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 a Fast Fourier Transform (FFT)--i.e., a high-speed algorithm for computing the DFT. FFTs are 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 started out as a series of readers for my introductory course in digital audio signal processing that I have 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 math background, including some calculus exposure.