9.4 Noise filtering

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §9.4. Rewritten in original wording — no listing dumps. Continues 9.3b. Next: 9.4b.

Goal

Measured data look like

$$y(t)=s(t)+n(t).$$

You want the frequencies that belong to $s$, then (optionally) an inverse transform to a cleaned $s(t)$. Two classical handles: autocorrelation and filters (sinc low-pass returns in §9.4.4 / wavelets later).

Correlation → autocorrelation → power

With means already subtracted, correlate $y$ with $x$ at lag $\tau$, move to Fourier space, then specialize to autocorrelation. For $y=s+n$ with uncorrelated noise, cross terms drop and you recover the pure-signal power spectrum. Pick $c(\tau)$, $C(\omega)$, and $A(\omega)\propto|S|^2$ among lookalikes.

Large $|c|$ at some lag means the shapes line up; random relative phase averages toward zero. Broadband high-frequency hash drops out of $A(\omega)$ relative to a direct $|Y|^2$ of the raw trace — often all you need without knowing the noise waveform.

Practical checklist

1. Sample a known $s(t)$ (e.g. $1/(1-0.9\sin t)$ — infinite overtones from the geometric expansion).
2. DFT → semilog $|S|^2$.
3. Form $A(\tau)$ (discrete sum on your grid) and DFT it; compare to $|S|^2$.
4. Add noise $y_i=s_i+\alpha(2r_i-1)$; raise $\alpha$ until the direct spectrum drowns but autocorrelation still shows peaks — then find where even $A$ fails.

Filters (multiply $Y(\omega)$ by a low-pass mask, inverse transform) are the complementary attack when you want a cleaned time trace, not only a power spectrum.

Quiz

Autocorrelation and noise

1 / 4

Autocorrelation A(τ) measures…

Pick an answer to see if you are right.

Next: 9.4b Noise filtering labs.