Introduction

The processing of signals whose domain is the 2-sphere or unit sphere1 has been an ongoing area of research in the past few decades and is becoming increasingly more active. Such signals are widely used in geodesy and planetary studies (Simons et al., 1997; Wieczorek and Simons, 2005; Simons et al., 2006; Audet, 2011). In many cases of interest flat Euclidean modeling of planetary and heavenly data does not work. Planetary curvature should be taken into account especially for small heavenly bodies such as the Earth, Venus, Mars, and the Moon (Wieczorek, 2007). Other applications, for the processing of signals on the 2-sphere, include the study of cosmic microwave background in cosmology (Wiaux et al., 2005; Starck et al., 2006; Spergel et al., 2007), 3D beamforming/sensing (Simons et al., 2006; Górski et al., 2005; Armitage and Wandelt, 2004; Ng, 2005; Wandelt and Górski, 2001; Rafaely, 2004; Wiaux et al., 2006), computer graphics and computer vision (Brechbühler et al., 1995; Schröder and Sweldens, 2000; Han et al., 2007), electromagnetic inverse problems (Colton and Kress, 1998), brain cortical surface analysis in medical imaging (Yu et al., 2007; Yeo et al., 2008), and channel modeling for wireless communication systems (Pollock et al., 2003; Abhayapala et al., 2003). This type of processing exhibits important differences from the processing of signals on Euclidean domains—such as time-based signals whose domain is the real line R, or 2D or 3D signals and images, whose domain is multi-dimensional, but still Euclidean.

Introduction to Hilbert spaces

The basic idea

Hilbert spaces are the means by which the “ordinary experience of Euclidean concepts can be extended meaningfully into the idealized constructions of more complex abstract mathematics” (Bernkopf, 2008).

If our global plan is to abstract Euclidean concepts to more general mathematical constructions, then we better think of what it is in Euclidean space that is so desirable in the first place. An answer is geometry — in geometry one talks about points, lines, distances and angles, and these are familiar objects that our brains are well-adept to recognize and easily manipulate. Through imagery we use pictures to visualize solutions to problems posed in geometry. We may still follow Descartes and use algebra to furnish a proof, but typically through spatial reasoning we either make the breakthrough or see the solution to a problem as being plausible. Contrary to any preconception you may have, Hilbert spaces are about making obtuse problems have obvious answers when viewed using geometrical concepts.

The elements of Euclidean geometry such as points, distance and angle between points are abstracted in Hilbert spaces so that we can treat sets of objects such as functions in the same manner as we do points (and vectors) in 3D space. Hilbert spaces encapsulate the powerful idea that in many regards abstract objects such as functions can be treated just like vectors.

To others, less fond of mathematics, Hilbert spaces also encapsulate the logical extension of real and complex analysis to a wider sphere of suffering.

Evolution towards mobile broadband communications

Explosive growth of wireless communications services and products in the past three decades or so has fundamentally changed the way by which the majority of the world's population exchange, distribute and access information. As a strong driver of the growth, cellular telephony has so far been the most successful application of wireless communications. All forms of wireless communications utilize the radio spectrum, a scarce natural resource. Spectrum access is a general term of the technologies by which users utilize the radio spectrum. Cellular telephony uses a cellular concept, which provides an effective spectrum access solution to improve the efficiency of radio spectrum utilization.

In a cellular system, many base stations are deployed to cover a large service area. The service area is divided into a number of cells, each served by a base station, as shown in Figure 1.1. When a user makes a call, it is connected to the base station with the best RF propagation. The base stations are connected to the operator's core networks via backhaul connections such as T1 or fiber optics. Spectrum is reused among the cells. This is possible because a signal decays fast as it travels through the wireless channel. If a signal utilizing some spectrum in a cell is sufficiently attenuated in another cell, then the same spectrum can be reused.

In this chapter, we will study power and bandwidth allocation in a multi-cell scenario where inter-cell interference dominates. As in Chapter 5, we assume that the channel gain does not vary over time or frequency.

Recall from Section 1.1 that spectrum reuse among cells is the key to increasing overall spectrum utilization and that spectrum reuse leads inter-cell interference to be managed. In a conventional cellular deployment, there are two basic tools to manage inter-cell interference. One is cell planning, including carefully choosing base station locations and fine tuning antenna patterns to maximize service quality. In an ideal world with homogeneous wireless channel propagation, base stations are placed in the hexagonal grids as shown in Figure 6.1. Practical considerations such as local terrain characteristics affect cell planning choices. The second tool is handoff. A user switches to a new base station as it moves across the boundary between two cells. Under so-called unrestricted association where the user is allowed to connect to any base station, handoff ensures that the user is always connected to the “best” base station, which is usually the closest one. As a result, the interference from an adjacent base station does not exceed the desired signal from a serving base station.

