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Noise-subtracted multiscale temporal power

haar_excess_power measures how a burst profile's temporal power is distributed across timescales, with the noise contribution removed empirically rather than assumed.

It exists because the obvious statistic — "the smallest significant timescale in this burst" — is not comparable between bursts. That quantity is a detection threshold, so it moves with signal-to-noise: the same burst observed twice as brightly appears to contain finer structure. Any population trend measured that way is partly a trend in S/N.

What it computes

At each scale \( s \) (a window width in samples), the estimator forms Haar differences between the means of adjacent windows, at every offset in the event window:

\[ d_s(i) = \frac{1}{s}\sum_{j=i}^{i+s-1} x_j \;-\; \frac{1}{s}\sum_{j=i+s}^{i+2s-1} x_j \]

The power at that scale is \( \langle d_s^2 \rangle \) over all offsets. A Haar difference responds to change on its own scale and cancels anything constant across the pair, so this decomposes the profile by how fast it varies.

The same calculation runs over each contiguous off-pulse run, and the pooled off-pulse power at each scale is subtracted from the event power:

\[ P_\mathrm{excess}(s) = P_\mathrm{event}(s) - P_\mathrm{off}(s) \]

Noise contributes power at every scale, so subtracting a measured off-pulse spectrum rather than a white-noise model removes both the radiometer term and whatever correlated noise the off-pulse region contains.

What comes back

HaarExcessPowerResult holds the event power, the noise power, the signed excess and the positive normalized excess at every retained scale, plus two summaries:

Field Meaning
characteristic_scale_ms Geometric mean scale, weighted by positive excess power. A single number describing where the burst's variability sits.
dominant_scale_ms The single scale carrying the most positive excess power.

excess_power_fraction_below integrates the normalized excess power at scales no larger than a given one — "what fraction of this burst's variability is faster than 1 ms".

from flits.analysis import excess_power_fraction_below, haar_excess_power

result = haar_excess_power(
    event_profile,
    [offpulse_before, offpulse_after],
    tsamp_ms=0.016,
)

print(result.characteristic_scale_ms, result.dominant_scale_ms)
print(excess_power_fraction_below(result, 1.0))

Using it across a population

The estimator is amplitude-weighted and noise-subtracted, which removes the first-order S/N dependence. It does not remove selection effects, and several remain:

  • Event-window selection. The window sets the largest measurable scale. Choose it consistently.
  • Finite resolution. No instrument reports power below its own sample time. When comparing instruments, restrict to the range of scales all of them resolve — otherwise the finer-sampled instrument appears to contain more fast structure by construction.
  • Scattering. Scattering suppresses fine structure genuinely, but it is a propagation effect, not an emission property.
  • Band occupancy and low S/N both alter the recovered distribution.

Two rules follow. Choose the comparison scale for the population, not per burst — otherwise you are comparing different questions. And calibrate with end-to-end injections plus bright bursts degraded to a common S/N, so the size of the residual bias is measured rather than hoped away.

Relationship to the other temporal diagnostics

Use this rather than the minimum-structure scan in Temporal and Spectral Analysis whenever the comparison is between bursts of unequal brightness. The scan answers "what is the finest structure I can demonstrate in this burst", which is the right question for a single burst and the wrong one for a population.