Thanks very much Ross for taking the time to look at this! There is a lot of reading and theory so until I get some more time I personally can't really take on board any of it to provide you with any useful comments, but I appreciate your time.
All the best, Andy -- cytomic - sound music software On 10 November 2013 23:58, Ross Bencina <rossb-li...@audiomulch.com> wrote: > Hi Everyone, > > I took a stab at converting Andrew's SVF derivation [1] to a state space > representation and followed Laroche's paper to perform a time varying BIBO > stability analysis [2]. Please feel free to review and give feedback. I > only started learning Linear Algebra recently. > > Here's a slightly formatted html file: > > http://www.rossbencina.com/static/junk/SimperSVF_BIBO_Analysis.html > > And the corresponding Maxima worksheet: > > http://www.rossbencina.com/static/junk/SimperSVF_BIBO_Analysis.wxm > > I had to prove a number of the inequalities by cut and paste to Wolfram > Alpha, if anyone knows how to coax Maxima into proving the inequalities I'm > all ears. Perhaps there are some shortcuts to inequalities on rational > functions that I'm not aware of. Anyway... > > The state matrix X: > > [ic1eq] > [ic2eq] > > The state transition matrix P: > > [-(g*k+g^2-1)/(g*k+g^2+1), -(2*g)/(g*k+g^2+1) ] > [(2*g)/(g*k+g^2+1), (g*k-g^2+1)/(g*k+g^2+1)] > > (g > 0, k > 0 <= 2) > > Laroche's method proposes two time varying stability criteria both using > the induced Euclidian (p2?) norm of the state transition matrix: > > Either: > > Criterion 1: norm(P) < 1 for all possible state transition matrices. > > Or: > > Criterion 2: norm(TPT^-1) < 1 for all possible state transition matrices, > for some fixed constant change of basis matrix T. > > norm(P) can be computed as the maximum singular value or the positive > square root of the maximum eigenvalue of P.transpose(P). I've taken a > shortcut and not taken square roots since we're testing for norm(P) > strictly less than 1 and the square root doesn't change that. > > From what I can tell norm(P) is 1, so the trapezoidal SVF filter fails to > meet Criterion 1. > > The problem with Criterion 2 is that Laroche doesn't tell you how to find > the change of basis matrix T. I don't know enough about SVD, induced p2 > norm or eigenvalues of P.P' to know whether it would even be possible to > cook up a T that will reduce norm(P) for all possible transition matrices. > Is it even possible to reduce the norm of a unit-norm matrix by changing > basis? > > From reading Laroche's paper it's not really clear whether there is any > way to prove Criterion 2 for a norm-1 matrix. He kind-of side steps the > issue with the norm=1 Normalized Ladder and ends up proving that > norm(P^2)<1. This means that the Normalized Ladder is time-varying BIBO > stable for parameter update every second sample. > > Using Laroche's method I was able to show that Andrew's trapezoidal SVF > (state transition matrix P above) is also BIBO stable for parameter update > every second sample. This is the final second of the linked file above. > > If anyone has any further insights on Criterion 2 (is it possible that T > could exist?) I'd be really interested to hear about it. > > Constructive feedback welcome :) > > Thanks, > > Ross > > > [1] Andrew Simper trapazoidal integrated SVF v2 > http://www.cytomic.com/files/dsp/SvfLinearTrapOptimised2.pdf > > [2] On the Stability of Time-Varying Recursive Filters > http://www.aes.org/e-lib/browse.cfm?elib=14168 > -- > 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
