[music-dsp] R: R: R: Trapezoidal integrated optimised SVF v2

```Hi Andrew,
you misinterpreted my words :) I know you are not intentionally hiding
anything.
The computation are intrinsically hidden themselves because of the nature of
matrixes). All the coeffs involved in input, output and state vars belong to
some matrix, which is why I say it's a statespace-like algo.

AFAIK the transient suppressor technique is a nice analysis of the problem
and suggest a fix (btw which is expensive). I am not aware of anything that
worked fine to keep a a given filter timevarying with theoretically no
artifacts, even if I am working on something about recently (just got some
ideas and I cant wait to test them).
Laroche suggested an analysis to understand if a filter is BIBO stable
during transients, but I remember in his paper he says that's not the
glitch-free problem test (of course).

M.

-Messaggio originale-
Da: music-dsp-boun...@music.columbia.edu
[mailto:music-dsp-boun...@music.columbia.edu] Per conto di Andrew Simper
Inviato: domenica 10 novembre 2013 13:18
A: A discussion list for music-related DSP
Oggetto: Re: [music-dsp] R: R: Trapezoidal integrated optimised SVF v2

On 10 November 2013 18:43, Marco Lo Monaco marco.lomon...@teletu.it wrote:
if you look at Yeh's work you can have an idea. The (D)KMethod is a
generalization/extension of the state space ABCD approach to analog
systems.
Vadim's and Andrew are basically the same thing and the inversion is
hidden in the calculation of the coeffs and also takes benefit of the
order 2 size of matrix A (which is very simple to invert).

I didn't mean to hide anything from you :) I have mentioned MNA or modified
nodal analysis, and in all the links to qucs it shows how to add entries to
the matrices involved in the solution of the circuit equations. Here is a
direct link to the MNA matrix formulation:

http://qucs.sourceforge.net/tech/node14.html

There is also a lot of good work made by the finnish guys (Valimaki et
al) about the usage of the so called transient suppressors.

Transient suppressors just screams to me of an underlying problem that
should be fixed.

Without telling too much (sorry I cant :) ) if I have time I will show
the similarity and the matrix inversion problem analyzing the SVF via
a statespace approach similar to Andrew's. I am unfortunately fully
loaded of work, but as I get some free time I will try to publish a pdf.

My 0.02EUR ;-)

Marco

No rush on any of this, whenever you get a chance it would be appreciated.

All the best,

Andy
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[music-dsp] Analog versus digital systems

```

Hi all,

Of course I'm aware of it this work probably won't give m a (bit late)
YUP existence in SanFrancisco or a well paid Berkeley professorship that
I like, but at least I don't really run the risk of looking like a
dumb-*ss when playing the unpaid professor a bit in this territory, and
hopefully cut down some Non-Giant Redwood trees that appear to create
more pollution than oxygen.

First, a repeat of what I've tried to communicate a number of times, as
it were to discourage the idea of taking interesting mathematical truths
from (lame or interesting) digital signal processing effects in music,
let's first consider the theoretical basics that defy all ignoration:

(A)
{Digital System} -- {Digital Sample stream} -- {Digital to Analog}
-- {Analog signal}

, versus:

(B)
{Analog System} -- {Analog Signal}

The main differences are on a short list:

(1) The only way in which the two analog output signals of graph (A)
and (B) are going to be (almost or perfectly) the same is when somehow
the digital system creates very well made samples (which *CAN* come from
simply playing back accurately sampled form of some frequency limited
signal), and the Digital to Analog convertor is very high quality (or to
achieve actual mathematical perfection: is a perfect reconstruction type)
(2) The digital system implemented as a filtering of any kind of
combination of FIR/IIR tap-connections is normally not coming close to
making analog-equivalent signals, by far, unless it is big, and there
explicit measures being taken (extremely high sampling frequencies and
vertical resolution, tuning of the DA-convertors always-present
transient behavior, medium long averaging effects control (hard problem)
seriously long sinc-based integral corrections (computationally intensive)).
(3) Reconstructing an analog signal from samples that isn't a
retarded subset of all possible signals, will require a DA convertor
design which has a serious signal delay, for all known normal and
industrial Audio convertors. So to prevent some very measurable (by over
see-ably simple traditional measurement techniques at the level of the
THD of a very moderate transistor radio) distortion, serious measures
would have to be taken, like outside the scope of this list. Even making
sure those distortions don't become multi-fold ugly and even a potential
danger to the hearing of the customers isn't easy (and thus far never
has been discussed, even though these distortions are almost incredibly
ugly, and host of unrealistic monitors have been invented which are
supposed to smooth some of this over, apparently through lack of
awareness of the impossibility to approach per-sample sinc functions by
any resonance or other mechanical or switching amp trick).
(4) It is quite possible to create a computer simulation of an
electronics circuit, like a Moog filter, even with serious accuracy, and
to state the output of such simulation in the form of a sequence of
equidistant digital samples with accurate vertical quantization. Even
this does not preclude you from having to take equal relevant care of
the above, except for point (2).

