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Low complexity MIMO equalization for long-haul SDM systems

Yuling Xue, Zhilong Zheng, Shuai Yuan, Liuzhu Wang, Hui Yan, Wendou Zhang, Shaohua Hu, Jing Zhang, and Kun Qiu

Open Access Open Access

Abstract

Space-division multiplexing (SDM) has been expected to support the continuous growth of transmission capacity. However, it suffers from high computation complexity that limits its physical implementations. In this paper, we propose and experimentally demonstrate a low-complexity MIMO equalization method to leverage the sparsity of weights and reduce the complexity by L1&L2-regularization in long-haul space-division multiplexing (SDM) systems. The L1-regularization finds the sparse solution of equalizer filters and substitutes it for optimal solution, reducing the complexity with performance degradation. On the other hand, the L2-regularization tends to produce a smoother estimation than L1 regularization and is therefore more robust to large variance. We conduct a 39.87-GBaud QPSK coherent optical transmission experiment based on a 4-core coupled-core fiber with the transmission distance from 1206-km to 7236-km. Comparisons on the equalization performance and computational complexity show that the sparse equalizer using L1&L2-regularization achieves a 30% reduction in complexity at the similar system performance, compared with the traditional time-domain MIMO.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Space-division multiplexing (SDM) has shown great potential in increasing transmission capacity to support the upgrade of transmission data rate and capacity [1]. Instead of using single mode fiber (SMF), SDM transmission systems employ few-mode or multi-core fibers (FMF or MCF) as the transmission medium to increase the transmission capacity by introducing an additional dimensionality. These fibers can be classified into two categories: weakly coupled fibers and strongly coupled fibers. For weakly coupled fibers, specially designed FMF and MCF with lower inter-modes/cores crosstalk provide a relatively simple scheme to migrate from current SMF systems, where the existing optical transceivers can be directly used [2,3]. For strongly coupled fibers, strongly coupled FMF has been demonstrated in long-haul transmission [4]. The multi-core few-mode fiber (MC-FMF) has attracted many research interests due to its ability to increase the number of spatial modes [5–7]. However, the long-haul transmission involving strongly-coupled few-mode fibers/cores face a trade-off between the complexity and performance. To reduce the complexity, one solution is to ignore the coupling between non-degenerate mode groups, leading to performance penalties [2,4,6]. However, the full MIMO equalization results in prohibitively larger complexity [5,7]. The coupled-core MCF (CC-MCF) exhibits potential for achieving lower MIMO complexity without performance degradation. In the strong-coupling regime, the channel impulse response increases linearly with the square root of the fiber length, resulting in smaller differential mode delays (DMDs) compared with weakly-coupled transmissions [8]. Moreover, the mode-dependent loss (MDL), which directly affects the channel capacity [9], decreases as the inter-mode coupling increases [10,11]. A strong linear coupling also mitigates the nonlinear impairment [12,13]. As a result, the CC-MCF possesses a smaller MDL and DMD as well as lower fiber nonlinearity compared to weakly coupled fibers. Several studies have demonstrated that the CC-MCF exhibits better per-core transmission capacity than current SMF systems [11,14]. However, the complexity of MIMO is an essential for the application of SDM systems in the evolution of high-speed and long-haul systems, as MIMO equalization is likely to be a major contributor to power consumption in the future SDM systems [15].

In situations where coupling between different spatial modes in a $D$-mode system cannot be neglected, a $D \times D$ MIMO equalizer has to be deployed to decouple modes. However, the computational complexity of an adaptive MIMO time domain equalization (TDE) algorithm scales quadratically with the number of modes and linearly with the DMD [16]. Therefore, a low-complexity MIMO equalization scheme is highly desirable. In [17], a DMD-independent least-mean-square (LMS) algorithm based on delay line has been demonstrated since the coupling in FMF is negligible and coupling only occurs in Mux/deMux. Frequency domain equalizers based on delay lines can further reduce the complexity [18]. In [19], a prior knowledge of the channel impulse response is obtained through the least square (LS) method, which has been used to determine the significant taps. An adaptive algorithm, i.e. the improved proportionate normalized LMS (IPNLMS), has also been demonstrated to adaptively suppress the insignificant taps [19]. However, the sparsity process introduces relatively large amount of calculations, thereby increasing the complexity. For strongly coupled fibers, where coupling occurs throughout the fiber and leads to a Gaussian-like impulse response [11,20], the sparsity of redundant coefficients in equalizers can be leveraged to reduce the equalization complexity.

