X- and Y-Codes for MIMO precoding

We consider a time division duplex (TDD) n_t × n_r multiple-input multiple-output (MIMO) system with known channel state information (CSI) at both transmitter and receiver. Using singular value decomposition (SVD) precoding at the transmitter, the MIMO channels are transformed into parallel subchannels. To improve the low diversity order, we propose X- and Y-Codes, prior to SVD precoding, to pair subchannels having different diversity orders. Specifically, a pair of channels is jointly encoded using a 2 × 2 real matrix, which is fixed a priori and does not change with each channel realization. Moreover, we propose X-, Y-Precoders with the same encoding matrices as X-, Y-Codes, which adapt to each channel realization. The optimal encoding matrices for X- and Y-Codes/Precoders are derived analytically to minimize the average error probability. Finally, we see that X-, Y-Codes/Precoders indeed achieve higher diversity gains at very low encoding/decoding complexity for both well- and ill-conditioned channels, respectively, when compared to other precoding schemes in the literature. We also observe that for the Rayleigh fading channel model X- and Y-Codes/Precoders exhibit the best average error performance.