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

In this crucial lecture we introduce the definition of the Lie algebra associated to a Lie group and its relation to that group. All three sections are logically necessary for what follows; §8.1 is essential. We use here a little more manifold theory: specifically, the differential of a map of manifolds is used in a fundamental way in §8.1, the notion of the tangent vector to an arc in a manifold is used in §8.2 and §8.3, and the notion of a vector field is introduced in an auxiliary capacity in §8.3. The Campbell-Hausdorff formula is introduced only to establish the First and Second Principles of §8.1 below; if you are willing to take those on faith the formula (and exercises dealing with it) can be skimmed. Exercises 8.27–8.29 give alternative descriptions of the Lie algebra associated to a Lie group, but can be skipped for now.

Lie Algebras and Lie Groups

http://ieeexplore.ieee.org/iel5/5/31341/01457276.pdf

Parameter estimation: Principles and problems

A simple procedure is provided to write the equations of motion of controlled mechanical systems with constraints as controlled Hamiltonian equations with respect to a "Poisson" bracket which does not necessarily satisfy the Jacobi-identity. Based on the Hamiltonian form a stabilization procedure is proposed. >

A Hamiltonian approach to stabilization of nonholonomic mechanical systems

A scroll chaotic system containing a HP memristor model and triangular wave sequence is proposed in this article. Because the memristor is both a nonlinear element and a memory element intrinsically, it is considered a potential candidate to reduce system power consumption and circuit size. A reasonable mathematical structure of triangular wave sequence and the selection of appropriate amplitude, balance point and turning point reduce the dynamic range of signal input caused by the integrator. The proposed system produces a wealth of chaos, just by changing one parameter. Circuit simulations are conducted and the chaotic attractors can be observed. Theoretical analysis, computer simulation and calculation of maximum Lyapunov exponent have been used to research the basic dynamics of this system. The consistency of circuit implementation and computer simulations verifies the effectiveness of the system design.

A Memristor-Based Scroll Chaotic System — Design, Analysis and Circuit Implementation

This paper deals with generalization of the Brayton–Moser network decomposition and related structural properties to a relatively large class of finite dimensional strictly causal systems, which can be described in the state-space representation form. The resulting energy-metric function is defined for dissipative systems and is induced by the output signal dissipation power. It is demonstrated that such a power-oriented approach determines both, the structure of a system representation as well as the corresponding system state space topology. A special form of physically correct internal structure of an equivalent state space representation has been derived as a natural consequence of strict causality, the state-space energy conservation, dissipativity assumption and the state minimality requirement.

On state-space energy based generalization of Brayton–Moser topological approach to electrical network decomposition

The presented contribution is motivated by some fundamental questions arising in theoretical analysis of nonlinear state space energy exchange. Knowledge of principles of nonlinear dynamics and synthesis of chaotic attractors is found to be very important also for other potential applications, such as encryption by secure communication, modeling of nonlinear phenomena in power networks, modeling in biomedical engineering, etc. This provides a strong motivation for the current research on exploiting some new chaotic attractors and their implementations. In this paper, an electronic circuit was designed and built to confirm typical behavior of a class of nonlinearly coupled chaotic oscillators. It was explained theoretically, as well as demonstrated by computer simulation combined with laboratory experiments that some typical chaotic phenomena can appear as a consequence of irregular energy exchange between two coupled oscillators. In our experiments, the system consists of a 2-nd order nonlinear antidissipative subsystem nonlinearly coupled with a linear oscillator with dissipation.

http://www.wseas.us/e-library/conferences/2010/Corfu/SYSTEMS/SYSTEMS1-21.pdf

Nonlinearly coupled oscillators and state space energy approach

The paper deals with structural properties of a class of strictly causal systems. It is shown that a special physically correct internal structure of a given system representation caled dissipation normal form can be derived as a natural consequence of strict causality, dissipativity, minimality and asymptotic stability requirements. A proper generalization of classic Tellegen's theorem together with a concept of bi-orthonormal basis of the state velocity space have been used as basic ingrediences expressing the signal energy conservation law for abstract system state space representations. It is demonstrated by examples that in continuous-time version the resulting structure represent a unifying tool for analysis and synthesis of a relatively general class of linear as well as nonlinear causal systems.

http://www.wseas.us/e-library/conferences/2005athens/cscc/papers/497-848.pdf

Dissipation normal form, conservativity, instability and chaotic behavior of continuous-time strictly causal systems

Observer design is one of large fields investigated in automatic control theory and a lot of articles have already been dedicated to it in technical literature. Non-linear observer design method based on dissipation normal form proposed in the paper represents a new approach to solving the observer design problem for a certain class of non-linear systems. As the theoretical basis of the approach the well known dissipative system theory has been chosen. The main achievement of the contribution consists in the fact that the error dynamics of the observer is priory chosen non-linear. It provides more flexibility in the sense of specifying error convergence properties to zero in comparison with other techniques. Lyapunov’s stability theory is the other basic point of the approach.

Non-linear observer design method based on dissipation normal form

The paper studies some connections between the main results of the well known Wiener-Kalman-Bucy stochastic approach to filtering problems based mainly on the linear stochastic estimation theory and emphasizing the optimality aspects of the achieved results and the classical deterministic frequency domain linear filters such as Chebyshev, Butterworth, Bessel, etc. A new non-stochastic but not necessarily deterministic (possibly non-linear) alternative approach called asymptotic filtering based mainly on the concepts of signal power, signal energy and a system equivalence relation plays an important role in the presentation. Filtering error invariance and convergence aspects are emphasized in the approach. It is shown that introducing the signal power as the quantitative measure of energy dissipation makes it possible to achieve reasonable results from the optimality point of view as well. The property of structural energy dissipativeness is one of the most important and fundamental features of resulting filters. Therefore, it is natural to call them asymptotic filters. The notion of the asymptotic filter is carried in the paper as a proper tool in order to unify stochastic and non-stochastic, linear and nonlinear approaches to signal filtering.

Josef Hrusak

Papers

On state-space energy based generalization of Brayton–Moser topological approach to electrical network decomposition

Nonlinearly coupled oscillators and state space energy approach

Dissipation normal form, conservativity, instability and chaotic behavior of continuous-time strictly causal systems

Non-linear observer design method based on dissipation normal form

Non-linear and signal energy optimal asymptotic filter design