf

Igor Simone Stievano

Papers

Stochastic Analysis of Multiconductor Cables and Interconnects

M/spl pi/log, macromodeling via parametric identification of logic gates

Parameters Variability Effects on Multiconductor Interconnects via Hermite Polynomial Chaos

Parametric macromodels of digital I/O ports

Stochastic transmission line analysis via polynomial chaos methods: an overview