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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Multirate Filter Banks
       Wavelet Filter Banks
          Geometric Signal Theory
             Natural Basis


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Natural Basis

The natural basis for a discrete-time signal $ x(n)$ is the set of shifted impulses:

$\displaystyle \varphi_k \isdefs [\ldots, 0,\underbrace{1}_{k^{\hbox{\tiny th}}},0,\ldots], $

or,

$\displaystyle \varphi_k (n) \isdefs \delta(n-k) $

for all integers $ k$ and $ n$. The basis set is orthonormal since $ \left<\varphi_k ,\varphi_l \right>=\delta(k-l)$. The coefficient of projection of $ x$ onto $ \varphi_k $ is given by

$\displaystyle \left<\varphi_k ,x\right> \eqsp x(k) $

so that the expansion of $ x$ in terms of the natural basis is simply

$\displaystyle x \eqsp \sum_{k=-\infty}^{\infty}x(k) \varphi_k , $

i.e.,
$\displaystyle x(n)$ $\displaystyle =$ $\displaystyle \sum_{k=-\infty}^{\infty}x(k) \varphi_k (n)$  
  $\displaystyle =$ $\displaystyle \sum_{k=-\infty}^{\infty}x(k) \delta(n-k) \isdefs (x \ast \delta)(n).$  

This expansion was used in Book II [247] to derive the impulse-response representation of an arbitrary linear, time-invariant filter.

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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.


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