Multipath medium identification using efficient sampling schemes

Time delay estimation arises in many applications in which a multipath channel has to be identified using pulses transmitted through the medium. Various approaches have been proposed in the literature to identify the time delays of the multipath components. However, these methods require high sampling rates. In this paper, we develop a unified approach to time delay estimation from low rate samples of the output of a multipath medium. Our approach results in a sampling theorem for analog signals defined over an infinite union of subspaces. The proposed method leads to perfect recovery of the multipath delays from samples of the channel output at the lowest possible rate, which depends only on the number of multipath components and the transmission rate, and not on the bandwidth of the probing signal. By properly manipulating the low-rate samples, we show that the time delays can be recovered using the well-known ESPRIT algorithm. Combining results from sampling theory with those obtained in the context of direction of arrival estimation methods, we develop necessary and sufficient conditions on the transmitted pulse and the sampling functions in order to ensure perfect recovery of the channel parameters at the minimal possible rate.