Shaping is a practical method to approximate the optimal Gaussian input distribution for a power-constrained AWGN channel. The Shaping Gain is the reduction in average constellation energy, compared to a uniform (equi-probable) cubical constellation. The maximal Shaping Gain at high SNR is 1.53dB, which can be achieved by a constellation shape of a high dimensional ball with a uniform distribution. By freeing ourselves from the idea of equally probable symbols, we can reach the full shaping gain even with a one dimensional constellation. This idea has been studied before within the high SNR regime. We study and improve the system performance for medium SNRs (rates of 1-3 bit/1D) by inserting dither, and using MAP decoding to take into account the unequal probabilities. We also incorporate coding, and compare the system to regular coded uniform QAM signaling. The results show improvements of up to 0.6dB with respect to older works, and a total shaping gain of about 1dB.
[1]
Frank R. Kschischang,et al.
Optimal nonuniform signaling for Gaussian channels
,
1993,
IEEE Trans. Inf. Theory.
[2]
G. David Forney.
Trellis shaping
,
1992,
IEEE Trans. Inf. Theory.
[3]
P. Kabal,et al.
Shaping Multidimensional Signal Spaces-Part I : Optimum Shaping , Shell Mapping
,
1998
.
[4]
Gottfried Ungerboeck,et al.
Channel coding with multilevel/phase signals
,
1982,
IEEE Trans. Inf. Theory.
[5]
Amir K. Khandani,et al.
Shaping multidimensional signal spaces - I: Optimum shaping, shell mapping
,
1993,
IEEE Trans. Inf. Theory.