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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Fourier Transforms for Continuous/Discrete Time/Frequency
       Spectral Interpolation
          Ideal Spectral Interpolation

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Ideal Spectral Interpolation

Ideally, the spectrum of any signal $ x(n)$ at any frequency $ \omega = 2\pi f$ is obtained by projecting the signal $ x$ onto the zero-phase, unit-amplitude, complex sinsuoid at frequency $ \omega$ [248]:

$\displaystyle X(\omega) \isdef \left<x,s_\omega\right>, $

where

\begin{eqnarray*} s_\omega(t) &\isdef & e^{j\omega t}\qquad\qquad\qquad\quad\;\,... ...^{j\omega t_n} \isdefs e^{j\omega n} \quad\qquad\;\mbox{(DTFT)}. \end{eqnarray*}

Thus, for signals in the DTFT domain which are time limited to $ n\in[-N/2,N/2-1]$, we obtain

$\displaystyle X(\omega) \isdefs \left<x,s_\omega\right> = \sum_{n=-\infty}^\infty x(n) e^{-j\omega n} = \sum_{n=-N/2}^{N/2-1} x(n) e^{-j\omega n}. $

This can be thought of as a zero-centered DFT evaluated at $ \omega\in[-\pi,\pi)$ instead of $ \omega_k = 2\pi k/N$ for some $ k\in[0,N-1]$. It arises naturally from taking the DTFT of a finite-length signal. Such time-limited signals may be said to have ``finite support'' [165].

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Next: Interpolating a DFT

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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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