The MISO free-space optical channel at low and moderate SNR

The capacity of the multiple-input single-output (MISO) free-space optical channel with a per-antenna peak-power constraint and a sum (over all antennas) average-power constraint is studied. The asymptotic low-signal-to-noise-ratio (low-SNR) capacity is determined exactly and close upper and lower bounds are presented in the low- and moderate-SNR regimes. The asymptotic low-SNR limit is achieved by having each transmit antenna signal either with zero or with the maximally allowed peak power, and the latter only if all stronger antennas also send at maximum peak power. In particular, for almost all channel gains, the input to achieve the asymptotic low-SNR capacity is such that its projection on the channel-gain vector has only two or three positive probability point masses, one of them being at zero. The lower bounds at finite SNR are numerical and are obtained using input distributions whose projection on the channel-gain vector has either two, three, or four positive probability masses. Finally, the paper presents two analytic upper bounds on the capacity of the MISO channel: the first one closely follows the proposed numerical lower bounds in the low-SNR regime, and the second one can improve on previous bounds in the moderate-SNR regime.