Equations to graph 4-pole LPF/HPF/BP/Notch?

[mikejm]

I have worked out equations for graphing One-Pole and Two-Pole basic filters here:

https://www.desmos.com/calculator/mjudrnexbo

I worked these out from various PDFs I found on the net and basic physics tutorials on filters.

What would be the equivalent Four-Pole equations? Those I can't figure out or find for the life of me. If anyone knows how to graph these in the same way, could you perhaps add them to the Desmos for me or post the equations here or how to find them?

I tried asking on StackExchange but got no useful answers there. I'd appreciate any help.

Thanks.

[egbert]

In basic terms, one pole is a linear decline with a slope of 20dB per decade or 6dB per octave. 4 pole is 4 times that - so 24 dB per octave. The quoted frequency for the filter is the frequency of the 3 dB point - ie 3 dB below the prefiltered level. When I say linear - it will be a straight line on a log-log graph where frequency and power are both log scales (dB vs octaves).

[mtytel]

You can reduce higher order filters into chains of one and two pole filters. So factor your 4 pole transfer function into two 2 pole functions multiplied together.

[juha_p]

For Butterworth LP and HP

https://www.desmos.com/calculator/dxcnvc63av

Play the variable n.

Sources: https://www.electronics-tutorials.ws/fi ... ter_8.html http://fourier.eng.hmc.edu/e84/lectures ... node6.html

[mystran]

If you have the transfer function (of whatever order), then for analog filters you can substitute s=i*w and for digital filters z=exp(i*w) where w=2*pi*f/fs typically, then compute the complex response directly. The complex magnitude gives you the amplitude response while the complex argument gives you the phase response. Simplification will get you the "real-valued" expressions if you need those, but honestly it's usually just easier to use the complex formulas and then take the magnitude at the end.

[mikejm]

For Butterworth LP and HP

https://www.desmos.com/calculator/dxcnvc63av

Play the variable n.

Sources: https://www.electronics-tutorials.ws/fi ... ter_8.html http://fourier.eng.hmc.edu/e84/lectures ... node6.html

Thanks man! That's great. But do you by any chance have a way of integrating Q into the equation? Ie. For resonant equation? Or what the equivalent would be for BP/BR?

[mikejm]

If you have the transfer function (of whatever order), then for analog filters you can substitute s=i*w and for digital filters z=exp(i*w) where w=2*pi*f/fs typically, then compute the complex response directly. The complex magnitude gives you the amplitude response while the complex argument gives you the phase response. Simplification will get you the "real-valued" expressions if you need those, but honestly it's usually just easier to use the complex formulas and then take the magnitude at the end.

I'm actually using these equations to modulate the partial levels in a modal additive synth directly, so the simplest basic equation (ie. a magnitude plot) is all I actually need. Unfortunately, I don't have a deep knowledge of filter design/practice. It tapped me out mentally just to get the first order and second order filters worked out.

I can't find any direct magnitude plot equations for any simple 4th order resonant filters anywhere.