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