> I want to know the distribution of the maximum of the absolute
> value of a vector of samples of a colored, zero-mean Gaussian
> process. Or, stated another way, I want to shove white noise
> into a filter, then examine a finite chunk of the filter output
> for it's maximum absolute value.
Look up the Gumbel distribution. My memory is hazy,
but it applies to extreme value, worst case applications.
As I recall, it's used in designs where there's
some quantization error of a system parameter -
not signal noise - and one needs to estimate the
worst case performance.
--
Rich
Reply by Tim Wescott●July 15, 20142014-07-15
On Sat, 12 Jul 2014 17:03:55 -0500, Tim Wescott wrote:
> I need a number, and I'm feeling lazy; has anyone worked this out
> recently?
>
> I want to know the distribution of the maximum of the absolute value of
> a vector of samples of a colored, zero-mean Gaussian process.
>
> Or, stated another way, I want to shove white noise into a filter, then
> examine a finite chunk of the filter output for it's maximum absolute
> value.
>
> Anyone know the answer? Or should I sharpen my pencil and get to work?
I broke down and got an answer by simulation.
It looks like -- at least for the vector lengths that I'm using -- using
the max(abs) metric to estimate noise level is nearly as good as using
the variance: the ratio of measured standard deviation to mean for the max
(abs) measure is about 5% worse than using the "normal" variance estimate.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
Reply by Tim Wescott●July 15, 20142014-07-15
On Tue, 15 Jul 2014 09:58:35 -0400, Randy Yates wrote:
> Tim Wescott <tim@seemywebsite.really> writes:
>> [...]
>> My goal is to know the probability distribution of a measure of the
>> "magnitude" of noise, said measure being the maximum absolute value
>> found in a vector of samples of a colored (i.e., filtered white noise,
>> autocorrelation != impulse, etc.) random process.
>>
>> It is the mean (and variance, and pdf if I can get it) of this measure
>> that I am seeking.
>
> Tim,
>
> What is this for?
I go over this elsewhere in this astonishingly long (to me) thread, but
basically I'm receiving an on-off keyed signal with unknown noise levels
and more or less unknown amplitude. I'm trying to set the threshold for
detection by measuring the noise in a known quiet period, and setting the
actual threshold proportional to that.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
Reply by Randy Yates●July 15, 20142014-07-15
Tim Wescott <tim@seemywebsite.really> writes:
> [...]
> My goal is to know the probability distribution of a measure of the
> "magnitude" of noise, said measure being the maximum absolute value found
> in a vector of samples of a colored (i.e., filtered white noise,
> autocorrelation != impulse, etc.) random process.
>
> It is the mean (and variance, and pdf if I can get it) of this measure
> that I am seeking.
> On Mon, 14 Jul 2014 19:03:30 +0000, Eric Jacobsen wrote:
>
>> On Mon, 14 Jul 2014 12:31:45 -0500, Tim Wescott
>> <tim@seemywebsite.really> wrote:
>>
>>>On Mon, 14 Jul 2014 07:59:57 -0700, julius wrote:
>>>
>>>> On Monday, July 14, 2014 12:54:00 AM UTC-4, Tim Wescott wrote:
>>>>> On Sun, 13 Jul 2014 18:35:48 -0700, julius wrote:
>>>>>
>>>>>
>>>>>
>>>>> > On Sunday, July 13, 2014 1:46:39 PM UTC-4, Tim Wescott wrote:
>>>>>
>>>>>
>>>>>
>>>>> >
>>>>>
>>>>> >> Maybe I'll Monte-Carlo it (blech).
>>>>>
>>>>>
>>>>>
>>>>> >>
>>>>>
>>>>> > Most of the time, people who go down this route end up discovering
>>>>> > not
>>>>>
>>>>> > the answer to their inquiry, rather the quality of their RNG (or
>>>>> > lack
>>>>>
>>>>> > thereof).
>>>>>
>>>>>
>>>>>
>>>>> RNG?
