In this chapter we discuss the use of multiple antennas by radios. While the radio links that we discussed up to this point in the text have assumed single-input single-output (SISO) channels, we now consider the use of multiple antennas at both the transmitter (source) and receiver (destination), as indicated in Figure 11.1. We introduce the channel model for a multiple-antenna receiver. We discuss channel estimation and spatial receive beamforming techniques. We introduce the multiple-input multiple-output (MIMO) channel model, define the capacity of this system under the assumptions that the transmitter is uninformed and the transmitter is informed of the channel matrix. Finally, we discuss the concept of space–time coding and present various approaches, including Alamouti’s space–time block code.

The role of the state in the modern world is a complex one. According to legal theory, each state is sovereign and equal. In reality, with the phenomenal growth in communications and consciousness, and with the constant pressure and fluidity of global rivalries, not even the most powerful of states can be entirely sovereign and independent. Interdependence and the close-knit character of contemporary international commercial and political society ensures that virtually any action of a state could well have profound repercussions upon the system as a whole and the decisions under consideration by other states. This has led to an increasing interpenetration of international law and domestic law across a number of fields, such as human rights, environmental and international investment law, where at the least the same topic is subject to regulation at both the domestic and the international levels (and indeed the regional level in the case of the European Union and others). With the rise and extension of international law, questions begin to arise paralleling the role played by the state within the international system and concerned with the relationship between the internal legal order of a particular country and the rules and principles governing the international community as a whole. Municipal law governs the domestic aspects of government and deals with issues between individuals, and between individuals and the administrative apparatus, while international law focuses primarily upon the relations between states. That is now, however, an overly simplistic assertion. There are many instances where problems can emerge and lead to difficulties between the two systems. In a case before a municipal court a rule of international law may be brought forward as a defence to a charge, as for example in R v. Jones, where the defence of seeking to prevent a greater crime (essentially of international law) was claimed with regard to the alleged offence of criminal damage (in English law), or where a vessel is being prosecuted for being in what, in domestic law, is regarded as territorial waters, but in international law would be treated as part of the high seas. Further, there are cases where the same situation comes before both national and international courts, which may refer to each other’s decisions in a complex process of interaction. For example, the failure of the United States to allow imprisoned foreign nationals access to consular assistance in violation of the Vienna Convention on Consular Relations 1963 was the subject of case-law before the International Court of Justice, the Inter-American Court of Human Rights and US courts, while there is a growing tendency for domestic courts to be used to address violations of international law.

The ascertainment of the sources of international law lies at the heart of the system. The discovery of the law on any given point in domestic legal orders is not usually too difficult a process. In the English legal system, for example, one looks to see whether the matter is covered by an Act of Parliament (or possibly a statutory instrument) and, if it is, the law reports are consulted as to how it has been interpreted by the courts. If the particular point is not specifically referred to in a statute, court cases will be examined to elicit the required information. In other words, there is a definite method of discovering what the law is. In addition to verifying the contents of the rules, this method also demonstrates how the law is created, namely, by parliamentary legislation or judicial case-law. This gives a degree of certainty to the legal process because one is able to tell when a proposition has become law and the necessary mechanism to resolve any disputes about the law is evident. It reflects the hierarchical character of a national legal order with its gradations of authority imparting to the law a large measure of stability and predictability.

In this chapter we discuss Fourier analysis. We categorize signals into energy or power signals. We introduce the foundational concept of the complex tone. We define the Fourier transform and identify it as a linear operator. We review energy and the power spectral densities. We survey a set of useful Fourier transform relationships such as time shift, frequency shift, scaling, time reversal, and conjugation. We evaluate the Fourier transform of the top hat function, real Gaussian function, convolution, functional derivative, and autocorrelation. We introduce the Fourier series and evaluate series coefficients for sawtooth functions and impulse train. We discuss the discrete-time Fourier transform and the discrete Fourier transform and related fast Fourier transform. Finally, we review digital filters.

No book on administrative law could sensibly omit an account of executive law-making, least of all one that purports to deal with the functions of law in the administration. Executive law-making falls within the purview of administrative law merely by virtue of being made by the executive; it is something that government does. The subject is rarely studied in much detail, however.

