Probabilistic Shaping for Thermal Noise-Limited Direct-Detection Channels with Peak and Average Power Constraints

We study probabilistic shaping for thermal noiselimited direct-detection channels under both peak and average power constraints. We use the high-rate continuous approximation (HCA) to derive induced marginal densities and shaping gains under varying peak power constraints. The HCA analysis finds that the exact induced marginal density is well approximated by a truncated beta distribution or a truncated exponential distribution for sufficiently long shaping block length N. We derive closed-form expressions for the shaping gain for finite N and finite peak-to-average (power) ratio (PAR) and show that shaping gains within 0.1 dB of the ultimate shaping gain can be achieved with N = 100 or a PAR as low as 4. We study shaping using finite, discrete constellations and find that a scaled geometric distribution obtained by sampling a truncated exponential distribution achieves a shaping gain within 0.22 dB of a numerically optimized distribution for modulation orders M = 4, 8, and 16 and varying constraints on the PAR. We also consider shaping for peak- and average-power-constrained direct-detection channels with limited extinction ratios. We find that the induced marginal density is still well approximated by a truncated beta distribution or a truncated exponential distribution and derive closed-form expressions for the shaping gain.