**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 length signal onto the set of sampled complex sinusoids generated by the roots of unity.Fourier Theorems for the DFT

This chapter derives variousFourier theoremsfor 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 forlinearity,time-invarianceand four basic representations of digital filters: thedifference equation coefficients, impulse response, transfer function,andfrequency response. Next the concepts ofphase delayandgroup delayare defined. This material is a subset of that in [1].