The previous result can be extended toward bandlimited interpolation
of
which includes all nonzero samples from an
*arbitrary* time-limited signal
(*i.e.*, going beyond the interpolation of only periodic bandlimited
signals given one or more periods
) by

- replacing the rectangular window
with a
*smoother spectral window*, and - 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 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 remains acyclic in the time domain).

Equation (7.8) 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
:

where now the lowpass filter frequency response must be close to zero for all . While such a sampling-rate conversion algorithm can be made more efficient by using an FFT in place of the DFT (see Appendix A), it is not necessarily the most efficient algorithm possible. This is because (1) out of output samples from the IDFT need not be computed at all, and (2) has many zeros in it which do not need explicit handling. For an introduction to time-domain sampling-rate conversion (bandlimited interpolation) algorithms which take advantage of points (1) and (2) in this paragraph, see,

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