[PD] 4-point interpolation changes timbre depending on sample rate

[Charles Z Henry]

On Wed, Apr 28, 2021 at 11:14 AM Miller Puckette <msp at ucsd.edu> wrote:
>
> On Wed, Apr 28, 2021 at 10:56:58AM -0500, Charles Z Henry wrote:
> My 2 cents...
>
> The 4-point interpolation scheme gets radically better if the signal it's
> used on is oversampled (error goes down asymptotically by 24 dB for each
> doubling of sample rate) - so my own strategy is simply to 4x upsample
> everything I send through tabread4~ or delread4~.  This moves the "problem"
> to that od designing an upsampling filter, which is much easier than a general
> interpolator.

I found a related problem this fall, teaching a University Physics I
class that got thrown my way at the last minute.... and I decided to
make a couple lessons about numerically integrating systems of
differential equations

The first one (Euler's method) worked just fine, but later on, I
wanted to show we could move past the 1st order derivative
approximation and get better results than just upsampling.
Then, my simulations started blowing up during class and I realized I
don't understand implicit methods as well as I thought I did.  Whoops

So, the related problem is what is the best truncated differentiation
kernel on [-a, 1]?  Once sampled, you'd get the coefficients of a
numerical derivative scheme that can be re-arranged into an implicit
method.

Either case, it goes back to the spectrum of the kernel.  The
derivative approximation used in Euler's method is only good over a
small range of frequencies near zero, so the signal has to be
upsampled in order to produce good results.  By making that kernel
longer, you can get higher frequencies and a spectrum that more
closely matches 2*pi*i*f (although I haven't found a best scheme for
it either).
I think this question is different from the one Runge-Kutta methods
answer.  I think there's something here to find that's relevant