  A scheme is proposed that combines probabilistic signal shaping with
bit-metric decoding. The transmitter generates symbols according to a
distribution on the channel input alphabet. The symbols are labeled by bit
strings. At the receiver, the channel output is decoded with respect to a
bit-metric. An achievable rate is derived using random coding arguments. For
the 8-ASK AWGN channel, numerical results show that at a spectral efficiency of
2 bits/s/Hz, the new scheme outperforms bit-interleaved coded modulation (BICM)
without shaping and BICM with bit shaping (i Fabregas and Martinez, 2010) by
0.87 dB and 0.15 dB, respectively, and is within 0.0094 dB of the coded
modulation capacity. The new scheme is implemented by combining a distribution
matcher with a systematic binary low-density parity-check code. The measured
finite-length gains are very close to the gains predicted by the asymptotic
theory.
