An important theme in this book is the study of systems y = Φx. We have looked carefully at the case when Φ is linear and can be taken as an m × n matrix or more generally a linear operator on a Hilbert space. In the various applications we have considered thus far, x is the signal or input, Φ is the transform and y is the sample or output. We have studied in detail the reconstruction of x from y and have developed many tools to deal with this case.

This chapter is concerned with the parsimonious representation of data x, a fundamental problem in many sciences. Here, typically x is a vector in some high-dimensional vector space (object space), and Φ is an operation (often nonlinear) we perform on x, for example dimension reduction, reconstruction, classification, etc. The parsimonious representation of data typically means obtaining accurate models y of naturally occurring sources of data, obtaining optimal representations of such models, and rapidly computing such optimal representations. Indeed, modern society is fraught with many high-dimensional highly complex data problems, critical in diverse areas such as medicine, geology, critical infrastructure, health and economics, to name just a few.

This is an introduction to some topics in this subject showing how to apply mathematical tools we have already developed: linear algebra and SVD, Fourier transforms/series, wavelets/infinite products, compressive sampling and linear filters, PCA and dimension reduction/clustering, compression methods (linear and nonlinear).

In this chapter we will provide some solutions to the questions of the existence of wavelets with compact and continuous scaling functions, of the L2 convergence of the cascade algorithm, and the accuracy of approximation of functions by wavelets. For orthogonal wavelets we use a normalized impulse response vector c = (c0, …, cN) that satis_es double-shift orthogonality, though for nonorthogonal wavelets we will weaken this orthogonality restriction on c. We will start the cascade algorithm at i = 0 with the Haar scaling function φ(0)(t), and then update to get successive approximations φ(i)(t), i = 0, 1, … The advantages of starting with the Haar scaling function in each case are simplicity and the fact that orthogonality of the scaling function to integer shifts is satisfied automatically. We already showed in Section 9.9 that if c satisfies double-shift orthogonality and φ(i)(t) is orthogonal to integer shifts of itself, then φ(i+1)(t) preserves this orthogonality. Our aim is to find conditions such that the φ(t) = limi→∞φ(i)(t) exists, first in the L2 sense and then pointwise. The Haar scaling function is a step function, and it is easy to see that each iteration of the cascade algorithm maps a step function to a step function. Thus our problem harkens back to the Hilbert space considerations of Section 1.5.1. If we can show that the sequence of step functions φ(0), φ(1), φ(2), … is Cauchy in the L2 norm then it will follow that the sequence converges.

We saw in Section 10.2 that significant progress has been made at the physical layer of fiber-wireless (FiWi) and in particular radio-over-fiber (RoF) networks. However, state-of-the-art radio-and-fiber (R&F) networks integrating Ethernet passive optical networks (EPONs) with a wireless local area network (WLAN)-based wireless mesh network (WMN) (see Fig. 10.4) suffer from a poor quality of video transmissions that sharply deteriorates for an increasing number of wireless hops. Therefore, a more involved investigation of the performance of integrated EPON/WLAN-based WMN networks, especially in the wireless segment, is needed.

In this chapter, we propose and investigate a FiWi network architecture that converges next-generation WLAN-based WMN and EPON networks. The considered FiWi network architecture enables existent EPON networks to be upgraded with wireless extensions in a pay-as-you-grow manner while providing backward compatibility with legacy infrastructure and protecting previous investment. Furthermore, the benefits of extending advanced frame aggregation techniques to EPON and their integrated operation across both optical and wireless segments are investigated. For more detailed information the interested reader is referred to (Ghazisaidi et al. [2010], Ghazisaidi and Maier [2011]).

Integration of next-generation WLAN and EPON

Figure 16.1 depicts our proposed network architecture and node structures for integrating a next-generation WLAN-based WMN with an EPON network. In this figure, an ONU represents a conventional EPON optical network unit (ONU), as described in Chapter 4. Some of the ONUs are upgraded with a mesh portal point (MPP) to interface with the WMN.

Basically, this is a book about mathematics, pitched at the advanced undergraduate/beginning graduate level, where ideas from signal processing are used to motivate much of the material, and applications of the theory to signal processing are featured. It is meant for math students who are interested in potential applications of mathematical structures and for students from the fields of application who want to understand the mathematical foundations of their subject. The first few chapters cover rather standard material in Fourier analysis, functional analysis, probability theory and linear algebra, but the topics are carefully chosen to prepare the student for the more technical applications to come. The mathematical core is the treatment of the linear system y = Φx in both finite-dimensional and infinite-dimensional cases. This breaks up naturally into three categories in which the system is determined, overdetermined or underdetermined. Each has different mathematical aspects and leads to different types of application. There are a number of books with some overlap in coverage with this volume, e.g., [11, 15, 17, 19, 53, 69, 71, 72, 73, 82, 84, 95, 99, 101], and we have profited from them. However, our text has a number of features, including its coverage of subject matter, that together make it unique. An important aspect of this book on the interface between fields is that it is largely self-contained. Many such books continually refer the reader elsewhere for essential background material.

This book covers fundamental concepts in probability, random processes, and statistical analysis. A central theme in this book is the interplay between probability theory and statistical analysis. The book will be suitable to graduate students majoring in information sciences and systems in such departments as Electrical and Computer Engineering, Computer Science, Operations Research, Economics and Financial Engineering, Applied Mathematics and Statistics, Biology, Chemistry and Physics. The instructor and the reader may opt to skip some chapters or sections and focus on chapters that are relevant to their fields of study. At the end of this preface, we provide suggested course plans for various disciplines.

Organization of the book

Before we jump into a mathematical description of probability theory and random processes, we will provide in Chapter 1, Introduction, specific reasons why the subjects of this book pertain to study and research across diverse fields or disciplines: (i) communications, information and control systems, (ii) signal processing, (iii) machine learning, (iv) bioinformatics and related fields, (v) econometrics and mathematical finance, (vi) queueing and loss systems, and (vii) other applications. We will then provide a brief but fascinating historical review of the development of (a) classical probability theory, (b) modern probability theory, (c) random processes, and (d) statistical analysis and inference. This historical review also serves as an overview of various topics discussed in this volume.

Broadband access involves various enabling technologies. The choice of broadband technology generally depends on a number of factors such as availability, price, location, service bundling, and technological requirements. In this chapter, we review the technical details of the most common types of fixed wireline, fixed wireless, and mobile wireless legacy broadband technologies, including, but not limited to, digital subscriber line, cable modem, and 3G systems. In our discussion, we try to highlight the benefits and limitations of available legacy broadband technologies.

Fixed wireline broadband technologies

Digital subscriber line

The local subscriber loop of most of today's telephone companies consists of unshielded twisted pair (UTP) copper wires. The length of the local loop depends on a number of factors such as population density and location of the connected residential or business customers. However, it is usually no longer than 4–6 km due to limitations stemming from legacy narrowband telephony. The copper wire pairs can be buried or aerial and are typically grouped together in so-called binders, also known as bundles, which may contain tens or even hundreds or thousands of twisted pairs (Czajkowski [1999]).

Traditional voiceband modems operate at the bottom frequencies (0–4 kHz) of the spectrum available on twisted pairs and offer data rates of no more than 56 kb/s. It is important to note, however, that it is not the UTP copper wires that prevent transport of broadband data signals but rather the bandwidth allocated by legacy telephone company switches to voice calls.