Time reversal and the inverse filter

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Abstract
To focus ultrasonic waves in an unknown inhomogeneous medium using a phased array, one has to calculate the optimal set of signals to be applied on the transducers of the array. In the case of time-reversal mirrors, one assumes that a source is available at the focus, providing the Green’s function of this point. In this paper, the robustness of this time-reversal method is investigated when loss of information breaks the time-reversal invariance. It arises in dissipative media or when the field radiated by the source is not entirely measured by the limited aperture of a time-reversal mirror. However, in both cases, linearity and reciprocity relations ensure time reversal to achieve a spatiotemporal matched filtering. Nevertheless, though it provides robustness to this method, no constraints are imposed on the field out of the focus and sidelobes may appear. Another approach consists of measuring the Green’s functions associated to the focus but also to neighboring points. Thus, the whole information characterizing the medium is known and the inverse source problem can be solved. A matrix formalism of the propagation operator is introduced to compare the time-reversal and inverse filter techniques. Moreover, experiments investigated in various media are presented to illustrate this comparison.

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