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Abstract
This paper presents an orbital angular momentum mode division multiplexing (OAM-MDM) ring-core fiber transmission method based on non-orthogonal multiband carrierless amplitude and phase (NMCAP) modulation with an extremely randomized trees-hidden Markov model (ExtraTrees-HMM)-based equalizer. The ExtraTrees-HMM equalizer uses the statistical characteristics of the received distorted signals to model the nonlinear channel of the system to classify these distorted signals into corresponding constellation classes. Experiments were conducted using a 216 Gbit/s OAM-MDM NMCAP modulation optical fiber communication system with 2 km ring-core fiber transmission and the results show that compared with a conventional Volterra nonlinear equalizer (VNE), the proposed ExtraTrees-HMM equalizer could improve the receiver sensitivity by 1 dB for OAM mode l = + 2, and 0.6 dB for OAM mode l = + 3. In addition, the computational complexity of the proposed equalizer was reduced by 43.94% compared with the VNE. In brief, the ExtraTrees-HMM is a promising equalization candidate for ultra-high-capacity inter-data-center interconnections.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The ubiquitous use of cloud services has triggered a demand for higher-speed optical links for data center interconnections and intra-data-center networks [1,2]. Intensity-modulation direct-detection (IM/DD) schemes are preferred for such short-reach and high-capacity scenarios due to their low costs, low power consumption, and small footprint [3]. In order to keep up with the fast growth in data traffic, there is an urgent need to expand the capacity of short-reach IM/DD-based optical communication systems. Space division multiplexing (SDM) technology offers an effective solution to alleviate the capacity crunch by employing the spatial modes of light as information carriers, and has shown promise for applications in optical fiber communications [4,5]. Recently, in SDM technology, orbital angular momentum (OAM) multiplexing [6–18] and linearly polarized (LP) modes [19,20] has attracted interest. An OAM-carrying beam is a helically phased beam with an azimuthal phase [21,22]. Due to the intrinsic orthogonality and unbounded states of OAM modes, OAM-mode division multiplexing (OAM-MDM) has shown significant potential in terms of increasing transmission capacity in short-reach IM/DD-based optical fiber communication systems [6–11].
In short-reach IM/DD, pulse amplitude modulation (PAM), carrierless amplitude and phase modulation (CAP), and discrete multi-tone (DMT) are the main modulation schemes. Both PAM and conventional CAP are sensitive to non-flat spectral channels, which inhibits their ability to achieve higher bitrates, while DMT, as a multicarrier modulation, segments a frequency fading channel into multiple relatively flat sub-bands and can fully exploit the spectral resources, which enables efficient, simple and one-dimensional equalization techniques. However, DMT suffers from a high peak-to-average ratio (PAPR), leading to severe nonlinear distortion [23]. Inspired by DMT, a novel scheme was proposed entitled multiband CAP (MultiCAP) [24], in which the CAP signals are broken onto independent sub-bands occupying different frequency bands. Each sub-band can then be assigned a tailored signal power and modulation order. MultiCAP yields similar OSNR performance and complexity with low PAPR when compared with DMT [25]. However, it suffers from low spectral efficiency (SE) due to the need for guard bands between sub-bands. Recently, a non-orthogonal MultiCAP (NMCAP) modulation scheme was proposed to improve the SE [26,27], in which the carrier frequencies were modified to compress the sub-bands into non-orthogonal by overlapping adjacent sub-bands. To sum up, OAM-MDM IM/DD systems with NMCAP modulation has the potential to enhance the transmission capacity and to improve the spectrum resource utilization to a significant extent.
In an OAM-MDM IM/DD system with NMCAP modulation, transmission performance is mainly limited by (i) unavoidable intrinsic inter-mode/mode group (MG) crosstalk (XT), due to the imperfections in the fabrication of the fibers and multiplexers/demultiplexers [16]; (ii) inter-band interference (IBI) caused by the overlap of the sub-bands in NMCAP modulation [28]; and (iii) nonlinear noise induced by electro/optical devices [29]. Although many equalizers, such as the Volterra nonlinear equalizer (VNE), have been shown to be effective in single-mode fiber (SMF) systems, it is still challenging for them to compensate for the complex distortions in OAM-MDM systems with NMCAP modulation. Novel deep learning-based equalization methods have also been reported, such as convolutional neural networks (CNNs) [30], bidirectional long short-term memory networks (Bi-LSTMs) [31], bidirectional recurrent neural networks (Bi-RNNs) [32], and Echo State Networks (ESNs) [27]. However, their high complexities make it difficult for these equalizers to compensate for the impairments effectively and efficiently, especially for IBI and inter-MG XT in OAM-MDM with NMCAP modulation.