In this paper, we propose and experimentally demonstrate a low-complexity MIMO TDE scheme for long-haul CC-MCF transmission systems. We add regularization terms in the cost function to leverage the sparsity of weights and reduce the complexity. We first use L1-regularization, i.e. L1 norm, to identify the redundant weights [21]. Therefore, the computational complexity of TDE is reduced by pruning the redundant weights. However, the L1-regularization leads to a performance penalty. We introduce the L2-regulation to enhance robustness and rectify negative effects such as noise enhancement by controlling the magnitude of weights and making the channel response more smoother. The L1&L2-regularization simultaneously reduces the complexity and guarantees the transmission performance. We conduct an experiment to transmit a long-haul four-core CC-MCF system with fiber lengths ranging from 1206 km to 7236 km to compare the performance of sparse equalizer with the traditional time-domain MIMO. The proposed L1&L2-regulation can achieve a reduction in total equalization complexity 30% without performance penalty, leading to a lower power consumption in the receiver-side DSP.

2. Principle of sparse equalizers

2.1 MIMO equalization

The SDM receivers use adaptive equalizers to demultiplex the coupled signals and compensate for linear channel effects such as DMD and modal crosstalk [22]. Figure 1 shows the schematic of the $D\times D$ MIMO equalizer.

 figure: Fig. 1.

Fig. 1. The schematic of a $D\times D$ MIMO equalizer using finite impulse response (FIR) filters

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The $n^{th}$ sample of the output of the $i^{th}$ channel is written as,

$$y_{i}(n)=\sum^{D}_{j=1}\boldsymbol{h}_{ij}(n)\boldsymbol{x}_{j}(n),\quad i=1,\dots,D,\quad j=1,\dots,D$$
where, $\boldsymbol {x}_{j}(n)$ is a vector of length $L$ and is constructed from $L$ consecutive samples received from the $j^{th}$ channel, $D$ is the total number of channels, and $\boldsymbol {h}_{ij}(n)$ is MIMO equalizer to mitigate the modal crosstalk between different channels and it is of the same length as $\boldsymbol {x}_{j}(n)$.

To adaptively adjust weight coefficients of FIR filters, the LMS algorithm is employed. The cost function and update equation are expressed as,

$$\begin{aligned}e_{i}(n)=d_{i}(n)-y_{i}(n)\end{aligned}$$
$$\begin{aligned}Loss(n)={\rm E}[e_{i}^{2}(n)]={\rm E}[(d_{i}(n)-y_{i}(n))^{2}] \end{aligned}$$
$$\begin{aligned} \boldsymbol{h}_{ij}(n+1)=\boldsymbol{h}_{ij}(n)+\mu e_{i}(n)\boldsymbol{\bar{x}}_{j}(n)\end{aligned}$$
where $d_{i}(n)$ is the training symbol and $\mu$ is the step-size parameter. The error signal in the $i^{th}$ channel is denoted by $e_{i}(n)$ and calculated as Eq. (2). The $Loss(n)$ is the cost function which is the mean square of the error signal $e_{i}(n)$. The LMS algorithm aims to minimize $Loss(n)$ by iterations.

2.2 Sparse equalizer using L1-regularization

Since the equalizer length is set according to the maximum DMD even though different cores coexist vary DMD, it is possible to select the most significant weights for equalization and discard the redundant ones. To achieve this, we first use a sparse equalizer with L1-regularization. In this approach, an L1-regularization is added to the cost function to search important tap coefficients. The block diagram of the sparse equalizer using L1-regularization is shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. Block diagram of the sparse equalizer using L1-regularization

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In the training stage, the cost function with L1-regularization is expressed as:

$$Loss_{1}=Loss+\lambda \sum^{D}_{i=1} \sum^{D}_{j=1} \left\|\boldsymbol{h}_{ij}\right\|_{1}$$
where the $Loss$ term is the same as Eq.  (3) and the second term in the right side is the L1-regularization term. The $\lambda$ is a positive parameter that controls the sparsity of weights. The column vector $\boldsymbol {h}_{ij}$ is composed of the weight coefficients of FIR filters. The L1-regularization term restricts the magnitude of coefficients, resulting in some weights shrinking to zero. Thus, the computational complexity can be reduced in the tracking stage [23]. As the L1-regularization term is added to the cost function, the weight coefficients are suppressed so that leading to a slight performance degradation. The computational complexity can be reduced by increasing $\lambda$ with slight performance degradation.

2.3 Sparse equalizer using L1&L2-regularization

The L1-regularization can identify the significant weights by forcing the unimportant coefficients to zero, but it sacrifices the system performance [24]. To achieve better performance, we use a sparse equalizer with L1&L2-regularization. Figure 3 shows the block diagram of the sparse equalizer using L1&L2-regularization. The proposed sparse scheme consists of three stages: training, pruning, and tracking stage.

 figure: Fig. 3.

