I posting here my problem, perhaps somebody can point me how to proceed further :)

[The challenge]

I have an electronic system that can be modeled as a Wiener process with a drift $\mu$:

$ X_t = \mu t + \sigma W_t $

$E[X_t] = \mu t$ , $Var[X_t] = \sigma \sqrt t$

where the value of the drift $\mu$ is unknown and can be either positive or negative. The index $t$ can be seen as the time evolving form the initial time $t_0$, where $X_0 = 0$.

In my physical system I can only observe the $X_t$ evolution over time $t$.

My goal is to minimize the amount of time $t_{detection}$, required to estimate the sign of the drift $\mu$, with a given confidence interval $CI$.

[The current (naive) approach]

The system signal-to-noise-ratio is: $SNR(t) = \frac{\mu \sqrt t}{3 \sigma}$ for a CI of $3 \sigma$ = 99.7%

Intuitively, if I wait long enough, I could reach any arbitrary SNR level as the signal power is proportional to $t$, whereas the noise power only to the $\sqrt t$.

However, in my scenario my $t_{detection}$ is constrained, and with the current values I am off by a factor of x100 outside specification with this approach.

[The question]

Is there a way to reduce the $t_{detection}$ time for the same $CI$ confidence interval?

I am not familiar with Kalman filter, but I suspect it may help here? Or perhaps by looking at the derivate of $X_t$?


You are talking to an analog designer. Please bear with me... and thank you for any help!

  • $\begingroup$ Is $\mu$ constant? And just to see if I got it right, you are after the CI of the estimate? $\endgroup$ – A_A Jun 24 '19 at 6:34
  • $\begingroup$ look at en.m.wikipedia.org/wiki/Sequential_probability_ratio_test. do you know the standard deviation? $\endgroup$ – user28715 Jun 24 '19 at 15:39
  • $\begingroup$ yes! $\mu$ is constant. yes, I'm looking for the estimate and its CI. Also the standard deviation is known. $\endgroup$ – groviere Jun 26 '19 at 20:28

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