Mathematics of the Discrete Fourier Transform (DFT) with Audio Applications

by Julius O. Smith III

Detailed derivation of the Discrete Fourier Transform (DFT) and its associated mathematics, including elementary audio signal processing applications and matlab programming examples.

Genres: Mathematics

322 pages, Paperback

First published April 13, 2007

Reviews

[Aykut Çağlayan · Rating 5 out of 5]

The book touches many fundamental corners of mathematics like complex numbers, Taylor series, vector products, Laplace and z-transforms etc. Studying it gave me the opportunity to dive into math and appreciate its truthful beauty. The author provides the reader with prerequsite mathematical details and algebraic proofs at each stage. One gets a clear insight into how the Fourier transform works so perfectly both ways between the time- and frequency-domain. High-school pre-calculus math gives you enough knowledge to follow most topics, although in order to apprehend Euler's identity, which is crucial to DFT, one needs to be familiar with concepts like limit, derivation and Taylor series expansions. I most enjoyed chapter 5 about the geometric signal theory, which utilizes linear algebra to represent the signal; this was inspiring from a musical perspective as well, i.e. how to translate different domains into each other and how to mingle them together.