Bandlimited Interpolation of Time-Limited Signals Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

#### Bandlimited Interpolation of Time-Limited Signals

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 FIR7.15 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

 (7.10)

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 :

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 computation. For an introduction to time-domain sampling-rate conversion (bandlimited interpolation) algorithms which take advantage of points (1) and (2) in this paragraph, see, e.g., Appendix D and [70].

Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work] [Order a printed hardcopy]