B Probability of seeing peak noise in a given time window

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The discussion revolves around calculating the probability of observing peak noise in an electric signal with an RMS noise value of 10uV, leading to a peak noise estimate of 66uV. The probability is influenced by the measurement apparatus's bandwidth, which determines the number of independent samples taken within a specific time window, such as 20 microseconds. The conversation highlights that if the noise is memoryless, the likelihood of reaching peak noise increases, while real-world noise sources exhibit dependencies that complicate this probability. A proposed model suggests using a differential equation to account for multiple independent noise sources with varying durations and amplitudes. The complexity of the model indicates that finding a practical solution may involve advanced mathematical techniques.
jaydnul
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Hi!

Say I have a electric signal that has an RMS noise value of 10uV, I would calculate peak noise by multiplying by 6.6, so 66uV. I am looking for an equation that describes the probability of seeing a noise voltage that reaches 66uV in a given viewing time window. For example if I look at the voltage signal for 20us, what is the probability of seeing 66uV?

Thanks!
 
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I think it depends upon the "bandwidth" of your measurement apparatus. For instance if the bandwidth is 1MHz then you are effectively taking 20 independent "samples" in 20 us. What is the probability that a single sample exceeds 6.6 σ for a (presumably) Gaussian distribution?

Consider this is from a non statistician, so corrections are invited!
 
It depends on your model for the source of the noise.

If it is completely memoryless, the noise at an instant being independent of all preceding levels, then you have an infinity of independent samples in any interval. You are guaranteed to get maximum signal in there somewhere.

In practice, noise is not like that. Any actual source of noise will have some duration. Your model could have a number, possibly infinite, of independent noise sources, each with a Poisson distribution of occurrence and some distribution of duration and amplitude (and randomly +/-). These parameters would rapidly tail off down the sequence so that the sum of the noise stays reasonable.

But do you really care about the peak across a continuous interval, or as @hutchphd suggests, only at certain instantaneous samples in the interval?

Edit:
I've thought of a model that might be tractable.
An infinite population of sources independently, with probability that one will start of ##\lambda\delta t## in each period ##\delta t##. Of those currently active, each stops with probability ##\mu\delta t## in each period ##\delta t##.
That yields a differential equation in the form of a recurrence relation. Using a generating function turns it into a PDE in two independent variables.
 
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Likes hutchphd and DaveE
Here's my attempt using the model I outlined:
Let ##P_n(t)## be the probability of n current sources at time t. For n>0:
##P_n(t+\delta t)=P_n(t)(1-\lambda\delta t-n\mu\delta t)+P_{n-1}(t)\lambda\delta t+P_{n+1}(t)(n+1)\mu\delta t##
and
##P_0(t+\delta t)=P_0(t)(1-\lambda\delta t)+P_{1}(t)\mu\delta t##.
Whence for n>0, in steady state:
##\dot P_n=-(\lambda+n\mu)P_n+\lambda P_{n-1}+(n+1)\mu P_{n+1}##
and
##\dot P_0=-\lambda P_0+\mu P_1##.
Using the generating function ##G(s)=\Sigma_{s=0}^\infty s^nP_n##, I get
##(1-s)G'=\sigma(1-s)G-\sigma P_0+P_1##, where ##\sigma=\lambda/\mu##.
Unfortunately, the solution appears to involve integrating a double exponential.
 
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