NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION: "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 ©
2017-09-28 by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University
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It is straightforward to show that the ''additive synthesis''
reconstruction method of the previous section actually works
exactly (in the periodic case) in the following sense:
- The reconstructed signal
is
band-limited to 
, i.e., its Fourier
transform 
is zero for all 
. (This is not quite true in the truncated case.)
- The reconstructed signal
passes through the samples
exactly. (This is true in both cases.)
Is this enough? Are we done? Well, not quite. We know by
construction that is a band-limited interpolation of.
But are band-limited interpolations unique? If so, then this must
be it, but are they unique? The answer turns out to be yes, based
on
Shannon's Sampling Theorem. The uniqueness follows from the
uniqueness of the inverse Fourier transform. We still have two
different cases, however, depending on whether we assume periodic
extension or zero extension beyond the range 
. In
the periodic case, we have found the answer; in the zero-extended
case, we need to use the sum-of-sincs
construction provided in the proof of Shannon's sampling
theorem.
Why do the DFT sinusoids
suffice for interpolation in the periodic case and not in the
zero-extended case? In the periodic case, the spectrum consists of
the DFT frequencies and nothing else, so additive synthesis
using DFT sinusoids works perfectly. A sum of 
DFT
sinusoids can only create a periodic signal (since each of
the sinusoids repeats after samples).
Truncating such a sum in time results in all frequencies
being present to some extent (save isolated points) from
to 
. Therefore, the truncated result is not band-limited, so
it must be wrong.
It is a well known Fourier fact that no function can be both
time-limited and band-limited. Therefore, any truly
band-limited interpolation must be a function which has infinite
duration, such as the sinc
function 
used in bandlimited
interpolation by a sum of sincs. Note that such a sum of sincs
does pass through zero at all sample times in the ''zero
extension'' region.
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