Mathematics of the Discrete Fourier Transform (DFT) by Julius O. Smith III
Mathematics of the Discrete Fourier Transform (DFT) - Table of Contents
1. Introduction to the DFT
2. Complex Numbers
3. Proof of Euler’s Identity
4. Sinusoids and Exponentials
5. Geometric Signal Theory
6. The DFT Derived
7. Fourier Theorems for the DFT
8. DFT Applications
9. Fast Fourier Transform (FFT)
10. Continuous and Discrete Transforms
11. Continuous Fourier Theorems
12. Sampling Theory
13. Taylor Series Expansions
14. Logarithms and Decibels
15. Digital Audio Number Systems
16. Matrices
17. MATLAB/Octave Examples
What You Will Learn in Mathematics of the Discrete Fourier Transform (DFT)
"Mathematics of the Discrete Fourier Transform (DFT): with Audio Applications" by Julius O. Smith III is a comprehensive guide to understanding the "Discrete Fourier Transform", a fundamental tool in "signal processing". The book explains the DFT from first principles, starting with "complex numbers", sinusoids, and Euler’s formula, gradually building up to its applications in analyzing and transforming discrete signals. It bridges the gap between abstract mathematical concepts and practical computational methods, making it ideal for both students and professionals.
The text covers essential topics such as the "shift theorem", convolution, correlation, and the inverse DFT, showing how signals behave in both time and frequency domains. It also addresses "sampling theory", spectral analysis, and the implementation of DFTs in software like MATLAB or Octave. Through clear derivations and practical examples, the book provides a strong foundation for anyone working with digital audio, communications, or other fields that rely on frequency analysis.
Designed for accessibility and rigor, Smith’s book combines theory with computation, offering both a technical reference and a learning resource. With its focus on "Fourier analysis", the DFT, and practical applications, "Mathematics of the Discrete Fourier Transform" equips readers with the tools to understand, analyze, and process discrete signals effectively.
Book Details & Specifications
Title:
Mathematics of the Discrete Fourier Transform (DFT) by Julius O. Smith III
Publisher:
W3K Publishing
Year:
2007
Pages:
328
Type:
PDF
Language:
English
ISBN-10 #:
097456074X
ISBN-13 #:
978-0974560748
License:
External Educational Resource
Amazon:
Amazon
About the Author: Julius O. Smith
The author Julius O. Smith
is an American engineer and professor emeritus at "Stanford University", specializing in "digital signal processing", audio technology, and music computing. He is a leading expert in "DSP algorithms" and has made pioneering contributions to "digital waveguide synthesis" and computer music research.
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