This chapter gives a brief overview of sampling based on sparsity. The idea is that a signal which is not bandlimited can sometimes be reconstructed from a sampled version if we have a priori knowledge that the signal is sparse in a certain basis. These results are very different from the results of Shannon and Nyquist, and are sometimes referred to as sub-Nyquist sampling theories. They can be regarded as generalizations of traditional sampling theory, which was based on the bandlimited property. Examples include sampling of finite-duration signals whose DFTs are sparse. Sparse reconstruction methods are closely related to the theory of compressive sensing, which is also briefly introduced. These are major topics that have emerged in the last two decades, so the chapter provides important references for further reading.

This concise and rigorous textbook introduces students to the subject of continuum thermodynamics, providing a complete treatment of the subject with practical applications to material modelling.

• Presents mathematical prerequisites and the foundations of continuum mechanics, taking the student step-by-step through the subject to allow full understanding of the theory.

• Introduces more advanced topics such as theories for the investigation of material models, showing how they relate to real-world practical applications.

• Numerous examples and illustrations, alongside end-of-chapter problems with helpful hints, help describe complex concepts and mathematical derivations.

This chapter presents mathematical details relating to the Fourier transform (FT), Fourier series, and their inverses. These details were omitted in the preceding chapters in order to enable the reader to focus on the engineering side. The material reviewed in this chapter is fundamental and of lasting value, even though from the engineer’s viewpoint the importance may not manifest in day-to-day applications of Fourier representations. First the chapter discusses the discrete-time case, wherein two types of Fourier transform are distinguished, namely, l1-FT and l2-FT. A similar distinction between L1-FT and L2-FT for the continuous-time case is made next. When such FTs do not exist, it is still possible for a Fourier transform (or inverse) to exist in the sense of the so-called Cauchy principal value or improper Riemann integral, as explained. A detailed discussion on the pointwise convergence of the Fourier series representation is then given, wherein a number of sufficient conditions for such convergence are presented. This involves concepts such as bounded variation, one-sided derivatives, and so on. Detailed discussions of these concepts, along with several illuminating examples, are presented. The discussion is also extended to the case of the Fourier integral.

This chapter introduces recursive difference equations. These equations represent discrete-time LTI systems when the so-called initial conditions are zero. The transfer functions of such LTI systems have a rational form (ratios of polynomials in z). Recursive difference equations offer a computationally efficient way to implement systems whose outputs may depend on an infinite number of past inputs. The recursive property allows the infinite past to be remembered by remembering only a finite number of past outputs. Poles and zeros of rational transfer functions are introduced, and conditions for stability expressed in terms of pole locations. Computational graphs for digital filters, such as the direct-form structure, cascade-form structure, and parallel-form structure, are introduced. The partial fraction expansion (PFE) method for analysis of rational transfer functions is introduced. It is also shown how the coefficients of a rational transfer function can be identified by measuring a finite number of samples of the impulse response. The chapter also shows how the operation of polynomial division can be efficiently implemented in the form of a recursive difference equation.

This concise and rigorous textbook introduces students to the subject of continuum thermodynamics, providing a complete treatment of the subject with practical applications to material modelling.

• Presents mathematical prerequisites and the foundations of continuum mechanics, taking the student step-by-step through the subject to allow full understanding of the theory.

• Introduces more advanced topics such as theories for the investigation of material models, showing how they relate to real-world practical applications.

• Numerous examples and illustrations, alongside end-of-chapter problems with helpful hints, help describe complex concepts and mathematical derivations.

This chapter discusses various approaches to demand response, including time-of-use pricing, critical peak pricing, interruptible service, and priority service pricing. A quantitative model of time-of-use pricing is presented, and its properties are analyzed using the KKT conditions of the model. A model of priority service pricing is presented, whereby an aggregator offers different levels of reliability in power service at different prices. The model is described as a leader-follower interaction, where the leader is the aggregator that prices contracts and the followers are residential customers who select menu options.

Gives a brief overview of the book. Notations for signal representation in continuous time and discrete time are introduced. Both one-dimensional and two-dimensional signals are introduced, and simple examples of images are presented. Examples of noise removal and image smoothing (filtering) are demonstrated. The concept of frequency is introduced and its importance as well as its role in signal representation are explained, giving musical notes as examples. The history of signal processing, the role of theory, and the connections to real-life applications are mentioned in an introductory way. The chapter also draws attention to the impact of signal processing in digital communications (e.g., cell-phone communications), gravity wave detection, deep space communications, and so on.

