Number/theoretic solutions to intercept time problems

A good radar warning receiver should observe a radar very soon after it begins transmitting, so in designing our radar warning receiver we would like to ensure that the intercept time is low or the probability of intercept after a specified time is high. We consider a number of problems concerning the overlaps or coincidences of two periodic pulse trains. We show that the first intercept time of two pulse trains started in phase is a homogeneous Diophantine approximation problem which can be solved using the convergents of the simple continued fraction (s.c.f.) expansion of the ratio of their pulse repetition intervals (PRIs). We find that the intercept time for arbitrary starting phases is an inhomogeneous Diophantine approximation problem which can be solved in a similar manner. We give a recurrence equation to determine the times at which subsequent coincidences occur. We then demonstrate how the convergents of the s.c.f. expansion can be used to determine the probability of intercept of the two pulse trains after a specified time when one or both of the initial phases are random. Finally, we discuss how the probability of intercept varies as a function of the PRIs and its dependence on the Farey points