Introduction

In layman's terms, a sufficient statistic for guessing M based on the observable Y is a random variable or a collection of random variables that contains all the information in Y that is relevant for guessing M. This is a particularly useful concept when the sufficient statistic is more concise than the observables. For example, if we observe the results of a thousand coin tosses Y1, …, Y1000 and we wish to test whether the coin is fair or has a bias of ¼, then a sufficient statistic turns out to be the number of “heads”among the outcomes Y1, …, Y1000. Another example was encountered in Section 20.12. There the observable was a two-dimensional random vector, and the sufficient statistic summarized the information that was relevant for guessing H in a scalar random variable; see (20.69).

In this chapter we provide a formal definition of sufficient statistics in the multihypothesis setting and explore the concept in some detail. We shall see that our definition is compatible with Definition 20.12.2, which we gave for the binary case. We only address the case where the observations take value in the d-dimensional Euclidean space ℝd. Extensions to observations consisting of a stochastic process are discussed in Section 26.3. Also, we only treat the case of guessing among a finite number of alternatives.

Introduction

In this chapter we finally address the detection problem in continuous time. The setup is described in Section 26.2. The key result of this chapter is that—even though in this setup the observation consists of a stochastic process (i.e., a continuum of random variables)—the problem can be reduced without loss of optimality to a finite-dimensional problem where the observation consists of a random vector. Before stating this result precisely in Section 26.4, we shall take a detour in Section 26.3 to discuss the definition of sufficient statistics when the observation consists of a continuous-time SP. The proof of the main result is delayed until Section 26.8. In Section 26.5 we analyze the conditional law of the sufficient statistic vector under each of the hypotheses. This analysis enables us in Section 26.6 to derive an optimal guessing rule and in Section 26.7 to analyze its performance. Section 26.9 addresses the front-end filter, which is a critical element of any practical implementation of the decision rule. Extensions to passband detection are then described in Section 26.10, followed by some examples in Section 26.11. Section 26.12 treats the problem of detection in “colored” noise, and the chapter concludes with a discussion of the detection problem for mean signals that are not bandlimited.

Signal processing deals with the analysis, interpretation, and manipulation of signals. Signals of interest include sound, images, biological signals, radar signals, and many others. Processing of such signals includes filtering, storage and reconstruction, separation of information from noise (e.g. aircraft identification by radar), compression (e.g. image compression), and feature extraction (e.g. speech-to-text conversion). In communications systems, signal processing is mostly performed at OSI layer 1, the physical layer (modulation, equalization, multiplexing, radio transmission, etc.), as well as at OSI layer 6, the presentation layer (source coding, including analog-to-digital conversion, and data compression). In cognitive radio networks, the major task of signal processing is spectrum sensing for detecting the unused spectrum and sharing it without harmful interference to other users. One important requirement in a cognitive radio network is sensing spectrum holes reliably and efficiently. Spectrum sensing techniques can be classified into three categories. First, cognitive radios must be capable of determining if a signal from a primary transmitter is locally present in a certain spectrum. Several approaches are used for transmitter detection, such as matched filter detection, energy detection, cyclostationary feature detection, and wavelet detection. Second, collaborative detection refers to spectrum sensing methods where information from multiple cognitive radio users is exploited for primary user detection. Third, the sensing devices can be separated from the secondary users and can be deployed into the cognitive network by the cognitive radio service provider. By doing this, the cost of the secondary user devices can be reduced and the hidden terminal problem/exposed terminal problem can be mitigated.

Cognitive radio is a new paradigm of designing wireless communications systems which aims to enhance the utilization of the radio frequency (RF) spectrum. The motivation behind cognitive radio is the scarcity of the available frequency spectrum, increasing demand, caused by the emerging wireless applications for mobile users. Most of the available radio spectrum has already been allocated to existing wireless systems, however, and only small parts of it can be licensed to new wireless applications. Nonetheless, a study by the Spectrum Policy Task Force (SPTF) of the Federal Communications Commission (FCC) has showed that some frequency bands are heavily used by licensed systems in particular locations and at particular times, but that there are also many frequency bands which are only partly occupied or largely unoccupied [110]. For example, spectrum bands allocated to cellular networks in the USA [111] reach the highest utilization during working hours, but remain largely unoccupied from midnight until early morning.

The major factor that leads to inefficient use of the radio spectrum is the spectrum licensing scheme itself. In traditional spectrum allocation based on the command-and control model, where the radio spectrum allocated to licensed users is not used, it cannot be utilized by unlicensed users and applications [5]. Due to this static and inflexible allocation, legacy wireless systems have to operate only on a dedicated spectrum band, and cannot adapt the transmission band according to the changing environment. For example, if one spectrum band is heavily used, the wireless system cannot change to operate on another more lightly used band.

Wireless communications technology has become a key element in modern society. In our daily life, devices such as garage door openers, TV remote controllers, cellular phones, personal digital assistants (PDAs), and satellite TV receivers are based on wireless communications technology. Today the total number of users subscribing to cellular wireless services have surpassed the number of users subscribing to the wired telephone services. Besides cellular wireless technology, cordless phones, wireless local area networks (WLANs), and satellites are being extensively used for voice- as well as data-oriented communications applications and entertainment services.

In 1895, Guglielmo Marconi demonstrated the feasibility of wireless communications by using electromagnetic waves. In 1906, the first radio broadcast was done by Reginald Fessenden to transmit music and voice over the air. In 1907, the commercial trans-Atlantic wireless transmission was launched. In 1946, the first public mobile telephone systems were introduced in several American cities. The first analog cellular system, the Nordic Mobile Telephone System (NMT), was introduced in Europe in 1981. In 1983, the first cellular wireless technology, the advanced mobile phone system (AMPS), was deployed for commercial use. During the last two decades there has been significant research and development in wireless communications technology. In fact, today it has emerged as the most flourishing branch of development in the area of telecommunications.

