#### Bandlimited Interpolation of Time-Limited Signals

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).

*frequency response*of the FIR

^{7.22}filter used as the bandlimited interpolation kernel in the time domain. The number of zeros needed in the zero-padding of in the time domain is simply length of minus 1, and the number of zeros to be appended to is the length of minus 1. With this much zero-padding, the cyclic convolution of and implemented using the DFT becomes equivalent to acyclic convolution, as desired for the time-limited signals and . Thus, if denotes the nonzero length of , then the nonzero length of is , and 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 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
:

*e.g.*, Appendix D and [72].

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Applying the Blackman Window

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Relation to Stretch Theorem