Statistics of the performance of time reversal in a lossy reverberating medium

Abstract

It has been proposed [J. De Rosny, Ph.D. thesis, Université Paris VI, 2000] that the performance of time reversal at recreating a coherent pulse in a strongly reverberating medium is directly proportional to the number of resonant modes M actively taking part at the transmission of energy. This idea is here tested against experimental results showing that as soon as losses are taken into account, the quality of the focused pulse is a sublinear function of M leading to a saturation phenomenon that was previously unacknowledged. This is here proven to be caused by mutual coupling between lossy resonant modes, thanks to a statistical modal description of the transmission of signals through the medium. Closed-form relationships are proposed for the first two moments of the pulse signal-to-noise ratio linking them to the occupied bandwidth, the number of active modes, and the degree of resonance of the medium. These formulas, supported by experimental and numerical results, prove that the performance of time reversal can be affected by a strong statistical dispersion. The proposed analysis also predicts that time reversal is a self-averaging process when applied to a reverberating medium, thus, allowing the use of models developed in an ensemble-average framework.