Paper on bandlimited analog waveform synthesis using FM

[Stephen Blinkhorn-2]

Hi Peter, I just found an old thread on Music-DSP discussing a paper  
you wrote regarding synthesizing sawtooth wave forms via FM.  I'd love  
to read it.  Is it available anywhere?

Thanks, Stephen.
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[Victor Lazzarini]

I also just checked out that thread, fascinating discussion on the
perceptual
merits of the method!

On that matter, we have developed a method to generate sawtooth &
other typical 'analogue' waves which is also (approximately) bandlimited
(very low aliasing, -90 dB) using what we call "modified FM". This is
 a version of FM whose expansion is based on modified Bessels
(which is not exactly FM anymore, but damn it, we like the name
and it can be traced back to the original FM/PM easily :).
The advantage of this is that modified Bessels, unlike their
cousins, provide a better behaved spectrum with regard to changes in the
index of modulation. Besides, we can get a very good sawtooth, that
not only sounds it, but looks like it too in the time-domain.

If you are coming to DAFx in Finland, you can check it out there,
otherwise, it'll be in the online proceedings.

Victor

----- Original Message -----
From: "Stephen Blinkhorn" <stephen.blinkhorn@...>
To: <music-dsp@...>
Sent: Saturday, May 31, 2008 3:48 AM
Subject: [music-dsp] Paper on bandlimited analog waveform synthesis using FM

> Hi Peter, I just found an old thread on Music-DSP discussing a paper  you
> wrote regarding synthesizing sawtooth wave forms via FM.  I'd love  to
> read it.  Is it available anywhere?
>
> Thanks, Stephen.
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[Peter Schoffhauzer]

Hi,

You can find it here:
http://scp.web.elte.hu/papers/synthesis1.pdf

Best,
Peter


Stephen Blinkhorn wrote:

> Hi Peter, I just found an old thread on Music-DSP discussing a paper
> you wrote regarding synthesizing sawtooth wave forms via FM.  I'd love
> to read it.  Is it available anywhere?
>
> Thanks, Stephen.
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[robert bristow-johnson]

> ----- Original Message -----
> From: "Peter Schoffhauzer" <scoofy@...>
> To: "A discussion list for music-related DSP" <music-dsp@...>
> Subject: Re: [music-dsp] Paper on bandlimited analog waveform synthesis using FM
> Date: Sun, 01 Jun 2008 06:20:48 +0200
>
>
> You can find it here:
> http://scp.web.elte.hu/papers/synthesis1.pdf
>

Peter, thank you for posting this.  i'm still looking at it, but can i ask you questions about it a little later?  either here or privately, i don't care.

the first thing i would say about it after first glance is that i wish the spectrums and spectrografs had their axis labeled (in terms of normalized frequency and time in sample units, i s'pose).  the explanation for "exotic" functions that you have for beta and such is still a mystery to me.  but i'll get it eventually, i think.

thanks.

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[Peter Schoffhauzer]

Hi,

robert bristow-johnson wrote:

>> ----- Original Message -----
>> From: "Peter Schoffhauzer" <scoofy@...>
>> To: "A discussion list for music-related DSP" <music-dsp@...>
>> Subject: Re: [music-dsp] Paper on bandlimited analog waveform synthesis using FM
>> Date: Sun, 01 Jun 2008 06:20:48 +0200
>>
>>
>> You can find it here:
>> http://scp.web.elte.hu/papers/synthesis1.pdf
>>
>
> Peter, thank you for posting this.  i'm still looking at it, but can i ask you questions about it a little later?  either here or privately, i don't care.

Sure, I'm open for discussion, both here and in private.

> the first thing i would say about it after first glance is that i wish the spectrums and spectrografs had their axis labeled (in terms of normalized frequency and time in sample units, i s'pose).  the explanation for "exotic" functions that you have for beta and such is still a mystery to me.  but i'll get it eventually, i think.

Yep, that's one thing I'm not satisfied with either, partly that's the
reason why the word "draft" is in the title. Well, this was my first
"paper" of this kind, it was more like just learning how to use Latex,
instead of actually trying to do something really serious.

