Fractional averaging of repetitive waveforms induced by self-imaging effects

We report the theoretical prediction and experimental observation of averaging of stochastic events with an equivalent result of calculating the arithmetic mean (or sum) of a rational number of realizations of the process under test, not necessarily limited to an integer record of realizations, as discrete statistical theory dictates. This concept is enabled by a passive amplification process, induced by self-imaging (Talbot) effects. In the specific implementation reported here, a combined spectral-temporal Talbot operation is shown to achieve undistorted, lossless repetition-rate division of a periodic train of noisy waveforms by a rational factor, leading to local amplification, and the associated averaging process, by the fractional rate-division factor.