1. Introduction

Future deep space exploration missions, whether they are located in geosynchronous orbit (GEO) or on the moon or other planets, must have the real-time communication capability to transmit a large amount of data back to the Earth without significantly impacting the size, weight, and power (SWaP) available for the primary mission payload. Free-space optical (FSO) communication has recently attracted extensive research attention as an alternative transport technology to radio-frequency wireless communication, due to their reduction of space-side terminal SWaP and the ability to exploit the unregulated and nearly unlimited amount of bandwidth available in the infrared portion of the electromagnetic spectrum [1]. For many years, the SWaP reduction efforts focused on improving the receiver sensitivity of ground terminal receivers. This was achieved by replacing direct detection [2,3] or self-coherent [4,5] with coherent reception technology [6,7], advanced forward error correction (FEC) [8], and high-performance erbium-doped fiber amplifier (EDFA) or Raman amplifiers with extremely low noise figure [9]. However, the experimental results have demonstrated that the improvement in receiver sensitivity has approached the theoretical limit [10]. One way to continue reducing the SWaP of the space-side terminal is to create a ground receiver with a large effective collection area.

Several technical approaches have been proposed in the literature to increase the effective collection area and mitigate the effects of atmospheric turbulence. The obvious means is to increase the diameter for a single optical aperture of the ground terminal. However, the optical lens with a diameter larger than tens of centimeters is difficult to manufacture and very expensive [11]. Furthermore, with the increase of aperture diameter, the spatial phase across the lens plane caused by atmospheric turbulence changes more dramatically, which makes it increasingly difficult for coupling light from the large lenses into the optical fiber and will require more complex adaptive optics (AO) systems [12,13]. However, the AO technique not only needs expensive optical devices such as a wavefront sensor and a deformable mirror but also their operation speed should be faster than the time scale of temporal fluctuations of the wavefront. Another approach called spatial diversity uses multiple optical apertures to increase the collected photon energy. Multi-aperture combining technology can be divided into optical fiber combining [14,15] and digital combining [16]. Optical fiber combining technology needs to accurately match each branch signal with strict skew, phase, and polarization state, and the system performance depends on the accuracy and stability of multistage synchronous closed-loop control. These techniques have difficulty scaling to a large number of signals due to the amount of analog hardware required [15]. Digital combining has the advantages of flexible design and simple control. The maturity of the devices needed in digital fiber communication systems provides favorable conditions for the application of digital combining technology. Several digital combining algorithms have been investigated and extensively used including maximal ratio combining (MRC) [15,17], relative phase recursive estimation-based equal gain combining (EGC) [18], and selection combining (SC) [15]. The above algorithms introduced from radio-frequency wireless communication do not take into account the unique characteristics of coherent receiving signals, including high dynamic time-varying phase rotation and state of polarization, and polarization crosstalk, which would reduce the combining efficiency. Two branches phase difference estimation designed for single-polarization signal combining has the advantage of low system complexity [19], but the system performance is sensitive to the selection of branch combining sequence and the combining efficiency is low under high noise scenarios. The branch block phase correction [20,21] is proposed to synthesize multi-branch phases specifically for polarization multiplexing signals. However, the introduction of training symbols increases the redundancy, and the accuracy of phase estimation decreases with the increase of inter-branch skew and turbulence-induced fading depth. Complex-valued MIMO 2N × 2 adaptive equalizer in our previous work [22] is proposed to combine multi-aperture polarization multiplexing (PM) QPSK signal, where the multi-aperture combining is transformed into a MIMO equalization design.

I/Q imbalance and I/Q skew induced by the imperfected coherent receiver front end will reduce the digital combining efficiency especially when the system scales to a larger number of apertures and under ultra-low optical signal-to-noise ratio (OSNR) scenarios. However, the MIMO 2N × 2 equalizer [22] cannot solve the above tricky problem. In this work, a general analytical model for multiple-aperture combining is derived based on widely linear (WL) complex analysis. It is indicated that multi-aperture channel equalization and combining operations can be achieved simultaneously by using a MIMO 4N × 2 WL equalizer. In addition, the feasibility of the proposed adaptive equalizer is validated experimentally with an offline digital combining of 2.5-GBaud data rate PM-QPSK and PM-16QAM signals over a four-aperture coherent test bed. This technique can relax the time alignment requirements of the receiver sampling channels and multi-aperture branch since the skew can be compensated electronically thanks to coherent detection and digital signal processing.

