Time Reversal Communication in Multi-Path Fading Channels with Pinholes

The paper presents an analysis of the time reversal in multi-path 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 of transmitters. It is shown that when the number of layers, $n-1$, is relatively low compared to the logarithm of numbers of pinholes $N_{\rm eff}$ is given by $n^{-1}$ times the harmonic mean of the number of transmitters and the numbers of pinholes at all layers. On the other hand, when the number of layers is relatively large the effective number of pinholes diminishes exponentially. The energy efficiency is shown to be optimal when the power supply is set to the noise level times $BN_{\rm eff}$ and that the maximal information rate is roughly $BN_{\rm eff}$ when the stability condition is violated.