Generating non-Gaussian signal

[DigitalSignal]

I wonder how to generate a non-Gaussian signal of which the Kurtosis
can be controlled.

James
www.go-ci.com

[julius]

On Jul 6, 7:30 pm, DigitalSignal <[email protected]> wrote:
> I wonder how to generate a non-Gaussian signal of which the Kurtosis
> can be controlled.
>
> Jameswww.go-ci.com

I wonder what you have in your disposal to go about your task.
I wonder how to help you if you don't say what you are starting
with :-P.

[DigitalSignal]

It is a general question. We make dynamic signal analyzers. In a
signal analyzer, the typical waveform signa sources include sine,
white noise, swept sine, saw tooth, square etc.. The white noise is
always Gaussian type (with Kuotosis 3). This signal is usually
generated by summing up a few uniformly distributed random numbers. In
order to simulate what is happenning in the physical world, some users
ask for a random signal of which the Kuotosis is larger than 3. We
don't have a easy way to generate such as signal.

James
www.go-ci.com

[dbell]

On Jul 7, 6:42*am, julius <[email protected]> wrote:
> On Jul 6, 7:30 pm, DigitalSignal <[email protected]> wrote:
>
> > I wonder how to generate a non-Gaussian signal of which the Kurtosis
> > can be controlled.
>
> > Jameswww.go-ci.com
>
> I wonder what you have in your disposal to go about your task.
> I wonder how to help you if you don't say what you are starting
> with :-P.

I usualy have food-related garbage in my disposal, but I turn on the
water, flip the switch, and the garbage is gone. Turn off the switch,
turn off the water, and I am on my way.

Dirk

[Piergiorgio Sartor]

DigitalSignal wrote:
> It is a general question. We make dynamic signal analyzers. In a
> signal analyzer, the typical waveform signa sources include sine,
> white noise, swept sine, saw tooth, square etc.. The white noise is
> always Gaussian type (with Kuotosis 3). This signal is usually
> generated by summing up a few uniformly distributed random numbers. In
> order to simulate what is happenning in the physical world, some users
> ask for a random signal of which the Kuotosis is larger than 3. We
> don't have a easy way to generate such as signal.

Well, since a Gaussian (noise) is fully defined by its
mean and variance, I guess you'll first to backtrace
the Kurtosis to the variance.

The, I still guess, multiplying the normal noise N(0,1)
by the square root of the wanted variance "s", will give
you N(0, s), hence the wanted Kurtosis.

bye,

--

piergiorgio

[dbd]

On Jul 6, 5:30 pm, DigitalSignal <[email protected]> wrote:
> I wonder how to generate a non-Gaussian signal of which the Kurtosis
> can be controlled.
>
> Jameswww.go-ci.com

A discussion of the motivation for and a claim of providing a kurtosis
knob on the signal generator can be found at:
http://www.testing-expo.com/usa/06conf/txna_pdfs/day_1/john_van_baron.pdf

This can be found by performing a Google on the mystical words:
controlled kurtosis.

Whether a useful method will be shared with you is up to you to find.

Dale B. Dalrymple
http://dbdimages.com

[Randy Yates]

dbd <[email protected]> writes:

> On Jul 6, 5:30 pm, DigitalSignal <[email protected]> wrote:
>> I wonder how to generate a non-Gaussian signal of which the Kurtosis
>> can be controlled.
>>
>> Jameswww.go-ci.com
>
> A discussion of the motivation for and a claim of providing a kurtosis
> knob on the signal generator can be found at:
>
> http://www.testing-expo.com/usa/06conf/txna_pdfs/day_1/john_van_baron.pdf

In a nutshell, some real-world random variables aren't really Gaussian?
--
% Randy Yates % "My Shangri-la has gone away, fading like
%% Fuquay-Varina, NC % the Beatles on 'Hey Jude'"
%%% 919-577-9882 %
%%%% <[email protected]> % 'Shangri-La', *A New World Record*, ELO
http://www.digitalsignallabs.com

[dbd]

