Faster than Nyquist (FTN) signaling is an attractive transmission technique that is capable of improving the spectral efficiency with additional detection complexity at the receiver. Semidefinite relaxation (SDR) based FTN detectors are appealing as they provide good performance with linear decoding complexity. In this paper, we propose a soft-output semidefinite relaxation (soSDR) based FTN detector which has a similar polynomial complexity order when compared to its counterpart that only produces hard-output decisions. The main complexity reduction lies in re-using the candidate sequences generated in the Gaussian randomization (GR) step to produce reliable soft-output values, which approximate the calculation of the log-likelihood ratio (LLR) inputs for the channel decoder. The effectiveness of the proposed soSDR algorithm is evaluated using polar codes with successive cancellation decoding (SCD) through simulations, and its performance is compared against the state-of-the-art techniques from the literature. Simulation results show that the proposed soSDR algorithm provides reliable LLR values and strikes a good balance between detection complexity and bit error rate (BER) performance.
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