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. First, however, we will ``get our bearings'' with respect to the DFT.

---

Subsections

• The DFT
• Signals as Vectors
  • An Example Vector View:
  
  
• Vector Addition
• Vector Subtraction
• Scalar Multiplication
• Linear Combination of Vectors
• Linear Vector Space
• Signal Metrics
  • Example:
  • Example:
  • Example:
  • Other Lp Norms
  • Norm Properties
  • Banach Spaces
  
  
• The Inner Product
  • Example:
  • 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
  • Changing Coordinates
    • An Example of Changing Coordinates in 2D
  • Projection onto Linearly Dependent Vectors
  • Projection onto Non-Orthogonal Vectors
  • General Conditions
  • Signal/Vector Reconstruction from Projections
  • Gram-Schmidt Orthogonalization
  
  
• Signal Projection Problems

---

Order a Hardcopy of Mathematics of the DFT

Previous: Sinusoid Problems
Next: The DFT

---

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.