In free space optical communication systems, multi-aperture coherent optical receivers based on digital coherent beam combining (D-CBC) technique can provide exceptionally high sensitivity and are more robust to the atmosphere turbulence compared with the single aperture receivers with the same collection area. D-CBC relies on the digital phase alignment algorithm (PAA) to align the different versions of signals in phase. However, due to the limited working frequency and space of the digital signal processing (DSP) circuits, the main obstacle to realizing real-time phase alignment of multiple high-speed optical signals is the computation complexity. Therefore, we need to minimize the computation complexity while guaranteeing a satisfactory performance. In this paper, we investigate the relationship between the computation complexity and the combining loss (CL) for both maximum ratio combining (MRC) and equal gain combining (EGC) based D-CBC. Universal analytical expressions are deduced that allow easy minimization of the computation complexity for both MRC and EGC based receivers according to the prescribed CL and input optical signal-to-noise ratios (OSNRs). The analytical expressions are validated by extensive numerical simulations. It is demonstrated that the computation complexity is mainly determined by the quality of the signal with a larger OSNR in MRC, while it is determined by the overall quality of the signals in EGC. When EGC is replaced with MRC, the computation complexity can be reduced by more than 55% at the same CL when the OSNR difference between the signals to be combined is above 10dB. The maximum computation complexity increases exponentially with decreasing input OSNR lower limit and the smaller the CL, the steeper the slope. Furthermore, when the prescribed CL is relaxed from 0.1 to 0.5dB, the maximum computation complexity can be reduced by about 80%. The results provide useful guidelines toward practical phase alignment on a real-time platform.INDEX TERMS Multi-aperture receiver, free-space optical communication, phase alignment, digital coherent combining, equal gain combing, maximum ratio combining, combining loss

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Free-space optical communication (FSOC) systems with variable data-rates can support different missions with different transmission distances and ensure reliable transmission in periods of unfavorable weather conditions. The temporal diversity coherent combining (TDCC) technique can adjust the data-rate in a very large range using a software-defined manner by taking advantage of the digital coherent receivers able to recover the signal field. In this paper, we investigate the relationships between the computational complexity, optical phase alignment error, combining loss (CL) and data-rate for the TDCCbased variable data-rate coherent optical receiver. We also propose a method to minimize the computational complexity while achieving the expected output optical signal-to-noise ratio (OSNR) and the highest datarate for an arbitrary input OSNR. Numerical simulations and experiments are carried out to validate the analytical expressions and proposed methods. The results provide an efficient tool and useful guidelines for the design of low computation complexity TDCC-based variable data-rate FSOC systems. INDEX TERMSTemporal diversity coherent combining, free-space optical communication, variable data rate, phase alignment, combining loss.