A One Bit Feedback Based Beamforming Scheme for FSO MISO System Over Gamma-Gamma Fading

In this paper, we study a beamforming scheme for multiple-input single-output (MISO) free space optical (FSO) system over Gamma-Gamma fading channels. The considered beamforming scheme requires feedback of only one information bit from the receiver about the channel state information. If the feedback bit is always received error free, then the beamforming technique reduces to best transmit aperture selection scheme. It is inferred by bit error rate (BER) analysis in a 2 × 1 FSO system over Gamma-Gamma fading links, that repetition coding is the best of all arbitrary beamforming schemes with non-zero transmit weights. Further, it is also analytically shown that the simple transmit aperture selection scheme (or one bit feedback based beamforming with error free feedback) has a coding gain advantage over repetition coding in 2 × 1 FSO system over Gamma-Gamma fading links. However, it is shown by analysis that the simple transmit aperture selection based beamforming scheme is very sensitive to errors in the feedback bit and loses diversity gain even for very small error in the feedback. Then we establish conditions under which a 2 × 1 FSO system, employing one bit based beamforming with erroneous feedback, achieves full diversity. A simplified asymptotic upperbound of BER of the proposed scheme is obtained by using the order statistics and an optimized beamforming vector is found by minimizing this upperbound. We also analyze the BER performance of a general FSO MISO system that employs equal partitioning of transmit apertures for beamforming, based on one bit feedback. It is shown by simulations that the proposed beamforming scheme