Origin of Rice´s formula

The origin of S.O. Rice´s formula for the average rate of level crossings of a rather general random process is traced to his 1936 private notes entitled Singing Transmission Lines. The notes formulate an engineering problem in terms of the probability distribution of the absolute maximum of an envelope process in a closed interval. Rice showed that if the envelope of the reflected wave on a loaded transmission line were to equal or exceed the maximum value in the closed interval, then `singing´ or oscillation could be expected to occur and the analog transmission system would become unstable and useless for the transmission of information. The formula and its generalizations are now basic for the analysis of level crossings and first and second passage times of random processes