>>>>>
>>>>>
>>>>>
>>>>> --
>>>>>
>>>>>
>>>>>
>>>>> Tim Wescott
>>>>>
>>>>> Wescott Design Services
>>>>>
>>>>> http://www.wescottdesign.com
>>>>
>>>> Random Number Generator. But you already have your answer, which is
>>>> that distribution has nothing to do with color of the process,
>>>> assuming stationarity etc.
>>>
>>>Actually I'm certain that the color of the process does impact the end
>>>measurement -- the measurement mean certainly rises the longer a vector
>>>of independent random variables you use; this indicates to me that the
>>>more correlated the variables are, the lower the mean is going to be,
>>>since the vector is going to act like it has fewer independent
>>>variables.
>>>
>>>The language is wishy-washy, but I hope you can get the sense through
>>>the hand waving.
>>
>> To be honest, I've had a very difficult time following what you've
>> written in this thread. I'm a little surprised that other people have
>> apparently been able to sort some of it out.
>>
>> What you wrote above doesn't parse for me at all.
>>
>> Mean of what? The vector? How is that at all dependent on the length
>> of the vector if the process is stationary?
>>
>> Correlation or dependence drives the mean?
>>
>> No grok.
>
> My goal is to know the probability distribution of a measure of the
> "magnitude" of noise, said measure being the maximum absolute value found
> in a vector of samples of a colored (i.e., filtered white noise,
> autocorrelation != impulse, etc.) random process.
>
> It is the mean (and variance, and pdf if I can get it) of this measure
> that I am seeking.
>
> The mean of such a metric DOES go up as the number of points in the
> vector go up, and you can see how it does intuitively: the more points
> you include, the more chances there are for the vector to have one really
> high-magnitude member.
>
> I'm pretty sure the variance falls with increasing vector length, but I'm
> not sure.
On Mon, 14 Jul 2014 19:03:30 +0000, Eric Jacobsen wrote:
> On Mon, 14 Jul 2014 12:31:45 -0500, Tim Wescott
> <tim@seemywebsite.really> wrote:
>
>>On Mon, 14 Jul 2014 07:59:57 -0700, julius wrote:
>>
>>> On Monday, July 14, 2014 12:54:00 AM UTC-4, Tim Wescott wrote:
>>>> On Sun, 13 Jul 2014 18:35:48 -0700, julius wrote:
>>>>
>>>>
>>>>
>>>> > On Sunday, July 13, 2014 1:46:39 PM UTC-4, Tim Wescott wrote:
>>>>
>>>>
>>>>
>>>> >
>>>>
>>>> >> Maybe I'll Monte-Carlo it (blech).
>>>>
>>>>
>>>>
>>>> >>
>>>>
>>>> > Most of the time, people who go down this route end up discovering
>>>> > not
>>>>
>>>> > the answer to their inquiry, rather the quality of their RNG (or
>>>> > lack
>>>>
>>>> > thereof).
>>>>
>>>>
>>>>
>>>> RNG?
>>>>
>>>>
>>>>
>>>> --
>>>>
>>>>
>>>>
>>>> Tim Wescott
>>>>
>>>> Wescott Design Services
>>>>
>>>> http://www.wescottdesign.com
>>>
>>> Random Number Generator. But you already have your answer, which is
>>> that distribution has nothing to do with color of the process,
>>> assuming stationarity etc.
>>
>>Actually I'm certain that the color of the process does impact the end
>>measurement -- the measurement mean certainly rises the longer a vector
>>of independent random variables you use; this indicates to me that the
>>more correlated the variables are, the lower the mean is going to be,
>>since the vector is going to act like it has fewer independent
>>variables.
>>
>>The language is wishy-washy, but I hope you can get the sense through
>>the hand waving.
>
> To be honest, I've had a very difficult time following what you've
> written in this thread. I'm a little surprised that other people have
> apparently been able to sort some of it out.
>
> What you wrote above doesn't parse for me at all.
>
> Mean of what? The vector? How is that at all dependent on the length
> of the vector if the process is stationary?
