# Constructing a lowpass digital differentiator such that maximum error is $n$ dB for passband

I want to construct a lowpass digital differentiator with uniform-interval samples such that error from ideal differentiator is $$n$$ dB in magnitude at maximum for the passband up to frequency $$\omega_c$$. (The reference for dB here is amplitude of $$1$$.)

This error magnitude is not magnitude of frequency response of a lowpass differentiator subtracted from magnitude of frequency response of an ideal differentiator. The error magnitude literally is magnitude of the subtraction of frequency response of a lowpass differentiator from frequency response of the ideal differentiator.

Also, I want the lowpass differentiator to roll off close to zero for frequency response of frequency from $$\omega_c$$ to $$\pi$$.

It is hard to find an article that shows how to construct such a filter as a function of $$n$$ and $$\omega_c$$. Can anyone show me any reference? There surely must be such a filter construction. I do not care if it is IIR or FIR filter, Savitzky-Golay or not, though the error magnitude restriction seems to favor zero-phase filters such as Savitzky-Golay.