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Optimal LLR Clipping Levels for Mixed Hard/Soft Output Detection

Consider a communications system where the detector generates a mix of hard and soft outputs, which becomes fed to a soft-input channel decoder. In such a setting, it is of interest to find the optimal soft representation for the hard detected bits, which minimizes the probability of error at the decoder output. In this contribution we prove that for repetition codes transmitted over the AWGN channel using antipodal signaling, the optimal soft representation is given by the error probability at the detector output. This provides an indication of how "LLR clipping levels" should be chosen, e.g., in the context of list based detection of multiple-input multiple-output (MIMO) signals.