There, that's a few New things, apparently for those not blessed with
either the intelligence, means or geographical or time opportunity to
follow a good EE university (or for most of this: bachelor level)
Sophomore year equivalent.

Of course going a bit further in the better EE education (say second or
third year of a serious education), you may want to practice yourself in
creating computer models of interesting non-linear electronics circuits,
and see if you computer simulations on the basis of these models and
some form of circuit-to-signal strategy, be it based on the frequency
domain or not, turn out to be accurate, and maybe invent some fun games
with this, like a Virtual Prophet-5 that everybody can run on their

I had done some (extremely low budget) preparatory work because of my
much longer standing personal interests for this (like owning various
synthesizers and samplers with digital filters like the TG500 in the
80s), see eg  http://theover.tripod.com/so1.html  and
http://theover.tripod.com/switch.html  , written before the year 1999.

Ir. T. Verelst
http://www.theover.org/Synth

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Re: [music-dsp] R: R: Trapezoidal integrated optimised SVF v2

```

Urs, I don't know if you're referring to music-dsp here, but this list
is specifically not meant to be in the academic realm, but rather a
place where people of all sorts with an interest in music and digital
signal processing can chat. So I encourage you to share your results here!

I'm sorry there has been unpleasantness on the list recently.

best,
douglas

On 11/10/13 10:00 AM, Urs Heckmann wrote:

We had planned to write a paper about our numerical method for the
non-linear case, including a pretty fast solving algorithm that's
fundamentally more precise than Newton-Raphson. But seeing how this
we might just share it in more practically oriented place (KVR dev
forum).

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Re: [music-dsp] Analog versus digital systems

```

Theo, please stop with the insults.

On 11/10/13 9:55 AM, Theo Verelst wrote:

Of course I'm aware of it this work probably won't give m a (bit late)
YUP existence in SanFrancisco or a well paid Berkeley professorship that
I like, but at least I don't really run the risk of looking like a
dumb-*ss when playing the unpaid professor a bit in this territory, and
hopefully cut down some Non-Giant Redwood trees that appear to create
more pollution than oxygen.

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Re: [music-dsp] Trapezoidal integrated optimised SVF v2

```
On 09.11.2013, at 01:57, Tom Duffy tdu...@tascam.com wrote:

To paraphrase, all are good enough if the frequency
is low ( pi/3); Simpsons is the best, but blows
up above 0.8 pi

Note that Andys approach uses trapezoidal integrators just as that -
integrators. While the whole structure of the SVF looks like a 2nd order
biquad, it isn't one. The z-1 sections are all in branches parallel to the
signal path.

The blow up problem of classical digital implementations of SVFs (e.g.
Chamberlin) stems from the unit delay in the feedback path. This unit delay
isn't present in Andy's approach and thus there is no blow up problem
anywhere below Nyquist.

The obvious advantage of Andy's approach is the numerical accurateness. Unlike
biquads which require double precision floating processing from 2nd order and
up, Andys filters don't suffer much from rounding errors. This should be
obvious because the biquad filter coefficients (if you really want to call
them that - I really prefer Andy's term) of the integrators are 1.0f

The thing about the recent approaches to eleminate the unit delay in the
feedback path is the vastly improved stability, the much closer match of phase
response towards analogue equivalents and the true independence of cutoff and
resonance that make these filters sweepable. The drawback of these approaches
is the higher computational effort, which has become less of an issue for
realtime processing on desktop computers in recent years.

- Urs
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Re: [music-dsp] Trapezoidal integrated optimised SVF v2

```Hi Douglas,

No worries, I couldn't have started my business without this list, it was an
important factor in my career - for the very reason that it isn't strictly
academic. I would love to give back if it's of anybody's help.

(if we ever find the time to write that damn paper ;-)

Thanks,

- Urs

On 10.11.2013, at 16:03, douglas repetto doug...@music.columbia.edu wrote:

Urs, I don't know if you're referring to music-dsp here, but this list is
specifically not meant to be in the academic realm, but rather a place where
people of all sorts with an interest in music and digital signal processing
can chat. So I encourage you to share your results here!

