Time Reversal Matched Field Processing: An Analytical Justification

The paper develops an analytical model to explain the phenomena of super resolution focusing achieved with time-reversal, matched field processing (TR/MFP). Our analytical model is based on multipaths introduced due to total internal reflections of the transmitted signal within a random medium. Under minor constraints, we show that the transfer function Htilde(y^rarr _sx^rarr _py^rarr _s, w) for the field observed at the location y^rarr _s of the scatterer contains a squared sine term. Since multiplication of two similar sine functions in the frequency domain is equivalent to the convolution of two rectangular functions with equal widths in the time domain, the result of convolution is a triangular function with a distinct peak at y ^rarr _s. In comparison, the transfer function Htilde(y^rarr _sx^rarr _py ^rarro, w) for the field observed at an arbitrary site y^rarro includes multiplication of two sine functions with different lobe widths, which leads to a trapezoidal function with no distinct peak in the time domain. In addition, the transfer function Htilde(y^rarr _sx^rarr _py ^rarro, w) at any arbitrary site y^rarro contains a third sine function, which peaks to its maximum value of (2P + 1) at the location of the scatterer y^rarr _s. This factor further amplifies super resolution focusing at the scatterer