Ideal Spectral Interpolation Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Ideal Spectral Interpolation

Let $ x\in{\bf C}^N$ include all nonzero samples from a time-limited (as opposed to periodic) signal $ x^\prime\in{\bf C}^\infty$, and define $ y=\hbox{\sc ZeroPad}_M(x)$. Then $ y\in{\bf C}^M$ with $ M\geq N$. Denote the original frequency index by $ k$, where $ \omega_k \isdeftext 2\pi
k/N$, and the new frequency index by $ k^\prime $, where $ \omega_{k^\prime }\isdeftext 2\pi
k^\prime /M$.

Definition: Given a sampled spectrum $ X(\omega_k )\isdeftext \hbox{\sc DFT}_{N,k}(x)$, for $ k\in[0,N-1]$, its ideal bandlimited interpolation at frequency $ \omega\in[-\pi,\pi)$ is defined as

$\displaystyle X(\omega)$ $\displaystyle \isdef$ $\displaystyle \sum_{n=0}^{N-1} x(n) e^{-j\omega n}
\protect$ (7.5)
  $\displaystyle =$ $\displaystyle \lim_{M\to\infty}\sum_{n=0}^{N-1} y(n) e^{-j\omega_{k^\prime(\omega)} n}.
\protect$ (7.6)

where we may define $ k^\prime(\omega)\isdeftext \lfloor\omega
M/(2\pi)\rfloor$. Note that this is just the definition of the DFT with $ \omega_k$ replaced by $ \omega$. That is, the spectrum is interpolated by projecting onto new sinusoids at arbitrary frequencies $ \omega$ exactly as if they were DFT sinusoids (see Chapter 6). Since the interval $ n\in [0,N-1]$ spans all nonzero samples from the time-limited signal $ x^\prime$, the inner product between $ x$ and any sampled sinusoid reduces to exactly Eq. (7.5) above. Thus, for time limited signals, this kind of spectral interpolation is ideal.

Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[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
CCRMA  [Automatic-links disclaimer]