On the full diversity property of a space-frequency code family with multiple carrier frequency offsets in cooperative communication systems

Space-frequency (SF) coded orthogonal frequency division multiplexing (OFDM) systems have been recently proposed for cooperative communications to achieve both full cooperative and full multipath diversities without the time synchronization requirement. In this paper, we show that the full diversity property still holds for a family of SF codes (rotation-based SF codes) when there are multiple carrier frequency offsets (CFOs) from relay nodes under the condition that the absolute values of the normalized CFOs are less than 0.5. We then prove that this full diversity property can be preserved if we seek to reduce the receiver complexity by using a zero forcing (ZF) method to equalize the multiple CFOs, before applying maximum likelihood (ML) decoding. Furthermore, by exploiting the properties of SF codes, we show that a specific permuted version of this family of SF codes can still achieve full diversity even when the inter-carrier interference (ICI) matrix is singular. This is possible as long as the maximum absolute value of the normalized CFOs is not less than 0.5. However, in this case the ZF method cannot be directly used to reduce the decoding complexity. To avoid the necessity of jointly considering all the subcarriers, two suboptimal detection methods are proposed in which full diversity is achieved even for singular ICI matrix. All these imply that the SF coded OFDM system is robust to both timing errors and frequency offsets.