Band limited Gaussian White noise generation

I need to generate a Gaussian White Noise which has the frequency content
of less than a constant frequency, cut-off frequency, for running a
vibration exciter. I thought to generate this signal in frequency domain
with the frequency content that I need and then transform it to time domain
by means of ifft. The point is that I need to have control over the time
domain content of my signal, more specifically, sigma of Gaussian PDF of
the signal in the time domain. I have no idea how the parameters of Fourier
transform of a Gaussian White noise represent in time domain. In other
words how sigma and mean of a time domain Gaussian white noise represent in
amplitude and phase of it's Fourier Transform in frequency domain.
Thanks in advance

On Wed, 12 Oct 2011 12:57:18 -0500, veccia wrote:

> I need to generate a Gaussian White Noise which has the frequency
> content of less than a constant frequency, cut-off frequency, for
> running a vibration exciter. I thought to generate this signal in
> frequency domain with the frequency content that I need and then
> transform it to time domain by means of ifft. The point is that I need
> to have control over the time domain content of my signal, more
> specifically, sigma of Gaussian PDF of the signal in the time domain. I
> have no idea how the parameters of Fourier transform of a Gaussian White
> noise represent in time domain. In other words how sigma and mean of a
> time domain Gaussian white noise represent in amplitude and phase of
> it's Fourier Transform in frequency domain. Thanks in advance

"Of less than a constant frequency, cut-off frequency" rings an alarm 
bell: are you trying to have a "brick wall" response, where the spectrum 
is something like

       {  1   f < f_0
S(w) = {
       {  0   f >= f_0

?

If you are, then you're doomed to failure, or at least to a lot of 
(probably useless) thrashing, because the sharper the cut-off, the longer 
the filter settling time, and it's awfully hard to perform an infinitely 
long ifft in one's lifetime.

The "easy" way to do this is to probably generate wideband white noise 
(that may not even need to be Gaussian, depending on the ratio of your 
sample rate and your bandwidth) and run it through a filter whose 
spectral response equals your desired spectrum.  If the filter's DC gain 
is unity and the power spectral density of your input noise is known, 
then you already know the PSD of your output in the filter's passband, 
and it'll be fairly easy to calculate the noise variance.

-- 
www.wescottdesign.com

veccia wrote:

> I need to generate a Gaussian White Noise which has the frequency content
> of less than a constant frequency, cut-off frequency, for running a
> vibration exciter.

Generate gaussian samples at regular time interval T. The spectrum will 
be sin(Pi*T*F)/(Pi*T*F), i.e. with the 3dB cutoff at F ~ 0.4/T.
This is probably good enough for the job.


Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com

On 10/12/2011 1:57 PM, veccia wrote:
> I need to generate a Gaussian White Noise which has the frequency content
> of less than a constant frequency, cut-off frequency, for running a
> vibration exciter. I thought to generate this signal in frequency domain
> with the frequency content that I need and then transform it to time domain
> by means of ifft. The point is that I need to have control over the time
> domain content of my signal, more specifically, sigma of Gaussian PDF of
> the signal in the time domain. I have no idea how the parameters of Fourier
> transform of a Gaussian White noise represent in time domain. In other
> words how sigma and mean of a time domain Gaussian white noise represent in
> amplitude and phase of it's Fourier Transform in frequency domain.
> Thanks in advance

Maybe I've been retired too long to get the big picture, but I just 
can't imagine why you need the signal that you plan to IFFT. I would 
generate the random signal in the time domain (i.e., as a series of 
numbers) filter it to get the desired PSD, and convert to analog, and 
blast the shaker with it. (The D-to-A needs to be followed by a 
[re]construction filter. You could adjust the PSD there.)

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

On Oct 12, 3:34&#4294967295;pm, Jerry Avins <j...@ieee.org> wrote:
>
> Maybe I've been retired too long to get the big picture, but I just
> can't imagine why you need the signal that you plan to IFFT. I would
> generate the random signal in the time domain (i.e., as a series of
> numbers) filter it to get the desired PSD, and convert to analog, and
> blast the shaker with it. (The D-to-A needs to be followed by a
> [re]construction filter. You could adjust the PSD there.)
>
> Jerry
> --
> Engineering is the art of making what you want from things you can get.

To the OP:

My mind too boggles at the idea of fooling around in the
frequency domain and using iFFTs when the *stated*
desire is

>>I need to have control over the time domain content
>>of my signal, more specifically, sigma of Gaussian
>>PDF of the signal in the ****time domain.*****
(emphasis added)

There is a standard description of white Gaussian noise
in the time domain that will allow you to "control" sigma to
eight significant digits if you so desire.  Why even think
of the frequency domain as the obvious setting?  Is it that
you have just acquired this nice hammer called iFFT and
so now everything looks like a nail?  Or to mix my metaphors,
why try to grab your right ear with your right hand after
wrapping your arm around your head first?

Dilip Sarwate

On 10/12/2011 6:06 PM, dvsarwate wrote:

   ...

> why try to grab your right ear with your right hand after
> wrapping your arm around your head first?

Ooo goody! I haven't lost it all yet!

Jerry
-- 
A voice-controlled dog is one of the greatest joys one can have.

On 10/12/2011 06:29 PM, Jerry Avins wrote:
> On 10/12/2011 6:06 PM, dvsarwate wrote:
>
> ...
>
>> why try to grab your right ear with your right hand after
>> wrapping your arm around your head first?
>
> Ooo goody! I haven't lost it all yet!
>
> Jerry

I doubt you ever will, Jerry.

