NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

This chapter provides an introduction to the elements of geometric signal theory, including vector spaces, norms, inner products,orthogonality, projection of one signal onto another, and elementary vector space operations.

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Subsections

• The DFT
• Signals as Vectors
  • An Example Vector View:
• Vector Addition
• Vector Subtraction
• Signal Metrics
  • Other Norms
• The Inner Product
  • Linearity of the Inner Product
  • Norm Induced by the Inner Product
  • Cauchy-Schwarz Inequality
  • Triangle Inequality
  • Triangle Difference Inequality
  • Vector Cosine
  • Orthogonality
  • The Pythagorean Theorem in N-Space
  • Projection
• Signal Reconstruction from Projections
  • An Example of Changing Coordinates in 2D
  • General Conditions
  • Gram-Schmidt Orthogonalization
• Appendix: Matlab Examples

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