Fig. 3. Block diagram of the sparse equalizer using L1&L2-regularization

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During the training stage, the cost function with both L1- and L2-regularization terms is utilized to update the weights. The cost function can be expressed as,

$$Loss_{2}=Loss+\lambda \{\alpha\sum^{D}_{i=1} \sum^{D}_{j=1} \left\|\boldsymbol{h}_{ij}\right\|_{1}+(1-\alpha)\sum^{D}_{i=1} \sum^{D}_{j=1} \left\|\boldsymbol{h}_{ij}\right\|_{2}^{2}\}$$
where, $\sum ^{D}_{i=1}\sum ^{D}_{j=1} \left \|\boldsymbol {h}_{ij}\right \|_{1}$ represents the L1-regularization and $\sum ^{D}_{i=1}\sum ^{D}_{j=1} \left \|\boldsymbol {h}_{ij}\right \|_{2}^{2}$ represents the L2 regularization. The parameter $\alpha$ determines the mixing proportion of the L1 and L2 norms of the weight coefficients, and $\lambda$ controls the influence of the combined regularization. The L2-norm improves the system robustness by decreasing the variability of the coefficients. Therefore, the sparse equalizer using L1&L2-Regularization has the potential to discard more coefficients, leading to further complexity reduction while maintaining system performance [25]. Since the L2-norm brings some coefficients near zero, we can prune the small-valued weight coefficients after the training stage. The detailed steps of the pruning stage are as follows: Firstly, a pruning factor, ranging from $0$ to $1$, denoted as $s$, is selected to remove a certain proportion of weight coefficients, which determines the number of coefficients used in the tracking stage and the complexity. Next, the weight coefficients are sorted in descending order as $\boldsymbol {w}=[w_{1}, w_{2},\ldots w_{j},\ldots w_{N}]$ according to their absolute value, i.e. $|w_{1}|\geq |w_{2}|\cdots |w_{j}|\cdots \geq |w_{N}|$, where $w_{j}$ represents the $j^{th}$ smallest coefficients, and $N$ is the number of weight coefficients that remain non-zero in all $D\times D$ filters after the training stage. The pruning symbol index is then calculated by $l=(1-s)\times N$. Finally, the threshold is determined as $\beta =\boldsymbol {w}[l]$, and the weight coefficients below this threshold are removed.

Note that the threshold is set by the pruning factor. A larger pruning factor leads to the removal of more weight coefficients. This can reduce the computation complexity. However, a higher pruning factor also results in the performance degradation.The tracking stage is the same as a traditional TDE MIMO.

2.4 Complexity analysis

The computational complexity of the linear equalizer can be quantified by the required number of complex multiplications per mode. Since the complexity reduction from sparsity is contributed to the tracking stage, we compare the computational complexity in the tracking stage. The computational complexity of a normal LMS MIMO equalizer is calculated in Table 1 where $L$ is the number of weights in the FIR. As the equalization process generates $D$ outputs in one step, the resulting total number of complex multiplication have to be divided by $D$ to represent the computational complexity per symbol per channel, thus it requires $2DL+1$ complex multiplications per symbol per channel in a normal LMS. In contrast, using the proposed two equalizers, after the training and pruning stages, the structure of the equalizer is simplified. As the number of the total non-zero coefficients of the MIMO equalizer is reduced from $D^2L$ to $M$, the two sparse equalizers require $2M/D+1$ complex multiplications, leading to a $1-M/(D^2L)$ complexity reduction.

Tables Icon

Table 1. The computational complexity of normal LMS

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3. Experimental setup and discussions

We conduct a 4-core MCF transmission experiment to evaluate the performance of the proposed MIMO FDE scheme. Figure 4(a) shows the experimental setup. At the transmitter (Tx) side, 39.87-Gbaud PDM QPSK signal was generated by an integrated coherent transmitter (ICT), and the center wavelength of the optical signal was 1550.04 nm. After 60 Hz polarization scrambling, the optical signal was split into four copies by splitter, and de-correlated with fixed and variable delay lines. Acoustic optical modulators (AOMs) and 10:90 couplers were used to expand the transmission link into the loop. The loop contains two spans of CC-MCF with length of 63 km and 57.6 km, respectively. Fan-In/Fan-out was used to connect MCF with SMF. The loop gain spectrum was controlled by optical equalizer (OEQ). Variable optical delay lines (VODLs) were used to compensate for the skew introduced by OEQ, AOM and patch cords. At the receiver side, the optical signal was filtered by WSS, and then injected into four integrated coherent receivers (ICRs) to convert the optical signals into electrical ones with four copies of local oscillator (LO). The electrical signals were captured by 16-channel digital storage oscilloscope (DSO), and processed with offline DSP. The structure of the offline DSP is shown in Fig. 4(b). We transmit a total number of 660882 symbols and use first 61440 symbols for the training step of the equalization.

 figure: Fig. 4.