This paper is organized as follows. In Section 2, the system architecture is proposed, and a MIMO widely linear model and a MIMO 4N × 2 WL equalizer are introduced. The tap coefficients update strategy is briefly given. The experimental setup of the four-aperture test bed is described in Section 3. The results and discussion are given in Section 4. Finally, conclusions are drawn in Section 5.

2. System architecture, modeling, and combining

2.1 System architecture

The multi-aperture coherent receiver architecture and digital signal processing (DSP) algorithm are shown in Fig. 1(a). Spatial optical signals are collected by N optical apertures. Each coherent receiver front end contains one EDFA, one ICR, and four high-speed analog-to-digital converters (ADCs). The premise of equalization and combining with a single adaptive equalizer is that the linear frequency response between multiple branches must meet the quasi-static characteristics. Therefore, a shared local oscillator (LO) is implemented to maintain the same phase deviation caused by frequency offset and laser linewidth of different aperture branches. After clock recovery, a 4N × 2 MIMO complex-valued equalizer is designed to realize weight optimization, phase synchronization, polarization demultiplexing, and linear distortion compensation. In addition, the interference at receivers, such as time mismatch between multiple apertures, I/Q imbalance, and I/Q skew of the coherent receiver front end, can also be compensated simultaneously to prevent the receiver sensitivity degradation caused by manufacturing imperfection of high complexity hardware architecture. The DSP algorithms before and after MIMO adopt 2× and 1× sampling per symbol processing, respectively. Frequency offset and phase offset compensation are performed after digital combining. Considering additive white Gaussian noise (AWGN) channel, the constellations of a 2.5-GBaud PM-QPSK signal before and after combining using single, two, and four-aperture with 4N × 2 equalizers are shown in the bottom right corner of Fig. 1(a). The signal quality is significantly improved as the increase of aperture number in a multi-aperture system even if each branch has an ultra-low OSNR.

 

Fig. 1. (a) Illustration of coherent DSP algorithm architecture for multi-aperture FSO communication receivers. Constellations of 2.5-GBaud data rate PM-QPSK before and after digital combining with single, two, and four-aperture schemes at OSNR = -2 dB for each aperture branch. (b) Block diagram of the complex-valued MIMO 4N × 2 WL equalizer. LO: local oscillator, ICR: integrated coherent receiver.

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2.2 Widely linear channel model

Real-valued continuous-time channel models with receiver I/Q imbalance and I/Q skew effects are derived. Then, compl-valued continuous-time channel models and complex-valued discrete-time equalizer models are given based on the theory of WL equalization. Frequency offset and phase offset are not included in the model derivations, although it is included in the simulation and experiment. No imbalance and skew are considered on the transmitter side. The transmitted complex-valued baseband signals of X and Y Pol. can be expressed as $x(t )$ and $y(t )$, and the corresponding real and imaginary parts are ${x_I}(t )= {{\cal R}e}({x(t )} )$, ${x_Q}(t )= {{\cal I}m}({x(t )} )$., ${y_I}(t )= {{\cal R}e}({y(t )} )$, and ${y_Q}(t )= {{\cal I}m}({y(t )} )$, respectively. ${{\cal R}e}(. )$ and ${{\cal I}m}(. )$ represent the operation of taking the real and image parts. The received optical filed signals after atmosphere turbulence for the i^th aperture branch are described by

2.3 Equalizer and tap coefficients update

To obtain the optimal N-aperture system performance, equalization for each aperture is first performed and then maximum ratio combining for multiple branches is implemented. Equalization is based on discrete-time domain processing. The complex-valued polarization signals for the i^th aperture branch after ADC are expressed as ${{\boldsymbol s}_{x,i}}$ and ${{\boldsymbol s}_{y,i}}$. ${{\boldsymbol s}_{x,i}}$ and ${{\boldsymbol s}_{y,i}}$ are the elements of complex vector spaces, ${{\boldsymbol s}_{x,i}}/{{\boldsymbol s}_{y,i}} \in {\mathrm{\mathbb{C}}^{K \times 1}}$. Here the same representation of [23] is used. The length of a complex vector is consistent with that of filter tap coefficient, K. The structure of a complex-valued MIMO 4N × 2 WL equalizer for the i^th polarization multiplexed receiver is given by

