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Geometric Signal Theory

In general, signals can be expanded as a linear combination of orthonormal basis signals $ \varphi_k $ [264]. In the discrete-time case, this can be expressed as

$\displaystyle x(n)$ $\displaystyle =$ $\displaystyle \sum_{k=-\infty}^{\infty}\left<\varphi_k ,x\right> \varphi_k (n) \protect$ (12.104)
    $\displaystyle n\in(-\infty,\infty), \quad x(n), \varphi_k (n) \in {\cal C}$  

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

$\displaystyle \left<\varphi_k ,x\right> \; \isdef \sum_{n=-\infty}^{\infty}\varphi_k ^\ast(n) x(n) \qquad \qquad \hbox{(inner product)}$ (12.105)

and the basis signals are orthonormal:

$\displaystyle \left<\varphi_k ,\varphi_l \right> \eqsp \delta(k-l) \eqsp \left\{\begin{array}{ll} 1, & k=l \\ 0, & k\neq l \\ \end{array} \right. \qquad \hbox{(orthonormal)}$ (12.106)

The signal expansion (11.104) can be interpreted geometrically as a sum of orthogonal projections of $ x$ onto $ \{\varphi_k \}$ , as illustrated for 2D in Fig.11.30.
Figure 11.30: Orthogonal projection in 2D.

A set of signals $ \{h_k,f_k\}_{k=1}^N$ is said to be a biorthogonal basis set if any signal $ x$ can be represented as

$\displaystyle x = \sum_{k=1}^N \alpha_k\left<x,h_k\right>f_k$ (12.107)

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

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 [264].

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University