Mathematics of the Discrete Fourier Transform (DFT)

Date: 08 May 2011, 00:40

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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 summary, the DFT is simpler mathematically, and more relevant computationally than the Fourier Transform. At the same time, the basic concepts are the same. Therefore, this book will begin with the DFT, and address FT-specific results in the appendices.
Intended Audience:
This book started out as a series of readers for an introductory course in digital audio signal processing that has been given at the Center for Computer Research in Music and Acoustics (CCRMA) since 1984. The course was created primarily for entering Music Ph.D. students in the Computer Based Music Theory program at CCRMA. As a result, the only prerequisite is a good high-school level algebra and trigonometry, some calculus, and prior exposure to complex numbers.
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