Re: [PD] tabread4~ "broken" interpolation algorithm - was Re:, Max Smoother Audio than Pd?
Is it really possible to express a cubic interpolator (such as Lagrange or Hermite, i.e. such as tabread4 or tabread4c) in terms of impulse response? Is it equivalent to a convolution? That is to ask: is it linear??? Or is that an approximation?
Yes, as far as I know it's identical -- when you do one of these interpolations with four points, you can either think of it in terms of a cubic polynomial formula involving those four points, or in terms of the sum of four scaled "basis functions" - the latter seems to me intuitively equivalent to direct convolution of an impulse response centered on the four points, and scaled to each. It will be limited by the quantization of the index, so in that sense it might not be linear..
MB
On Tue, Mar 30, 2010 at 6:45 AM, Matteo Sisti Sette matteosistisette@gmail.com wrote:
Is it really possible to express a cubic interpolator (such as Lagrange or Hermite, i.e. such as tabread4 or tabread4c) in terms of impulse response? Is it equivalent to a convolution? That is to ask: is it linear??? Or is that an approximation?
-- Matteo Sisti Sette matteosistisette@gmail.com http://www.matteosistisette.com
Matt Barber escribió:
Yes, as far as I know it's identical -- when you do one of these interpolations with four points, you can either think of it in terms of a cubic polynomial formula involving those four points, or in terms of the sum of four scaled "basis functions" - the latter seems to me intuitively equivalent to direct convolution of an impulse response centered on the four points, and scaled to each. It will be limited by the quantization of the index, so in that sense it might not be linear..
Hmm, by linear I mean that the interpolated version of the sum of any two discrete signals is equal to the sum of the interpotated versions of each one. It shouldn't be difficult to tell whether this is true or not by looking at the formulas but I'm too lazy right now :P
If the transformation is not linear, then there is no "impulse response" at all nor any "frequency response" strictly speaking, thought that doesn't mean that it doesn't make sense to do some reasoning in terms of impulse or frequency response under certain simplifications or assumptions - i.e. to speak less strictly....
On Tue, Mar 30, 2010 at 9:52 AM, Matteo Sisti Sette matteosistisette@gmail.com wrote:
Matt Barber escribió:
Yes, as far as I know it's identical -- when you do one of these interpolations with four points, you can either think of it in terms of a cubic polynomial formula involving those four points, or in terms of the sum of four scaled "basis functions" - the latter seems to me intuitively equivalent to direct convolution of an impulse response centered on the four points, and scaled to each. It will be limited by the quantization of the index, so in that sense it might not be linear..
Hmm, by linear I mean that the interpolated version of the sum of any two discrete signals is equal to the sum of the interpotated versions of each one. It shouldn't be difficult to tell whether this is true or not by looking at the formulas but I'm too lazy right now :P
Right, I think it should absolutely be linear in that sense, provided you used exactly the same index for both processes and ignored any rounding errors from adding samples. "In math" it should be linear, but I'm not 100% sure about what quantization of index or sample does to it (we often don't worry so much about it anyway).
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Matt Barber -
Matteo Sisti Sette