>well, that's what i mean. the Laroche point is that they might go crazy of coefficients are being continually modulated.
Yes indeed. In practice, I have found that theoretical guarantees of time-varying stability are not a whole lot of use, and that one is generally left with some combination of smoothing of the coefficient changes (possibly in some reparameterized domain), and output crossfade. Even with a guarantee of stability, you can still have large, ugly (but finite!) artifacts. But if you are smoothing/crossfading sufficiently that you don't get artifacts, you don't need to worry about stability separately. Probably the most interesting case here is modeling of analog filters that are designed to allow operation in an unstable range in the first place. Then we *want* the instability, but not the transition artifacts! >but, if in experiment A we have a stable BIBO DF1 filter with zero states, and >in other experiment we have the same filter (with same coefficients), i can with >only two samples x[0] and x[1], steer y[2] and y[3], to be whatever finite values >you specify. so that guy's question: *How do I check for stability of a filter with >non-zero initial conditions?* is superfluous, in my opinion. Ah, right, now I see where you were going with that. Indeed, that question is kind of beside the point. Although it seemed like that stack exchange post was more about the guy convincing himself of what he intuited to be true, than answering any really salient questions. E On Tue, Feb 3, 2015 at 5:13 PM, robert bristow-johnson < r...@audioimagination.com> wrote: > On 2/3/15 7:07 PM, Ethan Duni wrote: > >> well, the output states, y[n-1] and y[n-2], will change if coefs change. >>> >> No, those have already been computed and (presumably) output at that >> point. >> It's true that these states won't match what they would have been if you'd >> been running the new coefficients from the beginning of time, but of >> course >> they aren't supposed to in this context. >> > > well, that's what i mean. the Laroche point is that they might go crazy > of coefficients are being continually modulated. > > The point is that they retain their (pseudo)physical meaning ala the >> difference equations in a Direct Form I filter - you're still simply >> taking >> inputs and outputs and doing linear combinations with your coefficients, >> > > i know, but y[n] could blow up, in some cases, if coefs are being > continually modulated. > > i think you can model the initial conditions of the states at time n=0 as >>> >> a function of x[n-1] and x[n-2]. >> >> The states are a function of *all* previous inputs, generally. >> >> > but, if in experiment A we have a stable BIBO DF1 filter with zero states, > and in other experiment we have the same filter (with same coefficients), i > can with only two samples x[0] and x[1], steer y[2] and y[3], to be > whatever finite values you specify. so that guy's question: *How do I > check for stability of a filter with non-zero initial conditions?* is > superfluous, in my opinion. > >> so, if the thing is stable for zero initial states, it's stable for >>> >> arbitrary finite initial states. it's just super-position. >> >> I think that what you intend to say here is essentially correct, but we >> have to be careful how we define "stable" in this context. I.e., it is >> true >> that a given BIBO-stable LTI filter with zero state is also stable for any >> particular (finite) initial state you care to start it with (modulo >> numerical effects). Which is not to say that it won't produce a really bad >> transient artifact (i.e., much louder than the desired output signal). >> > > oh, yeah. and if these fictitious x[0] and x[1] required to get the > feedback states set to arbitrary values are huge, one can expect a filtered > click. > > But if we think about doing this continuously, it's easy to see how those >> huge switching transients could keep getting amplified and amplified on >> each coefficient change. >> > yup. i agree. i thought that's what Jean Laroche was writing about. > like what if you modulate your filter with some sinusoidal vibrato. that > could send it to hell and different forms (like SVF or Ladder or > DF-whatever) will behave differently. > > So the time-variant system isn't BIBO stable any >> more. Although in my view even one such big switching transient is >> unacceptable anyway, so stability is kind of beside the point. >> >> BTW, i think the Normalized Ladder has states that are both have L2 norms >>> >> that are the same as the input L2 norm. >> >>> they preserve r.m.s amplitude (or energy). no states blowing up once the >>> >> coefs stop changing. >> >> Yeah IIRC the state update is basically a rotation? >> > yup. > > These stability >> guarantees are what you get for the extra computational cost of normalized >> ladder topology. >> > > yup. another advantage is that the resonant frequency is determined fully > by the k1 coefficient. same for lattice. same for SVF. > > L8r, > > -- > > r b-j r...@audioimagination.com > > "Imagination is more important than knowledge." > > > > -- > 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
