1. Introduction

Mode-division multiplexing (MDM) in few-mode fibers (FMF) has received significant attention in recent years due to its potential to increase the per-fiber transmission capacity. However, MDM suffers from strong multimode interference in long-haul transmission due to random mode coupling. To recover information carried by each mode at the transmitter, multiple-input-multiple-output (MIMO) equalization is required. Due to large accumulated mode group delay (MGD), which grows with distance, the computational complexity of MIMO equalization can be unmanageable. To overcome this key challenge, frequency-domain equalization (FDE) has been proposed [1]. While the complexity of time-domain equalization (TDE) scales linearly with MGD, the complexity of FDE scales logarithmically [2]. Least mean square (LMS) FDE algorithm has been demonstrated for MDM, achieving excellent performance as well as low complexity [3].

Convergence speed of an MIMO algorithm is also an important consideration since it would determine the spectrum efficiency of the overall system, especially for the long-haul transmission. Generally, training symbols are employed for initial channel estimation in FDE before switching into a decision-directed algorithm for continuous tracking. Thus, slower convergence speed will lead to a longer length of the training sequence and thus lower spectral efficiency [4]. As compared to the LMS algorithm, the recursive least square (RLS) algorithm offers a superior convergence rate, especially for highly correlated input signals. The price to pay for this is an increase in the computational complexity. RLS has been applied for the MDM for achieving fast convergence speed [4,5]. In the simulation of [4], the implementation of RLS was realized by a cyclic prefix for each data block. The cyclic prefix can minimize the computational complexity per symbol but reduces the throughput efficiency. We have proposed an overlap-save based RLS-FDE algorithm for MDM with a fixed forgetting factor, resulting in a limited performance [6].

A fixed forgetting factor leads to a compromise between 1) the convergence speed and 2) the convergence error and stability. Therefore, a VFF-RLS algorithm can potentially resolve this compromise. Much effort has been directed to developing VFF-RLS algorithms for signal processing applications [7–9]. However, controlling the forgetting factor is rather expensive in terms of computational complexity due to extra operations such as division or extraction of a root. To circumvent this problem, we propose a strategy of choosing an exponential VFF. We apply this VFF-RLS-FDE algorithm to 1,000-km 6 × 6 MDM transmission system. Our experimental results show that the convergence speed of the proposed VFF-RLS/RLS algorithm is significantly improved over conventional LMS. The increase in convergence speed becomes more profound when transmission distance is increased. Moreover, the proposed VFF-RLS algorithm obtains better performance than conventional RLS (in terms of convergence speed and convergence error). Our analysis also indicates that the computational complexity of our proposed VFF-RLS-FDE algorithm is comparable with conventional LMS.

2. Theory

The adaptive recursive least square frequency-domain equalization is an extension of the time-domain RLS theory of adaptive filters [10]. The RLS-FDE algorithm is schematically shown in Fig. 1 for an m × m MDM system, which is implemented using the overlap-and-save method. Here, $y_i(n)$ and $x_i(n)$ represent the time-domain input signal vector and output signal vectors of mode $i$, $1\le i\le m$. After parallelization, two consecutive blocks are concatenated with an overlap rate 0.5 to obtain the input signal vector and the output signal vectors $Y(k)$ and $X(k)$ in the frequency domain using the overlap-and-save method. The procedure of this algorithm is similar to that of the LMS-FDE algorithm [1] except the updating part contained in the dotted box. The RLS algorithm aims to minimize an exponentially weighted cost function, which includes the assumption of mutual independence between frequency bins. The update process can be described mathematically as follows:

 

Fig. 1 Block diagram of the proposed RLS FDE.

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In this realization, $\overline{\overline{R(k)}}$ is an m × m inverse correlation matrix at the k-th frequency bin, which is time-varying and can be initialized with an identity matrix times a positive number proportional to the reciprocal of the average input signal power at this frequency bin; $0\ll \lambda <1$ is a forgetting factor and the superscript H denotes the Hermitian conjugate operator. Before updating the filter taps, Kalman vectors, $\overline{K}$, of each frequency is computed based on the signals in the current block and the inverse correlation matrix of the last block. It should be noted that the even or odd tributaries must be updated separately. In general, RLS can collect more information about the original signal by the Kalman vector compared with conventional LMS based on the steepest descent [11]. It should be noted that $\lambda$ can be time-varying. We propose an exponential VFF,

The computational complexity of the algorithm can be measured by the number of complex multiplications per symbol per mode. Let the FIR filter length be $N_f$ for each even/odd tributary, which equals to $2\Delta \tau L{R}_s$ where $\Delta \tau$ is the MGD of the fiber, $L$ is the link distance and $R_s$ is the symbol rate. Table 1 is the summary of the complexity of VFF-RLS/RLS and the conventional LMS algorithms. As an example, if we consider a 15 spatial modes transmission [12] where $m=30$ (including 2 polarizations) and $N_f=1024$, we can derive $C_{RLS}/C_{LMS}\approx 1.3$. While their complexities are comparable, VFF-RLS/RLS only requires a fraction of the training symbols as compared to that for the LMS algorithm, which leads to overall saving on system throughput and complexity. To optimize overall computational efficiency, the VFF-RLS/RLS algorithm can be switched to the LMS algorithm as soon as initial convergence is achieved.

