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I'd like to design a Digital Differentiator with a pre-specified lowpass characteristic.

I've tried Wiener Filters (using inverseFFT to transform from frequency to time-domain), Savitzky Golay and others. Intuitively, I've come to the conclusion, that by specifying DFT points, i.e. uniformly distributed samples of the frequency response, one can achieve only a maximal low-pass frequency. For example using $N$ points, the minimal lowpass frequency would be $1/N$ bins of the FFT, thus $\pi/N$ $rad/sample$.

Is this notion correct?

And further along this path: if it's true, is this a universal boundary, or are there methods, where one can implicitly or explicitly specify not-equally sampled frequency points and thus achieve lower stopband frequencies with FIR filters?

For IIR filters I think this is no problem, since with a simple $1/(z+a)$ one can choose $a$ to achieve a variable stopband frequency just as in the continuous case.

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The Remez exchange method isn't limited to DFT bin constraints so to answer your question, no FIR filters do not have the limitation you hypothesized.

If you can find METEOR, a program written in part by Tom Parks, you can be fairly free in your filter specifications provided they are feesable

and there is

http://gmeteor.sourceforge.net/

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  • $\begingroup$ Thanks, what remains to be stated is whether or not the limitation is true for methods using the DFT and then transform back to an impulse response. I suppose with such methods you only have so and so much freedom. $\endgroup$
    – mike
    Jul 9, 2017 at 7:10

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