Statistics of a synchronous binary message pulse train applied to a regenerative repeater are related to those of the original binary message, which is assumed ergodic. It is shown that the ensemble of possible message pulse trains is a nonstationary random process having a periodically varying mean and autocovariance. A spectral density is calculated which shows line-spectral components at harmonics of the pulse rate and a continuous density function, both with intensity proportional to the square of the absolute value of the Fourier transform of the standard pulse at the frequency considered. The continuous component has many properties similar to thermal noise but differs in that, under certain conditions described, it can exhibit regularly spaced axis crossings, can be exactly predicted over finite intervals and is capable of producing discrete components when nonlinear operations are performed on it, even though no line spectral terms are originally present. The analytical results are applied to the problem of deriving a timing wave from the message pulse train by shock-exciting a tuned circuit with impulses occurring at the axis crossings.