 Ghost-level consistency test for a delayed pulse replica

A ghost-level consistency test for a delayed pulse replica checks whether measured side lobes agree with the simple model of one real delayed pulse and its nonlinear correlation ghost. It can reject that model; agreement alone does not prove that no real prepulse exists.

Use the separated, identical-shape replica model in third-order pulse correlation ghost. Let $a$ be the positive-delay lobe and $b$ the opposite-delay lobe, each divided by the central correlation peak. For delayed-to-main intensity ratio $0<\epsilon <1$,Thus the measured ratio gives $\epsilon =b/a$, and the first equation supplies an additional consistency condition. The frequently useful approximation $b\simeq a^2$ omits the small central-peak correction; exactly, $b/a^2=1+{\epsilon }^3$.

For an original pair of measured relative heights $a=0.080$ and $b=0.020$, their ratio would require $\epsilon =0.25$. That value predicts $a=0.25/1.015625\approx 0.246$, not $0.080$. These two readings therefore cannot both come from the stated single-replica model with equal instrument sensitivity at the two delays. A real earlier pulse is one possible explanation, but background, unequal pulse shapes or additional replicas can also invalidate the model.

An independent check varies the delayed replica while holding the main pulse fixed. Before central-peak normalization, halving $\epsilon$ halves the real side lobe and quarters its ghost. This attenuation scaling of a third-order correlation ghost probes the nonlinear origin rather than only one accidental height match. The existing parent diagram shows a direct numerical example without an actual pulse at the ghost's delay.

The historical correlator's inserted-etalon experiment provided a controlled physical postpulse and a known temporal direction.^[ref] The algebraic height test here is an independent ideal calculation. Use unsaturated, background-corrected readings with comparable response and lobe resolution; otherwise the measured ratios need not represent the model's intensities.