Limited-Shift-Full-Rank Matrices With Applications in Asynchronous Cooperative Communications

Shift-full-rank (SFR) matrices are matrices that have full row rank no matter how their rows are shifted. SFR matrices have been used lately as generator matrices for a family of space–time trellis codes to achieve full diversity in asynchronous cooperative communications, where the numbers of columns of the SFR matrices correspond to the memory sizes of the trellis codes. A systematic construction of SFR matrices, including the shortest (square) SFR (SSFR) matrices, has been also previously proposed. In this paper, we study a variation of SFR matrices with a relaxed condition: limited-shift-full-rank (LT-SFR) matrices, i.e., the matrices that have full row rank no matter how their rows are shifted as long as the shifts are within some range called delay tolerance. As the generator matrices for the previously proposed space–time trellis codes, LT-SFR matrices can guarantee asynchronous full diversity of the corresponding codes when the timing errors are within the delay tolerance. Therefore, due to the relaxed condition imposed on LT-SFR matrices, more eligible generator matrices than SFR matrices become available.