is there an "ideal" derivator? or I am say something totally wrong?
Nope, there's not an "ideal" differentiator.
I take that back....I wrote too hastily. There is an ideal differentiator, related to the ideal interpolator.
For ideal interpolation, we have to have an infinitely long signal. We have a function defined on the set of real numbers to the set of real numbers, for instance. The Whittaker Cardinal function is the ideal interpolator. If we have a signal that is band-limited in the frequency domain, we can choose a sinc(k*t) for some k, that contains the frequency bands, we have in our function. sinc(x) = sin(pi*x)/pi*x For simplicity sake, we'll assume that our frequency spectrum is limited to (-1/2,1/2)...Then, we choose k=1 (this is a related bit to the sampling theorem, just replace k with fs, and (-fs/2,fs/2).
And we sample our function at the integers, -inf, ... -2, -1, 0, 1, 2, ..., inf The Whittaker Cardinal function is f(t) = sum( i= -inf to inf, f(i)*sinc(i-t) )
Also, this can be written as a convolution f(t) = sinc(t) -conv-with- sum( i= -inf to inf, f(i)*delta(t-i))
and the result is *exactly* the function we started with! and we can differentiate this function: d/dt (sin (pi*t) / (pi*t)) = ((pi*t)*cos(pi*t) - sin(pi*t) )/ (pi*t^2)
and, when we take this function and convolve it with our sampled function values, we get the derivative of the sampled function. There is a problem, here, namely that the sequence we need to convolve by is infinitely long....so, there's a problem.... but, we can truncate the series to as many samples as we need. For example, a length 11 sequence is: (-1/5 1/4 -1/3 1/2 -1 0 1 -1/2 1/3 -1/4 1/5)*fs
I'm not sure if I've done something wrong here, yet. Anyway, all of your great mathematicians just made it up as they went along, right?
Chuck