I'm sorry there has been unpleasantness on the list recently.

best,
douglas

On 11/10/13 10:00 AM, Urs Heckmann wrote:
We had planned to write a paper about our numerical method for the
non-linear case, including a pretty fast solving algorithm that's
fundamentally more precise than Newton-Raphson. But seeing how this
we might just share it in more practically oriented place (KVR dev
forum).

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

```
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, -1/1 ]

This matrix requires k  1/1 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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Re: [music-dsp] Analog versus digital systems

```i don't want to feed the troll, but...

a) of course we all know about the sampling theorem and sync
interpolation. truly!

b) though i enjoy reading ASM synth code, i don't see anything here
that is interesting. what am i supposed to look at? a character
display? a phasor? an in-place filter or chromatic tuning table in
BASIC? confused.

c) why are you posting a schematic of a switched capacitor filter to
music-dsp? are you obliquely proposing a direct-modelling algorithm
for this kind of circuit? thats an interesting thought but i'd love to
see more elaboration, and what kind of advantage it would have over
FIR/IIR structures.

my degrees are in music, not EE, so actually i can often use a little
help connecting theoretical dots and i certainly admit that. but i
know how to hold the pen, and i can see the picture when it's done, so
to speak.

i would like to read your posts in the future, in case they are
useful, but i'm no longer giving them the benefit of the doubt if they
start off with vitiriol and end with ancient links of dubious
relevance.

- ezra buchla ( in Berkeley CA )

On Sun, Nov 10, 2013 at 10:08 AM, douglas repetto
doug...@music.columbia.edu wrote:

Theo, please stop with the insults.

On 11/10/13 9:55 AM, Theo Verelst wrote:

Of course I'm aware of it this work probably won't give m a (bit late)
YUP existence in SanFrancisco or a well paid Berkeley professorship that
I like, but at least I don't really run the risk of looking like a
dumb-*ss when playing the unpaid professor a bit in this territory, and
hopefully cut down some Non-Giant Redwood trees that appear to create
more pollution than oxygen.

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Re: [music-dsp] Trapezoidal integrated optimised SVF v2

```as a longtime lurker, it's really nice to see this list heating up
again, and to see mr. simper's comments finally make it through.
cytomic is wonderful stuff and i for one certainly appreciate the
sharing.

there's a non-technical thread of discussion interleaved here, which
is about education and theoretical accuracy and so on. it's wearisome
and frequently insulting. creative technology has never advanced
solely by cleaving to strict numerical perfection, and will not do so
in the future. there are other fora for talking about DSP applications
in automotive or medical technology or other truly critical problem
domains. [or even AD/DA design, loudness limiting, other low-level and
certainly important problems.] but in music, we can afford to play

and i look forward to hearing any advice you have or things you've
learned, and i'd love to read that paper.

thanks again, and all the best

ezra buchla

On Sun, Nov 10, 2013 at 10:48 AM, Urs Heckmann u...@u-he.com wrote:
Hi Douglas,

No worries, I couldn't have started my business without this list, it was an
important factor in my career - for the very reason that it isn't strictly
academic. I would love to give back if it's of anybody's help.

(if we ever find the time to write that damn paper ;-)

Thanks,

- Urs

On 10.11.2013, at 16:03, douglas repetto doug...@music.columbia.edu wrote:

Urs, I don't know if you're referring to music-dsp here, but this list is
specifically not meant to be in the academic realm, but rather a place where
people of all sorts with an interest in music and digital signal processing
can chat. So I encourage you to share your results here!

I'm sorry there has been unpleasantness on the list recently.

best,
douglas

On 11/10/13 10:00 AM, Urs Heckmann wrote:
We had planned to write a paper about our numerical method for the
non-linear case, including a pretty fast solving algorithm that's
fundamentally more precise than Newton-Raphson. But seeing how this
we might just share it in more practically oriented place (KVR dev
forum).

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... http://music.columbia.edu/~douglas

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Re: [music-dsp] Analog versus digital systems (Ezra Buchla)

```...
a) of course we all know about the sampling theorem and sync
interpolation. truly!

b) though i enjoy reading ASM synth code, i don't see anything here
that is interesting.  ...

I'm sorry to say, but while of course I don't feel all too much of it, and of
course that isn't a reason to use my free speech necessarily for placing a
correction, but that little snippet is quite insulting, given the story thus
far. So outside of rethoric, that is a technical/scientific insult of the first
order that you're trying to force my direction.

I don't really take the insult, and am glad there's serious discussion, and
people feeling inspired to share maxima code, etc. and apparently not
overwhelmed or something, so they post about what interests them, so on the

I don't feel like scientifically defending the quotes I few simple quotes I
posted.