Fig. 4. (a) Experimental setup for long-haul SDM CC-MCF transmission with 4 space channels (b) Offline DSP

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We first optimize the MIMO parameters, including the linear memory length, pruning factor, and mixing factor. Figure 5(a) shows the MIMO impulse response and its Gaussian fitting after the 1206-km CC-MCF transmission, calculated based on the sum of the squares of the 8$\times$8 time-domain transmission channel matrix [20]. To successfully demultiplex the signal, the memory length should cover the $\pm 2\sigma$ range of the Gaussian fitting, which is 124. Based on the estimated memory length, the $Q^2$-factor is depicted as a function of the memory length after the 1206-km transmission as shown in Fig. 5(b). When the memory length is less than 110, the performance increased rapidly as the memory length increases. However, after the memory length exceeds 110, the performance gradually saturates. Since the complexity of the equalizer is determined by the memory length, all memory lengths of the traditional time-domain MIMO and sparse equalizers are set to 125 for the 1206-km transmission. For 4824-km, 6030-km and 7236-km transmission, we optimize the memory lengths accordingly to 301, 325 and 361. In the following analysis we use the traditional TDE equalizer as the baseline to investigate the performance and complexity of the sparse MIMO equalizer.

 figure: Fig. 5.

Fig. 5. (a) MIMO impulse responses after the 1206-km transmission (b) $Q^2$-factors as a function of memory length of the linear equalizer

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We then investigate the transmission performance and complexity of the sparse equalizer using L1-regularization. Figures 6 (a) and (b) illustrate the effects of the regularization parameter on the transmission performance and complexity, respectively. As shown in Fig. 6(a), the transmission performance first increases and then decreases as the regularization parameter increases. This is because when the regularization parameter $\lambda$ is small, the weight coefficients tend to converge to larger values, which amplifies the noise and causes misjudgment in the decision process. However, by increasing the $\lambda$, some coefficients shrink to zero, reducing the number of multiplications and improving the transmission performance. But as the $\lambda$ keeps increasing, important coefficients are discarded, resulting in degraded transmission performance. On the other hand, in Fig. 6(b), as the regularization parameter increases, more coefficients are forced to zero, decreasing the computational complexity. Notably, the least $Q^2$-factor penalty is smaller than 0.02 dB, and the complexity is reduced by 20% compared to the traditional TDEMIMO.

The L1&L2-regularization differs from the L1-regularization in which both the regularization parameter $\lambda$ and the mixing parameter $\alpha$ affect the complexity. We first investigate the influence of the two parameters without pruning stage. Figures 7(a) and 7(b) illustrate the $Q^2$-factor and complexity after a 1206-km transmission with respect to the regularization parameter under different mixing parameter. As the mixing parameter increases from 0 to 1 with a step of 0.25, the complexity decreases at the cost of system performance degradation. It should be noted that while L2-regularization cannot simplify the equalizer, it can stabilize the transmission performance, whereas L1-regularization provides the complexity reduction with performance penalty. To balance the transmission performance and complexity, we set $\alpha$ and $\lambda$ to 0.75 and 0.025, respectively. After optimizing the regularization and the mixing parameters, the system performance is slightly improved by 0.02 dB, while the complexity is reduced from 2001 to 1555 complex multiplications per mode per symbol, corresponding to a 22.2% reduction. Since the equalizer length is set equally even though DMD varies in different cores, there exists redundant tap coefficients in equalizer.

 figure: Fig. 6.

Fig. 6. (a) $Q^2$-factors curve and (b) complexity curve, calculated with respect to the regularization parameter $\lambda$

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 figure: Fig. 7.

Fig. 7. (a) $Q^2$-factors curve and (b) complexity curve, calculated with respect to the regularization parameter $\lambda$ for 1206-km transmission under different mixing parameters $\alpha$. $Q^2$-factors versus (c) pruning factor $s$ and (d) complexity

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The pruning factor $s$ in the pruning stage is applied to further reduce complexity by removing some redundant coefficients. Figures 7(c) and 7(d) show the performance versus pruning factor and complexity after the 1206-km transmission with the L1&L2-regularization based sparse equalizer. When $s=0.05$, the L1&L2-regularization achieves similar transmission performance compared to traditional TDE MIMO with 30%-complexity reduction.

