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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