  Gravitational waves are a sensitive probe into the structure of compact
astronomical objects and the nature of gravity in the strong regime.
Modifications of near-horizon physics can imprint on the late time ringdown
waveform, leaving behind a train of echoes, from which useful information about
new physics in the strong gravity regime can be extracted. We propose a novel
approach to compute the ringdown waveform and characterize individual echoes
perturbatively using the Fredholm determinants, which can be intuitively
represented via a diagrammatical scheme. Direct non-perturbative treatments can
also be easily implemented for some cases. Numerically, the method is also
effective and accurate even for relatively low resources.