Introduction

The second half of the twentieth century experienced an explosive growth in information technology, including data transmission, processing, and computation. This trend will continue at an even faster pace in the twenty-first century. Radios and televisions started in the 1920s and 1940s respectively, and involved transmission from a single transmitter to multiple receivers using AM and FM modulations. Baseband analog telephony, starting in the 1900s, was originally suited only for local area person-to-person communication. It became possible to have long-distance communication after using cascades of regeneration repeaters based on digital PCM modulation. Various digital modulations with and without coding, across microwave, satellite, and optical fiber links, allowed the explosive transmissions of data around the world starting in the 1950s–1960s. The emergence of Ethernet, local area net, and, finally, the World Wide Web in the 1980s–1990s allowed almost unlimited communication from any computer to another computer. In the first decade of the twenty-first century, by using wireless communication technology, we have achieved cellular telephony and instant/personal data services for humans, and ubiquitous data collection and transmission using ad hoc and sensor networks. By using cable, optical fibers, and direct satellite communications, real-time on-demand wideband data services in offices and homes are feasible.

This chapter first summarizes the system level benefits of using OFDMA as the underlying multiple access technology, and then qualitatively presents the basic system design principles of OFDMA mobile broadband communications in a cellular network. This chapter serves as a preview of many topics to be covered in the remaining chapters of the book. We emphasize the concepts and insights here and leave the quantitative modeling and analysis to the subsequent chapters.

In order to better understand the system design principles, throughout this section we often contrast OFDMA with CDMA. CDMA has been widely-used in cellular wireless communications from the second generation circuit-switched voice system to the third generation data system. A basic commonality between CDMA and OFDMA is that they are both wideband spread spectrum technologies. Therefore, comparisons between the two will help demonstrate the evolution of the system design ideas. CDMA is well studied in the literature. The readers can learn CDMA from textbooks such as [61, 159, 168].

System benefits of OFDMA

Recall that an OFDMA signal is the sum of tone signals, each being sinusoid at a given tone frequency. We next elaborate on the fundamental property of OFDMA, orthogonality, in three aspects.

When a receiver is synchronized to an OFDMA signal both in time and in frequency, the power of a tone signal is entirely contained on that tone and does not spill to others. Therefore two synchronized OFDMA signals on distinct sets of tones do not interfere with each other as long as the receiver is synchronized to them.

The medium of wireless communications is the wireless radio frequency channel. We are interested in the characteristics of the wireless channel, in particular, how the channel response varies over time and frequency, as well as over the distance between a transmitter and a receiver. The variation in the channel response is usually referred to as channel fading. For a given signal, channel fading depends on the particular propagation environment, such as buildings, walls, ground, vehicles, between the transmitter and the receiver, as well as the carrier frequency of the signal. To characterize channel fading, we often use a statistical approach based on measurements made in a large variety of environments. Statistically, channel fading can be characterized by the following two different types of behaviors:

1. • Large-scale fading, which varies in a slow time scale (on the order of seconds) or in a large distance of many wavelengths. Large-scale fading is mainly caused by path loss and shadowing. Path loss is caused by signal strength degradation as the electromagnetic (EM) wave of the signal propagates through space. Shadowing results from penetration or reflection of objects much larger than the wavelength of the EM wave.

2. • Small-scale fading, which varies in a fast time scale (on the order of tens of milliseconds depending on mobility) or in a distance on the same order of the wavelength. Small-scale fading is mainly caused by multipath, as multiple copies of the transmitted signal add constructively or destructively at the receiver. Thus, small-scale fading is also referred to as multipath fading.

In a cellular system where a base station communicates with multiple users in every cell, a key question is how to share the system resources (power and bandwidth) among cells and users. We will study two aspects related to this question. The first aspect is about how to create a large data pipe between the base stations and users. We focus on the intra-cell issues in this chapter and leave the study of the inter-cell issues in Chapter 6. Specifically, in this chapter we will investigate how different user multiplexing schemes affect data rates and what the Pareto-optimal user multiplexing schemes, are. For simplicity, we decouple user multiplexing and channel fading by assuming that the wireless channel gain does not vary over time or frequency. Communications over the wireless fading channel has been studied in Chapter 4. The second aspect is about how to share the data pipe among users in a fair manner taking into upper layer considerations such as quality of service, and will be studied in Chapter 8.

We compare different multiplexing schemes, including orthogonal and nonorthogonal ones, in terms of rate region. In orthogonal multiplexing, different users are allocated system resources non-overlapping in time and frequency. Time division multiplexing (TDM) and frequency division multiplexing (FDM, such as OFDMA) are two examples. As noted in Chapter 3, an important advantage of OFDMA lies in its orthogonality and flexibility in user multiplexing.