# How far can numerical differentiation be accurate for sinusoids?

Suppose you have samples of uniform interval of a sinusoidal signal that is some sum of sinusoids. Forget about aliasing, as the signal is bandlimited. If we have infinite number of samples at sampling rate right at Nyquist rate, we can technically infer the whole bandlimited function and of course differentiate. We cannot do that for finite number of samples.

Thus, my question is how far can numerical differentiation be accurate for each frequency - that is deviation from ideal frequency response - given $n$ samples of uniform interval. One can assume that the filter being used is FIR filter.

Parks-McClellan is often used to obtain an optimal filter, so my question can be rephrased as how required maximum frequency response error magnitude (relative to ideal frequency response of a differentiator) corresponds to upper bound on number of samples/taps/order required for the derived filter.

• Is the signal in question a band-limited signal or a sinusoid? This is not clear because both are mentioned in the question. – Olli Niemitalo Sep 14 '18 at 9:08
• Is your question: "Given a specified (FIR) filter order, can I know in advance the maximum approximation error for the design of a differentiator?" – Matt L. Sep 14 '18 at 13:33
• Hi Park! As the other two comments indicate, it would be great if you could rephrase your question with a focus on clarity; please clearly detail your exact conditions and also indicate exactly which parameter(s) you want to compute or estimate under those conditions. – Fat32 Sep 14 '18 at 22:08