[music-dsp] Article: antialiased sine waveform syncing

Martin Eisenberg martin.eisenberg at udo.edu
Mon Jun 8 10:01:25 EDT 2009


george bezerianos wrote:

> There is a perfect analogy here to the well known Taylor's
> Theorem. One interpretation of this theorem is that we can
> approximate a function (let's stick to one variable without
> loss of generality) by knowing the derivatives at one single
> point. In a similar manner we can bandlimit a generic
> hard-sync case by bandlimiting higher order derivatives of a
> Cn discontinuity through integration but also by convolving a
> BLIT tuned tot he master frequency with sufficiently long
> iterations of the original waveform!!. I'm pretty sure that
> one can even find a formula connecting the sufficient length
> of the slave waveform to the order of the discontinuity and
> that's to my eyes a clear analogy to the Taylor case!!

It has been noted in the field of interpolation that the
sign-flipped monomials sgn(x)*x^n, in a sense the prototypical
C^n discontinuities, are related to B-splines by differencing.
It's something of another direction but perhaps you get something
out of the lead. See e.g. reference
http://dx.doi.org/10.1109/TIP.2003.818018 available at
http://www.ee.cuhk.edu.hk/~tblu/monsite/pdfs/blu0302.pdf


Martin



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