As a serious remark about the content of many of the musical and signal
processing subjects: it's a great idea to use well known *analog* synthesizer
designs as the basis for (partial) digital simulation, which I though already
before people like Dave Smith were writing award winning software to that
effect, and which interested me long before the advent of a number of software
companies that occupy themselves with the subject. As the suggestion is from
some of my quotes, it would be good to have a potent, 64 bit circuit simulator
which allows audio output, and explicit (parts with curves for parameter
changes) or implicit (driven sources in the network, OTAs in replacement
circuits, etc) time dependencies, possibilities for storing/continuing network
states, and a choice of accuracy feedbacks that I've been hinting at, and which
clearly isn't understood by most, which doesn't make me continue.

Also, I've suggested signal improvements, but they won't work without some
fundamental changes to the most used algorithms, which I would prefer to be
applied to some musical software, preferably Open Source. Those things are very
audible, and it surprises me that people who may feel the need for improvements
are so numb. Must be some limited musicians trying to rule the show, which in
broader circles, which also can benefit from DSP for musical purposes is
getting in demand. Just saying. Sounds like a simple statement of truth to me,
and I think I'm qualified to judge that, so if you feel a bit humble, don't
confuse that with feeling insulted. I do feel slandered, regulaly, and that
*is8 a real issue for me, and the law.

T.V.

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Re: [music-dsp] Analog versus digital systems (Ezra Buchla)

```i'm sorry mr. vereslt, i do apologize for insulting you. to be honest,
i made a mistake with the e-mail and thought this was still under that
very interesting and informative to me (since i am not nearly at your
level as an engineer), it shows some fundamental aspects of
high-resolution sample reconstruction rendered in a highly-efficient
way ] - but it is not so directly relevant to coefficient changing
within the filter, is it?

if it is, can you put the relationship a little more plainly so that
dummies like me could understand?

for example, would you bandlimit an SVF coefficient change signal
using sinc or something? perhaps by offline-processing it and using
buffer playback?

is this stuff only relevant at 64 bits (high vertical resolution)?
what if we obtain much greater processing efficiency at 32?

switched capacitors is interesting to me too because i have been
thinking a lot about tunable sampling rates. this schematic itself is
old news though, is it the precise behavior of these integrators while
changing the clock rate that you wand to point out?

you see, i really respect your intelligence and erudition on this
stuff, so i woud love to hear more explanation, just without being so
angry. i have of course read the code on your website before now, but
still need to know more specifically what part you refer to.

i mean to try and starve troll in all of us, i am glad that your
pursue correctness aggressively, it is a benefit for all. and so
telling you that i'm from a certain area of the world that you make
fun of, and so is dave smith for that matter, and that it is easier to
pay attention when the tone is more civil. thank you for reminding me
of it as well.

sincerely, and thanks,
ezra b

On Sun, Nov 10, 2013 at 3:31 PM, Theo Verelst theo...@telfort.nl wrote:
...
a) of course we all know about the sampling theorem and sync
interpolation. truly!

b) though i enjoy reading ASM synth code, i don't see anything here
that is interesting.  ...

I'm sorry to say, but while of course I don't feel all too much of it, and of
course that isn't a reason to use my free speech necessarily for placing a
correction, but that little snippet is quite insulting, given the story thus
far. So outside of rethoric, that is a technical/scientific insult of the
first order that you're trying to force my direction.

I don't really take the insult, and am glad there's serious discussion, and
people feeling inspired to share maxima code, etc. and apparently not
overwhelmed or something, so they post about what interests them, so on the

I don't feel like scientifically defending the quotes I few simple quotes I
posted.

As a serious remark about the content of many of the musical and signal
processing subjects: it's a great idea to use well known *analog* synthesizer
designs as the basis for (partial) digital simulation, which I though already
before people like Dave Smith were writing award winning software to that
effect, and which interested me long before the advent of a number of
software companies that occupy themselves with the subject. As the suggestion
is from some of my quotes, it would be good to have a potent, 64 bit circuit
simulator which allows audio output, and explicit (parts with curves for
parameter changes) or implicit (driven sources in the network, OTAs in
replacement circuits, etc) time dependencies, possibilities for
storing/continuing network states, and a choice of accuracy feedbacks that
I've been hinting at, and which clearly isn't understood by most, which
doesn't make me continue.

Also, I've suggested signal improvements, but they won't work without some
fundamental changes to the most used algorithms, which I would prefer to be
applied to some musical software, preferably Open Source. Those things are
very audible, and it surprises me that people who may feel the need for
improvements are so numb. Must be some limited musicians trying to rule the
show, which in broader circles, which also can benefit from DSP for musical
purposes is getting in demand. Just saying. Sounds like a simple statement of
truth to me, and I think I'm qualified to judge that, so if you feel a bit
humble, don't confuse that with feeling insulted. I do feel slandered,
regulaly, and that *is8 a real issue for me, and the law.

T.V.

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Re: [music-dsp] Analog versus digital systems (Ezra Buchla)

