The previous result can be applied to bandlimited interpolation of
*arbitrary* time-limited signals
(*i.e.*, not just
periodic signals) by (1) replacing the rectangular window
with a *smoother spectral window*
,
and (2) using extra zero-padding in the time domain to convert the
*cyclic* convolution between
and into an
*acyclic* convolution between them (recall §7.2.4).
The smoother spectral window can be thought of as the *frequency
response* (sampled) of the FIR^{7.15} filter used as the
bandlimited interpolation kernel in the time domain. The number of
extra zeros appended to in the time domain is simply length of
minus 1, and the number of zeros appended to is the length of
minus 1. If denotes the nonzero length of ,
then the nonzero length of
is
. Thus, we
require the DFT length to be
, where is the
filter length. In operator notation, we can express bandlimited
sampling-rate up-conversion by the factor for time-limited signals
by

The approximation symbol `' approaches equality as the spectral window approaches (the frequency response of the ideal lowpass filter passing only the original spectrum ), while at the same time allowing no time aliasing (convolution is still acyclic in the time domain).

Equation (7.10) can provide the basis for a high-quality
sampling-rate conversion algorithm. Arbitrarily long signals can be
accommodated by breaking them into segments of length , applying
the above algorithm to each block, and summing the up-sampled blocks using
*overlap-add*. That is, the lowpass filter ``rings''
into the next block and possibly beyond (or even into both adjacent
time blocks when is not causal), and this ringing must be summed
into all affected adjacent blocks. Finally, the filter can
``window away'' more than the top copies of in , thereby
preparing the time-domain signal for *downsampling*, say by
:

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