Noise & the limits of averaging

A time series → its power spectrum (PSD) → its Allan deviation. Mix three kinds of noise and watch when averaging stops helping.

The instrument — mix noise, read three views

White noise

flat spectrum · averaging always helps (τ^−1/2)

Pink / flicker (1/f)

−1 slope · sets a floor averaging can't beat

Random walk / drift

−2 slope · averaging eventually hurts (τ^+1/2)

Samples

record length N = 2^k points at f_s = 100 Hz

the bundled trace used in the judgement task — one click, no download

…or load your own: one number per line; resets on any preset/slider

① Time series — Δg vs time (mean removed). White looks fuzzy; drift wanders off.

④ Histogram of Δg vs a matched Gaussian. White ≈ a clean bell; drift goes lumpy.

② Power spectral density (log–log) — slope tells the type: white ≈ 0 pink ≈ −1 walk ≈ −2

③ Allan deviation (log–log vs τ) — slopes white −½ pink 0 walk +½; red line = where averaging stops helping.

Try this Mix the noise, watch all three views

The question: you averaged your phone’s accelerometer for longer and longer — why did the reading stop getting better? Build a recipe with the presets and sliders (or load the exercise trace), then watch plot ③ — the Allan deviation.

Got a feel for it? Step to 2 · Notice →

Notice this What the curves are telling you

…

Explain this

You just saw the Allan deviation fall, flatten, then rise as you add pink and random-walk noise. In your own words: why does averaging eventually stop helping — and why doesn’t more data fix it?

Reveal a one-paragraph answer

Averaging beats down independent fluctuations: N independent white-noise samples shrink the mean’s uncertainty as 1/√N, so the Allan deviation falls with slope −½. But pink noise is correlated across time — long stretches drift together, so averaging over a longer window can’t cancel them; the curve flattens into a flicker floor. Random-walk noise is worse: the longer you wait, the further the value has wandered from where it started, so averaging over a long window actually adds error — the curve turns up with slope +½. The minimum between the two is the optimal averaging time: average that long and no longer. More raw data doesn’t move a drift-limited floor; only a better sensor (or modelling/removing the drift) does.

Show the math — why −½, 0, +½?

White noise: samples are independent, so averaging m of them shrinks the mean’s standard deviation by √m. With τ = m/f_s, that is σ(τ) ∝ τ^−1/2; the flat power spectrum (slope 0) is the frequency-domain face of the same fact. This −½ line is the standard quantum limit 1/√N when the “samples” are independent quanta (→ anchor 3).
Flicker / pink: power ∝ 1/f means fluctuations are correlated across all timescales — a longer average can’t cancel what drifts with it, so the Allan deviation goes flat (the floor).
Random walk: the value is the running sum of little kicks (power ∝ 1/f²); the longer you wait, the further it has wandered, so averaging over a long window adds error and σ(τ) ∝ τ^+1/2. The bathtub minimum is where the −½ and +½ lines cross.

Now connect it

Back to the lecture: this is the “systematic offset & noise” story behind the smartphone-g measurement — mean & standard error tell you precision only if the noise is white; PSD and Allan deviation reveal when it is not.
Forward to anchors 3, 4 & 5: this white-noise −½ slope has the same 1/√N scaling as the standard quantum limit — and is the SQL exactly when the white noise is the shot / projection noise of quanta (not, say, an amplifier's classical white floor). Squeezing (anchor 5) and entanglement bend that slope — but they can do nothing about a flicker floor or a drift. First know which wall you are hitting.

What good looks like:

Reads the bathtub minimum off plot ③ and names that τ as the optimal averaging time — the point where more data stops helping — not as the “best” setting in any absolute sense.
Identifies the branch on each side: a falling −½ (white) slope below the minimum, a rising +½ (drift / random-walk) slope above it — and reads the crossover as where the two meet.
Stops there. The minimum fixes when to stop averaging; it does not by itself name the physical noise source (that needs the PSD slope and domain knowledge).

Logbook (one entry): Tried · Stuck (where/how long) · Hub resource used · Changed my mind because… · Still unclear.

Exercise check. Set a recipe with all three noises on. On plot ③, read off the τ where the Allan deviation bottoms out, and say in one line which noise dominates just below it and just above it. (That τ is exactly where “more averaging stops helping.”)

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