Well, beta functions were a bit of an experimental. Generally, the
higher beta you have, the higher bandwith the osc will have because of
the increased modulation. The most elegant would have be to investigate
the bandwith formulas and derive a nice beta function from those, but
instead I just tried guess some ad hoc forumas, and see what works best.
This was satisfactory for my purposes.

Finally I didn't work on this method more, because the costs for making
the asymmetrical waveform look and sound more like an ideal wave are
high, and other waveform generation methods, like mip-mapped linear
interpolation suited my needs better. I think recursive FM wave
generation may be better suited for systems with very limited amount of
memory.

Peter

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[Andy Farnell]

On Mon, 02 Jun 2008 18:13:10 +0200
Peter Schoffhauzer <scoofy@...> wrote:

> I think recursive FM wave
> generation may be better suited for systems with very limited amount of
> memory.

Or more generally, where you just want to avoid doing memory access.
An application is to parallelise many sound objects without
knowing/caring what thread/core they run on, being able to keep
everything in registers becomes rather nice. So a comapct FM blosc
is useful, thanks for sharing.

For the graphs, I find a combo of Octave + Gnuplot does just about
everything I need vis scales/axes/labels and plays nice with LaTeX.

a.



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[robert bristow-johnson]

> ----- Original Message -----
> From: "Andy Farnell" <padawan12@...>
> To: "A discussion list for music-related DSP" <music-dsp@...>
> Subject: Re: [music-dsp] Paper on bandlimited analog waveform synthesis using FM
> Date: Mon, 2 Jun 2008 17:58:57 +0100
>
...
>
> For the graphs, I find a combo of Octave + Gnuplot does just about
> everything I need vis scales/axes/labels and plays nice with LaTeX.
>

boy, Andy, if you could tell us what you do with Gnuplot, how you use it.  i am stuck using a Linux machine, and although i appreciate some of the command-line UNIX tool making ability, i have great trouble in using *other's* tools.  and gnuplot was one of them.


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[Victor Lazzarini]

I think what Andy means  is that gnuplot is the plotting
engine for octave. Works well, for those inclined to
 matlab-like programming,

I was once an  user of  gnuplot (straight, no octave),
but now I am moving into scipy & matplotlib, which
suits me, since I like python a lot.

Gnuplot was good for simple stuff, but I found it
difficult to do more complex stuff.

Victor
----- Original Message -----
From: "robert bristow-johnson" <rbj@...>
To: "A discussion list for music-related DSP" <music-dsp@...>
Sent: Monday, June 02, 2008 7:04 PM
Subject: Re: [music-dsp] Paper on bandlimited analog waveform synthesis
using FM



> ----- Original Message -----
> From: "Andy Farnell" <padawan12@...>
> To: "A discussion list for music-related DSP"
> <music-dsp@...>
> Subject: Re: [music-dsp] Paper on bandlimited analog waveform synthesis
> using FM
> Date: Mon, 2 Jun 2008 17:58:57 +0100
>
...
>
> For the graphs, I find a combo of Octave + Gnuplot does just about
> everything I need vis scales/axes/labels and plays nice with LaTeX.
>

boy, Andy, if you could tell us what you do with Gnuplot, how you use it.  i
am stuck using a Linux machine, and although i appreciate some of the
command-line UNIX tool making ability, i have great trouble in using
*other's* tools.  and gnuplot was one of them.


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[Andy Farnell]

Sure Robert... First I have to agree that both Gnuplot and Octave are
hardly "easy" programs. But I suspect most of us DSP coders can get around
them.

Gnuplot now lives here
http://www.gnuplot.info/

And Octave here
http://www.gnu.org/software/octave/

Octave, which is basically a free MATLAB, uses Gnuplot as the default
visualisation back-end (although you can import other more sophisticated
2D and 3D plotters/renderers)

This (not so) frequently asked questions collection is good imho.
http://t16web.lanl.gov/Kawano/gnuplot/index-e.html

As quick examples I've a few files on my site;


http://obiwannabe.co.uk/temp/plotting/plotspec.plt
http://obiwannabe.co.uk/temp/plotting/spectrum.dat

These first two show the basics. The script file can have any
extension, I choose .plt, YMMV.

