On the time reversal of noise signals

This paper addresses the question of whether, and under what conditions, a noise trace changes its statistical properties when the time axis is reversed in direction. The autocorrelation function of the noise or its power spectrum cannot be used to identify the directionality of time in a noise signal since both are always the same for the signal and for its time reverse, regardless of the characteristics of the noise. However, the autocorrelation function can be generalized to represent the average of the products of powers of the signal at pairs of time instances separated by a given interval. If the powers are not the same for the first and second time instances, the generalized autocorrelation function can detect whether the statistical properties of a noise signal change upon the reversal of the direction of the time axis. We show that noise generated by systems that obey microscopic reversibility, i.e., that are at thermodynamic equilibrium, show the same statistical properties when evaluated forward and backward in time. A noise signal that does not demonstrate such time-reversal behavior discloses that the system that generates it is not at thermodynamic equilibrium. Several model examples are presented for illustration.