Lattice codes for amplified direct-detection optical systems

Theories of shaping for lattice codes have been developed for systems (optical or non-optical) using coherent detection with additive white Gaussian noise (AWGN) and for direct-detection optical systems with AWGN. This paper considers shaping for amplified direct-detection optical systems in which signal-spontaneous beat noise, a form of signal-dependent noise, is dominant. An A-dimensional (A-D) signal is formed by modulating the intensities(squares of field magnitudes) of a sequence of N time-disjoint pulses. In field magnitude coordinates, signal energy is represented by a L² norm, and the optimal constellation bounding region is the nonnegative orthant bounded by an N-sphere. Under a continuous approximation, as Nrarrinfin, the ultimate shape gain is 1.53 dB and the induced signaling distribution on the constituent 1-D constellation becomes half- Gaussian. In practice, the ultimate shape gain can be approached when the 1-D constellation follows a truncated half-Gaussian distribution. We investigate the tradeoffs between shape gain and increases in constellation expansion ratio or peak-to-average power ratio. We compare our shaping results with those for coherent detection systems and direct-detection optical systems with AWGN.