I couldnt wait to put my hands on it :) (the creativity/curiosity spark is
Here is my MuPad notebook :
I basically demonstrate what I already said in my previous posts.
The standard state-space approach leads to identical results to your
algorithm, I would say even without the trick of the TPT, because of course
we are talking about an instantaneous _linear_ feedback.
That makes sense: my ABCD statespace matrixes are identical to yours,
same thing applies for CPU load in terms of MULs/ADDs.
Of course the main purpose of my analysis was to keep in mind that you will
_always_ have to deal with an implicit/hidden inversion of a matrix A of
the analog system (actually (I-A*h/2)) of the same order of your system,
which in this SVF fortunate case is 2 and also easy. For a moment think
about an analog system who has 10 capacitors and you want it to change it at
audio rate: you will have to deal with a0,a1,a2... coeffs that will be
rationaly polynomials, meaning lots of DIVs at runtime). Generally speaking,
inverting a matrix at audio rate is not a good idea :)
As I already said, we agree that these approaches are dated a long ago.
Ross showed the same results of matrix A,B (he didn't showed C,D because
they are not needed for the Laroche BIBO analysis). His results match my
Sorry if I omitted some code in collecting state/in/out variables in your
original algo, but I already got 6 pages of pdf this way and I thought it
would have been a good idea to keep it simple.
Hope to have helped you clear my my point of view.
[mailto:music-dsp-boun...@music.columbia.edu] Per conto di Ross Bencina
Inviato: domenica 10 novembre 2013 16:58
A: A discussion list for music-related DSP
Oggetto: [music-dsp] Time Varying BIBO Stability Analysis of Trapezoidal
integrated optimised SVF v2
I took a stab at converting Andrew's SVF derivation  to a state space
representation and followed Laroche's paper to perform a time varying BIBO
stability analysis . Please feel free to review and give feedback. I only
started learning Linear Algebra recently.
Here's a slightly formatted html file:
And the corresponding Maxima worksheet:
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:
The state transition matrix P:
[-(g*k+g^2-1)/(g*k+g^2+1), -(2*g)/(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:
Criterion 1: norm(P) 1 for all possible state transition matrices.
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 :)
 Andrew Simper trapazoidal integrated SVF v2
 On the Stability of Time-Varying Recursive Filters