--Randy

Tim Wescott <tim@seemywebsite.com> wrote:
> On Wed, 12 Oct 2011 12:57:18 -0500, veccia wrote:

>> I need to generate a Gaussian White Noise which has the frequency
>> content of less than a constant frequency, cut-off frequency, for
>> running a vibration exciter. I thought to generate this signal in
>> frequency domain with the frequency content that I need and then
>> transform it to time domain by means of ifft. The point is that I need
>> to have control over the time domain content of my signal, more
>> specifically, sigma of Gaussian PDF of the signal in the time domain. I
>> have no idea how the parameters of Fourier transform of a Gaussian White
>> noise represent in time domain. In other words how sigma and mean of a
>> time domain Gaussian white noise represent in amplitude and phase of
>> it's Fourier Transform in frequency domain. Thanks in advance

> "Of less than a constant frequency, cut-off frequency" rings an alarm 
> bell: are you trying to have a "brick wall" response, where the spectrum 
> is something like

>       {  1   f < f_0
> S(w) = {
>       {  0   f >= f_0

> ?

> If you are, then you're doomed to failure, or at least to a lot of 
> (probably useless) thrashing, because the sharper the cut-off, the longer 
> the filter settling time, and it's awfully hard to perform an infinitely 
> long ifft in one's lifetime.

But it doesn't need to be infinitely long, only long enough to
do the desired experiment.  Say you generate some number of random
uniform distributed values, pad with zeros, and iFFT.  You then
have a periodic signal with frequency components only up to a certain
point and zero after that.  The cutoff won't be infinitely sharp,
but will be as sharp as it can be given the frequency resolution.

The old story is that you can't do an experiment longer than the
time for one graduate student to graduate, but most likely
in the range of hours.  

> The "easy" way to do this is to probably generate wideband white noise 
> (that may not even need to be Gaussian, depending on the ratio of your 
> sample rate and your bandwidth) and run it through a filter whose 
> spectral response equals your desired spectrum.  If the filter's DC gain 
> is unity and the power spectral density of your input noise is known, 
> then you already know the PSD of your output in the filter's passband, 
> and it'll be fairly easy to calculate the noise variance.

I like this way better, but it won't give such a sharp cutoff.

-- glen

On Thu, 13 Oct 2011 02:29:27 +0000, glen herrmannsfeldt wrote:

> Tim Wescott <tim@seemywebsite.com> wrote:
>> On Wed, 12 Oct 2011 12:57:18 -0500, veccia wrote:
> 
>>> I need to generate a Gaussian White Noise which has the frequency
>>> content of less than a constant frequency, cut-off frequency, for
>>> running a vibration exciter. I thought to generate this signal in
>>> frequency domain with the frequency content that I need and then
>>> transform it to time domain by means of ifft. The point is that I need
>>> to have control over the time domain content of my signal, more
>>> specifically, sigma of Gaussian PDF of the signal in the time domain.
>>> I have no idea how the parameters of Fourier transform of a Gaussian
>>> White noise represent in time domain. In other words how sigma and
>>> mean of a time domain Gaussian white noise represent in amplitude and
>>> phase of it's Fourier Transform in frequency domain. Thanks in advance
> 
>> "Of less than a constant frequency, cut-off frequency" rings an alarm
>> bell: are you trying to have a "brick wall" response, where the
>> spectrum is something like
> 
>>       {  1   f < f_0
>> S(w) = {
>>       {  0   f >= f_0
> 
>> ?
> 
>> If you are, then you're doomed to failure, or at least to a lot of
>> (probably useless) thrashing, because the sharper the cut-off, the
>> longer the filter settling time, and it's awfully hard to perform an
>> infinitely long ifft in one's lifetime.
> 
> But it doesn't need to be infinitely long, only long enough to do the
> desired experiment.  Say you generate some number of random uniform
> distributed values, pad with zeros, and iFFT.  You then have a periodic
> signal with frequency components only up to a certain point and zero
> after that.  The cutoff won't be infinitely sharp, but will be as sharp
> as it can be given the frequency resolution.

I was hoping that the OP would think about what he _really_ needs and 
come back to us.

> The old story is that you can't do an experiment longer than the time
> for one graduate student to graduate, but most likely in the range of
> hours.
> 
>> The "easy" way to do this is to probably generate wideband white noise
>> (that may not even need to be Gaussian, depending on the ratio of your
>> sample rate and your bandwidth) and run it through a filter whose
>> spectral response equals your desired spectrum.  If the filter's DC
>> gain is unity and the power spectral density of your input noise is
>> known, then you already know the PSD of your output in the filter's
>> passband, and it'll be fairly easy to calculate the noise variance.
> 
> I like this way better, but it won't give such a sharp cutoff.
> 
> -- glen





-- 
www.wescottdesign.com

>> sigma of Gaussian PDF
first of all, a periodic signal should be fine. So it just loops around.
Periodic means you can simply use fft() and ifft() to go back and forth
between frequency and time domain.

Frequency domain filtering is trivial (calculate bin frequencies, calculate
|H(f)| and multiply element-by-element (and / or set regions to zero, as
needed).

>> sigma of Gaussian PDF
isn't that simply the power of your signal?