Geometric Signal Theory

In general, signals can be expanded as a linear combination of orthonormal basis signals [248]. In the discrete-time case, this can be expressed as

where the coefficient of projection of onto is given by

and the basis signals are orthonormal:

The signal expansion (10.13) can be interpreted geometrically as a sum of orthogonal projections of onto , as illustrated for 2D in Fig.10.31.

A set of signals is said to be a biorthogonal basis set if any signal can be represented as

where is some normalizing scalar dependent only on and/or . Thus, in a biorthogonal system, we project onto the signals and resynthesize in terms of the basis .

The following examples illustrate the Hilbert space point of view for various familiar cases of the Fourier transform and STFT. A more detailed introduction appears in Book I [248].

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Subsections

• Natural Basis
• Normalized DFT Basis for
• Normalized Fourier Transform Basis
• Normalized DTFT Basis
• Normalized STFT Basis
• Continuous Wavelet Transform
• Discrete Wavelet Transform
• Discrete Wavelet Filterbank
• Dyadic Filter Banks
• Dyadic Filter Bank Design
• Generalized STFT

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Next: Natural Basis

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About the Author: 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.