Separating detector noise from pulse-energy variance ( $Var (X) = Var (U) + Var (n) + 2 Cov (U, n)$ )

Separating detector noise from pulse-energy variance estimates source variability using a model of the readout's own fluctuations. In energy-calibrated units, write

X = U + n

. The exact identity is

Var (X) = Var (U) + Var (n) + 2 Cov (U, n) .

(1)

Only when the covariance vanishes can a separately estimated noise variance be subtracted directly. Covariance terms are part of measurement-model uncertainty propagation, rather than an optional correction.^[ref]

For an original independent-noise example, suppose the observed energy standard deviation is

0.50 nJ

and a matched noise-only measurement gives

0.30 nJ

. The inferred source standard deviation is

0.5 0^{2} - 0.3 0^{2} = 0.40 nJ

, not

0.50 - 0.30 = 0.20 nJ

. The source contributes

64%

of the observed variance in this stipulated model.

A noise-only record should use comparable gain, integration gate and reset conditions. Background subtraction removes an estimated mean, not the random variance remaining around it. Amplifier clipping, unmatched integrator reset or signal-dependent detector noise can invalidate an additive independent-noise model. A negative variance difference is not a physical negative source variance; it can indicate estimation uncertainty or mismatched assumptions.

Splitting pulses between two independently characterized readouts can provide a two-channel covariance estimate of pulse-energy variance under additional noise assumptions. Shared electronics noise and common drift must still be assessed; independence cannot be established just by using two detectors.

Table of contents
- Two-channel covariance estimate of pulse-energy variance Separating detector noise from pulse-energy variance

Two-channel covariance estimate of pulse-energy variance ( $Var (U) = Cov (X, Y) / (ab)$ )

 0  0

A two-channel covariance estimate of pulse-energy variance uses the component shared by paired readouts to estimate source fluctuations. In a stipulated linear model let

X = a U + n_{X}

and

Y = b U + n_{Y}

, with calibrated nonzero gains. If both noise terms are uncorrelated with

U

and with each other,

Cov (X, Y) = ab Var (U) .

(1)

This follows by expanding the covariance. Correlation assumptions are essential measurement-model inputs; the uncertainty guide retains covariance terms explicitly.^[ref]

For an original example, choose

a = 2 V/nJ

b = 4 V/nJ

and measured covariance

0.08 V^{2}

. The inferred source variance is

0.01 n J^{2}

, giving a standard deviation of

0.10 nJ

. Independent readout-noise variances

0.36

and

0.64 V^{2}

would give total variances

0.40

and

0.80 V^{2}

while preserving that cross-covariance.

But a shared noise covariance of

0.04 V^{2}

raises the observed cross-covariance to

0.12 V^{2}

. Ignoring it would infer a source standard deviation of

0.1225 nJ

, about

22.5%

too high.

A split-beam experiment records both channels for the same pulses, checks gains and measures dark or independent-input controls for correlated electronics. NIST's calibrated splitting apparatus supplies an example of the optical geometry, not a claim that it used this covariance estimator.^[ref] Matching pulse indices, bandwidth and integration windows is part of the proposed model.