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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 sinusoid at frequency $ \omega$ [264]:

$\displaystyle X(\omega) \isdef \left<x,s_\omega\right>,$ (3.43)


s_\omega(t) &\isdef & e^{j\omega t}\qquad\qquad\qquad\quad\;\,\mbox{(Fourier Transform)}\\
s_\omega(t_n) &\isdef & e^{j\omega_k t_n} \isdefs e^{j2\pi nk/N} \quad\mbox{(DFT)} \\
s_\omega(t_n) &\isdef & e^{j\omega t_n} \isdefs e^{j\omega n} \quad\qquad\;\mbox{(DTFT)}.

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

$\displaystyle X(\omega) \isdefs \left<x,s_\omega\right> = \sum_{n=-\infty}^\infty x(n) e^{-j\omega n} = \sum_{n=-N/2}^{N/2-1} x(n) e^{-j\omega n}.$ (3.44)

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'' [175].

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2020-07-26 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University