```or, oh! would you suggest that we stick to an analog design like the
switched capacitors? but there are still lots of reasons to try and
keep everything on DSP in terms of part count in a device.

again, i think of this space as very informal discussion, s please no
hard feelings,

ezra b

On Sun, Nov 10, 2013 at 4:06 PM, Ezra Buchla ezra.buc...@gmail.com wrote:
i'm sorry mr. vereslt, i do apologize for insulting you. to be honest,
i made a mistake with the e-mail and thought this was still under that
very interesting and informative to me (since i am not nearly at your
level as an engineer), it shows some fundamental aspects of
high-resolution sample reconstruction rendered in a highly-efficient
way ] - but it is not so directly relevant to coefficient changing
within the filter, is it?

if it is, can you put the relationship a little more plainly so that
dummies like me could understand?

for example, would you bandlimit an SVF coefficient change signal
using sinc or something? perhaps by offline-processing it and using
buffer playback?

is this stuff only relevant at 64 bits (high vertical resolution)?
what if we obtain much greater processing efficiency at 32?

switched capacitors is interesting to me too because i have been
thinking a lot about tunable sampling rates. this schematic itself is
old news though, is it the precise behavior of these integrators while
changing the clock rate that you wand to point out?

you see, i really respect your intelligence and erudition on this
stuff, so i woud love to hear more explanation, just without being so
angry. i have of course read the code on your website before now, but
still need to know more specifically what part you refer to.

i mean to try and starve troll in all of us, i am glad that your
pursue correctness aggressively, it is a benefit for all. and so
telling you that i'm from a certain area of the world that you make
fun of, and so is dave smith for that matter, and that it is easier to
pay attention when the tone is more civil. thank you for reminding me
of it as well.

sincerely, and thanks,
ezra b

On Sun, Nov 10, 2013 at 3:31 PM, Theo Verelst theo...@telfort.nl wrote:
...
a) of course we all know about the sampling theorem and sync
interpolation. truly!

b) though i enjoy reading ASM synth code, i don't see anything here
that is interesting.  ...

I'm sorry to say, but while of course I don't feel all too much of it, and
of course that isn't a reason to use my free speech necessarily for placing
a correction, but that little snippet is quite insulting, given the story
thus far. So outside of rethoric, that is a technical/scientific insult of
the first order that you're trying to force my direction.

I don't really take the insult, and am glad there's serious discussion, and
people feeling inspired to share maxima code, etc. and apparently not
overwhelmed or something, so they post about what interests them, so on the

I don't feel like scientifically defending the quotes I few simple quotes I
posted.

As a serious remark about the content of many of the musical and signal
processing subjects: it's a great idea to use well known *analog*
synthesizer designs as the basis for (partial) digital simulation, which I
though already before people like Dave Smith were writing award winning
software to that effect, and which interested me long before the advent of a
number of software companies that occupy themselves with the subject. As the
suggestion is from some of my quotes, it would be good to have a potent, 64
bit circuit simulator which allows audio output, and explicit (parts with
curves for parameter changes) or implicit (driven sources in the network,
OTAs in replacement circuits, etc) time dependencies, possibilities for
storing/continuing network states, and a choice of accuracy feedbacks that
I've been hinting at, and which clearly isn't understood by most, which
doesn't make me continue.

Also, I've suggested signal improvements, but they won't work without some
fundamental changes to the most used algorithms, which I would prefer to be
applied to some musical software, preferably Open Source. Those things are
very audible, and it surprises me that people who may feel the need for
improvements are so numb. Must be some limited musicians trying to rule the
show, which in broader circles, which also can benefit from DSP for musical
purposes is getting in demand. Just saying. Sounds like a simple statement
of truth to me, and I think I'm qualified to judge that, so if you feel a
bit humble, don't confuse that with feeling insulted. I do feel slandered,
regulaly, and that *is8 a real issue for me, and the law.

T.V.

