Information transfer in disordered media by broadband time-reversal: stability, resolution and capacity

We analyse rigorously the time reversal of a multiple-input-multiple-output system in a strongly fluctuating disordered medium described by the stochastic Schr¨ odinger equation with a random potential under the sub-Gaussian assumption. We prove that in a broadband limit the conditions for stable super-resolution are the packing condition such that the spacing among the N transmitters and M receivers be more than the coherence lengthc and the consecutive symbols in the datum streams are separated by more than the inverse of the bandwidth B −1 and the multiplexing (or stability) condition such that the number of the degrees of freedom per unit time at the transmitters (∼ NB) be much larger than the number of the degrees of freedom (∼ MC) per unit time in the ensemble of intended messages. Here C is the number of symbols per unit time in the datum streams intended for each receiver. When the two conditions are met, all receivers simultaneously receive streams of statistically stable, sharply focused signals intended for them, free of fading and interference. We show that an O(P/ν) information rate P/ νcan be achieved with statistical stability under the condition NMP /(νB) where P is the average total power constraint and ν the noise power per unit bandwidth. The packing condition then implies that γ −d β −d cP /(νB) in the optimal transfer regime where γ is the Fresnel number, βc the coherence bandwidth and d the transverse dimension. Our results should be valid for diffusive waves with βc = Thouless frequency. Therefore, under the ideal packing and multiplexing conditions time reversal communications result in a high signal-to-interference ratio and low probability of intercept and is an effective means for achieving the information capacity of disordered media in the presence of multiple users.