The various wireless communications systems available today differ in terms of data rate of transmission, geographical coverage area, transmission power, and mobility support for users.

Learning algorithms are used to build knowledge about a cognitive radio network so that the cognitive radio users can dynamically adapt their decisions on spectrum access. Learning algorithms are useful for cognitive radio networks with either collaborative or non-collaborative behavior among the network entities. In a non-collaborative scenario, there is no exchange of information in the network and a cognitive radio user has to learn from the local observations only. This chapter deals with learning-based schemes for distributed dynamic spectrum access. Different protocols to support distributed dynamic spectrum access are also discussed. These protocols can be used to facilitate spectrum handoff, exchange the information between cognitive radio users, and synchronize the transmission between cognitive radio transmitter and receiver.

Learning-based distributed dynamic spectrum access

Distributed resource management in multihop cognitive radio networks

In a multihop cognitive radio network, the channel selection algorithm plays an important role in optimizing the transmission performance and avoiding interference. To achieve an optimal channel selection, information of the channel availability is required at each node. Although an information exchange mechanism can be developed for this channel selection algorithm, this can incur a significant cost in the system. This information exchange not only reduces the available resources for data transmission, but also increases the delay of data transmission. Therefore, with the constraint of available information in a multihop cognitive radio network, a distributed minimum delay routing and channel selection algorithm based on learning was proposed for delay-sensitive traffic (e.g. multimedia) [540].

In the system model, there are multiple licensed users occupying the channels. In this multihop cognitive radio network, the licensed users require an interference-free environment.

The ideas underlying game theory have appeared throughout history, in the Bible, the Talmud, the works of Descartes and Sun Tzu, and the writings of Chales Darwin. Modern game theory, however, can be considered as an outgrowth of three seminal works:

• Augustin Cournot's Research into the Mathematical Principles of the Theory of Wealth in 1838 gives an intuitive explanation of what would eventually be formalized as the Nash equilibrium, as well as provides an evolutionary, or dynamic notion of best-response to the actions of others.

• Francis Ysidro Edgeworth's Mathematical Psychics demonstrated the notion of competitive equilibria in a two-person (as well as two-type) economy; Emile Borel, in “Algèbre et calcul des probabilités,” Comptes Rendus Académie des Sciences, vol. 184, 1927, provided the first insight into mixed strategies that randomization may support a stable outcome.

• While many other contributors hold a place in the history of game theory, it is widely accepted that modern analysis began with John von Neumann and Oskar Morgenstern's book, Theory of Games and Economic Behavior. Then building on von Neumann and Morgenstern's results John Nash developed the modern framework for methodological analysis.

Depending on the nature of the different approaches, there are different possible applications of game theory. If the information is strictly limited to local information, the non-cooperative game might be the only choice for each individual to play. However, such a game might have a very low-efficiency outcome. To overcome this problem, pricing or referee approaches have been proposed. If the users care about long-term benefits, the repeated game can be employed to enforce cooperation by the threat of future punishment from others.

To design efficient and effective dynamic spectrum access techniques for a cognitive radio network, the related technical aspects (e.g. channel allocation, power control) as well as economic aspects (e.g. pricing, spectrum auction) need to be considered. The economic issues are crucial for cognitive radio networks operating under the exclusive-use spectrum access model, since they define the incentive for licensed users to yield the right of spectrum access to the unlicensed users. Economic issues are also important for dynamic spectrum access based on the shared-use and commons-use models, because they determine the competition and cooperation between the licensed and unlicensed users.

In this chapter, we describe the different economic aspects of dynamic spectrum access in cognitive radio networks. First, the concept of spectrum trading is presented, which involves spectrum selling by single or multiple licensed users and spectrum buying by unlicensed users. A taxonomy of the spectrum trading models is presented. The pricing issue for spectrum trading as well as authentication, authorization, and accounting (AAA) issues are discussed. Then, an overview of the economic theories for spectrum trading in a dynamic spectrum access environment is given. These include utility theory, the concept of market-equilibrium, competition in an oligopoly market, and auction theory. A survey on the spectrum trading models based on the above theories is then presented.

Introduction to spectrum trading

Generally, license for spectrum access is provided to a primary user or service provider through an auction process in a primary market (Figure 11.1). When the allocated spectrum is under-utilized, the licensed user can lease the spectrum in a secondary market to an unlicensed user which temporarily demands the spectrum for a particular service.

In a wireless network (more specifically, in a cognitive wireless network), the available radio resources such as bandwidth are very limited. On the other hand, the demands for the wireless services are exponentially increasing. Not only are the number of users booming, but also more bandwidth is required for new services such as video telephony, TV on demand, wireless Internet, and wireless gaming. Finding a way to accommodate all these requirements has become an emergent research issue in wireless networking. Resource allocation and its optimization are general methods to improve network performance, but there are tradeoffs for resource usage. One of the major research goals is to present these tradeoffs so that better implementations can be put into practice.

This chapter will focus on how to formulate cognitive wireless networking problems as optimization problems from the perspective of resource allocation. Specifically, this chapter discusses what the resources, parameters, practical constraints, and optimized performances are across the different layers. In addition, it addresses how to perform resource allocation in multiuser scenarios under the presence of the primary users. The tradeoffs between the different optimization goals and different users interests are also investigated. The goal is to provide a new perspective of wireless networking and resource allocation problems from the optimization point of view.

This chapter is organized as follows: Section 4.1 discusses the basic formulation of the cognitive radio resource allocation as a constrained optimization problem. Section 4.2 studies linear programming and the simplex algorithm as its solution. Section 4.3 investigates how to define a convex optimization problem and some variations. Then the solutions are discussed.