Ping-pong beam training for reciprocal channels with delay spread

We present an iterative beam training procedure for a reciprocal channel with delay spread between two massive antenna arrays. The primary application is communication at millimeter wave frequencies. The procedure is based on ping pong transmissions. For each matricial channel coefficient, it tracks the left and right singular vectors associated to the maximal singular value. Based on the set of singular vectors, we build transmit and receive equalizers. A simple equalization based on matched filtering is showed to be optimal in the case of a Gaussian i.i.d. channel and a sparse channel.