# Re: [music-dsp] Time Varying BIBO Stability Analysis of Trapezoidal integrated optimised SVF v2

```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
```
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
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