On Jul 10, 4:56 pm, Randy Yates <[email protected]> wrote:
> dbd <[email protected]> writes:
> > On Jul 6, 5:30 pm, DigitalSignal <[email protected]> wrote:
> >> I wonder how to generate a non-Gaussian signal of which the Kurtosis
> >> can be controlled.
>
> >> Jameswww.go-ci.com
>
> > A discussion of the motivation for and a claim of providing a kurtosis
> > knob on the signal generator can be found at:
>
> >http://www.testing-expo.com/usa/06conf/txna_pdfs/day_1/john_van_baron...
>
> In a nutshell, some real-world random variables aren't really Gaussian?
> --
> % Randy Yates % "My Shangri-la has gone away, fading like
> %% Fuquay-Varina, NC % the Beatles on 'Hey Jude'"
> %%% 919-577-9882 %
> %%%% <[email protected]> % 'Shangri-La', *A New World Record*, ELOhttp://www.digitalsignallabs.com

See p7 of the citation or ponder why anyone would care about Gaussian
mixture models or 'long-tailed' distributions if all real data were
Gaussian.

Dale B. Dalrymple

[[email protected]]

On Jul 10, 7:56*pm, Randy Yates <[email protected]> wrote:
> dbd <[email protected]> writes:
> > On Jul 6, 5:30 pm, DigitalSignal <[email protected]> wrote:
> >> I wonder how to generate a non-Gaussian signal of which the Kurtosis
> >> can be controlled.
>
> >> Jameswww.go-ci.com
>
> > A discussion of the motivation for and a claim of providing a kurtosis
> > knob on the signal generator can be found at:
>
> >http://www.testing-expo.com/usa/06conf/txna_pdfs/day_1/john_van_baron...
>
> In a nutshell, some real-world random variables aren't really Gaussian?


Hello Randy,

Not everything is Gaussian. Radioactive decay is as good example of a
non Gaussian process. Each particle (nucleus) has associated with it a
negative exponential decay density. Then if you look at the interval
times between decays in an ensemble of particles, then you will
observe a nearly Poisson process. This you have likely heard as the
audible "clicks" from a Geiger-Muller counter.

Clay

[Randy Yates]

[email protected] writes:

> On Jul 10, 7:56*pm, Randy Yates <[email protected]> wrote:
>> dbd <[email protected]> writes:
>> > On Jul 6, 5:30 pm, DigitalSignal <[email protected]> wrote:
>> >> I wonder how to generate a non-Gaussian signal of which the Kurtosis
>> >> can be controlled.
>>
>> >> Jameswww.go-ci.com
>>
>> > A discussion of the motivation for and a claim of providing a kurtosis
>> > knob on the signal generator can be found at:
>>
>> >http://www.testing-expo.com/usa/06conf/txna_pdfs/day_1/john_van_baron...
>>
>> In a nutshell, some real-world random variables aren't really Gaussian?
>
>
> Hello Randy,
>
> Not everything is Gaussian. Radioactive decay is as good example of a
> non Gaussian process. Each particle (nucleus) has associated with it a
> negative exponential decay density. Then if you look at the interval
> times between decays in an ensemble of particles, then you will
> observe a nearly Poisson process. This you have likely heard as the
> audible "clicks" from a Geiger-Muller counter.
>
> Clay

Hi Clay,

I didn't state my point very well. Yes, of course not all real-world
random variables are Gaussian; some are not even close, as you point
out.

What I meant was that some real-world RV's that may appear to be
Gaussian (or assumed to be) really aren't.
--
% Randy Yates % "Remember the good old 1980's, when
%% Fuquay-Varina, NC % things were so uncomplicated?"
%%% 919-577-9882 % 'Ticket To The Moon'
%%%% <[email protected]> % *Time*, Electric Light Orchestra
http://www.digitalsignallabs.com

[rip pelletier]

In article
<[email protected]>,
DigitalSignal <[email protected]> wrote:

> It is a general question. We make dynamic signal analyzers. In a
> signal analyzer, the typical waveform signa sources include sine,
> white noise, swept sine, saw tooth, square etc.. The white noise is
> always Gaussian type (with Kuotosis 3). This signal is usually
> generated by summing up a few uniformly distributed random numbers. In
> order to simulate what is happenning in the physical world, some users
> ask for a random signal of which the Kuotosis is larger than 3. We
> don't have a easy way to generate such as signal.
>
> James
> www.go-ci.com

there are other ways, one of which is completely general, to get a
normal variate from a uniform distribution. more importantly, if you
need to generate some non-gaussian variate, you need to know the general
transformation. you could look in "numerical recipes".

for a normal (i.e. gaussian) distribution, the (moment coefficient
of) kurtosis is always 3, independent of mean and variance. your title *
non-gaussian * suggests that you know this.

