This chapter discusses the Fourier series representation for continuous-time signals. This is applicable to signals which are either periodic or have a finite duration. The connections between the continuous-time Fourier transform (CTFT), the discrete-time Fourier transform (DTFT), and Fourier series are also explained. Properties of Fourier series are discussed and many examples presented. For real-valued signals it is shown that the Fourier series can be written as a sum of a cosine series and a sine series; examples include rectified cosines, which have applications in electric power supplies. It is shown that the basis functions used in the Fourier series representation satisfy an orthogonality property. This makes the truncated version of the Fourier representation optimal in a certain sense. The so-called principal component approximation derived from the Fourier series is also discussed. A detailed discussion of the properties of musical signals in the light of Fourier series theory is presented, and leads to a discussion of musical scales, consonance, and dissonance. Also explained is the connection between Fourier series and the function-approximation property of multilayer neural networks, used widely in machine learning. An overview of wavelet representations and the contrast with Fourier series representations is also given.

Building on the discussion of suprasegmentals in Chapters 6and 7, this chapter introduces the concept of intonation. The chapter begins by defining intonation both articulatorily and acoustically, with a focus on the relationship among pitch, tone, and intonation. The chapter then provides an overview of methods of analysing intonation, including acoustic analysis and theTone and Break Indices (ToBI) System. The chapter next examines intonational systems in different varieties of English by first describing GAmE and SSBE sentence intonation and pitch accent patterns, to provide a framework through which to examine other varieties. The intonation and pitch accent patterns from different varieties is then discussed, as is the pitch range of different varieties. The chapter then examines two global phenomena in detail – High Rising Terminal on declarative statements, or uptalk; and creaky voice, also commonly called vocal fry. The chapter ends with exercises to check your understanding of the content of the chapter, as well as to expand your knowledge through an analysis of the intonation and pitch accent patterns in your own variety of English.

Voice is the production of a vibrating tone through the vibration of the vocal folds under pulmonary airflow. The glottis is the space between the vocal folds. Phonation is produced through the myoelastic aerodynamic theory, which takes into account the actions of muscles, tissue elasticity, and aerodynamic (airflow) principles. Fundamental frequency is the frequency of vibration of a talker’s vocal folds (as distinguished from other frequencies in speech). An individual’s “normal” fundamental frequency is called their modal frequency. Voice onset time (VOT) is the time (duration) between the release of a plosive and the start (onset) of voicing in the following segment. The vocal folds are complex anatomical structures involving superficial and deep layers; this structure influences the nature of vocal fold vibration and the mucosal wave. In order to explain and illustrate the complex vibration, the two-mass model and the cover--body model are used. Whisper and nasality are phenomena that are counted under the heading "voice" by some authors.

The objective of this chapter is to extend the ad hoc least squares method of somewhat arbitrarily selected base functions to a more generic method applicable to a broad range of functions – the Fourier series, which is an expansion of a relatively arbitrary function (with certain smoothness requirement and finite jumps at worst) with a series of sinusoidal functions. An important mathematical reason for using Fourier series is its “completeness” and almost guaranteed convergence. Here “completeness” means that the error goes to zero when the whole Fourier series with infinite base function is used. In other words, the Fourier series formed by the selected sinusoidal functions is sufficient to linearly combine into a function that converges to an arbitrary continuous function. This chapter on Fourier series will lay out a foundation that will lead to Fourier Transform and spectrum analysis. In this sense, this chapter is important as it provides background information and theoretical preparation.

This chapter introduces the fast Fourier Transform (FFT) for discrete Fourier Transform, beginning with the discretization of the Fourier Transform to its digital expression with constant time intervals. When the integral in Fourier Transform is replaced by a summation, the continuous Fourier Transform is changed to discrete. The discrete Fourier Transform and its inverse are exact relations. An example of the discrete Fourier Transform is discussed for a simple rectangular window function which results in the sinc function, useful for the interpretation of finite sampling effect. A technique of zero-padding is introduced with the discrete Fourier Transform for better visualization of the spectrum. But the computation of discrete Fourier Transform of a long time series can be quite “labor intensive” or costly in computer time with a direct computation. However, since the base functions are periodic, a direct computation can have many duplications in multiplications of terms. Algorithms can be designed to reduce the duplications so that the speed of computation is increased. The reduction of duplicated computations can be repeatedly done through an FFT algorithm. In MATLAB, this is done by a simple command fft. The efficiency of FFT is discussed.

This chapter provides an overview of research on heritage language (HL) sound systems, with a focus on areas of convergence and divergence among heritage speakers (HSs), native speakers (NSs) who continue to be dominant in the language, and second language learners (L2ers) who acquired the language later in life. Drawing on data from a wide range of HLs, the chapter addresses both phonetic (articulatory, acoustic, perceptual) and phonological (phonemic, distributional, phonotactic) aspects of the HL sound system, as well as that of the majority language, in light of theories of bilingual speech and variables previously studied as predictors of HSs’ linguistic behavior. Despite the diversity of results reviewed, several recurring themes emerge, including intermediate patterning between NSs and L2ers, a higher level of performance in perception than production, and individual variability. In particular, the depth and the accessibility of HSs’ knowledge of the HL sound system show considerable variation related to structural linguistic factors, demographic and sociolinguistic factors, input and usage-based factors, and methodological factors. In addition to summarizing the areas in which there is an emerging consensus, the chapter points out a number of remaining questions that pave the way for future research on HL sound systems.

In humans, differences in voice quality influence sexual attraction and perceived dominance, but little is known about such mechanisms in Pan. Here, we investigated the degree of sexual dimorphism in chimpanzee vocalizations. We expected the sexes to differ in mean fundamental (F0) and peak frequencies (pF) according to the source filter theory, which predicts that differences in body size affect vocal characteristics. Using data from nine chimpanzees of the South Group, we analysed call rates and measured F0, pF and durations from pant–hoots and food calls. We also tested for sex differences in bout duration and the number of food grunts per bout. Males and females had equal rates of call production for all signals except pant–hoots and buttress-drumming, with males producing these significantly more often. Additionally, contrary to predictions based on body size, female food grunts were lower in dominant frequency bands and peak frequencies while males produced longer food-call bouts with a greater number of tonal grunts. Further analyses should address whether these sex differences in acoustic parameters could play a role in female choice and male competition in chimpanzees.