This chapter introduces ancillary services, focusing specifically on reserves. Reserves are classified in terms of their response time between primary (frequency containment, automatic generation control, load frequency control, regulation), secondary (frequency restoration reserve, operating reserve, spinning reserve, non-spinning), and tertiary (or replacement reserve). Contingency reserve and flexible ramp products are also discussed. Interactions between reserves are discussed. A co-optimization model of energy and reserves is presented, and its optimal solution is characterized using KKT conditions. The security-constrained economic dispatch model and the N-1 reliability criterion are introduced. A centralized optimization model for simultaneous auctioning of energy and reserves is introduced, and its equivalence to a market equilibrium is established. The sequential clearing of energy and reserves is presented. Market models for multiple types of reserves are introduced, and the effect of substitutability is discussed. Operating reserve demand curves are introduced, and their effect on energy prices is discussed. ORDCs based on loss of load probability and value of lost load are discussed. Balancing markets are defined, and the notions of balancing service providers and balancing responsible parties are discussed in the context of the balancing model.

This chapter introduces state-space descriptions for computational graphs (structures) representing discrete-time LTI systems. They are not only useful in theoretical analysis, but can also be used to derive alternative structures for a transfer function starting from a known structure. The chapter considers systems with possibly multiple inputs and outputs (MIMO systems); systems with a single input and a single output (SISO systems) are special cases. General expressions for the transfer matrix and impulse response matrix are derived in terms of state-space descriptions. The concept of structure minimality is discussed, and related to properties called reachability and observability. It is seen that state-space descriptions give a different perspective on system poles, in terms of the eigenvalues of the state transition matrix. The chapter also revisits IIR digital allpass filters and derives several equivalent structures for them using so-called similarity transformations on state-space descriptions. Specifically, a number of lattice structures are presented for allpass filters. As a practical example of impact, if such a structure is used to implement the second-order allpass filter in a notch filter, then the notch frequency and notch quality can be independently controlled by two separate multipliers.

This chapter introduces the medium-term hydrothermal planning problem. Two-stage stochastic linear programs are introduced first, and subsequently generalized to multi-stage stochastic linear programs. Various representations of multi-stage stochastic linear programs are presented, including representations on scenario trees, representations on lattices (for Markov processes), as well as representations with stagewise independent uncertainty. These models are applied to a running example of hydrothermal planning, which is used as the basis for introducing the notion of dynamic programming value functions. The value of water is defined, and demonstrated on a hydrothermal planning problem. The chapter then proceeds to focus on the performance of stochastic programs. The wait-and-see and here-and-now value are introduced for two-stage stochastic programs. These are used to define the expected value of perfect information. The expected value solution is also defined, and used in order to introduce the value of the stochastic solution. Sampling is discussed briefly in the context of sample average approximation and importance sampling.

This is a detailed chapter on digital filter design. Specific digital filters such as notch and antinotch filters, and sharp-cutoff lowpass filters such as Butterworth filters are discussed in detail. Also discussed are allpass filters and some of their applications, including the implementation of notch and antinotch filters. Computational graphs (structures) for allpass filters are presented. It is explained how continuous-time filters can be transformed into discrete time by using the bilinear transformation. A simple method for the design of linear-phase FIR filters, called the window-based method, is also presented. Examples include the Kaiser window and the Hamming window. A comparative discussion of FIR and IIR filters is given. It is demonstrated how nonlinear-phase filters can create visible phase distortion in images. Towards the end, a detailed discussion of steady-state and transient components of filter outputs is given. The dependence of transient duration on pole position is explained. The chapter concludes with a discussion of spectral factorization.

This chapter introduces some basic mathematical notions that are used throughout the book. Convex sets and functions, optimization problems, feasible solutions, and optimal solutions are first defined. The chapter then covers duality theory, including the definition of the Lagrangian function and the dual function, which are used to derive the duals of linear programs. Weak and strong duality are then defined and related to certain classes of optimization problems. The Karush–Kuhn–Tucker (KKT) conditions are defined, and their relation to the optimal solution of mathematical programs is discussed. KKT conditions are a fundamental concept used extensively in the book in order to understand the properties and economic interpretations of the various economic models encountered. Subgradients are subsequently defined in order to establish the relation between Lagrange multipliers and the sensitivity of an optimization model with respect to changes in the right-hand side parameters of its constraints. These sensitivity results are also used repeatedly in the book, for instance in order to derive locational marginal prices in chapter 5.