Many of the problems that exercised Greek mathematicians and their successors were geometric, and in particular concerned constructions using ruler (straight edge) and compasses. Here are the three most important.

In this chapter, we consider a field 𝐹 with a finite number 𝑞 of elements, and its Galois theory. Its prime subfield is isomorphic to ℤ𝑝 for some prime number 𝑝, and we identify it with ℤ𝑝. 𝐹 is a finish-dimensional vector space over ℤ𝑝, of dimension 𝑛, say, so that 𝑞 = 𝑝𝑛.

The preamble to the Universal Declaration of Human Rights adopted on 10 December 1948 emphasises that ‘recognition of the inherent dignity and of the equal and inalienable rights of all members of the human family is the foundation of freedom, justice and peace in the world’. While there is widespread acceptance of the importance of human rights in the international structure, there is considerable confusion as to their precise nature and role in international law. The question of what is meant by a ‘right’ is itself controversial and the subject of intense jurisprudential debate. Some ‘rights’, for example, are intended as immediately enforceable binding commitments, others merely as specifying a possible future pattern of behaviour. The problem of enforcement and sanctions with regard to human rights in international law is another issue which can affect the characterisation of the phenomenon. There are writers who regard the high incidence of non-compliance with human rights norms as evidence of state practice that argues against the existence of a structure of human rights principles in international law. Although sight must not be lost of violations of human rights laws, such an approach is not only academically incorrect, but also profoundly negative. The concept of human rights is closely allied with ethics and morality. Those rights that reflect the values of a community will be those with the most chance of successful implementation. Positive rights may be taken to include those rights enshrined within a legal system, whether or not reflective of moral considerations, whereas a moral right is not necessarily enforceable by law. One may easily discover positive rights. Deducing or inferring moral rights is another matter entirely and will depend upon the perception of the person seeking the existence of a particular right.

If 𝑓 ∈ ℚ[𝑥] we can consider 𝑓 as an element of ℂ[𝑥], and then 𝑓 splits over ℂ. We therefore have the comforting conclusion that, whenever 𝑓 ∈ ℚ[𝑥], we can find a splitting field extension for 𝑓 which is a subfield of the fixed field ℂ.

As every student of government should know, the administrative process is shaped not only by executive and legislature but also by courts. This chapter focuses on the judicial contribution in the form of procedural review, classically associated with the two Latin tags: audi alteram partem (hear the other side) and nemo iudex in causa sua (no one a judge in his own cause). Suitably hallowed, even hackneyed, the precept that ‘justice must not only be done but be seen to be done’ is of the essence of the rule of law. Ridge v. Baldwin famously ushered in a more vigorous concept of procedural fairness in the context of the extended administrative state (see p. XXX above). Running alongside the rise of managerialism and marketisation and latterly austerity, the development has picked up pace in the last few decades with successive and overlapping forms of groundbreaking intervention.

The impetus to create a world court for the international community developed as a result of the atmosphere engendered by The Hague Conferences of 1897 and 1907. The establishment of the Permanent Court of Arbitration, although neither permanent nor, in fact, a court, marked an important step forward in the consolidation of an international legal system. However, no lasting concrete steps were taken until after the conclusion of the First World War. The Covenant of the League of Nations called for the formulation of proposals for the creation of a world court and in 1920 the Permanent Court of International Justice (PCIJ) was created. It stimulated efforts to develop international arbitral mechanisms. Together with arbitration, the Permanent Court was intended to provide a reasonably comprehensive system serving the international community. It was intended as a way to prevent outbreaks of violence by enabling easily accessible methods of dispute settlement in the context of a legal and organisational framework to be made available.

In this chapter, we discuss the concepts of amplification that we use to overcome noise. We review the idea of power amplifiers that are used by the transmitter and introduce metrics for nonlinear contributions in signal amplification. We discuss the concept of the low-noise amplifier (LNA) that is typically the first amplifier in the receiver chain. We motivate this type of amplifier by introducing the idea of the noise figure. Finally, we review the idea of automatic gain control.