This appendix gives a brief guide to the probability theory needed at various stages in the book. The following is too brief to be intended as a first exposure to probability, but rather is here to act as a reference. Good introductory books on probability include Bishop, and Duda, Hart and Stork.

Discrete probabilities

Discrete events are the simplest to interpret. For example, what is the probability of

• it raining tomorrow?

• a 6 being thrown on a die?

Probability can be thought of as the chance of a particular event occurring. We limit the range of our probability measure to lie in the range 0 to 1, where

• lower numbers indicate that the event is less likely to occur, 0 indicates it will never occur;

• higher numbers indicate that the event is more likely to occur, 1 indicates that the event will definitely occur.

We like to think that we have a good grasp of both estimating and using probability. For simple cases such as “will it rain tomorrow?” we can do reasonably well. However, as situations get more complicated things are not always so clear. The aim of probability theory is to give us a mathematically sound way of inferring information using probabilities.

Discrete random variables

Let some event have have M possible outcomes. We are interested in the probability of each of these outcomes occurring.

The previous chapter introduced the concept of coding optimality, as based on variable-length codewords. As we have learnt, an optimal code is one for which the mean codeword length closely approaches or is equal to the source entropy. There exist several families of codes that can be called optimal, as based on various types of algorithms. This chapter, and the following, will provide an overview of this rich subject, which finds many applications in communications, in particular in the domain of data compression. In this chapter, I will introduce Huffman codes, and then I will describe how they can be used to perform data compression to the limits predicted by Shannon. I will then introduce the principle of block codes, which also enable data compression.

Huffman codes

As we have learnt earlier, variable-length codes are in the general case more efficient than fixed-length ones. The most frequent source symbols are assigned the shortest codewords, and the reverse for the less frequent ones. The coding-tree method makes it possible to find some heuristic codeword assignment, according to the above rule. Despite the lack of further guidance, the result proved effective, considering that we obtained η = 96.23% with a ternary coding of the English-character source (see Fig. 8.3, Table 8.3). But we have no clue as to whether other coding trees with greater coding efficiencies may ever exist, unless we try out all the possibilities, which is impractical.

This mathematically intensive chapter takes us through our first steps in the domain of quantum computation (QC) algorithms. The simplest of them is the Deutsch algorithm, which makes it possible to determine whether or not a Boolean function is constant for any input. The key result is that this QC algorithm provides the answer at once, whereas in the classical case it would take two independent calculations. I describe next the generalization of the former algorithm to n qubits, referred to as the Deutsch–Jozsa algorithm. Although they have no specific or useful applications in quantum computing, both algorithms represent a most elegant means of introducing the concept of quantum computation parallelism. I then describe two most important QC algorithms, which nicely exploit quantum parallelism. The first is the quantum Fourier transform (QFT), for which a detailed analysis of QFT circuits and quantum-gate requirements is also provided. As will be shown in the next chapter, a key application of QFT concerns the famous Shor's algorithm, which makes it possible to factor numbers into primes in terms of polynomials. The second algorithm, no less famous than Shor's, is referred to as the Grover quantum database search, whose application is the identification of database items with a quadratic gain in speed.

Deutsch algorithm

Our exploration of quantum algorithms shall begin with the solution of a very basic problem: finding whether or not a Boolean function f(x) is a constant.

This chapter is about coding information, which is the art of packaging and formatting information into meaningful codewords. Such codewords are meant to be recognized by computing machines for efficient processing or by human beings for practical understanding. The number of possible codes and corresponding codewords is infinite, just like the number of events to which information can be associated, in Shannon's meaning. This is the point where information theory will start revealing its elegance and power. We will learn that codes can be characterized by a certain efficiency, which implies that some codes are more efficient than others. This will lead us to a description of the first of Shannon's theorems, concerning source coding. As we shall see, coding is a rich subject, with many practical consequences and applications; in particular in the way we efficiently communicate information. We will first start our exploration of information coding with numbers and then with language, which conveys some background and flavor as a preparation to approach the more formal theory leading to the abstract concept of code optimality.

Coding numbers

Consider a source made of N different events. We can label the events through a set of numbers ranging from 1 to N, which constitute a basic source code. This code represents one out of N! different possibilities. In the code, each of the numbers represents a codeword.

The speech-production process was qualitatively described in Chapter 7. There we showed that speech is produced by a source, such as the glottis, which is subsequently modified by the vocal tract acting as a filter. In this chapter, we turn our attention to developing a more-formal quantitative model of speech production, using the techniques of signals and filters described in Chapter 10.

