Introductory resources to mathematical properties of "dynamically recalculable" filters in audio/musical equalizers?

Particularly, I want to understand

  • What mathematical features (e.g. monotonicity of coefficients) of filters are suited for real-time modulated filters?
  • How to analyze filters for the required features?

But I want to also understand things such as "filter instability" and the ways they are implemented in real-time (does one apply them in buffers or what?)?

Dynamic recalculation: That the filter's cutoff and resonance can be recalculated on the fly. Or that the parameters can be adjusted while the filter is running. W/o the filter "blowing up" or something else.

  • $\begingroup$ hm, again, it would be easier if you could at least try to narrow down the field of application; reconfigurable filters are a relatively rare need, and a lot of material will deal with reconfigurable filters for a specific purpose. (I feel like I'm really repeating myself, or what I said in the comments of your previous question, sorry) $\endgroup$ Jun 10, 2018 at 10:28
  • $\begingroup$ aahh this makes a lot more sense, thank you! Now, I'm no expert in audio processing, but there's enough audio experts around here; I think your edit makes it much more likely you'll get an answer :) $\endgroup$ Jun 10, 2018 at 10:47
  • 1
    $\begingroup$ Book: mdpi.com/2076-3417/6/5/129 (source codes: code.soundsoftware.ac.uk/projects/allaboutaudioeq/repository ) . EqualizerAPO project: sourceforge.net/p/equalizerapo/discussion/?source=navbar $\endgroup$
    – Juha P
    Jun 10, 2018 at 11:01
  • $\begingroup$ @JuhaP Hey I cannot find anything from the book by quickly searching it using terms like: "recalc", "modulat". So I'm not sure where in it does it present something related to the question. $\endgroup$
    – mavavilj
    Jun 10, 2018 at 11:08
  • $\begingroup$ @mavavilj seriously, that book seems to be a fine resource about how to design parametric equalizers, and how to adjust them. You'll need to work with the things you get when asking such specific questions. You cannot expect the world to have written a text that answers, and only answers, your very specific needs! I'd thank Juha P for giving me such a nice source, and, try to find the things in that book I want to adapt, and maybe follow up with questions on that, rather than demanding that people have texts exactly in the form you want them. $\endgroup$ Jun 10, 2018 at 12:01

1 Answer 1


If you want to modulate in time your filter parameters (like cut frequency), you shouldn't use the usual DF1/TDF2 implementations derived from a polynomial fraction in z. The reason is that they are originally LTI systems (linear time invariant), which is no longer the case now. The state that they have internally, whether an actual state or a function of the past inputs/outputs, is not reliable when the polynomial coefficients change. Instead, you should use things like TPT schemas (see the Art of VA filter design at https://www.kvraudio.com/forum/viewtopic.php?f=33&t=350246) that will keep a state that is compatible with parameter changes. Basically, if you map an existing analog system, you model the state of the capacitors and coils and keep them continuous. This doesn't happen when using the z transform.


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