Hi Andrew/Dominque, The DK-Method is a systematic way to implement on automatic systems the much more breathru approach called KMethod, who was (how fun) discovered by my professor in the late 1990s. I cant either tell the all the story (it would require a lot of time because it's a well defined and robust theory analogous to the state space one), but if you look at Yeh's work you can have an idea. The (D)KMethod is a generalization/extension of the state space ABCD approach to analog systems. Vadim's and Andrew are basically the same thing and the inversion is hidden in the calculation of the coeffs and also takes benefit of the order 2 size of matrix A (which is very simple to invert). What I want to point out is that it is an intrinsic state space matrix formulation where time variance of the system demands a matrix inversion either at audio or control rate, almost the same how you convert on Matlab via bilinear() the analog ABCD matrix in the equivalent digital one (via bilinear). Beware that using statespace for timevarying systems is better but not the best. In ANY system of course you preserve the values of the state variables (which is good compared to TF) but that doesnt mean that you wont have artifacts or transients at all. There is also a lot of good work made by the finnish guys (Valimaki et al) about the usage of the so called "transient suppressors".
Without telling too much (sorry I cant :) ) if I have time I will show the similarity and the matrix inversion problem analyzing the SVF via a statespace approach similar to Andrew's. I am unfortunately fully loaded of work, but as I get some free time I will try to publish a pdf. My 0.02EUR >;-) Marco -----Messaggio originale----- Da: music-dsp-boun...@music.columbia.edu [mailto:music-dsp-boun...@music.columbia.edu] Per conto di Dominique Würtz Inviato: domenica 10 novembre 2013 11:13 A: A discussion list for music-related DSP Oggetto: Re: [music-dsp] R: Trapezoidal integrated optimised SVF v2 Am Freitag, den 08.11.2013, 11:03 +0100 schrieb Marco Lo Monaco: > Being in the linear modeling field, I would rather have analized the > filter in the classic virtual analog way, reaching an s-domain > transfer function which has the main advantage that is ready to many > discretization > techniques: bilinear (trapezoidal), euler back/fwd, but also multi > step like AdamsMoulton etc. Once you have the s-domain TF you just > need to push in s the correct formula involving z and simplify the new > H(z) which is read to be implemented in DF1/2. I think a crucial point is that besides replicating steady state response of your analog system, you also want to preserve the time-varying behavior (modulating cutoff frequency) in digital domain. To achieve the latter, your digital system must use a state space representation equivalent to the original circuit, or, how Vadim puts it, "preserve the topology". By starting from an s-TF, however, all this information is lost. This is in particular visible from the fact that implementing different direct forms yields different modulation behavior. BTW, in case you all aren't aware: a work probably relevant to this discussion is the thesis of David Yeh found here: https://ccrma.stanford.edu/~dtyeh/papers/pubs.html When digging through it, in particular the so-called "DK method", you will find many familiar concepts incorporated in a more systematic and general way of discretizing circuits, including nonlinear ones. Can't say how novel all this really is, still it's an interesting read anyway. Dominique -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
