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Draft of "Mathematics of the Discrete Fourier Transform (DFT)," by J.O. Smith, CCRMA, Stanford, Winter 2002. The latest draft as linked HTML version are available on-line at http://www-ccrma.stanford.edu/~jos/mdft/.
Book Excerpts:
The Discrete Fourier Transform (DFT) can be understood as a numerical approximation to the Fourier Transform. However, the DFT has its own exact Fourier theory, which is the main focus of this book. The DFT is normally encountered in practice as the Fast Fourier Transform (FFT) -- a high-speed algorithm for computing the DFT. The FFT is used extensively in a wide range of digital signal processing applications, including spectrum analysis, high-speed convolution (linear filtering), filter banks, signal detection and estimation, system identification, audio compression (e.g., MPEG-II AAC), spectral modeling sound synthesis, and many other applications; some of these will be discussed in Chapter 8.
This book chooses to discuss DFT over the FT since the FT demands readers to use calculus right off the bat, while the DFT, on the other hand, replaces the infinite integral with a finite sum of various quantities. Calculus is not needed to define the DFT (or its inverse, as this book will show), and with finite summation limits, readers cannot encounter difficulties with infinities. Moreover, in the field of digital signal processing, signals and spectra are processed only in sampled form, so that the DFT is what readers really need anyway. In s Read more...
The providing a comprehensive overview of the mathematical theory of Fourier analysis, with straightforward verifications of its results and formulas, and clear indications of the limitations of those results of formulas. Designed to help readers to handle more sophisticated mathematics. DLC: Fourier analysis.
Fourier analysis is one of the most useful and widely employed sets of tools for the engineer, the scientist, and the applied mathematician. As such, students and practitioners in these disciplines need a practical and mathematically solid introduction to its principles. They need straightforward verifications of its results and formulas, and they need clear indications of the limitations of those results and formulas. Principles of Fourier Analysis furnishes all this and more. It provides a comprehensive overview of the mathematical theory of Fourier analysis, including the development of Fourier series, "classical" Fourier transforms, generalized Fourier transforms and analysis, and the discrete theory. Much of the author's development is strikingly different from typical presentations. His approach to defining the classical Fourier transform results in a much cleaner, more coherent theory that leads naturally to a starting point for the generalized theory. He also introduces a new generalized theory based on the use of Gaussian test functions that yields an even more general -yet simpler -theory than usually presented. Principles of Fourier Analysis stimulates th Read more...
The providing a comprehensive overview of the mathematical theory of Fourier analysis, with straightforward verifications of its results and formulas, and clear indications of the limitations of those results of formulas. Designed to help readers to handle more sophisticated mathematics. DLC: Fourier analysis.
Fourier analysis is one of the most useful and widely employed sets of tools for the engineer, the scientist, and the applied mathematician. As such, students and practitioners in these disciplines need a practical and mathematically solid introduction to its principles. They need straightforward verifications of its results and formulas, and they need clear indications of the limitations of those results and formulas. Principles of Fourier Analysis furnishes all this and more. It provides a comprehensive overview of the mathematical theory of Fourier analysis, including the development of Fourier series, "classical" Fourier transforms, generalized Fourier transforms and analysis, and the discrete theory. Much of the author's development is strikingly different from typical presentations. His approach to defining the classical Fourier transform results in a much cleaner, more coherent theory that leads naturally to a starting point for the generalized theory. He also introduces a new generalized theory based on the use of Gaussian test functions that yields an even more general -yet simpler -theory than usually presented. Principles of Fourier Analysis stimulates th Read more...