Invoke thus; > gnuplot elasticity-plot.plt, and it will generate
a postscript file in plotoutput.ps

It shows the basic two column data format in a separate .dat file
and the use of xticks (to set arbitrary range) and axes labels.


If you want to _generate_ data parametrically in Gnuplot this
example shows how to do a simple square root in a normalised range

http://obiwannabe.co.uk/temp/plotting/plotsqrt.plt

This next pair show how to do horizontal range bars using
a four column data set.

http://obiwannabe.co.uk/temp/plotting/elasticity-plot.plt
http://obiwannabe.co.uk/temp/plotting/elasticity.dat


For simple 3D plots this next example shows how to use degree based
trig and 3 axis plots with 'set parametric' and 'set angle degree'
Uncomment the penultimate line for a sphere example instead.

http://obiwannabe.co.uk/temp/plotting/plotcylinder.plt

Moving on to Octave/Matlab. The syntax is a bit different.

http://obiwannabe.co.uk/temp/plotting/phon-ISO226.m

This one demonstrates a plot of ISO226 phon curves which I trivially
translated from MATLAB version by Jeff Tackett. Notice that most of the
formatting is done through gset commands, although gset is tagged as
deprecated so there are 'better' ways.

http://obiwannabe.co.uk/temp/plotting/drum.m

This one shows a surface plot for modes of a skin. It generates a X-window
output so may not work on some systems ... which brings us to .....

http://www.tcptrace.org/jPlot/

Jplot is a Gnuplot-like substitute for x-platform (I hear it works
okay with Win).

Hope these are useful pointers.

a.












On Mon, 2 Jun 2008 14:04:18 -0400
"robert bristow-johnson" <rbj@...> wrote:

>
> > ----- Original Message -----
> > From: "Andy Farnell" <padawan12@...>
> > To: "A discussion list for music-related DSP" <music-dsp@...>
> > Subject: Re: [music-dsp] Paper on bandlimited analog waveform synthesis using FM
> > Date: Mon, 2 Jun 2008 17:58:57 +0100
> >
> ...
> >
> > For the graphs, I find a combo of Octave + Gnuplot does just about
> > everything I need vis scales/axes/labels and plays nice with LaTeX.
> >
>
> boy, Andy, if you could tell us what you do with Gnuplot, how you use it.  i am stuck using a Linux machine, and although i appreciate some of the command-line UNIX tool making ability, i have great trouble in using *other's* tools.  and gnuplot was one of them.
>
>
> --
>
> r b-j                  rbj@...
>
> "Imagination is more important than knowledge."
>
> --
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[robert bristow-johnson]

> ----- Original Message -----
> From: victor <Victor.Lazzarini@...>
> To: "A discussion list for music-related DSP" <music-dsp@...>
> Subject: Re: [music-dsp] Paper on bandlimited analog waveform synthesis using FM
> Date: Mon, 02 Jun 2008 19:16:39 +0100
>
>
> I think what Andy means  is that gnuplot is the plotting
> engine for octave. Works well, for those inclined to
> matlab-like programming,
>
> I was once an  user of  gnuplot (straight, no octave),
> but now I am moving into scipy & matplotlib, which
> suits me, since I like python a lot.
>
> Gnuplot was good for simple stuff, but I found it
> difficult to do more complex stuff.
>

okay, if i'm using gnuplot with Octave (and i have plotted things with Octave, the poor man's MATLAB), is there a way to save the plot as a gif or even in a .pdf or something?  how i long for the simple "cut-and-paste" days of the 90s.  i really feel that we have been moving backwards along the axis of computer utility and simplicity.  i could do more with my Mac in the 1990s using system 9.2 than i can do with *anything*, OSX, Windoze, Linux, nowadays.  here's a cute U-Tube to spell out what i mean:

http://video.google.com/videosearch?q=Mac+vs.+PC+Linux#

why is it that computers are getting more sophisticated in design, but devolving in their utility?

