[music-dsp] bandlimited interpolation of non-uniformly-spaced samples?

[Sampo Syreeni]

On 2009-11-27, Spencer Russell wrote:

> Implementations that I'm familiar with have all just interpolated 
> linearly, but I'm interested in exploring other interpolation methods, 
> especially looking to generate bandlimited waveforms to eliminate 
> aliasing problems.

There is a reason why those implementations use linear interpolation and 
risk aliasing in the process. That is because there is no general theory 
of higher order polynomial interpolation for completely arbitrary nodes. 
Least of all *any* theory that would guarantee bandlimited output given 
an arbitrary set of nodes (polynomials are of course bandlimited, but as 
soon as you start doing anything but summin them together, i.e. 
modulating their endpoints as in a spline, you can get just about any 
spectral feature). Nor is there a proper analytic solution to how a 
finite number of Markov random variables on the real line which are 
bounded not to cross each other asymptotically behave, AFAIK.

> Can anyone point me towards some efficient methods for interpolating 
> non-uniformly spaced samples to generate bandlimited(or 
> quasi-bandlimited) waveforms?

No, they cannot, because such a theory does not exist. In fact I seem to 
remember that sort of thing can be shown to be impossible at any finite 
order of the interpolating polynomial, and even in the case of a finite, 
rather well-behaved basis of infinite order polynomial approximations to 
real functions, the search problem can be proven NP-complete. (As usual, 
no references. But you can check e.g. the literature on finite 
approximations to arbitrary well-behaved functions, in the L^1 norm.)
-- 
Sampo Syreeni, aka decoy - decoy at iki.fi, http://decoy.iki.fi/front
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