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Re: [music-dsp] R: Trapezoidal integrated optimised SVF v2

```
On 11/8/13 5:20 PM, Andrew Simper wrote:

On 8 November 2013 18:03, Marco Lo Monacomarco.lomon...@teletu.it  wrote:

...

biquad, you not only have to prewarp the cutoff f0 but also the Q. In such

Are you talking about bell filters here? For a low pass resonant
filter it is hard to warp the Q since there is no extra degree of
freedom to keep its gain down, so I'm not sure how prewarping the Q is
possible in this case, but I'd love to hear if it can be done.

all depends on how Q is defined (which gets a little iffy for the bell
filters).

for an analog BPF or notch, defining the bandwidth as the number of
octaves between the -3 dB bandedges, then Q and BW are directly related as

1/Q = 2 * sinh( ln(2)/2 * BW )

if you use the Bilinear Xform to convert analog to digital, you can
prewarp bandwidth (which otherwise gets squished as f0 gets above, say,
Fs/4) a little using this adjustment

1/Q = 2 * sinh( ln(2) * BW * pi*(f0/Fs)/sin(2*pi*(f0/Fs)) )

i like to maintain that same relationship between Q and BW even for
filters that are not BPF or notch.  right now, the only other filter
that i can think of that makes sense is the bell filter.  if you do
that, it appears that the Q prewarping looks like

1/Q --  2 * sinh( arcsinh(1/2Q) * 2*pi*(f0/Fs)/sin(2*pi*(f0/Fs)) )

cases I typically use the analog s-domain TF and then also compensate the Q
via the very famous RBJ cookbook (compute the analog Q and redesign the
digital biquad with the f0, Q and gain params). Compensating the Q is
important not only because you prevent the stretching as your cutoff reaches
Nyquist but also because it minimizes the same stretch at different sampling
frequency.

not stretching, but the opposite problem.

You can do all the same warping no matter if you go through the
laplace space or directly integrate the circuits, and it doesn't
matter what realisation you are using,

yes.

L8r,

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Re: [music-dsp] Trapezoidal integrated optimised SVF v2

```
On 11/8/13 6:47 PM, Andrew Simper wrote:

On 9 November 2013 08:57, Tom Duffytdu...@tascam.com  wrote:

Having worked with Direct-Form I filters for half of my
career, I've been glossing over this discussion as
not relevant to me.

It depends if you value numerical performance, cutoff accuracy, dc
performance etc etc, DF1 scores badly on all these fronts,

nope.

and this is even in the case where you keep your cutoff and q unchanged.

you can, for a lot fewer instructions than a lattice or a ladder or an
SVF, do a DF1 with noise shaping with a zero in the quantization noise
TF at z=1 that obliterates any DC error.  infinite S/N at f=0.  in a
fixed-point context, this DF1 with noise shaping (sometimes called
fraction saving), has *one* quantization point, not two, not three.

you can also rewrite equations to get rid of the cosine problem, which
is at the root of problems regarding cutoff accuracy.  you do it by
replacing, in your equations, every occurrence of cos() with this:

cos(2*pi*f0/Fs) --  1  -  2*( sin(pi*f0/Fs) )^2

as you can see, even if you have floating point, all of the information
concerning f0 is in the difference that cos() is from 1.  so, assuming
f0Fs, even floating point doesn't help.  all of the information
concerning f0 is in the mantissa bits that are falling offa the edge as
f0 gets lower and lower.  double precision floats *do* help out here,
but it's a numerical problem.  and all of the designs using
tan(pi*f0/Fs) have their own numerical problems regarding ranging, which
is why i would suggest to move away from using tan() as soon as possible

you'll get something a little different, but pretty strongly related to
the standard DF1.

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Re: [music-dsp] Trapezoidal integrated optimised SVF v2

```
On 11/8/13 11:50 PM, David Reaves wrote:

I think the distinction is that SCIENCE is open-minded.

Scientists, OTOH, are only open-minded if they choose to be.

But then, the closed-minded ones aren't really scientists, now are they? ;-)

oh, c'mon.  *anyone* can be close-minded.  being skilled in science or
any other discipline is no immunity for close-mindedness.

e.g. Richard Dawkins.

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Re: [music-dsp] R: R: Trapezoidal integrated optimised SVF v2

```

On 11/9/13 6:52 PM, Andrew Simper wrote:

Marco writes:

Well, of course the s = (T/2)(z-1)/(z+1) conversion comes from discretizing
a differential equation.

but remember, that's not the only way to convert s to z.  that's the
bilinear way to do it.

...

intrinsicly a biquad, you not only have to prewarp the cutoff fc but
also the Q. In such

Are you talking about bell filters here? For a low pass resonant filter it

is hard to warp the Q since there is no

extra degree of freedom to keep its gain down, so I'm not sure how

prewarping the Q is possible in this case, but I'd love to hear if it can be
done.