--

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"Imagination is more important than knowledge."

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[Andy Farnell]

On Mon, 02 Jun 2008 19:16:39 +0100
victor <Victor.Lazzarini@...> wrote:

> scipy & matplotlib, which
> suits me, since I like python a lot.

Agreed. IMHO Python will evenually replace/augment most of
these tools - it's just more flexible. But there's
a lot of MATLAB/Octave code out there and it's ability
with matrices/solvers, curve fitting etc will take some
time to catch up in Python. So worth learning all of them
if you are doing research/papers etc. Octave/MATLAB is more
mathematical while scipi is more programmatic I would say.
Another wonderful free scientific data tool is R (for
statistics).

a.



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[Victor Lazzarini]

On Windows, with gnuplot, you can paste the whole graph onto
the clipboard. I guess that might be the same in Linux.

But you can print to GIF etc straight from the program (just
set the output type)

Victor
----- Original Message -----
From: "robert bristow-johnson" <rbj@...>
To: "A discussion list for music-related DSP" <music-dsp@...>
Sent: Monday, June 02, 2008 8:10 PM
Subject: Re: [music-dsp] Paper on bandlimited analog waveform synthesis
using FM

> ----- Original Message -----
> From: victor <Victor.Lazzarini@...>
> To: "A discussion list for music-related DSP"
> <music-dsp@...>
> Subject: Re: [music-dsp] Paper on bandlimited analog waveform synthesis
> using FM
> Date: Mon, 02 Jun 2008 19:16:39 +0100
>
>
> I think what Andy means  is that gnuplot is the plotting
> engine for octave. Works well, for those inclined to
> matlab-like programming,
>
> I was once an  user of  gnuplot (straight, no octave),
> but now I am moving into scipy & matplotlib, which
> suits me, since I like python a lot.
>
> Gnuplot was good for simple stuff, but I found it
> difficult to do more complex stuff.
>

okay, if i'm using gnuplot with Octave (and i have plotted things with
Octave, the poor man's MATLAB), is there a way to save the plot as a gif or
even in a .pdf or something?  how i long for the simple "cut-and-paste" days
of the 90s.  i really feel that we have been moving backwards along the axis
of computer utility and simplicity.  i could do more with my Mac in the
1990s using system 9.2 than i can do with *anything*, OSX, Windoze, Linux,
nowadays.  here's a cute U-Tube to spell out what i mean:

http://video.google.com/videosearch?q=Mac+vs.+PC+Linux#

why is it that computers are getting more sophisticated in design, but
devolving in their utility?

--

r b-j                  rbj@...

"Imagination is more important than knowledge."

--
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[Eric Brombaugh-2]

robert bristow-johnson wrote:
> okay, if i'm using gnuplot with Octave (and i have plotted things with Octave, the poor man's MATLAB), is there a way to save the plot as a gif or even in a .pdf or something?

You can dump Octave/Gnuplot output to a variety of formats, including PS
(which can then be converted to PDF with the 'ps2pdf' command on Linux).
Here's the online man page for it:

http://www.gnu.org/software/octave/doc/interpreter/Printing-Plots.html#Printing-Plots

I tried this - getting the syntax right was a bit tricky, but try this
example:

plot(sin(0:0.1:6.3));
print("sine.ps", "-solid", "-landscape", "-color", "-dps");

which generates a Postscript file titled 'sine.ps'.

Eric
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[Scott Lehman-2]

Not really an answer to the original question, but
someone pointed me to Grace as an alternative to
Gnuplot a while back.  It can export to PDF and svg,
though it's only 2D and I don't think it has the same
level of integration with Octave.