In this manuscript, an IBI components extraction-based ExtraTrees-HMM equalizer is presented for OAM-MDM ring-core fiber (RCF) communication systems with high spectral efficiency NMCAP modulation. Experiments were conducted to verify the effectiveness and efficiency of the proposed equalizer compared with conventional alternatives.
2. Principle of operation
2.1 Principle of operation of NMCAP modulation in OAM-MDM RCF optical communication
This section outlines the theoretical framework for NMCAP modulation in an OAM-MDM RCF optical communication system, as shown in Fig. 1.
First, the number of multiplexed OAM modes is defined as Θ, the number of sub-bands in NMCAP modulation is N, and n is the sub-band index, where $n = 1,2,\ldots ,N$, and the modulation order of the $n\textrm{th}$ sub-band is defined as ${M_n}$.
At the transmitter, N independent pseudorandom binary sequences (PRBSs) are generated in non-return to zero (NRZ) format, denoted by ${d_n}(t)$. Then, ${d_n}(t)$ is mapped into ${M_n}\textrm{ - ary}$ quadrature amplitude modulation (QAM) symbols, denoted by ${d_{QAM[n]}}(t)$.
${d_{QAM[n]}}(t)$ is then split into in-phase and quadrature components, denoted as $d_n^I(t)$ and $d_n^Q(t)$, respectively. Both components are unsampled and pulse shaped by filters $f_n^I(t)$ and $f_n^Q(t)$. The pulse shaping filter pair $\{ f_n^I(t),f_n^Q(t)\}$ is given as:
All the outputs of the shaping filters are then summed to generate an NMCAP signal as follows:
The NMCAP signal is subsequently modulated onto a Gaussian beam using a Mach-Zehnder modulator (MZM), and then converted into an OAM beam with different OAM modes using a spatial light modulator (SLM). Following this, the modulated OAM beams are multiplexed using polarization beam combiners (PBCs) and coupled to an RCF for transmission.
At the receiver, each OAM mode is demultiplexed and demodulated into a Gaussian beam using a vortex phase plate (VPP) with the corresponding vortex pattern, converted into an electrical signal $\hat{s}(t)$ via a photodetector (PD), and recorded by an oscilloscope (OSC).
2.2 Principle of operation of the ExtraTrees-HMM equalizer
The proposed ExtraTrees-HMM method mainly consists of two procedures: (i) extraction of IBI components and construction of feature vectors, and (ii) equalization of ExtraTrees-HMM. The details are as follows.
(i) Extraction of IBI components and construction of feature vectors
In the proposed ExtraTrees-HMM equalizer, the IBI components are first extracted as additional characteristic values from the OAM-MDM NMCAP signal besides the demodulated CAP signal for every sub-band. Figure 2 shows the extraction process of two adjacent IBIs, i.e., $IBI_{n - 1,n}^{}(t)$ and $IB{I_{n + 1,n}}(t)$, as well as the CAP demodulation for the $n\textrm{th}$ sub-band. The received OAM-MDM NMCAP signal $\hat{s}(t)$ is filtered using matched filters, as shown in Fig. 2(a). For the $n\textrm{th}$ sub-band, the matched filters, $m_n^I(t)$ and $m_n^Q(t)$, are used to demodulate the NMCAP signal to obtain the recovered QAM signal. This process can be expressed as:
As shown in Fig. 2(b), the IBI components from adjacent sub-bands can be extracted as [33]:
Next, as shown in Fig. 3, the recovered symbol for the $n\textrm{th}$ sub-band, i.e., $r_n^I(t)$ and $r_n^Q(t)$, and its IBI components, i.e., $IBI_{n + 1,n}^I(t)$, $IBI_{n + 1,n}^Q(t)$, $IBI_{n - 1,n}^I(t)$ and $IBI_{n - 1,n}^Q(t)$, are combined with their L preceding and L succeeding counterparts to form an input feature vector ${{\boldsymbol x}_t}$, which is expressed as:
(ii) Equalization of the ExtraTrees-HMM