Zero-forcing (ZF) algorithm based on training symbols can be used to calculate the tap coefficients in Eq. (9) after establishing the channel and equalization models. In addition, the estimation of signal-to-noise ratio (SNR) and relative phase differences [19] of the N-aperture branch are performed to calculate ${c_i}$ in Eq. (10). However, in order to make a blind dynamic update of tap coefficients independent of the carrier phase, the update rules for the MIMO 4N × 2 WL equalizer used in this work are implemented based on stochastic gradient descent optimization. Experiment results show that the constant modulus algorithm (CMA) criterion is suitable for the QPSK format, and the cascade of CMA for pre-convergence and radially directed equalizer (RDE) for accurate tracking is adequate for the 16QAM format. The tap coefficients can be updated using a blind algorithm without the need for training symbols. The filter tap updates for the i^th aperture branch are calculated as,

3. Experiment setup of the four-aperture receiver

To demonstrate the proposed MIMO 4N × 2 WL equalizer, an all-fiber experiment is performed in which a four-aperture array is emulated by a cascade of fiber beam splitters, as shown in Fig. 2(a). A total of 64 K bits PM-QPSK or 128 K bits PM-16QAM signals are repeatedly transmitted through field-programmable gate array (FPGA). The PM-16QAM signal is generated by driving PM-I/Q modulator with four-channel 4-ary pulse amplitude modulation (PAM4) signals, where the PAM4 signal is generated by electrical combining two on-off keying (OOK) signals with one of which is attenuated by 6 dB through adjusting the swing control register in the FPGA. The laser wavelength and linewidth are 1550.32 nm and 20 kHz, respectively. An optical power divider with a splitter ratio of 25% is used to simulate two and four-aperture receivers. The laser output is provided to each receiver as a shared local oscillator after polarization maintaining fiber coupler with a splitter ratio of 25%. Each coherent receiver is equipped with one manual polarization controller (MPC), two optical attenuators, one EDFA, one ICR, four ADCs acquisition (with the sampling rate and resolution of 5 GSa/s and 8 bits), and FPGA processing board. The first attenuator, $AT{T_{i1}}$, of the i^th aperture branch is used to adjust the received power, and the second one, $AT{T_{i2}}$, optimizes the signal optical power input to ICR. The clock distribution circuit provides a 25-MHz reference both for one transmitter and four receivers, thus simplifying the requirement of the clock recovery algorithm in the verified experiment. 1 M sampling points of received signals for each aperture are extracted into MATLAB for the following DSP algorithm processing. The maximum cross-correlation pulse search method is adopted for integer sampling sequence synchronization to deal with the uncertainty of hardware trigger delay. The I/Q imbalance and I/Q skew are first estimated and compensated through an in-service calibration method [24]. Then, the skew of different branches and skew between X and Y Pol. are compensated, where the delay alignment criterion is to ensure that the decimal interpolation time in the Gardner-based clock recovery algorithm is equal. Then, hardware imperfections are regenerated in a digitally controllable manner to evaluate the impact on system performance. Pre-decision-based angle differential estimator (PADE) based feedback equalizer [25] is employed to compensate frequency offset. Viterbi-Viterbi (V-V) [26] based feedforward phase recovery algorithm is employed to compensate residual phase noise. Four-aperture coherent test bed and four ADCs acquisition cards are shown in Fig. 2(b) and (c). To clearly demonstrate the improvement effect of the combining algorithm on signal quality, Fig. 2(d) shows the constellations of PM-QPSK and PM-16QAM before PADE with single-aperture and four-aperture digital combining at single-aperture OSNR of −2 and 4 dB, respectively.

 

Fig. 2. (a) The experimental setup of the 2.5-Gbaud data rate PM-QPSK and PM-16QAM modulations four-aperture coherent receiver. (b) Photo of the four-aperture coherent test bed. (c) Four ADCs acquisition cards and 25-MHz reference clock generation. (d) Constellations of PM-QPSK and PM-16QAM before frequency offset compensation with single-aperture and four-aperture digital combining at single-aperture OSNR of −2 and 4 dB, respectively. ABC: automatic bias point controller.

