With reference to my previous message:
It looks like there is a change of basis matrix T that can be used to
satisfy Laroche's Criterion 2 (time varying BIBO stability at full audio
rate), at least for k > 0.
T:
[ 0, 1 ]
[ 1, -1/10000 ]
This matrix requires k > 1/10000 but it seems that the lower bound on k
can approaches zero as the 2,2 entry approaches zero from below.
Hopefully I'm not imagining things.
Ross.
On 11/11/2013 2:58 AM, Ross Bencina 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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