Figure 8 shows the $Q^2$-factors and complexity of the two sparse equalizers for different transmission distances. In Figs. 8(a-c), the L1-regularization can reduce complexity with slight performance degradation. The L1&L2-regularization can reduce more computational complexity when its performance is comparable to traditional TDE MIMO. With fiber length increasing from 1206 km to 7236 km, the L1&L2-regularization can gradually improve the transmission performance from 0.018 dB to 0.08 dB, which is benefited from the robustness of L2-regularization to large variance. From Fig. 8, the L1-regularization can reduce complexity by an average of 14.7% with only 0.02 dB $Q^2$-factor penalty. Compared to traditional TDE MIMO, the sparse equalizer using L1&L2-regularization offers an average of 30% reduction in complexity without any performance penalty. Furthermore, L1&L2-regularization can further reduce complexity by 19.3% on average, compared with L1-regularization.

To demonstrate the superiority of sparse equalizers, we introduce the IPNLMS algorithm as a comparison, where the IPNLMS is applied in the training stage to find locations of essential weights and the LMS is used afterwards. Fig. 9 shows the complexity comparisons between the L1&L2-regularization and IPNLMS with the same performance after different transmission distances. The results show that both equalizers can reduce the complexity compared with the traditional TDE. The sparse equalizer using L1&L2-regularization outperforms IPNLMS in complexity by 5% on average.

 figure: Fig. 8.

Fig. 8. $Q^2$-factors as a function of complexity for different equalizers after (a) 1206-km, (b) 4824-km, (c) 6030-km, and (d) 7236-km transmission

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 figure: Fig. 9.

Fig. 9. The complexity at different transmission distances

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4. Conclusion

We have proposed a low complexity MIMO TDE using L1&L2-regularization to reduce the computational complexity of traditional MIMO TDE for long-haul CC-MCF transmissions. We have experimentally verified the sparse equalizer in a QPSK transmission system based on 4-core CC-MCF ranging from 1206 km to 7236 km. The sparse equalizer using L1&L2-regularization achieves about 30% reduction in complexity with the same system performance.

Funding

National Natural Science Foundation of China (U22A2086).

Ackonwledgements

This work was supported by National Science Foundation of China (NSFC) (U22A2086)

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (9)
Fig. 1.
Fig. 1. The schematic of a $D\times D$ MIMO equalizer using finite impulse response (FIR) filters

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Fig. 2.
Fig. 2. Block diagram of the sparse equalizer using L1-regularization

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Fig. 3.
Fig. 3. Block diagram of the sparse equalizer using L1&L2-regularization

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Fig. 4.
Fig. 4. (a) Experimental setup for long-haul SDM CC-MCF transmission with 4 space channels (b) Offline DSP

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Fig. 5.
Fig. 5. (a) MIMO impulse responses after the 1206-km transmission (b) $Q^2$-factors as a function of memory length of the linear equalizer

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Fig. 6.
Fig. 6. (a) $Q^2$-factors curve and (b) complexity curve, calculated with respect to the regularization parameter $\lambda$

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Fig. 7.
Fig. 7. (a) $Q^2$-factors curve and (b) complexity curve, calculated with respect to the regularization parameter $\lambda$ for 1206-km transmission under different mixing parameters $\alpha$. $Q^2$-factors versus (c) pruning factor $s$ and (d) complexity

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Fig. 8.
Fig. 8. $Q^2$-factors as a function of complexity for different equalizers after (a) 1206-km, (b) 4824-km, (c) 6030-km, and (d) 7236-km transmission

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Fig. 9.
Fig. 9. The complexity at different transmission distances

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Tables (1)

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Table 1. The computational complexity of normal LMS

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Equations (6)

Equations on this page are rendered with MathJax. Learn more.

y i ( n ) = ∑ j = 1 D h i j ( n ) x j ( n ) , i = 1 , … , D , j = 1 , … , D
e i ( n ) = d i ( n ) − y i ( n )
L o s s ( n ) = E [ e i 2 ( n ) ] = E [ ( d i ( n ) − y i ( n ) ) 2 ]
h i j ( n + 1 ) = h i j ( n ) + μ e i ( n ) x ¯ j ( n )
L o s s 1 = L o s s + λ ∑ i = 1 D ∑ j = 1 D ‖ h i j ‖ 1
L o s s 2 = L o s s + λ { α ∑ i = 1 D ∑ j = 1 D ‖ h i j ‖ 1 + ( 1 − α ) ∑ i = 1 D ∑ j = 1 D ‖ h i j ‖ 2 2 }
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