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Hi,

What is the most efficient way to generate a noise process that is Gaussian
correlated in time from a white noise process?

I'm having trouble coming up with a simple FIR filter that will do this.

Any suggestions?

Thanks,
~mohan

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msarovar wrote:
> Hi,
> 
> What is the most efficient way to generate a noise process that is Gaussian
> correlated in time from a white noise process?
> 
> I'm having trouble coming up with a simple FIR filter that will do this.
> 
> Any suggestions?

You _do_ mean that you want to take white noise and color it in 
frequency, not that you want to take noise with a non-Gaussian 
distribution and make it Gaussian, right?

Just run it through a FIR filter with a Gaussian shape.  Bim-bam-boom, 
you'll have noise with a Gaussian PSD.  Deciding where to truncate the 
Gaussian is up to you, of course.

-- 
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com

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On Mar 9, 6:22=A0am, "msarovar" <mo...@grommit.com> wrote:
> Hi,
>
> What is the most efficient way to generate a noise process that is Gaussi=
an
> correlated in time from a white noise process?
>
> I'm having trouble coming up with a simple FIR filter that will do this.
>
> Any suggestions?
>
> Thanks,
> ~mohan

By "Gaussian correlated" do you mean anything other than correlated
and Gaussian distributed?

What have you tried and why do you think it hasn't worked?

"most efficient" is a context dependent term. Are you concerned with
implementing in a 8-bit micro-controller, a supercomputer, an FPGA or
a PC? Do you mean efficient in your time or the processor's time or
memory space?

Dale B. Dalrymple

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>On Mar 9, 6:22=A0am, "msarovar" <mo...@grommit.com> wrote:
>> Hi,
>>
>> What is the most efficient way to generate a noise process that is
Gaussi=
>an
>> correlated in time from a white noise process?
>>
>> I'm having trouble coming up with a simple FIR filter that will do
this.
>>
>> Any suggestions?
>>
>> Thanks,
>> ~mohan
>
>By "Gaussian correlated" do you mean anything other than correlated
>and Gaussian distributed?
>
>What have you tried and why do you think it hasn't worked?
>
>"most efficient" is a context dependent term. Are you concerned with
>implementing in a 8-bit micro-controller, a supercomputer, an FPGA or
>a PC? Do you mean efficient in your time or the processor's time or
>memory space?
>
>Dale B. Dalrymple
>
>

Hi,
Thanks for the quick answers. And sorry for the lack of detail in my last
message.

First, by "Gaussian correlated" I mean colored noise for which the temporal
correlations are Gaussian.

What I have tried so far is to form a Gaussian FIR filter and convolve the
white noise with it to get output that is Gaussian correlated in time. I am
running this on a PC but need to do it many, many times and so efficiency
(in time, not memory) is important. Currently, the convolution is the
limiting step in my code and I was wondering if there was an easier way to
do this. 

For example, is there a recursive (IIR) filter for generating Gaussian
correlations that might decrease the number of convolution steps?

Thanks again for the help,
~mohan

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msarovar wrote:
>> On Mar 9, 6:22=A0am, "msarovar" <mo...@grommit.com> wrote:
>>> Hi,
>>>
>>> What is the most efficient way to generate a noise process that is
> Gaussi=
>> an
>>> correlated in time from a white noise process?
>>>
>>> I'm having trouble coming up with a simple FIR filter that will do
> this.
>>> Any suggestions?
>>>
>>> Thanks,
>>> ~mohan
>> By "Gaussian correlated" do you mean anything other than correlated
>> and Gaussian distributed?
>>
>> What have you tried and why do you think it hasn't worked?
>>
>> "most efficient" is a context dependent term. Are you concerned with
>> implementing in a 8-bit micro-controller, a supercomputer, an FPGA or
>> a PC? Do you mean efficient in your time or the processor's time or
>> memory space?
>>
>> Dale B. Dalrymple
>>
>>
> 
> Hi,
> Thanks for the quick answers. And sorry for the lack of detail in my last
> message.
> 
> First, by "Gaussian correlated" I mean colored noise for which the temporal
> correlations are Gaussian.
> 
> What I have tried so far is to form a Gaussian FIR filter and convolve the
> white noise with it to get output that is Gaussian correlated in time. I am
> running this on a PC but need to do it many, many times and so efficiency
> (in time, not memory) is important. Currently, the convolution is the
> limiting step in my code and I was wondering if there was an easier way to
> do this. 
> 
> For example, is there a recursive (IIR) filter for generating Gaussian
> correlations that might decrease the number of convolution steps?
> 

"I'm having trouble coming up with a simple FIR filter that will do this"

Thus, no one suggests the obvious.

