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.

- The DFT
- Signals as Vectors

- Vector Addition
- Vector Subtraction
- Scalar Multiplication
- Linear Combination of Vectors
- Linear Vector Space
- Signal Metrics

- 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
- Projection onto Linearly Dependent Vectors
- Projection onto Non-Orthogonal Vectors
- General Conditions
- Signal/Vector Reconstruction from Projections
- Gram-Schmidt Orthogonalization

- Signal Projection Problems

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University