The acoustic theory of speech production

Such models often come under the heading of the acoustic theory of speech production, which refers both to the general field of research in mathematical speech-production models and to the book of that title by Fant. Although considerable previous work in this field had been done prior to its publication, this book was the first to bring together various strands of work and describe the whole process in a unified manner. Furthermore, Fant backed his study up with extensive empirical studies with X-rays and mechanical models to test and verify the speech-production models being proposed. Since then, many refinements to the model have been made, as researchers have investigated trying to improve the accuracy and practicalities of these models. Here we focus on the single most widely accepted model, but conclude the chapter with a discussion on variations on this.

As with any modelling process, we have to reach a compromise between a model that accurately describes the phenomena in question and one that is simple, effective and suited to practical needs.

This chapter gives an outline of the related fields of phonetics and phonology. A good knowledge of these subjects is essential in speech synthesis because they help bridge the gap between the discrete, linguistic, word-based message and the continuous speech signal. More-traditional synthesis techniques relied heavily on phonetic and phonological knowledge, and often implemented theories and modules directly from these fields. Even in the more-modern heavily data-driven synthesis systems, we still find that phonetics and phonology have a vital role to play in determining how best to implement representations and algorithms.

Articulatory phonetics and speech production

The topic of speech production examines the processes by which humans convert linguistic messages into speech. The converse process, whereby humans determine the message from the speech, is called speech perception. Together these form the backbone of the field know as phonetics.

Regarding speech production, we have what we can describe as a complete but approximate model of this process. That is, in general we know how people use their articulators to produce the various sounds of speech. We emphasise, however, that our knowledge is very approximate; no model as yet can predict with any degree of accuracy how a speech waveform from a particular speaker would look like given some pronunciation input.

This chapter is concerned with the issue of synthesising acoustic representations of prosody. The input to the algorithms described here varies but in general takes the form of the phrasing, stress, prominence and discourse patterns which we introduced in Chapter 6. Hence the complete process of synthesis of prosody can be seen as one whereby we first extract a prosodic form representation from the text, as described in Chapter 6, and then synthesize an acoustic representation of this form, as described here.

The majority of this chapter focuses on the synthesis of intonation. The main acoustic representation of intonation is the fundamental frequency (F0), such that intonation is often defined as the manipulation of F0 for communicative or linguistic purposes. As we shall see, techniques for synthesizing F0 contours are inherently linked to the model of intonation used, so the whole topic of intonation, including theories, models and F0 synthesis, is dealt with here. In addition, we cover the topic of predicting intonation form from text, which was deferred from Chapter 6 since we first require an understanding of theories and models of intonational phenomena before explaining this.

Timing is considered the second important acoustic representation of prosody. Timing is used to indicate stress (phones are longer than normal), phrasing (phones become noticeably longer immediately prior to a phrase break) and rhythm.

Intonation overview

As a working definition, we will take intonation synthesis to be the generation of an F0 contour from higher-level linguistic information.

Speech and hearing are closely linked human abilities. It could be said that human speech is optimised toward the frequency ranges that we hear best, or perhaps our hearing is optimised around the frequencies used for speaking. However whichever way we present the argument, it should be clear to an engineer working with speech transmission and processing systems that aspects of both speech and hearing must often be considered together in the field of vocal communications. However, both hearing and speech remain complex subjects in their own right. Hearing particularly so.

In recent years it has become popular to discuss psychoacoustics in textbooks on both hearing and speech. Psychoacoustics is a term that links the words psycho and acoustics together, and although it sounds like a description of an auditory-challenged serial killer, actually describes the way the mind processes sound. In particular, it is used to highlight the fact that humans do not always perceive sound in the straightforward ways that knowledge of the physical characteristics of the sound would suggest.

There was a time when use of this word at a conference would boast of advanced knowledge, and familiarity with cutting-edge terminology, especially when it could roll off the tongue naturally. I would imagine speakers, on the night before their keynote address, standing before the mirror in their hotel rooms practising saying the word fluently. However these days it is used far too commonly, to describe any aspect of hearing that is processed nonlinearly by the brain. It was a great temptation to use the word in the title of this book.

This chapter contains a number of final topics, which have been left until last because they span many of the topics raised in the previous chapters.

Databases

Data-driven techniques have come to dominate nearly every aspect of text-to-speech in recent years. In addition to being affected by the algorithms themselves, the overall performance of a system is increasingly dominated by the quality of the databases that are used for training. In this section, we therefore examine the issues in database design, collection, labelling and use.

All algorithms are to some extent data-driven; even hand-written rules use some “data”, either explicitly or in a mental representation wherein the developer can imagine examples and how they should be dealt with. The difference between hand-written rules and data-driven techniques lies not in whether one uses data or not, but concerns how the data are used. Most data-driven techniques have an automatic training algorithm such that they can be trained on the data without the need for human intervention.

