# FIR filter digital differentiator with low cutoff

I'm trying to design a digital differentiator FIR Filter. It features a lowpass, such that above the cutoff frequency the amplification is very low. I get the coefficients by a linear program minimizing the chebychef error of desired and actual frequency response.

It works really well, but I cannot place the cutoff frequency below some 0.1*pi rad/sample. Small cutoff frequencies still have very steep rising amplitude responses in low frequencies and thus need a broad transition band. The picture shows such a design and the very broad transition band. The red is the desired and blue obtained frequency response. I've weighted the bands accordingly. Also I'm not talking about bandpass, nor lowpass, I design a differentiator - thus the linear slew rate in low frequencies.

There are limits to the possible lowpass frequency, correct? How can I make the cutoff even smaller, or even better: why is this degradation happening?

I know, that the frequency response in my formulation has the form $$H(e^{j\omega}) = 2\sum_{k=0}^M j \,h(k)\, \sin(k \omega)$$ where $M$ is $(N-1)/2$ with order $N$ filter. And thus the shape can be better traced by having longer filters. But the gain actually is very small.

Also I read, that with a derived then sampled Blackman Window (without control over cutoff frequency) one obtains a cutoff of around $\omega_C \approx 0.005$, while I struggle with $0.1$... I want to know why exactly.

This document suggests a method for a first order differentiator, where "only" the derivatives at $\omega = 0$ are matched to the ideal one. It results in an earlier drop off. However, as I understand it, this cannot be achieved with higher order differentiators, since second order is a quadratic function and I am not sure if a (basically taylor) approx in derivatives is sufficient for that. Let alone even higher orders.

• Can you clarify what are the red and blue plots (looks like desired and obtained responses?). This looks more like a bandpass filter than lowpass, can you clarify? Have you tried increasing the filter order? – MBaz Jul 20 '17 at 15:05
• To further approximate the red plot (ideal filter response?), the order of your filter will need to increase a lot. Which is not a good idea in general. You are already around order 50 for your FIR. – Juancho Jul 20 '17 at 16:11
• I remember seeing on Selesnick's page something similar(?), but for the life of me I cannot find it now. It was about smooth differentiators with very low cutoff frequency, if I'm not mistaken. – a concerned citizen Jul 21 '17 at 5:59