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4. Experimental results and discussion

4.1 I/Q imbalance and I/Q skew

The impact of I/Q imbalance and I/Q skew on the convergence of equalizers in terms of normalized mean square error (MSE) of the training iteration and the bit error rate (BER) in terms of Q²-factor is evaluated based on the four-aperture system. Normalized MSE is defined as the dB form of the average error energy after the tap converges, and the formula is $MS{E_{dB}} = 20lo{g_{10}}\left[ {\frac{1}{M}\mathop \sum \nolimits_{k = 1}^M \frac{{|{R_d^2(k )- {{|{\widehat {x(k )}} |}^2}} |}}{{R_d^2(k )}}} \right]$, where M is the number of symbols participating in calculation. The Q²-factor, in dB, is calculated from the BER according to $Q_{dB}^2 = 20lo{g_{10}}\left[ {\sqrt 2 erfcinv({2BER} )} \right]$. Each point corresponds to the MSE calculated by the averaging operation of 1024 symbols. Equivalent OSNR is defined as the dB form of the ratio of the total power of all aperture signals to the total noise power within an optical bandwidth of 0.1 nm. Both complex-valued MIMO 8 × 2 strictly linear (SL) [22] and MIMO 16 × 2 WL equalizers are configured with 31 taps time response. The convergence of SL equalizer and WL equalizer with I/Q skew of 0, 100, 200, and 400 ps for all branches are shown in Fig. 3(a) and (b), where the OSNR of the received PM-QPSK signals for each aperture branch is set to be −2 dB (four-aperture equivalent OSNR of 4 dB). The numbers of symbols required for convergence with SL and WL equalizers are no more than 50 K (∼20 us). The performance of WL equalizer has a small improvement with the increase of the number of iterations. Normalized MSE of SL equalizer converges to −8.9, −8.5, −7.8, and −5.6 dB for I/Q skew of 0, 100, 200, and 400 ps, respectively. However, the same floor of −8.9 dB is obtained for WL equalizer regardless of the presence or not of I/Q skew. It is also found that the speed of iteration convergence decreases with the increase of skew. The convergence process for PM-16QAM with I/Q skew of 200 ps at single-aperture OSNR of 4 dB (equivalent OSNR of 10 dB) is shown in Fig. 3(c). To increase the probability of convergence, CMA is used for the first 64 × 1024 iterations, and RDE for later. The convergence both for SL and WL equalizer becomes extremely slow, and 1000*1024 symbols (∼0.4 ms) are required, which is caused by the incorrect update of tap coefficient induced by decision error of ${R_d}$ in Eq. (13). Q²-factor as a function of training iterations is shown in the right coordinate system. Even after 1000*1024 times of convergence, the symbol error rate is as high as 0.9 × 10⁻³ (Q²-factor = 9.3 dB). The converged tap coefficient weights of MIMO 16 × 2 WL equalizer without and with I/Q skew of 200 ps under the same condition in (c) are shown in Fig. 3(d) and (e). The energy of the converged tap coefficient for the conjugate butterfly branch is relatively weak in the absence of I/Q skew, but relatively strong in the presence of delay. 16 × 2 WL equalizer can effectively compensate the interference induced by I/Q imbalance and I/Q skew without an additional imbalance compensation algorithm. CMA can meet the tap update requirement of SL and WL equalizers in the PM-QPSK four-aperture system under strong turbulence. However, it is necessary to find a tap update strategy with faster convergence speed for PM-16QAM to improve the dynamic channel tracking ability under strong turbulence.

 

Fig. 3. Normalized MSE evolution during the training iterations of (a) complex-valued MIMO 8 × 2 SL and (b) complex-valued MIMO 16 × 2 WL equalizers for PM-QPSK signal at four-aperture equivalent OSNR of 4 dB with I/Q skew of 0, 100, 200 and 400 ps, respectively. (c) Normalized MSE for PM-16QAM modulation with complex-valued 8 × 2 SL and complex-valued 16 × 2 WL equalizers at equivalent OSNR of 10 dB and I/Q skew of 200 ps, and Q²-factor as a function of training iterations. Converged tap coefficient weights of MIMO 16 × 2 WL equalizer (d) without and (e) with I/Q skew of 200 ps under the same condition of (c).

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As shown in Fig. 4, the impacts of I/Q gain imbalance, phase imbalance, and I/Q skew on PM-QPSK and PM-16QAM system performance are evaluated both for SL and WL equalizers. The performances in terms of Q²-factor and the convergence probability of tap coefficients are evaluated in stationary regimes after the convergence of CMA and RDE algorithms. Both equalizers are configured with 31 taps. The performance of the MIMO 8 × 2 SL equalizer is degraded in presence of any impairments since the SL structure cannot handle I/Q mismatch. As it can be expected that the penalty in Q²-factor increases with the order of modulation formats. As it can be seen that the optimal performance is maintained using MIMO 16 × 2 WL equalizer for I/Q gain imbalance range from −5 to 5 dB, I/Q phase imbalance range from −30 to 30 degrees, and I/Q skew range from −0.5 to 0.5 symbol period. The system performance of the SL equalizer in absence of any I/Q mismatch is used as an evaluation benchmark, i.e. “optimal”. The convergence probability versus I/Q skew with MIMO 16 × 2 WL equalizer for PM-QPSK and PM-16QAM are shown in Fig. 4(c) and (f), respectively. The convergence probability is calculated over 100 independent experiments and random polarization rotation. Obviously, there has a decreasing probability of convergence with the increase of I/Q skew, and the convergence becomes more difficult for high-order modulation formats. For PM-16QAM, CMA and RDE algorithms fail to converge when ${\tau _{RX,i}} = 1$ and ${\tau _{RY,i}} = 1$.

