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

Charles Henry czhenry at gmail.com
Thu Apr 1 18:05:22 CEST 2010

```I get what you're saying too, and I'm at least a little skeptical
myself.  But as I think about it generally, my entire approach to
looking at these problems has been very similar.

I basically thought that when comparing interpolators, I could
disregard the signals involved and just look at the properties of the
impulse responses (or convolution kernels or spectra, etc...).  So, if
I can't do that, I really have to rethink what I know.

On Thu, Apr 1, 2010 at 10:44 AM, Matteo Sisti Sette
<matteosistisette at gmail.com> wrote:

>> Here, I want to make the
>> *convenient* assumption that the spectrum of x is flat
>
> Stated this way, it sounds reasonable, doesn't it. If it does, then it means
> that by "flat spectrum" you mean the _power spectrum_ of x considered as a
> _stochastic process_ rather than a deterministic signal.

When it comes to the general class of functions with flat spectra, the
only difference is in phase, right?
But the error is the same in time domain as in frequency domain thanks
to the isometric property of the Fourier transform.  Our interpolation
is the same as a convolution, so we're still just multiplying our
spectra and the phase comes out differently in each frequency.

So, when we integrate the error^2 in the frequency domain, the phase
makes no contribution, and then, it's really just the same thing as
the error in the time domain.  Then, all flat spectra are equivalent
for this problem.  I really am enjoying this math discussion, and I do
want to be corrected or shown something I don't see yet.  Please let
me know if there's something wrong with what I'm saying.

> Brought to the domain of time, assuming x has a flat power spectrum means
> assuming x is white noise. (btw a closer-to-reality assumption would be that
> it is pink noise - but that's not the point here) Not a dirac delta.
>
> So minimizing the error would be to minimize the power, or probably energy,
> of the error meant as a stochastic process.
>
> Though I should have the notions to go a bit further in at least
> _formulating_ (not solving) the problem, those notions are a bit oxidated,
> if not completely gone from my head :(
>
> But I'm sure it is not equivalent to minimizing the integral of the
> difference between the operators applied to a delta function.
>
>
>
>
> --
> Matteo Sisti Sette
> matteosistisette at gmail.com
> http://www.matteosistisette.com
>

```