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Geometric Signal Theory

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

$\displaystyle x(n)$ $\displaystyle =$ $\displaystyle \sum_{k=-\infty}^{\infty}\left<\varphi_k ,x\right> \varphi_k (n) \protect$ (11.13)
    $\displaystyle n\in(-\infty,\infty), \quad x(n), \varphi_k (n) \in {\cal C}$  

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)}

and the basis signals are orthonormal:

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

The signal expansion (10.13) can be interpreted geometrically as a sum of orthogonal projections of $ x$ onto $ \{\varphi_k \}$, as illustrated for 2D in Fig.10.31.
Figure 10.31: 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

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 [248].

Previous: Wavelet Filter Banks
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 (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 for details.


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