On Wed, Mar 31, 2010 at 5:12 PM, Matteo Sisti Sette matteosistisette@gmail.com wrote:
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
I may be missing something but I'm afraid the E in your formula is not the error that is supposed to be minimized.
Sorry, I often go kind of fast-and-loose with the math, but I think you'll see it's true within a certain context, which you may or may not accept.
The ideally interpolated signal (which is the one in reference to which the error has to be minimized) is not just a sinc: it is the sum of an infinite series of sinc's centered at the sampled points and scaled with the sampled values.
Let x be the series of samples, each multiplied by Dirac-delta functions at the sample times. Let S be the convolution operator which convolves a function by sinc(t) and let F be our arbitrary convolution operator which convolves by an interpolation function f(t).
Then, the quantities we need to compare are Sx and Fx where we want to minimize the L2 norm, the integral of the squared error (Sx - Fx)^2
|Sx-Fx|^2 = |(S-F)x|^2
The error depends on x the signal. Here, I want to make the *convenient* assumption that the spectrum of x is flat, since we want some kind of generality and we want to minimize average error across frequencies. This would make the problem equivalent to using just *one* dirac-delta in place of x and we would get the problem to reduce back to just the difference of the impulse responses
|sinc(t)-f(t)|^2
For a little while, I was going in circles on how to minimize operator norms, but it's not quite the right problem for that and I'd probably spend all day on it, that way :)
(I won't try to write it in a latex-like fashon, I would certainly get it wrong - not because of latex syntax, I mean I would get it wrong even if I tried to write it down manually)
I'm reluctant to try latex because it looks like too much work, but I think lyx (a wysiwyg latex editor) is more my speed.