The spectrum, where flat is the good answer
Radioactive decay is a Poisson process, and the power spectrum of a Poisson process is flat — white noise, every frequency carrying the same expected power. A healthy counter watching background therefore produces no shape at all, and that featureless strip is the useful result: a statement that nothing periodic is happening.
It earns its place on the other case. A peak means something is arriving on a schedule, and decay does not have a schedule — mains hum on the tube's supply, a fan carrying a source past, a loose connector, firmware that batches its reporting. In the time domain every one of those looks exactly like more counts.
It accumulates. One periodogram of a Poisson process is flat in expectation and violently noisy in fact — every bin an exponential variable whose standard deviation equals its own mean. Averaging N of them divides that scatter by √N, so a real line climbs out of the grass while the grass settles. Windows are half-overlapped (Welch rather than Bartlett), which gets two averages out of each window's data instead of one.
Sigma alone is not a reason to believe anything, and this is the trap the panel is most likely to fall into. Sigma is computed for one bin, but the eye picks the tallest of 127, and the largest of many draws is far bigger than any single draw. One bin of an N-window average is Gamma(N)/N; asking how high B draws of it reach means solving N(r − 1 − ln r) = ln B, and expanding that gives
chance max ≈ 1 + √(2·ln B / N) + ⅔·ln B / N
At two windows that is 5.4×. So a bin at 4.9×, reading as a confident five sigma, is below what a perfectly healthy counter produces every time you look. At twenty-eight windows the same arithmetic gives 1.86×, and 4.9× is then overwhelming. The headline compares against that:
spectrum flat -- arrivals look random, as decay should (28 windows)
spectrum peak at 8s, 9.4x the mean (chance gives 1.86x), 7.2 sigmaThe leading term is the square root, and leaving it out cost a year of false alarms. This used to compute the bar as 1 + ln(B)/N, which is the correct answer for one periodogram — a single bin of white noise is exponential, and the largest of B of those does land near ln(B) above the mean — and the wrong one for an average of N, whose tail is not exponential at all. The two agree only at N = 1. Everywhere else the old form was too low, and it got worse the longer you watched: the bar sinks like 1/N while the real peak only sinks like 1/√N, so the two cross over.
Measured against a null built from this counter's own recorded counts, resampled i.i.d. so the spectrum is flat by construction and every flag is a false one:
| Watching for | Old bar | Real median peak | Called suspect |
|---|---|---|---|
| 30 minutes | 2.13× | 2.51× | 33% |
| 1 hour | 1.52× | 1.97× | 64% |
| 2 hours | 1.25× | 1.64× | 80% |
The longer the session, the more reliably it cried wolf — and a health check that fires four times in five on healthy hardware is not a health check, it is a decoration. "Called suspect" is the rule the panel actually applies, which wants a peak at 1.25× the bar rather than merely above it; that margin is the only reason the first row is a third and not a half, given that the median peak has already overtaken the old bar. With the square-root term restored the same null stays under 1% at every length.
Raising the bar did not cost the detection it is there for. The same null with a period-8 s source added on top — Poisson arrivals at a fraction of the background rate, which for this counter is 0.73 counts a second, about 44 CPM:
| Source at | Peak after 30 min | Called at 30 min | at 1 h | at 2 h |
|---|---|---|---|---|
| 0.2 × background | 2.7× | 2% | 2% | 11% |
| 0.3 × background | 3.0× | 19% | 50% | 92% |
| 0.4 × background | 4.5× | 65% | 98% | 100% |
| 0.5 × background | 6.4× | 95% | 100% | 100% |
The detection floor is a real number, and it sits between a third and a half of background. Half the counts arriving on a schedule is unmissable inside half an hour. A fifth of them is invisible at every length, and no threshold would fix that: half an hour at 44 CPM is thirteen hundred arrivals, and the line is not in them to be found. radbeeper and the Rust build were both wrong in the same way and are both fixed; chance_max()'s two quoted values, 5.4× and 1.86×, have a test pinning them in each.
Resolution grows with time, because it has to. Frequency resolution is 1/T for an observation of length T — you cannot resolve a 512-second period in 128 seconds of listening. Rather than pick one, RadBeeper runs a ladder: 128, 256 and 512 seconds side by side, fed the same samples. The 128 answers after two minutes; the 512 takes eight and a half but resolves four times as finely, and by the time it has anything to say it is the better answer. It costs about ten kilobytes, fixed for the life of the process.
The axis runs from long periods on the left to short on the right (2 s, the Nyquist limit), and the bars are coloured by significance rather than height.