Table 1. Complexity comparison between RLS and LMS

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3. Experimental results

The experiment setup for loop transmission is the same as that described in [13]. We transmitted 6 × 28-Gbaud QPSK signals at 1557.3 nm. The 28-Gbaud signal was generated from a 2²³-1 pseudo-random binary sequence (PRBS), and then the signal was loaded onto six spatial/polarization modes of the fiber after polarization/mode multiplexing and decorrelation delays. The recirculating loop was made up by a 25-km spool FMF and a few-mode EDFA, whose gain had been equalized. At the receiver, a mode de-multiplexer (MDMUX) was used to extract the spatial/polarization mode components of the received electric field. Three tributaries of signal finally reach three synchronized coherent receivers, whose waveforms were captured by three synchronized sampling scopes and processed offline.

Figure 2 shows the exponential envelope of VFF-RLS. To obtain a stable performance, we fixed ${\lambda }_{\max }$ to 0.99. After sweeping through$\gamma$ and $\tau$parameter space, we found their optimum values to be 0.28 and 0.008, respectively.

 

Fig. 2 Variable forgetting factor based on exponential envelope.

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In order to highlight the improvement of the convergence speed of VFF-RLS/RLS compared to LMS, we define the ratio of convergence speed (RCS) as the ratio of training symbols required by LMS to that by VFF-RLS/RLS for initial convergence to achieve a fixed level of mean squared error (MSE). Figure 3(a) shows the learning curves of the VFF-RLS/RLS algorithm and the conventional LMS algorithm at a transmission distance of 600 km. The inset shows the details of initial convergence. In Fig. 3(a), the green dotted line indicates the desired MSE level. For fair comparison, we fix the target MSE of 1.9 × 10⁻⁵, which is chosen because it can be reached by both the VFF-RLS/RLS and LMS algorithms, up to the maximum transmission distance of 1000 km. It should be noted that the improvement in convergence speed resulting from VFF-RLS-FDE compared with RLS-FDE is 15.3% at 600 km, which is still significant. Figure 3(b) shows the RCSs as functions of transmission distance where the solid and dotted lines are third-order polynomial fits. The advantage in convergence speed for VFF-RLS/RLS increases with the transmission distance. The reason behind this is that when link length becomes longer, lower SNR requires more robust performance and greater tracking capabilities, which can be provided by VFF-RLS/RLS. In particular, compared with conventional LMS, the VFF-RLS/RLS algorithm could improve the convergence speed by over 90% after 800 km. There is also a significant improvement of the convergence of the proposed VFF-RLS algorithm over the conventional constant-forgetting-factor RLS algorithm.

 

Fig. 3 Convergence speed comparison between VFF-RLS/RLS and LMS: (a) MSE convergence at 600-km vs. symbols; (b) Ratio of convergence speed of LMS and VFF-RLS/RLS vs. Distance.

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Figure 4 compares the transmission performance in terms of the Q-factor of VFF-RLS/RLS and the conventional LMS algorithms as the transmission distance was swept from 0 to 1,000 km. For a fair comparison, we use the same equalizer length (512) and the same initial equalizer matrix for each implementation. About 2.5 × 10⁵ symbols were used for Q² measurement. As can be seen, the Q² factors for RLS have a penalty of up to 0.2 dB on average compared with conventional LMS for transmission distance shorter than 700 km, and become superior to LMS when the transmission distance becomes longer. The Q² factors of VFF-RLS are up to 0.5 dB higher than those of the conventional LMS. The performances of LP₁₁ channels are better than the LP₀₁ channel likely because of the combined effects of the mode-dependent gain/loss and noise figure in our experimental setup.

 

Fig. 4 Comparison between Conventional LMS and VFF-RLS/RLS: Q² vs. Distance.

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4. Conclusion

An adaptive VFF-RLS-FDE equalization algorithm for long-distance MDM transmission has been demonstrated experimentally. The VFF-RLS algorithm improves the convergence speed significantly with superior convergence error performance compared with the LMS algorithm. The increase in convergence speed grows as the transmission distance increases.