Full diversity under multiple carrier frequency offsets of a family of space-frequency codes

A cooperative system may have both timing errors and multiple carrier frequency offsets (CFOs). To combat timing errors, space-frequency (SF) coded OFDM systems have been recently proposed for cooperative communications to achieve both full cooperative and full multipath diversities without time synchronization requirement. In this paper, we study the effect of multiple CFOs from relay nodes on a family of rotation based SF codes. We find that they can still achieve full diversities under the condition that the absolute values of normalized CFOs are less than 0.5. We further show that this full diversity property still holds for a complexity-reduced two-stage zero forcing (ZF) aided maximum likelihood (ML) decoding method, for which a ZF method is used to equalize multiple CFOs before ML decoding.