# Best Methods for Interpolating Short Non-Periodic Bandlimited Signals

I need advice on interpolating a very short (N < 10) discrete, bandlimited signal that is sampled above the Nyquist rate. I understand that technically finite length signals have infinite bandwidth, but for practical purposes, this signal is considered bandlimited because the energy at higher frequencies is low.

Typically, I would use a windowed sinc filter for bandlimited interpolation. However, given the very short length of this signal, most samples are near the edges. This proximity to the edges causes issues during convolution as the filter runs over the signal's edges.

An alternative method I'm considering is zero-padding the DFT of the signal (in the middle) and then taking the IDFT. I believe this approach is equivalent to using a Dirichlet kernel for the interpolation filter, which is known to have worse spectral characteristics compared to the windowed sinc filter. However, this method might mitigate the edge problem mentioned above.

Could anyone provide insights or recommendations on the best approach to interpolate such short non-periodic bandlimited signals? Are there other methods or techniques that could be more effective in this scenario?

Additional links: This question and answer are also relevant. And also this answer.

• Have you considered mirroring your signal, applying your windowed sinc interpolation and discarding the padded portions after?
– Jdip
Commented Jun 7 at 3:58
• Also, Lanczos kernel
– Jdip
Commented Jun 7 at 4:02
• Good thought @Jdip, I have considered that. It should be on the list. One issue with mirroring is there will likely be a sharp peak/corner at the mirror point, creating high frequencies that weren't there before. Any ideas on how to fix that? Commented Jun 7 at 4:04
• Maybe a tapering window at the transition? Or some sort of overlap-add?
– Jdip
Commented Jun 7 at 4:09
• Are these 10 samples assumed to be zero-padded (out to $\pm \infty$) on both left and right sides? In any case, I don't see why the interpolation should be any different than for a long sequence of samples from a bandlimited signal. I would recommend a Kaiser-windowed sinc function. Commented Jun 7 at 19:21

## 1 Answer

Fundamentally, it appears you have no choice but to make an assumption about what the signal is doing outside your window of data, or use windowing.

If you claim that the signal is periodic, then use the fft to identify frequency components and interpolate based on those.

If you claim that the signal is zero outside the window (or a constant), then use sinc interpolation.

If you know nothing about what the signal is doing outside the window, then use (Barycentric) Lagrange interpolation, which, for a finite dataset, is the equivalent of sinc interpolation with a binomial window (it is also the equivalent of using a Taylor series expansion of all datapoints with as many terms as datapoints then solving the system of equations). Since the degree of your polynomial will be low (that is, less than 10), it is highly unlikely that you will have any issues, especially if you use the Barycentric form. In papers I've come across on various topics, Lagrange interpolation has been the most common approach used by researchers to prove various results.