Coherent Free-Space Optical Transmission with Diversity Combining for Gamma-Gamma Atmospheric Turbulence Mingbo Niu, Student Member, IEEE, Julian Cheng, Member, IEEE, Jonathan F. Holzman, Member, IEEE, and Robert Schober, Fellow, IEEE Abstract— The error rate performance of coherent free-space optical transmission is studied for Gamma-Gamma turbulent environments. The moment generating function of Gamma-Gamma distributed irradiance is obtained, along with a closed-form expression for the characteristic func- tion for the square root of Gamma-Gamma distributed irradiance. Bit- error rate expressions are presented for binary phase-shift keying multiple- input multiple-output free-space optical transmission with maximum ratio combining and equal gain combining receptions. These coherent optical systems are shown to greatly improve the system error rate performance. I. I NTRODUCTION As a cost-effective and license-free wireless technique with reduced time-to-market [1], [2], free-space optical (FSO) com- munications is a viable alternative to optical fibre in dense ur- ban areas without fibre infrastructure. However, convection and wind or temperature gradients can induce atmospheric turbu- lence, which leads to fluctuations of the optical signals. Sta- tistical models have emerged in the FSO literature to describe this atmospheric turbulence by way of log-normal, K, and Gamma-Gamma models. Weak turbulence is generally mod- eled by log-normal distributions [3]–[5], K distribution models are well-suited to stronger turbulence conditions [6], [7], and the Gamma-Gamma model can be used to describe a wide range of turbulence conditions [8]. Compared to single-input single-output (SISO) transmission systems, FSO systems with spatial diversity reception have been particularly successful at mitigating atmospheric turbulence. As a result, FSO system performance with diversity reception im- proves the FSO system performance and capacity, as has been shown for independent and correlated log-normal turbulence in [3] and [4], respectively. Links employing Q-ary pulse-position- modulation [5] and equal gain combining (EGC) reception [9], [10] led to similar performance improvements. The FSO literature has mainly considered turbulence with irradiance modulation with direct detection (IM/DD) schemes [3]–[5], [9], [10]. However, coherent homodyne/heterodyne FSO systems [7], [11]–[13] have great potential to improve the spectral efficiency and data rates compared to IM/DD systems. With this in mind, we present in this work an analysis of coher- ent FSO transmission in Gamma-Gamma distributed turbulence. We analyze coherent FSO transmission with Gamma-Gamma atmospheric turbulence and derive the moment generating func- tion (MGF) for Gamma-Gamma distributed irradiance. The characteristic function (CHF) for the square root of Gamma- This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC). M. Niu, J. Cheng, and J. F. Holzman are with the School of Engineering, University of British Columbia, Okanagan, Kelowna, British Columbia, Canada (e-mail: mingbo.niu@gmail.com). R. Schober is with the Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, British Columbia, Canada (e-mail: rschober@ece.ubc.ca). Gamma distributed irradiance is then derived. The MGF and CHF are used to obtain the bit-error rate (BER) for the SISO case. We ultimately derive new BER results for coherent FSO with maximum ratio combining (MRC) and EGC receptions. II. COHERENT FSO SYSTEM MODEL The coherent FSO system considered in this paper is a bi- nary phase-shift keying (BPSK) modulated heterodyne detec- tion scheme with N receiver antennas. In the following analysis, we assume that the effects of background noise are negligible, there is no phase noise, and the polarization state of the local oscillator and optical signals are matched with a polarizer at the photodetector surface. The coherent detection is then ob- tained by mixing the optical signal and local oscillator beams. Given that the received optical beam and local oscillator beam are mixed in perfect spatial coherence over a sufficiently small photodetector area, the photocurrent at the output of the nth pho- todetector is given by i n (t )= i dc,n + i ac,n (t )+ ν n (t ) [13] where i dc,n = R (P s,n + P LO ) and i ac,n (t )= 2R  P s,n P LO cos(ω IF t + φ ) are the DC and AC current, respectively, R denotes the pho- todetector responsivity, and ν n (t ) is a zero-mean additive white Gaussian noise (AWGN) process due to shot noise 1 . Here, P s,n is the received optical signal power incident on the beam split- ter at the nth receiver antenna, P LO denotes the local oscillator power which is assumed to be the same for all receiver branches, φ ∈{0, π } is the phase information, and ω IF  ω 0 - ω LO is the intermediate frequency, where ω 0 and ω LO denote the carrier frequency and local oscillator frequency, respectively. To facil- itate our analysis, we assume that the turbulence in each trans- mission link is independent and identically distributed (i.i.d.). The signal-to-noise ratio (SNR) of an optical receiver is de- fined as the ratio of the time-averaged AC photocurrent power to the total noise variance [14]. For coherent FSO with MRC reception, the instantaneous SNR at the combiner output can be shown as γ MRC = R N ∑ n=1 P s,n qΔ f (1) where q is the electronic charge and Δ f denotes the noise equiv- alent bandwidth of the photodetector. With P s,n = AI s,n , where A denotes the photodetector area, and I s,n is the instantaneous received optical irradiance at the nth receiver antenna, we can 1 In coherent optical detection, which should not be confused with coherent detection in RF literature, a sufficiently large local oscillator power P LO can make shot noise to be the dominant noise source, and the shot noise can be modeled as AWGN with high accuracy [7], [13], [14]. 978-1-4244-5711-3/10/$26.00 ©2010 IEEE 217 25th Biennial Symposium on Communications