Linear Precoding for Faster-Than-Nyquist Signaling with Imperfect Channel State Information

In faster-than-Nyquist (FTN) transmission over frequency-selective fading channels, it is essential to eliminate both FTN-induced intersymbol-interference (ISI) (FTN-ISI) and channel-induced ISI (channel-ISI). Although frequency domain equalization (FDE) techniques can mitigate channel-ISI at the receiver, their performance is often inadequate. In contrast, cyclic prefix/suffix (CPS)-G-to-minus-half (GTMH) precoding shows superior effectiveness in mitigating channel-ISI. However, existing CPS-GTMH precoding techniques assume the transmitter obtains perfect channel state information (CSI) estimated at the receiver. This is impractical due to the high latency in satellite communication links. Therefore, this paper explores precoding for FTN signaling under imperfect CSI (ICSI) and proposes CPS-minimum mean squared error (MMSE) and linear pre-equalization (LPE)-MMSE precoding schemes based on the MMSE criterion. Both schemes employ a cyclic prefix to transform the channel-ISI matrix into the circular matrix, thereby avoiding inter-block interference. The primary distinction is that CPS-MMSE precoding handles FTN-ISI and channel-ISI together using MMSE precoding. In contrast, LPE-MMSE precoding addresses FTN-ISI with LPE and channel-ISI with MMSE precoding separately. Simulation results indicate that CPS-MMSE and LPE-MMSE precodings outperform CPS-FDE and LPE-FDE in mitigating channel-ISI when the estimation error is 0.005 and 0.01. For instance, with a packing ratio of 0.9, a roll-off factor of 0.25, and an estimation error of 0.01, CPS-MMSE and LPE-MMSE precodings achieve bit error rate (BER) gains of approximately 0.8 dB and 1 dB at a BER of 1e4, respectively. However, when the estimation error increases to 0.1, the BER performance of CPS-FDE and LPE-FDE surpasses that of the CPS-MMSE and LPE-MMSE precodings, indicating the proposed precoding schemes' unsuitability for larger channel estimation errors.