Haven't actually used it myself, but been meaning to
give it a shot:
http://plasma-gate.weizmann.ac.il/Grace/

Scott

--- Eric Brombaugh <ebrombaugh1@...> wrote:

> robert bristow-johnson wrote:
> > okay, if i'm using gnuplot with Octave (and i have
> plotted things with Octave, the poor man's MATLAB),
> is there a way to save the plot as a gif or even in
> a .pdf or something?
>
> You can dump Octave/Gnuplot output to a variety of
> formats, including PS
> (which can then be converted to PDF with the
> 'ps2pdf' command on Linux).
> Here's the online man page for it:
>
>

http://www.gnu.org/software/octave/doc/interpreter/Printing-Plots.html#Printing-Plots

>
> I tried this - getting the syntax right was a bit
> tricky, but try this
> example:
>
> plot(sin(0:0.1:6.3));
> print("sine.ps", "-solid", "-landscape", "-color",
> "-dps");
>
> which generates a Postscript file titled 'sine.ps'.
>
> Eric
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[david.lowenfels]

Hi Peter,
   Thanks for sharing your paper.
When was it written BTW, as there is no date of authorship?
I never played with a high shelf compensation filter, that's a neat  
idea.
It would be easier to compare frequency plots if you had a log(f) x-
axis.

I'm surprised there are not more references, as the basics of this  
technique have been around for a long time.
You should at least cite the original Yamaha patent: US#4249447  
(pat2pdf.com comes in handy)

Cheers,
-David

On May 31, 2008, at 9:20 PM, Peter Schoffhauzer wrote:

> Hi,
>
> You can find it here:
> http://scp.web.elte.hu/papers/synthesis1.pdf
>
> Best,
> Peter



On Jan 17, 2007, at 11:15 PM, David Lowenfels wrote:

> I like to call it Recursive Phase Modulation (RPM)
> c.f. yamaha patent #4249447, and my 8/22/06 post:
>
>> From:  dfl alum mit edu
>> Subject: Re: [music-dsp] different waveforms from a single
>> Date: August 22, 2006 11:48:14 AM PDT
>> To:  music-dsp@...
>>
>> My personal favorite for oscillator coolness is Yamaha-flavored  
>> feedback FM (actually phase modulation):
>>
>> Take a phasor (naive sawtooth aka wrapped accumulator) and feed it  
>> into a sine lookup table (linear interpolation is fine). Feed back  
>> the output with a gain control, and sum it back with the phasor  
>> before table lookup.
>>
>> (industry practice is to put a 2-point running average in the  
>> feedback chain to control wild self-oscillation at nyquist... the  
>> yammy patent calls it a "hunting filter" ??! )
>>
>> No feedback yields a regular sinewave.
>> The feedback gain controls the harmonic rolloff slope, kind of like  
>> a lowpass filter.
>> Increasing feedback gain approximates a quasi-bandlimited sawtooth-
>> like wave.
>> If you square (and invert?) the feedback you get a quasi-
>> bandlimited square-like wave.
>>
>> You can approximate a triangle wave by using the a low feedback  
>> gain on the square wave. Or just integrate (not sure if it needs to  
>> be leaky as there should be no DC in the original waveform?)
>> Use comb filter tricks on the sawtooth to get PWM, etc.
>>
>> -David
>>
>> On Aug 22, 2006, at 2:50 AM, kernel wrote:
>>
>>
>>> I've used it lots for low frequency stuff not necessarily LFOs but  
>>> low pitched sounds.  I'm after something I can use for both  
>>> regular oscillators and LFOs which provides a wide range of  
>>> traditional (tri/pulse etc) as well as more unusual waveforms.
>
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[Peter Schoffhauzer]

Hi David,

david.lowenfels@... wrote:
> When was it written BTW, as there is no date of authorship?

I originally posted about this on this list at 2007. jan 17, so I guess
I wrote it the previous days.

> I'm surprised there are not more references, as the basics of this
> technique have been around for a long time.
> You should at least cite the original Yamaha patent: US#4249447
> (pat2pdf.com comes in handy)

The reason that there are no references is because most of this is own
experimental research. I did not know about the Yamaha paper, I just
discovered about it later (actually because you sent it to me if you
remember), nor did I know before that they used a similar technique to
handle Nyquist-ringing. I just read a few introductory web pages about
FM / recursive FM before experimenting, that's all I used. Why would I
have cited works that I haven't read and didn't know about?

 > PS I just was poking around, and found your SilverBox synth. Very nice
 > sound, congrats!

Thanks!

Peter

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