Time Varying BIBO Stability Analysis of Trapezoidal Integrated Optimised SVF v2

In November 2013 I posted a draft of “Time Varying BIBO Stability Analysis of Trapezoidal integrated optimised SVF v2” to the music-dsp mailing list.[1] The draft applied Jean Laroche’s state-space method [2] to prove the time varying stability of the SVF filter that Andy Simper had recently derived using trapazoidal integration.[3] I didn’t quite finish the stability proof. You can read the discussion in the mailing list thread to see how close we got.

I recently learnt that Aaron Wishnick had completed the stability proof and had also published a number of useful coefficient formulas for the SVF.[4] Aaron kindly cited my partial proof. Thanks Aaron!

For future reference the maxima source file for my draft analysis is now archived on github.

[1] Bencina, R. (2013) “Time Varying BIBO Stability Analysis of Trapezoidal integrated optimised SVF v2,” music-dsp mailing list, Nov. 2013, Available at www.mail-archive.com/music-dsp@music.columbia.edu/msg02467.html

[2] Laroche, J. (2007) “On the Stability of Time-Varying Recursive Filters,” J. Audio Eng. Soc., vol. 55, no. 6, pp. 460-471, June 2007.

[3] Simper, A. (2013) “Solving the Continuous SVF equations Using Trapezoidal Integration and Equivalent Currents,” Nov. 2013, Available at www.cytomic.com/files/dsp/SvfLinearTrapOptimised2.pdf

[4] Wishnick, A. (2014) “Time-Varying Filters for Musical Applications” Proc. of the 17th Int. Conference on Digital Audio Effects (DAFx-14), Erlangen, Germany, September 1-5, 2014. Available at www.dafx14.fau.de/papers/dafx14_aaron_wishnick_time_varying_filters_fo r_.pdf

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Discretization of an RC Lowpass Filter

Over recent years Andy Simper has been posting novel digital audio filters derived from analog circuits using standard numerical techniques. I am a neophyte at numerical circuit modelling, so it was no surprise that I found some of Andy’s maths a bit terse and difficult to digest. In an effort to understand the basics I wrote up my own “pre-digested” version of the steps required to take an RC lowpass filter circuit and apply trapezoidal integration to generate the source code to model it (approximately).

You can read my article “Discretization of an RC Lowpass Filter” here [pdf].

Or go read Andy’s articles.

The source .tex file for my article is on github. Please feel free to post bugs, comments or corrections there, or send me an email.

Many thanks to Andy and the kind folks on #musicdsp for helping me through this.