NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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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.

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Subsections

• Sinusoids
  • Example Sinusoids
  • Why Sinusoids are Important
  • In-Phase and Quadrature Sinusoidal Components
  • Sinusoids at the Same Frequency
  • Constructive and Destructive Interference of Sinusoids
• Exponentials
  • Why Exponentials are Important
  • Audio Decay Time ()
• Complex Sinusoids
  • Circular Motion
  • Projection of Circular Motion
  • Positive and Negative Frequencies
  • The Analytic Signal and Hilbert Transform Filters
  • Generalized Complex Sinusoids
  • Sampled Sinusoids
  • Powers of
  • Why (Generalized) Complex Sinusoids are Important
  • Comparing Analog and Digital Complex Planes
• Mathematica for Selected Plots
• Acknowledgement

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