>
> Correlation or dependence drives the mean?
>
> No grok.
My goal is to know the probability distribution of a measure of the
"magnitude" of noise, said measure being the maximum absolute value found
in a vector of samples of a colored (i.e., filtered white noise,
autocorrelation != impulse, etc.) random process.
It is the mean (and variance, and pdf if I can get it) of this measure
that I am seeking.
The mean of such a metric DOES go up as the number of points in the
vector go up, and you can see how it does intuitively: the more points
you include, the more chances there are for the vector to have one really
high-magnitude member.
I'm pretty sure the variance falls with increasing vector length, but I'm
not sure.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
Reply by Randy Yates●July 14, 20142014-07-14
Tim Wescott <tim@seemywebsite.please> writes:
> [...]
> But, clearly you haven't done the math on this any time recently either.
It's not my math, it's my reading. My mind skipped the word
"distribution" in your original post. I.e., the DISTRIBUTION of the
maximum value of blah blah.
Which is admittedly a pretty serious error to make. Meh, maybe I
was tired..
--
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com
Reply by Eric Jacobsen●July 14, 20142014-07-14
On Mon, 14 Jul 2014 12:31:45 -0500, Tim Wescott
<tim@seemywebsite.really> wrote:
>On Mon, 14 Jul 2014 07:59:57 -0700, julius wrote:
>
>> On Monday, July 14, 2014 12:54:00 AM UTC-4, Tim Wescott wrote:
>>> On Sun, 13 Jul 2014 18:35:48 -0700, julius wrote:
>>>
>>>
>>>
>>> > On Sunday, July 13, 2014 1:46:39 PM UTC-4, Tim Wescott wrote:
>>>
>>>
>>> >
>>>
>>> >
>>> >> Maybe I'll Monte-Carlo it (blech).
>>>
>>>
>>> >>
>>>
>>> >>
>>> > Most of the time, people who go down this route end up discovering
>>> > not
>>>
>>> > the answer to their inquiry, rather the quality of their RNG (or lack
>>>
>>> > thereof).
>>>
>>>
>>>
>>> RNG?
>>>
>>>
>>>
>>> --
>>>
>>>
>>>
>>> Tim Wescott
>>>
>>> Wescott Design Services
>>>
>>> http://www.wescottdesign.com
>>
>> Random Number Generator. But you already have your answer, which is that
>> distribution has nothing to do with color of the process, assuming
>> stationarity etc.
>
>Actually I'm certain that the color of the process does impact the end
>measurement -- the measurement mean certainly rises the longer a vector
>of independent random variables you use; this indicates to me that the
>more correlated the variables are, the lower the mean is going to be,
>since the vector is going to act like it has fewer independent variables.
>
>The language is wishy-washy, but I hope you can get the sense through the
>hand waving.
To be honest, I've had a very difficult time following what you've
written in this thread. I'm a little surprised that other people
have apparently been able to sort some of it out.
What you wrote above doesn't parse for me at all.
Mean of what? The vector? How is that at all dependent on the
length of the vector if the process is stationary?
Correlation or dependence drives the mean?
No grok.
On Mon, 14 Jul 2014 11:49:20 -0700, gyansorova wrote:
> On Sunday, July 13, 2014 5:46:15 PM UTC+12, Tim Wescott wrote:
>> On Sat, 12 Jul 2014 21:47:51 -0400, robert bristow-johnson wrote:
>>
>>
>>
>> > On 7/12/14 6:03 PM, Tim Wescott wrote:
>>
>> >> I need a number, and I'm feeling lazy; has anyone worked this out
>>
>> >> recently?
>>
>>
>> >>
>> >> I want to know the distribution of the maximum of the absolute value
>> >> of
>>
>> >> a vector of samples of a colored, zero-mean Gaussian process.
>>
>>
>> >>
>> >> Or, stated another way, I want to shove white noise into a filter,
>> >> then
>>
>> >> examine a finite chunk of the filter output for it's maximum
>> >> absolute
>>
>> >> value.
