Aperture formation on reduced-channel arrays using the transmit-receive apodization matrix

In this paper, we present the transmit-receive apodization matrix (T/R matrix) as a tool for aperture synthesis on reduced-channel arrays. Reduced-channel arrays have a small number of transmit/receive channels multiplexed to a conventional dense array. For a 1D array of N elements, the T/R matrix is the N x N matrix of apodization values, where the rows correspond to transmit element positions and the columns correspond to receive element positions. We show that the round-trip beam pattern may be obtained from this matrix simply as the Fourier transform of its cross-diagonal sum. The aperture synthesis process consists of choosing the T/R matrix under certain constraints. If the T/R matrix has rank one, then a single transmit with parallel receive forms the beam (conventional case), and the round-trip beam can be separated into the product of the transmit beam and the receive beam. As the rank of the T/R matrix increases, greater beamforming flexibility is achieved, allowing for tradeoffs in SNR, frame-rate, and system complexity.