Hilbert transform is a quite sensitive topic here, since Gabor's paper, Theory of Communication, J. Inst. Electr. Engineering, London, 1946. Perhaps even more important than the Fourier transform.
There are short Gaussian FIR approximations, smoothing filters, or polynomial fits.
What would be methods for very short (say, less than $16$ taps) Hilbert tranform approximations of a signal, possibly including a weighting factor? I am intending to use them on small chunks of signals, in moving frames that progress along data as new samples are acquired. Signals can be considered freed from a potential low-frequency trend (e.g. polynomial).
Fey features would be:
- a management or reduction of small-sized frame artifacts (I am aware of those for short Fourier frames, much less with short Hilbert transforms),
- potential asymmetric windowing to add a forgeting factor to older samples,
- a recursive formulation, in the spirit of the sliding DFT, would be a plus.
Finally, an IIR formulation, such as the one of the recursive least squares (RLS) adaptive filters, or the exponentially-weighted moving average filter, would be very interesting as well.