>>
>>
>> >>
>> >> Anyone know the answer? Or should I sharpen my pencil and get to
>> >> work?
>>
>>
>> >>
>>
>> >>
>> > well, one thing, Tim, is that without some additional restriction
>> > about
>>
>> > how the process that colors the Gaussian noise and how the gaussian
>>
>> > p.d.f. random variable is generated. like we know that adding
>> > together
>>
>> > 12 independent uniform p.d.f. (1 unit width) random variables
>> > together
>>
>> > get a binomial distribution that well approximates a gaussian of 1
>> > unit
>>
>> > variance. and (assuming zero mean), it never gets beyond -6 and +6.
>>
>>
>> >
>> > so maybe you wanna know where the 99% likelihood limit level is. or
>>
>> > something like that.
>>
>>
>>
>> No, I really am looking for mean and variance. And I think the only
>>
>> missing restriction, which ought to be obvious, is that the coloration
>> on
>>
>> the noise must have a finite bandwidth (i.e., if you model the process
>> as
>>
>> white noise feeding a filter, the filter must have a finite noise
>>
>> bandwidth).
>>
>>
>>
>> But, clearly you haven't done the math on this any time recently
>> either.
>>
>>
>>
>> --
>>
>> Tim Wescott
>>
>> Control system and signal processing consulting
>>
>> www.wescottdesign.com
>
> It's mean is easy to find - just like a step input, the variance depends
> on whether the colouring process is IIR or FIR. For FIR it's easy, the
> variance is just the sum of squares of the filter coefficients times the
> variance of the input. For IIR you need one of those contour integrals
> or a table of integrals around the unit circle. For 2nd order it is
> quite easy to find.
The metric I'm taking is the maximum of the absolute value of the
elements of the vector. Not the variance of each element.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
Reply by ●July 14, 20142014-07-14
On Sunday, July 13, 2014 5:46:15 PM UTC+12, Tim Wescott wrote:
> On Sat, 12 Jul 2014 21:47:51 -0400, robert bristow-johnson wrote:
>
>
>
> > On 7/12/14 6:03 PM, Tim Wescott wrote:
>
> >> I need a number, and I'm feeling lazy; has anyone worked this out
>
> >> recently?
>
> >>
>
> >> I want to know the distribution of the maximum of the absolute value of
>
> >> a vector of samples of a colored, zero-mean Gaussian process.
>
> >>
>
> >> Or, stated another way, I want to shove white noise into a filter, then
>
> >> examine a finite chunk of the filter output for it's maximum absolute
>
> >> value.
>
> >>
>
> >> Anyone know the answer? Or should I sharpen my pencil and get to work?
>
> >>
>
> >>
>
> > well, one thing, Tim, is that without some additional restriction about
>
> > how the process that colors the Gaussian noise and how the gaussian
>
> > p.d.f. random variable is generated. like we know that adding together
>
> > 12 independent uniform p.d.f. (1 unit width) random variables together
>
> > get a binomial distribution that well approximates a gaussian of 1 unit
>
> > variance. and (assuming zero mean), it never gets beyond -6 and +6.
>
> >
>
> > so maybe you wanna know where the 99% likelihood limit level is. or
>
> > something like that.
>
>
>
> No, I really am looking for mean and variance. And I think the only
>
> missing restriction, which ought to be obvious, is that the coloration on
>
> the noise must have a finite bandwidth (i.e., if you model the process as
>
> white noise feeding a filter, the filter must have a finite noise
>
> bandwidth).
>
>
>
> But, clearly you haven't done the math on this any time recently either.
>
>
>
> --
>
> Tim Wescott
>
> Control system and signal processing consulting
>
> www.wescottdesign.com
It's mean is easy to find - just like a step input, the variance depends on whether the colouring process is IIR or FIR. For FIR it's easy, the variance is just the sum of squares of the filter coefficients times the variance of the input. For IIR you need one of those contour integrals or a table of integrals around the unit circle. For 2nd order it is quite easy to find.