Boxcar Integrator
Gated signal averaging of a repetitive pulse: gate-averaged signal, gate noise e_n·√NEB, SNR after N averages and the optimum gate width.
Gate-averaged signal
0.991 mV
Gate noise 221 µV rms
Single-shot SNR
4.49
Input noise 1.00 mV rms
SNR after 100 averages
44.9
Measurement time 100.0 ms
Optimum gate (centred)
593 ns
1.186 × FWHM at this B (white-noise limit 1.189); gate NEB now 5.00 MHz
⟨s⟩ = S·σ·√(π/2)/T · [erf((t_g + T/2)/σ√2) − erf((t_g − T/2)/σ√2)], σ = FWHM / 2.355
σ_gate = e_n · √NEB with NEB = 1/(2T) when B ≫ 1/T; the full single-pole form tends to σ_in = e_n·√B for B ≪ 1/T
SNR_N = SNR₁ · √N, measurement time = N / f_rep
Single-shot SNR vs Gate Width
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Gaussian pulse averaged over a rectangular gate; white input noise behind a single-pole input bandwidth, so the gate noise tends to e_n/√(2T) for wide gates. Shots average independently (√N). The optimum gate is found for the entered B; the pulse itself is not band-limited, so results need B·FWHM ≫ 1. Baseline drift, gate jitter, exponential (moving) averaging and 1/f noise are left out.
- Papoulis A., Pillai S. U. (2002). Probability, Random Variables and Stochastic Processes, 4th ed. McGraw-Hill. Variance of a time average of a stationary process, (2/T)∫(1 − u/T)R(u)du.
- Stanford Research Systems. SR250 Gated Integrator and Boxcar Averager, operating manual. Gated integration and shot averaging. thinksrs.com