NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University
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Outline
Below is an overview of the chapters.
- Introduction to the DFT
This chapter introduces the Discrete Fourier Transform (DFT) and points out the elements which will be discussed in this reader.- Introduction to Complex Numbers
This chapter provides an introduction to complex numbers, factoring polynomials, the quadratic formula, the complex plane, Euler's formula, and an overview of numerical facilities for complex numbers in Matlab and Mathematica.- Proof of Euler's Identity
This chapter for Music 320 outlines the proof of Euler's Identity which is an important tool for working with complex numbers. It is one of the critical elements of the DFT definition that we need to understand.- Logarithms, Decibels, and Number Systems
This chapter provides an introduction to logarithms (real and complex),decibels, and number systems such as binary integer fixed-point, fractional fixed-point, one's complement, two's complement, logarithmic fixed-point,-law, and floating-point number formats.
- Sinusoids and Exponentials
This chapter provides an introduction to sinusoids, exponentials, complex sinusoids,, in-phase and quadrature sinusoidal components, theanalytic signal, positive and negative frequencies, constructive anddestructive interference, invariance of sinusoidal frequency in linear time-invariant systems, circular motion as the vector sum of in-phase and quadrature sinusoidal motions, sampled sinusoids, generating sampled sinusoids from powers of
, and plot examples using Mathematica.
- The Discrete Fourier Transform (DFT) Derived
This chapter derives the Discrete Fourier Transform (DFT) as a projection of a lengthsignal
onto the set of
- Fourier Theorems for the DFT
This chapter derives various Fourier theorems for the case of the DFT. Included are symmetry relations, the shift theorem, convolution theorem,correlation theorem, power theorem, and theorems pertaining to interpolation and downsampling. Applications related to certain theorems are outlined, including linear time-invariant filtering, sampling rate conversion, andstatistical signal processing.- Example Applications of the DFT
This chapter goes through some practical examples of FFT analysis in Matlab. The various Fourier theorems provide a ''thinking vocabulary'' for understanding elements of spectral analysis.- A Basic Tutorial on Sampling Theory
This appendix provides a basic tutorial on sampling theory. Aliasing due to sampling of continuous-time signals is characterized mathematically. Shannon's sampling theorem is proved. A pictorial representation of continuous-time signal reconstruction from discrete-time samples is given.- Introduction to Digital Filter Theory
This appendix provides an introduction to the most basic elements ofdigital filter theory. Definitions are given for linearity,time-invariance and four basic representations of digital filters: the difference equation coefficients, impulse response, transfer function, and frequency response. Next the concepts of phase delay and group delay are defined. This material is a subset of that in [1].