Time Reversal Communication in Rayleigh-Fading Broadcast Channels with Pinholes

The paper presents an analysis of the time reversal in independent-multipath Rayleigh-fading channels with $N$ inputs (transmitters) and $M$ outputs (receivers). The main issues addressed are the condition of statistical stability, the rate of information transfer and the effect of pinholes. The stability condition is proved to be $MC\ll N_{\rm eff}B$ for broadband channels and $M\ll N_{\rm eff}$ for narrowband channels where $C$ is the symbol rate, $B$ is the bandwidth and $N_{\rm eff}$ is the {\em effective} number (maybe less than 1) of transmitters. It is shown that when the number of screens, $n-1$, is relatively low compared to the logarithm of numbers of pinholes $N_{\rm eff}$ is given by the {\em harmonic} (or {\em inverse}) {\em sum} of the number of transmitters and the numbers of pinholes at all screens. The novel idea of the effective number of time reversal array (TRA) elements is introduced to derive the stability condition and estimate the channel capacity in the presence of multi-screen pinholes. The information rate, under the constraints of the noise power $\nu$ per unit frequency and the average total power $P$, attains the supremum $P/\nu$ in the regime $M\wedge N_{\rm eff}\gg P/(\nu B)$. In particular, when $N_{\rm eff}\gg M\gg P/(B\nu)$ the optimal information rate can be achieved with statistically stable, sharply focused signals.