# Highly damped IIR/FIR digital filter

I want a highly damped highpass filter (damping of at least $2$), with a cutoff somewhere around $1\textrm{ Hz}$ ($51.2\textrm{ Hz}\quad f_s$)

• How do i go about designing the filter with adequate control over the damping?

My best guess was to use a standard second order response and frequency transform it, giving

$$\frac{s^2 \omega_0}{\omega _0^2 s^2 + 2\zeta\omega_0 ^2 + 1}$$ This gives me the required damping (after using MATLAB to transform it to the discrete domain using Tustin's method with tustin), but the frequency response is terrible (the $\rm dB$ point is much, much higher than it should be).

• Is there any way to solve this?
• Can I design a FIR filter with control over the damping?
• Is there a higher order $s$-domain function that still gives me control over the damping?

Notes: filter order and phase delay are not overly important to my design.

• What do mean by: The frequency response is terrible? – Stephen Rauch Jan 20 '17 at 16:13
• fair point - what i meant was that if i specify a cuttoff frequency (say 1hz) - the actual cut off is nowhere near that - the 3db point is usually closer to 15 hz or so – Mauvai Jan 20 '17 at 16:39
• If this isn't strictly a learning opportunity and you're just trying to find a filter design that works, you should check out Matlab's built in filter design tool (I believe it's located under apps in Matlab 2015 onwards). It essentially gives you a lot of control of specifying filter criteria (IIR vs FIR, linear phase, filter order, cutoff freq, etc.) – Izzo Jan 20 '17 at 16:44
• So isn't that the point of highly damped? Damping is one way to express the gain at the oscillatory frequency (location of poles). The equation only has two degrees of freedom. If you want the 3db point to be a particular frequency and yet want the oscillatory frequency to be highly damped you will need to lower the oscillatory frequency. – Stephen Rauch Jan 20 '17 at 16:45
• @StephenRauch I don't understand your argument - if i have two degrees of freedom, and i want to control two things, what is the issue with trying to influence both? – Mauvai Jan 23 '17 at 12:35

$$H(s)=\frac{s^2}{s^2+2\zeta\omega_0 s+\omega_0^2}$$