Let
include all nonzero samples from a *time-limited*
(as opposed to periodic) signal
, and
define
. Then
with .
Denote the original frequency index by , where
, and the new frequency index by , where
.

**Definition: **
Given a sampled spectrum
, for
,
its *ideal bandlimited interpolation* at
frequency
is defined as

where we may define . Note that this is just the definition of the DFT with replaced by . That is, the spectrum is interpolated by projecting onto new sinusoids at arbitrary frequencies exactly as if they were DFT sinusoids (see Chapter 6). Since the interval spans all nonzero samples from the time-limited signal , the inner product between and any sampled sinusoid reduces to exactly Eq. (7.5) above. Thus, for time limited signals, this kind of spectral interpolation is ideal.

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