Unit-selection databases

Unit selection is arguably the most data-driven technique because little or no processing is performed on the data, rather it is simply analysed, cut up and recombined in different sequences. As with other database techniques, the issue of coverage is vital, but in addition we have further issues concerning the actual recordings.

This chapter describes the principle of compression in quantum communication channels. The underlying concept is that it is possible to convey “faithfully” a quantum message with a large number of qubits, while transmitting a compressed version of this message with a reduced number of qubits through the channel. Beyond the mere notion of fidelity, which characterizes the quality of quantum message transmission, the description brings the new concept of typicality in the space defined by all possible “quantum codewords.” The theorem of Schumacher's quantum compression states that for a qubit source with von Neumann entropy S, the message compression factor R has S − ε for the lower bound, where ε is any nonnegative parameter that can be made arbitrarily small for sufficiently long messages (hence, R ≈ S is the best possible compression factor). An original graphical and numerical illustration of the effect of Schumacher's quantum compression and the evolution of the typical quantum-codeword subspace with increasing message length is provided.

Quantum data compression and fidelity

In this chapter, we have reached the stage where it is possible to start addressing the issues that are central to information theory, namely, “How efficiently can we code information in a quantum communication channel?” both in terms of economy of means – the concept of data compression – and accuracy of transmission – the concept of message integrity or minimal data error, referred to here as fidelity.

In effect, the concept of information is obvious to anyone in life. Yet the word captures so much that we may doubt that any real definition satisfactory to a large majority of either educated or lay people may ever exist. Etymology may then help to give the word some skeleton. Information comes from the Latin informatio and the related verb informare meaning: to conceive, to explain, to sketch, to make something understood or known, to get someone knowledgeable about something. Thus, informatio is the action and art of shaping or packaging this piece of knowledge into some sensible form, hopefully complete, intelligible, and unambiguous to the recipient.

With this background in mind, we can conceive of information as taking different forms: a sensory input, an identification pattern, a game or process rule, a set of facts or instructions meant to guide choices or actions, a record for future reference, a message for immediate feedback. So information is diversified and conceptually intractable. Let us clarify here from the inception and quite frankly: a theory of information is unable to tell what information actually is or may represent in terms of objective value to any of its recipients! As we shall learn through this series of chapters, however, it is possible to measure information scientifically. The information measure does not concern value or usefulness of information, which remains the ultimate recipient's paradigm.

Speech-processing technology has been a mainstream area of research for more than 50 years. The ultimate goal of speech research is to build systems that mimic (or potentially surpass) human capabilities in understanding, generating and coding speech for a range of human-to-human and human-to-machine interactions.

In the area of speech coding a great deal of success has been achieved in creating systems that significantly reduce the overall bit rate of the speech signal (from of the order of 100 kilobits per second to rates of the order of 8 kilobits per second or less), while maintaining speech intelligibility and quality at levels appropriate for the intended applications. The heart of the modern cellular industry is the 8 kilobit per second speech coder, embedded in VLSI logic on the more than two billion cellphones in use worldwide at the end of 2007.

In the area of speech recognition and understanding by machines, steady progress has enabled systems to become part of everyday life in the form of call centres for the airlines, financial, medical and banking industries, help desks for large businesses, form and report generation for the legal and medical communities, and dictation machines that enable individuals to enter text into machines without having to type the text explicitly.

This chapter is concerned with the measure of information contained in qubits. This can be done only through quantum measurement, an operation that has no counterpart in the classical domain. I shall first describe in detail the case of single qubit measurements, which shows under which measurement conditions “classical” bits can be retrieved. Next, we consider the measurements of higher-order or n-qubits. Particular attention is given to the Einstein–Podolsky–Rosen (EPR) or Bell states, which, unlike other joint tensor states, are shown being entangled. The various single-qubit measurement outcomes from the EPR–Bell states illustrate an effect of causality in the information concerning the other qubit. We then focus on the technique of Bell measurement, which makes it possible to know which Bell state is being measured, yielding two classical bits as the outcome. The property of EPR–Bell state entanglement is exploited in the principle of quantum superdense coding, which makes it possible to transmit classical bits at twice the classical rate, namely through the generation and measurement of a single qubit. Another key application concerns quantum teleportation. It consists of the transmission of quantum states over arbitrary distances, by means of a common EPR–Bell state resource shared by the two channel ends. While quantum teleportation of a qubit is instantaneous, owing to the effect of quantum-state collapse, it is shown that its completion does require the communication of two classical bits, which is itself limited by the speed of light.