[PD] tabread4~ "broken" interpolation algorithm - was Re: Max Smoother Audio than Pd?

[Charles Henry]

> I don't know either.  We have the formulas for each, so we can
> calculate squared error vs. sinc(x), but there also appears to be
> differences in which frequencies the distortion occurs and some could
> be more audible.

It occurs to me that there exists one very obvious function for which
the squared error is minimized for a 4-point interpolator.  4-point
interpolator impulse functions have to be 0 outside the interval
[-2,2].

So,
E=|f(x)-sinc(x)|^2 is minimized when

f(x)={sinc(x) -2<x<2  ,    0 elsewhere

Note that since the Fourier transform is isometric and linear, we know
that a function which minimizes the error in the time domain also
minimizes error in the frequency domain.  This spectrum ought to have
a lot of ringing in the upper frequency range, and in the stopband.

So, if we only considered the squared error in the reconstruction and
not the smoothness of the result, this would be the clear choice.  How
would we factor in the smoothness as well?