Difficulty bounds in a white noise model

Question

Suppose we make a measurement in a real/virtual experiment for which the noise is given by a white noise (Gaussian) model. Suppose also that we have a very good filter or machine learning (ML) model to fit this data. Can the filter (or ML model) accurately cancel out the noise? I think there should be a hard region of the white noise model where the filter (or ML model) will have difficulty predicting? If we consider a Gaussian model, we find that the probability of the noise being in a range $[-0.5\sigma_n, 0.5\sigma_n]$, where $\sigma_n$ is the standard deviation of the Gaussian (noise), is about $40\%$ (see for example here ). I believe that this represents the best region to delineate, where the filer (or the ML model) will find the lowest estimate of the noise to signal ratio. As we go outside, i.e., for the rest of the 60%, the noise to signal ratio will gradually increase making the distinguishability more and more difficult. I am wondering if it is possible to derive some theoretical predictions (some kinds of difficulty bounds) with respect to $\sigma_n$, where the signal represents a Gaussian distribution of standard deviation $\sigma_s$.

Answer 1

When you assume that your noise is white normal, then its spectrum (spectral density) $S_0(u)$ is constant frequency independent, $S_0(u)=\frac{1}{2}\mathcal N _0$ for all $u$, and if passed through a linear time invariant filter characterized by the impulse response $h(t)$ whose Fourier Transform is $H(u)$, the transfer function, then when $h(t)$ is driven by white noise the output spectrum of the filter will be $S_1(u)=|H(u)|^2S_0(u)$ and its total noise power (variance) is $$P_1 = \int_{-\infty}^{\infty} |H(u)|^2S_0(u)du \\ =\mathcal N _0\int_{0}^{\infty} |H(u)|^2du$$

For a real normalized $h(t)$, $H(u)=\bar H(-u)$ and $\max_{u} |H(u)|=1$, we define its "noise bandwidth" by $B=\int_0^\infty |H(u)|^2du$. Then the variance $P_1$ of the output noise is $P_1 = \mathcal{N}_0 B$ and we see that for fixed input noise density $\mathcal {N}_0$ the the output noise variance $P_1$ is directly proportional with filter's noise bandwidth $B$.

For white normal input noise that is measured in an infinitesimal bandwidth $\Delta u$ around the frequency $u$ has normal probability distribution $\mathcal {G}(0,\sigma ^2):p(X)= \rm{exp}\big(\frac{-X^2}{2\sigma ^2}\big)$ with $\sigma ^2 =\mathcal {N}_0 |H(U)|^2 \Delta u $. The total output distribution is also normal $\mathcal {G}(0,P_1)$ with $P_1=\mathcal {N}_0 B$