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Interpolation Operator

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

\hbox{\sc Interp}_{L,k^\prime }(X) &\isdef & X(\omega_{k^\prim...
...i k^\prime /M,\; k^\prime =0,1,2,\dots,M-1,\;\\
M&\isdef & LN.

Since $ X(\omega_k )\isdeftext \hbox{\sc DFT}_{N,k}(x)$ is initially only defined over the $ N$ roots of unity, 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.

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

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[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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