 

Fig. 4. The effect of I/Q imbalance and I/Q skew on Q²-factor and the convergence probability of tap coefficient with MIMO 8 × 2 SL equalizer and MIMO 16 × 2 WL equalizer for PM-QPSK modulation with four-aperture equivalent OSNR of 4 dB. Q²-factor versus (a) I/Q gain imbalance (${\alpha _{RX,i}}$ and ${\alpha _{RY,i}}$), I/Q phase imbalance (${\varphi _{RX,i}}$ and ${\varphi _{RY,i}}$), and (b) I/Q skew (${\tau _{RX,i}}$ and ${\tau _{RY,i}}$). (c) The convergence probability versus I/Q skew with MIMO 16 × 2 WL equalizer. The same evaluations are performed for PM-16QAM modulation with an equivalent OSNR of 10 dB. Q²-factor versus (d) I/Q gain imbalance, phase imbalance, and (e) I/Q skew. (f) The convergence probability versus I/Q skew with MIMO 16 × 2 WL equalizer.

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4.2 Skew of multi-aperture branch and number of tap coefficients

The multi-aperture coherent digital combining system has excellent anti-turbulence performance at the cost of increasing hardware and algorithm complexity. The skew of multi-aperture branches caused by the unequal length of several optical/electrical components, optical fibers, and cables in series is difficult to align in engineering implementations. Thus, the impact of multi-aperture branch skew, ${\tau _i}$, and the number of tap coefficients, K, on MIMO 8 × 2 SL equalizer and MIMO 16 × 2 WL equalizer would be evaluated based on four-aperture PM-QPSK system with equivalent OSNR of 4 dB, and PM-16QAM system with equivalent OSNR of 10 dB, respectively. The delays of two branches are set to zero (${\tau _1} = {\tau _2} = 0$), and the other two branches are adjusted. Q²-factor versus the multi-aperture branch skew and the number of tap coefficients for SL and WL equalizers are shown in Fig. 5(a) and (b). The same Q²-factor of ∼10.6 dB is obtained with SL and WL equalizers for ${\tau _3} = {\tau _4} = 0$, and for the number of tap coefficients, K, range from 3 to 31. The length of K needs to be increased to maintain the system performance with the increase of branch skew. Take the case with ${\tau _3} = {\tau _4} = 4$ symbols for example, the low limit of Q²-factor is 7.6 dB in the range of K from 3 to 15. Only the 1^st and the 2^nd branch signals contribute to the combining, and the 3^rd and 4^th branch sub-equalizers cannot complete skew compensation with fewer tap coefficients. The convergence probability of tap coefficient versus ${\tau _i}$ and K for MIMO 16 × 2 WL equalizer is shown in Fig. 5(c). The singularity of CMA and RDE algorithms is not considered when calculating the convergence probability, that is, convergence to the same polarization is also considered successful. The convergence probability is greater than 90% when Q²-factor is larger than 10.3 dB. The same evaluations are performed for PM-16QAM modulation with an equivalent OSNR of 10 dB. The impacts of multi-aperture branch skew and the number of tap coefficients for MIMO 8 × 2 SL equalizer and MIMO 16 × 2 WL equalizer are shown in Fig. 5(d) and (e), and the convergence probability with MIMO 16 × 2 WL equalizer is shown in Fig. 5(f). Compared with PM-QPSK, the convergence probability of PM-16QAM decrease under the same skew of multi-aperture branches and the same tap coefficient length. Therefore, the number of tap coefficients is determined according to the maximum branch skew in engineering implementation. Incorrect convergence of tap coefficients and fewer tap coefficients will seriously reduce system performance.

