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? -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Max absolute value of colored Gaussian noise
Started by ●July 12, 2014
Reply by ●July 12, 20142014-07-12
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. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●July 12, 20142014-07-12
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. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●July 12, 20142014-07-12
Tim Wescott <tim@seemywebsite.really> writes:> 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?Tim!!! It's infinite! Filtering an identically-distributed random process doesn't change its distribution, only its correlation properties. I think... am I missing something? -- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com
Reply by ●July 13, 20142014-07-13
On 7/12/14 10:39 PM, Randy Yates wrote:> Tim Wescott<tim@seemywebsite.really> writes: > >> 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? > > Tim!!! It's infinite! Filtering an identically-distributed random > process doesn't change its distribution, only its correlation > properties.dunno exactly what you're saying, Randy. if the random process has a gaussian p.d.f., then filtering it will result in something like a Markov process with gaussian p.d.f. and you can use the joint distributions (or joint p.d.f.) to calculate the autocorrelation from a probabilistic ensemble average. so with white (up to Nyquist) gaussian noise going in, you can add up a whole pile of gaussian random varibles (scaled by h[n]) to get another gaussian r.v. but the output r.v. will have joint p.d.f. p( y[n], y[n-i] ) that is not equal to p( y[n] ) * p( y[n-N] ) but, Randy, you're right about the max value of a true gaussian random variable. for some reason, i think Tim knows that, but then i am curious to the nature of the question.> > I think... am I missing something?i dunno. i might be. i think if he defines his max value as some percentile range of likelihood, then i think we can come up with a number that isn't infinite. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●July 13, 20142014-07-13
On 7/12/14 11:04 PM, robert bristow-johnson wrote: meant to say> > p( y[n], y[n-i] ) > > that is not equal to > > p( y[n] ) * p( y[n-i] )-- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●July 13, 20142014-07-13
On Sat, 12 Jul 2014 22:39:08 -0400, Randy Yates wrote:> Tim Wescott <tim@seemywebsite.really> writes: > >> 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? > > Tim!!! It's infinite! Filtering an identically-distributed random > process doesn't change its distribution, only its correlation > properties. > > I think... am I missing something?Well, a quick disproof can be had by looking at the output of a lowpass or bandpass filter on an oscilloscope. Since any such filter made with real components is going to have an output signal as the one I describe, you can see if it's infinite or not. -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com
Reply by ●July 13, 20142014-07-13
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
Reply by ●July 13, 20142014-07-13
Hi,>> examine a finite chunk of the filter outputif I think of the observation interval as an OFDM symbol, the maximum is finite, relative to the total energy in the observation interval. A maximum occurs when all the "subcarriers" are in phase. A white power spectrum after the filter maximizes the sum, if energy is normalized for the "colored" signal (that is, after the "filter"). The question is now, how do I normalize the energy. If I think of the worst possible "colored" noise chunk, it is white and a pulse ("Gauss" doesn't matter as I know the exact signal vector, except absolute phase). So it may be possible to simplify the problem, stating "the highest possible peak of x seconds of signal bandlimited to y Hz is z". Don't know if that's what you want. In OFDM we try to avoid that kind of symbol (and put lightning rods on the towers...). _____________________________ Posted through www.DSPRelated.com
Reply by ●July 13, 20142014-07-13
On 2014-07-13 07:43, Tim Wescott wrote: [...]> Well, a quick disproof can be had by looking at the output of a lowpass > or bandpass filter on an oscilloscope. Since any such filter made with > real components is going to have an output signal as the one I describe, > you can see if it's infinite or not.You're mixing reality with theory. A Gaussian random process (white or colored) is *unlimited*, that is it max/min are +/- infinity. And it stays like that after (linear) filtering. Of course, such thing does not really exists, it is only convenient for modeling. Different story is mean and *variance*. For this, it was long time ago and I'm not anymore sure. Maybe you're preferred search engine can help. bye, -- piergiorgio
