This chapter examines discrete-time LTI systems in detail. It shows that the input–output behavior of an LTI system is characterized by the so-called impulse response. The output is shown to be the so-called convolution of the input with the impulse response. It is then shown that exponentials are eigenfunctions of LTI systems. This property leads to the ideas of transfer functions and frequency responses for LTI systems. It is argued that the frequency response gives a systematic meaning to the term “filtering.” Image filtering is demonstrated with examples. The discrete-time Fourier transform (DTFT) is introduced to describe the frequency domain behavior of LTI systems, and allows one to represent a signal as a superposition of single-frequency signals (the Fourier representation). DTFT is discussed in detail, with many examples. The z-transform, which is of great importance in the study of LTI systems, is also introduced and its connection to the Fourier transform explained. Attention is also given to real signals and real filters, because of their additional properties in the frequency domain. Homogeneous time-invariant (HTI) systems are also introduced. Continuous-time counterparts of these topics are explained. B-splines, which arise as examples in continuous-time convolution, are presented.

This chapter discusses many interesting properties of bandlimited signals. The subspace of bandlimited signals is introduced. It is shown that uniformly shifted versions of an appropriately chosen sinc function constitute an orthogonal basis for this subspace. It is also shown that the integral and the energy of a bandlimited signal can be obtained exactly from samples if the sampling rate is high enough. For non-bandlimited functions, such a result is only approximately true, with the approximation getting better as the sampling rate increases. A number of less obvious consequences of these results are also presented. Thus, well-known mathematical identities are derived using sampling theory. For example, the Madhava–Leibniz formula for the approximation of π can be derived like this. When samples of a bandlimited signal are contaminated with noise, the reconstructed signal is also noisy. This noise depends on the reconstruction filter, which in general is not unique. Excess bandwidth in this filter increases the noise, and this is quantitatively analyzed. An interesting connection between bandlimited signals and analytic functions (entire functions) is then presented. This has many implications, one being that bandlimited signals are infinitely smooth.

This chapter presents the generation capacity expansion planning problem and provides an economic analysis of the model. Scarcity rents in energy-only markets are defined, and the equivalence of the centralized expansion planning problem to a decentralized long-term economic equilibrium is established. The missing money problem is discussed, and various approaches for overcoming it are analyzed through models. The value of lost load pricing mechanism remunerates units when the system is scarce at the estimated value of lost load. Capacity mechanisms introduce a separate revenue stream for paying investors to build or maintain capacity. The cost of new entry is defined, and is related to the loss of load expectation. A model for capacity auctions is introduced and the shape of the capacity auction demand curve and role of capacity credit is discussed. The role of reliability options in capacity auctions is discussed. Alternative mechanisms such as installed capacity obligations, capacity payments, decentralized capacity mechanisms, and strategic reserves are discussed. The operating reserve demand curve mechanism remunerates units for offering flexible capacity in the form of reserve. The ORDC model of chapter 6 is revisited in the context of a long-term equilibrium.

This chapter discusses the Fourier series representation for continuous-time signals. This is applicable to signals which are either periodic or have a finite duration. The connections between the continuous-time Fourier transform (CTFT), the discrete-time Fourier transform (DTFT), and Fourier series are also explained. Properties of Fourier series are discussed and many examples presented. For real-valued signals it is shown that the Fourier series can be written as a sum of a cosine series and a sine series; examples include rectified cosines, which have applications in electric power supplies. It is shown that the basis functions used in the Fourier series representation satisfy an orthogonality property. This makes the truncated version of the Fourier representation optimal in a certain sense. The so-called principal component approximation derived from the Fourier series is also discussed. A detailed discussion of the properties of musical signals in the light of Fourier series theory is presented, and leads to a discussion of musical scales, consonance, and dissonance. Also explained is the connection between Fourier series and the function-approximation property of multilayer neural networks, used widely in machine learning. An overview of wavelet representations and the contrast with Fourier series representations is also given.

This chapter introduces the discrete Fourier transform (DFT), which is different from the discrete-time Fourier transform (DTFT) introduced earlier. The DFT transforms an N-point sequence x[n] in the time domain to an N-point sequence X[k] in the frequency domain by sampling the DTFT of x[n]. A matrix representation for this transformation is introduced, and the properties of the DFT matrix are studied. The fast Fourier transform (FFT), which is a fast algorithm to compute the DFT, is also introduced. The FFT makes the computation of the Fourier transforms of large sets of data practical. The digital signal processing revolution of the 1960s was possible because of the FFT. This chapter introduces the simplest form of FFT, called the radix-2 FFT, and a number of its properties. The chapter also introduces circular or cyclic convolution, which has a special place in DFT theory, and explains the connection to ordinary convolution. Circular convolution paves the way for fast algorithms for ordinary convolution, using the FFT. The chapter also summarizes the relationships between the four types of Fourier transform studied in this book: CTFT, DTFT, DFT, and Fourier series.

Chemical nucleation involves cluster growth by chemical reactions. In the case where clusters grow via a simple sequence of reversible chemical reactions, a summation expression for the steady-state nucleation rate can be derived. However, in many cases the chemical pathway to cluster growth is more complicated, and requires solving a set of species population balance equations that depend on the specific chemical system. Two examples are considered: soot nucleation in hydrocarbon combustion and nucleation of silicon particles in thermal decomposition of silane. In both cases, chemical kinetic mechanisms have been developed that allow for numerical simulations of particle formation. Soot nucleation is believed to proceed through the formation of polycyclic aromatic hydrocarbons. Models have been developed for the formation of the first aromatic ring and for subsequent growth, either through reaction with small molecules or by coagulation. Silicon nucleation from silane involves a large set of silicon hydride species, which can be grouped into classes according to their structure and reactivity, facilitating estimates of their free energies and reaction rate constants.

In single-component homogeneous nucleation, the summation expression for the steady-state nucleation rate requires values of the forward rate constants and Gibbs free energies of cluster formation. If atomistic data are available for these quantities, then these could be used instead of CNT. In an atomistic approach, clusters are treated as distinct molecular species, rather than as a small piece of the bulk condensed phase. Examples are presented of atomistic data generated by means of computational chemistry for water clusters up to size 10, and for aluminum clusters up to size 60. In both cases, the free energy of cluster formation is found to be a multimodal function of cluster size, both quantitatively and qualitatively different than in CNT. Condensation rate constants can be affected by the need for a third body as a collision partner, and by attractive intermolecular forces in collisions between clusters and monomers. An approach is suggested for constructing a “master table” of free energies of cluster formation, based on a hybrid of atomistic data, experimental values inferred by means of the nucleation theorem, and extrapolations to larger cluster sizes based on CNT.

Particle nucleation in plasmas occurs under a wide range of conditions. In some cases, such as thermal plasma synthesis of metal nanoparticles, nucleation may follow the conventional scenarios of single-component homogeneous or ion-induced nucleation. In other cases, such as dust formation in nonthermal plasmas of the type used in semiconductor processing, the paths to nucleation are specific to the chemistry of the gases introduced into the processing chamber. In such cases, nucleation typically involves a mix of phenomena that combine chemical nucleation with plasma physics, with the chemistry being driven by electron impact, and the charging of small clusters by free electrons and ions playing an important role in cluster growth. The charging and transport of clusters and particles affect the electric field profile, causing the plasma and the aerosol phase to be strongly coupled. An example is considered of silicon particle nucleation in silane-containing plasmas, the most studied system because of its importance in semiconductor processing. Cluster growth in this system is dominated by reactions between anion clusters and neutral molecules.