  Spectrum scarcity necessitates innovative, spectral-efficient strategies to
meet the ever-growing demand for high data rates. Faster-than-Nyquist (FTN)
signaling emerges as a compelling spectral-efficient transmission method that
pushes transmit data symbols beyond the Nyquist limit, offering enhanced
spectral efficiency (SE). While FTN signaling maintains SE with the same energy
and bandwidth as the Nyquist signaling, it introduces increased complexity,
particularly at higher modulation levels. This complexity predominantly arises
from the detection process, which seeks to mitigate the intentional intersymbol
interference generated by FTN signaling. Another challenge involves the
generation of reliable log-likelihood ratios (LLRs) vital for soft channel
decoders. In this study, we introduce a lookup table (LUT) aided soft output
semidefinite relaxation (soSDR) based sub-optimal FTN detector, which can be
extended to higher modulation levels. This detector possesses polynomial
computational complexity, given the negligible complexity associated with soft
value generation. Our study assesses the performance of this soft output
detector against that of the optimal FTN detector, Bahl, Cocke, Jelinek and
Raviv (BCJR) algorithm as the benchmark. The likelihood values produced by our
LUT aided semidefinite relaxation (SDR) based FTN signaling detector show
promising viability in coded scenario.