 

Fig. 5. The effect of the multi-aperture branch skew and the number of tap coefficients on Q²-factor with (a) MIMO 8 × 2 SL equalizer and (b) MIMO 16 × 2 WL equalizer for PM-QPSK modulation with four-aperture equivalent OSNR of 4 dB. (c) The convergence probability of tap coefficient for WL equalizer in (b). The same evaluations are performed for PM-16QAM modulation with an equivalent OSNR of 10 dB. Q²-factor versus the multi-aperture branch skew and the number of tap coefficients for (d) MIMO 8 × 2 SL equalizer and (e) MIMO 16 × 2 WL equalizer. (f) The convergence probability with MIMO 16 × 2 WL equalizer.

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4.3 Number of multi-aperture branches

Then, the impact of multi-aperture branch number on the performance of MIMO 4N × 2 WL equalizer is investigated, where the results of 1, 2, and 4-aperture are obtained from the experiment, and these of 8, 16, and 32-aperture are from simulation. The Q²-factors of PM-QPSK and PM-16QAM as functions of equivalent OSNR for different aperture numbers are shown in Fig. 6(a) and (c). The curves of 1, 2, and 4-aperture are almost overlapped at the FEC limit (1 × 10⁻³ for BER, 9.8 dB for Q²-factor) for PM-QPSK. However, there are 0.2, 0.6, and 1.1-dB OSNR penalties are observed for 8, 16, and 32-aperture combining. The curves of 1 and 2-aperture are nearly overlapped for PM-16QAM, while the OSNR penalties of 0.2 and 1.2-dB are obtained for 4 and 8-aperture. Number simulations show that MIMO 4N × 2 WL equalizers with 16 and 32-aperture have poor performance. The OSNR penalty and the calculated combining efficiency for PM-QPSK and PM-16QAM versus the number of multi-aperture branches under different equivalent OSNR conditions are shown in Fig. 6(b) and (d), respectively. The experimental results show that the four-aperture combining efficiency of PM-QPSK exceeds 96% even at an equivalent OSNR of 0-dB (single-aperture OSNR of −6 dB), and 80% for PM-16QAM at an equivalent OSNR of 6 dB (single-aperture OSNR of 0 dB). In addition, the combining efficiency decreases with the increase of aperture number and noise, as the accuracy of tap coefficient convergence decreases with the increase of the MIMO scale, while noise increases the difficulty of CMA and RDE convergences. Therefore, enhancing the accuracy of tap coefficient convergence is a challenge that needs to be addressed in the future, especially in massive aperture systems under single-aperture ultra-low OSNR scenarios.

 

Fig. 6. The Q²-factor versus N-aperture equivalent OSNR for (a) PM-QPSK and (c) PM-16QAM with a different number of aperture branches. The theoretical curve is shown by a black dotted line. The OSNR penalty and combining efficiency as a function of the number of multi-aperture branches for (b) PM-QPSK and (d) PM-16QAM.

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Obviously, from the perspective of equalizer architecture design, the complexity of 4N × 2 WL equalizer is twice that of 2N × 2 SL equalizer. When the number of apertures and tap coefficients increase simultaneously, it is very difficult to use a commercial FPGA chip with a limited number of multipliers to implement 4N × 2 WL equalizer or 2N × 2 SL equalizer. The frequency domain implementation of WL equalizer [27] will be a feasible solution, which is not within the scope of discussion.

5. Conclusions

As the main contribution of this paper, a theoretical model of the multi-aperture coherent digital combining system is derived based on WL complex analysis. It is shown that the tricky problem of digital phase alignment for multi-aperture branches can be transformed into the design of large-scale MIMO equalizers and the optimization selection of tap coefficients. In addition, a complex-valued MIMO 4N × 2 WL equalizer is proposed to realize polarization demultiplexing, sampling deviation compensation, and digital combining of multi-aperture branch signals. The feasibility of the proposed 4N × 2 WL equalizer is verified by a 2.5-GBaud PM-QPSK and PM-16QAM modulations four-aperture offline experiment. Analysis shows that compared to SL equalizers, WL equalizers can not only compensate for inter-aperture skew mismatch but also can deal with I/Q imbalance and I/Q skew. Moreover, the four-aperture combining efficiency of PM-QPSK exceeds 96% even at a single-aperture extremely low OSNR of −6 dB, and 80% for PM-16QAM at a single-aperture OSNR of 0 dB. Therefore, multi-aperture coherent digital combining technology with a large-scale MIMO equalizer is expected to be an effective solution for ground terminal receivers in future deep space laser communication due to their excellent turbulence suppression performance and effective increase in the collection area.