Not only, but wait I could be wrong on this. I always took RBJ cookbook as a
bible and he doesn't really say that the Q cant be prewarped for LPF/HPF
starting from the analog Q. Maybe RBJ can correct me :)

frequency warping is a consequence of using BLT.  the cool thing about
BLT that other methods that convert s to z do not have is that it maps
the j*omega axis in the s-plane to the unit circle in the z-plane,
similarly to how the *exact* transformation does (which is z =
e^(s/Fs)).  and the frequency mapping is strictly increasing and close
to the identity function for fFs.  so every bump, every feature that
you see in the analog frequency response has a corresponding bump in the
digital filter frequency response.  but slightly (at low f) moved in
frequency.  at higher frequencies the effect is much more pronounced,
which is why compensating for frequency warping (what we call
pre-warping) is necessary.

it is an effect on *frequency*, not on amplitude.  if you have a 3 dB
bump in the analog frequency response, there will be a 3 dB bump in the
BLT-designed digital filter, but that bump might not be at the same
place.  so that's why the cookbook says nothing about prewarping Q.  i
am assuming that for a LPF, if you want to design for a resonant lip
or bump of 1 dB, you want the same 1 dB bump in the digital filter.  so
no compensation of that parameter.

but bandwidth is different.  bandwidth is about frequency or log of
frequency or something else (bark scale??).  for every degree of freedom
you have in the frequency anchor points, you can apply pre-warping and
independently compensate each one regarding their relocation done by
BLT.  that's why it's appropriate to pre-warp bandwidth when using the
BLT to convert s to z.

I would love it if you are right. If RBJ can pull it off I would love
to see how as it would be ultra ultra useful!

pull exactly *what* off?

I am however loath to use the dreaded words delay-free since they
are pretty meaningless. Did you know that every single of the famous
RBJ audio eq cookbook filters are delay-free? Yes, that's right,
everyone using RBJ cookbook eqs and low pass filters can now happily
slap on their software the hollow marketing term featuring delay-free
filter technology!

i have absolutely no idea what this is about.

So delay-free is a pointless expression to me,

it has been used to discuss or advertise delay-free feedback which, to
me, still remains an impossibility for discrete-time systems.

and i've seen the papers.  when it all boils down to it in a real-time
filter, you are defining your current output sample in terms of the
current input sample and previous input samples and, if it's recursive,
the previous output samples.  but you cannot define your current output
sample in terms of the current output sample.

a more meaningful
thing to mention could possibly be that (non-linear) implicit
functions are being solved. For example something that has a
non-linear voltage to current mapping, or something that solves for
non-linearities placed in feedback loops, anything else is just linear
algebra that can be solved in a single step. But even then who cares
as long as it sounds good?

yup.  and at Fs=96kHz, you can make it sound very good.  and that
single-sample delay is half of what it would be at 48 kHz.

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Re: [music-dsp] R: R: Trapezoidal integrated optimised SVF v2

```
On 11.11.2013, at 01:33, robert bristow-johnson r...@audioimagination.com
wrote:

but you cannot define your current output sample in terms of the current
output sample.

But that, with all due respect, is what has been done for quite a while. It
isn't the major ingredient of great sound, but it arguably has its perks,
albeit cpu smoking ones.

I agree with Andy though that the main advantage of preserving the topology is
the possibility to insert the non-linearities at the right point. When using
BLT, how would you insert the non-linear effects of a diode or an OpAmp into
your model, where these things are desirable for a musical result?

- Urs
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Re: [music-dsp] R: R: Trapezoidal integrated optimised SVF v2

```
On 11/10/13 5:12 PM, Urs Heckmann wrote:

On 11.11.2013, at 01:33, robert bristow-johnsonr...@audioimagination.com
wrote:

but you cannot define your current output sample in terms of the current output
sample.

But that, with all due respect, is what has been done for quite a while.

it's been reported or *reputed* to be done for quite a while.

but when the smoke and dust clear, logic still prevails.

