Sinusoids: Theory and Technological Applications

Book Preface

The classical Fourier transform has been used in solving boundary value problems in a variety of applications including continuum mechanics, potential theory, geophysics, physics, biology, and mathematical economics. The discrete Fourier transform evolved into the fast Fourier transform technique that helped numerically investigate boundary value problems. In general, the Fourier transform carries over a function of time into a new function whose argument is the frequency in units of cycles per second or radians per second. This new function is the frequency spectrum of the given function, or vice versa. There are theorems that determine both of these functions, one from the other, i.e., one in the time domain and the other in the frequency domain. In fact, the Fourier transform is an extension of the Fourier series in the case of periodic functions where the period is allowed to tend to infinity.

Overview

The sinusoids, which are periodic sine or cosine functions, are explained in Chapter 1, with some well-known examples from wave theory, especially the traveling and standing waves, from continuous musical rhythms, and from the field of medicine in the human liver. In all cases the Fourier transform is used to calculate the discrete set of complex amplitudes that involve the Fourier series. After sampling a time-domain function to computer-processing, the discrete Fourier transform yields the original Fourier transform by applying the Poisson summation formula. The Fourier series and the Fourier transform are discussed in both continuous and discrete cases in Chapters 2 and 3, along with an analysis of the Dirichlet kernel and the Gibbs phenomenon. The amplitudes, phase, and frequency of periodic functions (signals) are studied in the case of deterministic continuous and discrete signals. Invertibility and periodicity of Fourier transforms are used in the development of signals and filters, which are discussed in Chapter 4. These two topics are not only useful in signal processing but also in subsequent different technological applications

The general concept of communication systems is discussed in Chapter 5. It includes the process of quantization of analog/digital signals, interference, and data transmission. Space exploration, including the Mars project and SETI, are included. The general direction of current research in space travel seems to harness nuclear fusion and antimatter as an energy source to reach the nearest star in a few decades provided the machinery could last that long. The Alcuberre warp drive provides an interesting topic, which is based on the premise that it could travel faster than light if negative mass existed.

Although software-defined radio communication systems that use software for the modulation and demodulation of radio signals are known and have been used since 1995, the first Global Positioning System (GPS) was implemented in 1997. The Global Navigation Satellite Systems now include both GPS and Galileo systems. Although GPS is a complicated device, it can be understood as a single-frequency receiver, which describes its amplitude spectrum, autocorrelation function, and Fourier transform (power spectrum). This is done in Chapter 6. The linear time-invariant systems, already introduced in Chapter 4, together with L1 and L2 signals and binary offset carrier modulation, cyclic redundancy check, forward error correction, and block interleaving, are presented to explain the functioning of the GPS receiver, all based on the code division multiple access principle.