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Fourier analysis is one method of investigating the origin of functions and their properties by using Fourier series and Fourier transform(s). The Fourier series and Fourier transforms were introduced by the mathematician Jean Baptiste Joseph Fourier at the beginning of the nineteenth century they are widely applied in the engineering and science fields. One-dimensional waveforms (as functions of time or position) and two-dimensional images (as functions of two positional axes) are the main subjects studied nowadays using Fourier analysis. Fourier analysis is an important method used to analyze complex sound waveforms in the field of acoustical engineering. This is the reason why this book was originally listed in the series of books published by the Acoustical Society of Japan. However, since Fourier analysis is also applicable in many other engineering sciences, these two books, Digital Fourier Analysis: Fundamentals, and Digital Fourier Analysis: Advanced Techniques, are useful to readers in broader fields.
The Fourier transform itself does not fit well with analog processing because it requires too much numerical processing, such as multiplications and summations. That is the reason waveform analysis during the analog age could not make full use of the Fourier method of analysis. Fourier analysis was more valuable as the basis of theoretical analysis than for its practical applications during the analog era. However, the digitally processed Fourier transform became a reality with the emergence of the digital computer in the middle of the twentieth century. The later development of the Fast Fourier transform (FFT) algorithm in 1965 and the subsequent inventions of microchips for signal processing accelerated the application of Fourier analysis based signal processing.
In the twenty-first century, Fourier analysis technology is widely used in our daily activities, and as a natural consequence, the technology is hidden in a black box in most of its applications. Even experts in the field use this technology without knowing details of how Fourier techniques are implemented. However, engineers, who wish to play important roles in developing future technologies, must do more than just deal with black boxes. Engineers must understand the basis of the present Fourier technology in order to create and build up new technologies based on it.
This book is intended so that high-school graduates or first or second grade college students with the basic knowledge of mathematics can learn the Fourier analysis without too much difficulty. In order to do that, explanations of equations start from the very beginning and details of derivation steps are also described. This is very rare for this kind specialized book.
This book also deals with advanced topics so that engineers who are presently involved in signal processing work can get hints to solve their own specific problems. Ways of thinking that lie behind or lead to theories are also described, that are a must to apply theories to practical problems.
This work comprises two volumes. Seven chapters are included in Volume I, titled Digital Fourier Analysis: Fundamentals. Volume II, titled Digital Fourier Analysis: Advanced Techniques contains six chapters. As the titles indicate, more advanced topics are included in Volume II. In this sense, the former may be classified as a text for undergraduate course and the latter for graduate course. Notice, however, that Volume I includes some advanced topics, whereas Volume II contains necessary items needed for a better understanding of Volume I.
Sine and Cosine Waves
Fourier Series Expansion
Numerical (Digitized) Waveforms
Discrete Fourier Transform
Fast Fourier Transform
DFT and Spectrum
Time Window