Mathematics Of The Discrete Fourier Transform (DFT) - With Audio Applications

Covers all about the Discrete Fourier Transform formula and its constituents, with frequent references to audio applications.

**Tag(s):**
Signal Processing

**Publication date**: 13 Apr 2007

**ISBN-10**:
097456074X

**ISBN-13**:
978097456074

**Paperback**:
322 pages

**Views**: 21,438

Mathematics Of The Discrete Fourier Transform (DFT) - With Audio Applications

Covers all about the Discrete Fourier Transform formula and its constituents, with frequent references to audio applications.

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.

Tweet

About The Author(s)

Julius O. Smith III teaches a music signal-processing course sequence and supervises related research at the Center for Computer Research in Music and Acoustics (CCRMA). He is formally a professor of music and (by courtesy) electrical engineering at Stanford University. Prof. Smith is a Fellow of the Audio Engineering Society and the Acoustical Society of America. He is the author of four online books and numerous research publications in his field.

Book Categories

Computer Science
Introduction to Computer Science
Introduction to Computer Programming
Algorithms and Data Structures
Artificial Intelligence
Computer Vision
Machine Learning
Neural Networks
Game Development and Multimedia
Data Communication and Networks
Coding Theory
Computer Security
Information Security
Cryptography
Information Theory
Computer Organization and Architecture
Operating Systems
Image Processing
Parallel Computing
Concurrent Programming
Relational Database
Document-oriented Database
Data Mining
Big Data
Data Science
Digital Libraries
Compiler Design and Construction
Functional Programming
Logic Programming
Object Oriented Programming
Formal Methods
Software Engineering
Agile Software Development
Information Systems
Geographic Information System (GIS)

Mathematics
Mathematics
Algebra
Abstract Algebra
Linear Algebra
Number Theory
Numerical Methods
Precalculus
Calculus
Differential Equations
Category Theory
Proofs
Discrete Mathematics
Theory of Computation
Graph Theory
Real Analysis
Complex Analysis
Probability
Statistics
Game Theory
Queueing Theory
Operations Research
Computer Aided Mathematics

Supporting Fields
Web Design and Development
Mobile App Design and Development
System Administration
Cloud Computing
Electric Circuits
Embedded System
Signal Processing
Integration and Automation
Network Science
Project Management

Operating System
Programming/Scripting
Ada
Assembly
C / C++
Common Lisp
Forth
Java
JavaScript
Lua
Microsoft .NET
Rexx
Perl
PHP
Python
R
Rebol
Ruby
Scheme
Tcl/Tk

Miscellaneous
Sponsors