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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...