about all that comes to mind is: look at families of distributions,
say the Pearson's, to see if something has a plausible kurtosis and
shape. once you have the probability distribution, you can apply the
general method of generating variates from a uniform distribution.

vale,
rip

--
NB eddress is r i p 1 AT c o m c a s t DOT n e t

[Uwe Schmitt]

On 10 Jul., 22:21, DigitalSignal <[email protected]> wrote:
> It is a general question. We make dynamic signal analyzers. In a
> signal analyzer, the typical waveform signa sources include sine,
> white noise, swept sine, saw tooth, square etc.. The white noise is
> always Gaussian type (with Kuotosis 3). This signal is usually
> generated by summing up a few uniformly distributed random numbers. In
> order to simulate what is happenning in the physical world, some users
> ask for a random signal of which the Kuotosis is larger than 3. We
> don't have a easy way to generate such as signal.
>
> Jameswww.go-ci.com

You could sample values according the following distribution:

http://en.wikipedia.org/wiki/Pearson_distribution#The_Pearson_type_VII_distribu tion

In this description the parameter m is connected to the kurtosis, and
if m->infty
you get a normal-distribution.

One method to sample values to an arbitrariy distribtution f(x) is:

http://en.wikipedia.org/wiki/Rejection_sampling

Greetings, Uwe

[Uwe Schmitt]

On 12 Jul., 12:03, Uwe Schmitt <[email protected]> wrote:
> On 10 Jul., 22:21, DigitalSignal <[email protected]> wrote:
>
> > It is a general question. We make dynamic signal analyzers. In a
> > signal analyzer, the typical waveform signa sources include sine,
> > white noise, swept sine, saw tooth, square etc.. The white noise is
> > always Gaussian type (with Kuotosis 3). This signal is usually
> > generated by summing up a few uniformly distributed random numbers. In
> > order to simulate what is happenning in the physical world, some users
> > ask for a random signal of which the Kuotosis is larger than 3. We
> > don't have a easy way to generate such as signal.
>
> > Jameswww.go-ci.com
>
> You could sample values according the following distribution:
>
> * *http://en.wikipedia.org/wiki/Pearson_distribution#The_Pearson_type_VI...
>
> In this description the parameter m is connected to the kurtosis, and
> if m->infty
> you get a normal-distribution.
>
> One method to sample values to an arbitrariy distribtution f(x) is:
>
> * * *http://en.wikipedia.org/wiki/Rejection_sampling
>

Look at GNU Scientific Library (GSL) which has lots of random
generators.

Greetings, Uwe

[DigitalSignal]

Thank you all for the insights and comments. They are really helpful.
Let me go further for this question:

How to generate a random process of which both its PSD (Power Spectral
Density) and Kurtosis can be controlled? A traditional method is that
to apply a uniformed randomized phase to the defined PSD and conduct
inverse FFT to create the time signal. Unfortunately this method
always create Gaussian distributed time signal.

James
www.go-ci.com

[dbd]

On Jul 12, 8:18 am, DigitalSignal <[email protected]> wrote:

> Thank you all for the insights and comments. They are really helpful.
> Let me go further for this question:

> How to generate a random process of which both its PSD (Power Spectral
> Density) and Kurtosis can be controlled? A traditional method is that
> to apply a uniformed randomized phase to the defined PSD and conduct
> inverse FFT to create the time signal. Unfortunately this method
> always create Gaussian distributed time signal.
>
> Jameswww.go-ci.com

You could try:

http://www.engineers.auckland.ac.nz/~aste127/Jies97.pdf

Of course, to find this you would have had to Google on controlled
kurtosis.

Dale B. Dalrymple

[DigitalSignal]

Thank you again. I met with the author a while back and we had an
argument. He uses multiple sine tones to create a random-look
deterministic signal of which the Kurtosis is controlled. I said it is
not a real random signal...
But his approach is certainly one worth considering...

James
www.go-ci.com

[Jerry Avins]

DigitalSignal wrote:
> Thank you again. I met with the author a while back and we had an
> argument. He uses multiple sine tones to create a random-look
> deterministic signal of which the Kurtosis is controlled. I said it is
> not a real random signal...
> But his approach is certainly one worth considering...
>
> James
> www.go-ci.com

No sequence that can be reproduced without being copied is truly random.
What did you mean?

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