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Interpolation Operator

The interpolation operator $ \hbox{\sc Interp}_L()$ interpolates a signal by an integer factor $ L$ using bandlimited interpolation. For frequency-domain signals $ X(\omega_k)$ , $ k=0,1,2,\ldots,N-1$ , we may write spectral interpolation as follows:

\begin{eqnarray*}
\hbox{\sc Interp}_{L,k^\prime }(X) &\isdef & X(\omega_{k^\prime }), \mbox{ where}\\
\omega_{k^\prime }&=& 2\pi k^\prime /M,\; k^\prime =0,1,2,\dots,M-1,\;\\
M&\isdef & LN.
\end{eqnarray*}

Since $ X(\omega_k )\isdeftext \hbox{\sc DFT}_{N,k}(x)$ is initially only defined over the $ N$ roots of unity in the $ z$ plane, while $ X(\omega_{k^\prime })$ is defined over $ M=LN$ roots of unity, we define $ X(\omega_{k^\prime })$ for $ \omega_{k^\prime }\neq\omega_k $ by ideal bandlimited interpolation (specifically time-limited spectral interpolation in this case).

For time-domain signals $ x(n)$ , exact interpolation is similarly bandlimited interpolation, as derived in Appendix D.


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-10-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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