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Re: [music-dsp] R: R: Trapezoidal integrated optimised SVF v2

```i hope it's ok if i sum up the main themes here with background links,
as i understand it. the thread is just such a mess...

here is the state variable filter from hal chamberlin 1985 or so:
http://t.co/SVJp7iAgqb

pierre dutilleux wrote up a nice discretized and tuned version in dafx 1998:
http://iua.upf.edu/dafx98/papers/
presumably drawing on (it can be shown that...) this much earlier
AES paper, which i have not purchased
https://secure.aes.org/forum/pubs/conventions/?elib=5937

andrew simper's suggestion is to integrate the capacitor functions
with trapezoidal approximation, allowing for some refactoring and
simplication of the dif. eq.s and  faster code.
http://cytomic.com/files/dsp/SvfLinearTrapOptimised.pdf

there seems to be some concern about distortion introduced by the
trapezoidal integration. i've tried the algo in both fixed 32 ands
float, and it seems to sound and look ok to but i have not done a
proper analysis either numerically or physically. has anyone else?

theo vereslt seems to think the distortion would be unacceptable, and
so i tend to believe that in a really high-fidelity environment it
could be an issue. additionally, he seems to imply (maybe? i'm not
sure) that not only is the optimization not worth it, but you can't
adequately suppress artifacts during coefficient change at all (which
i've always understood as the main raison for the digital SVF) without
analog components (referenced switched caps as variable R's, an
interesting trick), or more bandlimiting somewhere, or something.

the KeepTopology paper by vadim zavalishin (http://t.co/SVJp7iAgqb)
proposes modelling the SVF with digital integrators. it seems like
these would follow the behaviors of the caps pretty closely and be
amenable to parameter change at the cost of some expense of course...
wouldn't that satisfy the purists? this last exchange seems to
indicate not... but i'm not sure why. again i should just try this
myself but it's a bit more work! ha.

thank you all for the discussion and sorry for my noise

ezra b
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Re: [music-dsp] R: R: Trapezoidal integrated optimised SVF v2

```
Hi Ezra,

On 11/11/2013 3:19 PM, Ezra Buchla wrote:

there seems to be some concern about distortion introduced by the
trapezoidal integration. i've tried the algo in both fixed 32 ands
float, and it seems to sound and look ok to but i have not done a
proper analysis either numerically or physically. has anyone else?

All single-step digital integration procedures will necessarily diverge

Here's a graph of the amplitude response of various integrators:

Trapezoidal is good because it is stable but as you can see it has
high-frequency roll-off compared to some others. As noted earlier
Simpsons rule is not usable if you want your cutoff to go near Nyquist
but is more accurate up to pi/4, which means that if you're using 2x
oversampling.

Trapezoidal is equivalent to the warping introduced by BLT (bilnear
transform) I believe. One way to think about it is that a lowpass filter
in the analog domain has gain 0 at infinite frequency, and infinite
frequency maps to Nyquist after BLT/Trapezoidal -- so you will get some
warping.

theo vereslt seems to think the distortion would be unacceptable, and
so i tend to believe that in a really high-fidelity environment it
could be an issue.

The usual suggestion is to oversample by 2x. Many implementations do
this and it has been discussed here many times over the years. It is
possible to do other tricks to match the amplitude response at 1x but
the phase response can get screwed up and you really want something
close to analog for both amplitude and phase responses.

My take is that for synthesis it's all about your aesthetic -- if you
want it to sound like an analog synth that's different from if you want
it to sound good. There's whole music genres based on commodore 64 SID
chips or destorted noise.. so anything goes really.

additionally, he seems to imply (maybe? i'm not
sure) that not only is the optimization not worth it, but you can't
adequately suppress artifacts during coefficient change at all (which
i've always understood as the main raison for the digital SVF) without
analog components (referenced switched caps as variable R's, an
interesting trick), or more bandlimiting somewhere, or something.

I think the main point in the current discussion is that if you want to
do *audio rate* modulation (think filter FM) then most tricks won't
save you. You need a filter that can be modulated at audio rate and
doesn't introduce spurious artifacts. Of course audio-rate FM will
alias, but that's a separate issue that can be addressed by oversampling
etc.

the KeepTopology paper by vadim zavalishin (http://t.co/SVJp7iAgqb)
proposes modelling the SVF with digital integrators. it seems like
these would follow the behaviors of the caps pretty closely and be
amenable to parameter change at the cost of some expense of course...'

All filters use digital integrators. There's just a lot of different
types of integrators.

wouldn't that satisfy the purists? this last exchange seems to
indicate not... but i'm not sure why. again i should just try this
myself but it's a bit more work! ha.

I can recommend the following book as a gentle and no-hype low-math
introduction to numerical integration. Written by a guy who cut his
teeth computing trajectories on the Apollo project:

Math Toolkit for real-time programming
Jack W. Crenshaw
CMP Books.

Cheers,

Ross.

thank you all for the discussion and sorry for my noise

ezra b
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Re: [music-dsp] R: R: Trapezoidal integrated optimised SVF v2

```
On 11/11/2013 12:21 PM, robert bristow-johnson wrote:

but you cannot define your current output sample in terms of the
current output sample.

But that, with all due respect, is what has been done for quite a while.

it's been reported or *reputed* to be done for quite a while.

but when the smoke and dust clear, logic still prevails.

I presume Urs (hi Urs!) is talking about using implicit solvers. Which
some people have been using for a while.

I guess it depends how you define current output sample. Is it the
trial output sample that you're refining, or the sample you actually output?

Seems logic could go either way.

Ross.
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