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 :
Applying the Blackman Window
Relation to Stretch Theorem