Geometric Signal Theory

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Geometric Signal Theory

We will now approach filter bank derivation from a ``Hilbert space'' point of view. This is the most natural setting for the study of wavelet filter banks.

Signals can be expanded as a linear combination of orthonormal basis signals $\varphi_k$ :

$\displaystyle x(n)$ $\displaystyle =$ $\displaystyle \sum_{k=-\infty}^{\infty}\left<\varphi_k ,x\right> \varphi_k (n)$

$\displaystyle n\in(-\infty,\infty), \quad x(n), \varphi_k (n) \in {\cal C}$

where the coefficient of projection of onto $\varphi_k$ is given by

$\displaystyle \left<\varphi_k ,x\right> \; \isdef \sum_{n=-\infty}^{\infty}\varphi_k ^\ast(n) x(n) \qquad \quad \hbox{(Inner Product)}$

$\displaystyle \left<\varphi_k ,\varphi_l\right> = \delta(k-l) = \left\{\begin{a... ...{ll} 1, & k=l \\ 0, & k\neq l \\ \end{array} \right. \; \hbox{(Orthonormal)}$
Signal expansion can be viewed as sum of geometric projections of onto $\{\varphi_k \}$ , as illustrated for 2D in Fig.12.1.

**Figure 12.1:** Orthogonal projection in 2D.
$\includegraphics[scale=0.8]{eps/proj}$

Subsections

Example Basis Signals

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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.

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Spectral Audio Signal Processing
Wavelet Filter Banks
Geometric Signal Theory