Yes, there are IIR filters that will approximate a Gaussian filter.  In 
continuous time these are referred to as "Bessel filters"; I don't know 
how they've acquired a different name in the sampled time domain.

No matter what, you'll only get an approximation.

The FFT of a white Gaussian noise process is itself white Gaussian noise 
with uniformly distributed phase.  If you need finite-length vectors 
with your Gaussian autocorrelation, you can make sequences with white 
noise, shape them with the appropriate Gaussian envelope, then take the 
inverse FFT to get a sequence with your desired time-domain properties.

-- 
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com

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On 9 Mrz., 15:22, "msarovar" <mo...@grommit.com> wrote:
> Hi,
>
> What is the most efficient way to generate a noise process that is Gaussian
> correlated in time from a white noise process?
>
> I'm having trouble coming up with a simple FIR filter that will do this.
>
> Any suggestions?
>
> Thanks,
> ~mohan

This link is three years old but still works:

http://math.mit.edu/~stevenj/preprints/iir-colored-noise-2007-03-13.pdf

It describes the use of IIR filters to efficiently (wrt to computer
time and memory) generate correlated sequences from white sequences.

Regards,
Andor

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>On 9 Mrz., 15:22, "msarovar" <mo...@grommit.com> wrote:
>> Hi,
>>
>> What is the most efficient way to generate a noise process that is
Gaussian
>> correlated in time from a white noise process?
>>
>> I'm having trouble coming up with a simple FIR filter that will do
this.
>>
>> Any suggestions?
>>
>> Thanks,
>> ~mohan
>
>This link is three years old but still works:
>
>http://math.mit.edu/~stevenj/preprints/iir-colored-noise-2007-03-13.pdf
>
>It describes the use of IIR filters to efficiently (wrt to computer
>time and memory) generate correlated sequences from white sequences.
>
>Regards,
>Andor
>

Great, thanks guys.
I will read more about Chebyshev filters. They sound like my solution.
Cheers,
~mohan

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Andor wrote:
> On 9 Mrz., 15:22, "msarovar" <mo...@grommit.com> wrote:
>> Hi,
>>
>> What is the most efficient way to generate a noise process that is Gaussian
>> correlated in time from a white noise process?
>>
>> I'm having trouble coming up with a simple FIR filter that will do this.
>>
>> Any suggestions?
>>
>> Thanks,
>> ~mohan
> 
> This link is three years old but still works:
> 
> http://math.mit.edu/~stevenj/preprints/iir-colored-noise-2007-03-13.pdf
> 
> It describes the use of IIR filters to efficiently (wrt to computer
> time and memory) generate correlated sequences from white sequences.

Any low-pass filter's output is correlated at high-enough frequencies.

Jerry
-- 
Physics is like sex: sure, it may give some practical results, but
that's not why we do it.   -- Richard P. Feynman (Nobel Prize, Physics)
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

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On 10 Mrz., 05:35, Jerry Avins <j...@ieee.org> wrote:
> Andor wrote:
> > On 9 Mrz., 15:22, "msarovar" <mo...@grommit.com> wrote:
> >> Hi,
>
> >> What is the most efficient way to generate a noise process that is Gaussian
> >> correlated in time from a white noise process?
>
> >> I'm having trouble coming up with a simple FIR filter that will do this.
>
> >> Any suggestions?
>
> >> Thanks,
> >> ~mohan
>
> > This link is three years old but still works:
>
> >http://math.mit.edu/~stevenj/preprints/iir-colored-noise-2007-03-13.pdf
>
> > It describes the use of IIR filters to efficiently (wrt to computer
> > time and memory) generate correlated sequences from white sequences.
>
> Any low-pass filter's output is correlated at high-enough frequencies.

Of course. The trick is to get the correlation you want as cheap as
possible.

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On Mar 10, 3:22=A0am, "msarovar" <mo...@grommit.com> wrote:
> Hi,
>
> What is the most efficient way to generate a noise process that is Gaussi=
an
> correlated in time from a white noise process?
>
> I'm having trouble coming up with a simple FIR filter that will do this.
>
> Any suggestions?
>
> Thanks,
> ~mohan

Bit confused here. If you pass Guassian white noise through a LTI
filter the output is still Guassian but coloured.

Hardy

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