A hidden Markov model (HMM) is a statistical Markov model, composed of multiple observation variables and multiple hidden state variables, as shown in Fig. 4. In an HMM, a hidden state variable can be obtained through the observed variable and the previous hidden state variable. An HMM can be defined as $hmm(A,B,{\boldsymbol \pi })$. $A = {[{a_{i,j}}]_{{M_n} \times {M_n}}}$ is the state transition probability matrix, with ${a_{i,j}} = P({y_t} = {c_j}|{y_{t - 1}} = {c_i}),\,\textrm{ }i,j = 1,2,\ldots ,{M_n}$ being the state transition probability. ${a_{i,j}}$ represents the probability of moving from the constellation class ${c_i}$ at timestep $t - 1$ to the constellation class ${c_j}$ at timestep t. ${\boldsymbol \pi } = [{\pi _m}]$ is the initial hidden state distribution, where ${\pi _m} = P({y_1} = {c_m}),\textrm{ }m = 1,2,\ldots ,{M_n}$ is defined as the initial probability which indicate the probability of ${y_1}$ belonging to the constellation class ${c_m}$. $B = [{b_j}(t)]$ is the emission probability matrix, where ${b_j}(t) = P({{\boldsymbol x}_t}|{y_t} = {c_j}),\textrm{ }j = 1,2,\ldots ,{M_n}$ is the probability of emitting from the constellation class ${c_j}$ at timestep t to the observed feature vector ${{\boldsymbol x}_t}$.
The equalization of distorted OAM-MDM NMCAP signals using an HMM is a decoding problem. The observation sequence $[{{\boldsymbol x}_1},{{\boldsymbol x}_2},\ldots ,{{\boldsymbol x}_{{T_{eq}}}}]$ consists of input feature vectors after IBI extraction and construction, and the hidden state sequence $[{y_1},{y_2},\ldots ,{y_{{T_{eq}}}}]$ is the constellation class sequence, where ${T_{eq}}$ is the size of the dataset to be equalized. Decoding the observation sequence into hidden state sequence of constellation classes can be achieved via the Viterbi algorithm, which is a recursive optimization method for obtaining the maximum a posteriori probability [34]:
The ExtraTrees algorithm is a decision tree-based ensemble method that can be used effectively for classification tasks. In the hybrid ExtraTrees-HMM, ExtraTrees is used to obtain the emission probabilities ${b_n}(t)$ in the Viterbi decoding process. An ExtraTrees model consists of W independent ExtraTree branches, as shown in Fig. 5(a). In each ExtraTree branch, a random subset of ${{\boldsymbol x}_t}$ is selected, defined as ${\hat{{\boldsymbol x}}_t}$ with ${\hat{{\boldsymbol x}}_t} \subseteq {{\boldsymbol x}_t}$. The orange node represents the root node, the pink nodes are the split nodes, and the light green nodes represent the leaf nodes. In each node among root node and split nodes, a cutpoint is defined as ${f_{cut(j)}}$ [35]. If ${{\boldsymbol x}_t}$ is larger than ${f_{cut(j)}}$, the root or split nodes assign the input feature vector ${{\boldsymbol x}_t}$ to the left split, and otherwise, assign ${{\boldsymbol x}_t}$ to the right split, as shown in the area enclosed by a dashed line in Fig. 5(b). All cutpoints can be obtained through the training process. The splitting process is repeated until one of the two conditions is reached: (i) all feature vectors in the subset of the training dataset belong to the same constellation class, or (ii) the tree depth reach the maximum depth, denoted by $Dept{h_{\max }}$. After one branch of the ExtraTree are completed, the constellation class which ${{\boldsymbol x}_t}$ is belonged to can be determined by the ExtraTree branch, which is represented by ${y_t} = O({{\boldsymbol x}_t}) = {c_n}$.
Fig. 5. (a) Schematic diagram of an ExtraTrees model; (b) structure of an ExtraTree; (c) Viterbi decoding process of ExtraTrees-HMM.
Another random subset is then selected again, which is processed with another ExtraTree with different structures and cutpoints. After all branches have been processed, the ExtraTrees model outputs all the classification results of ${{\boldsymbol x}_t}$. Noted that in different branches, a given ${{\boldsymbol x}_t}$ may be classified into different constellation classes. Hence the probability of an input feature vector ${{\boldsymbol x}_t}$ belonging to one constellation class can be calculated as:
Before executing the Viterbi algorithm, two parameters are defined. The first parameter ${\rho _t}(i)$ represents the joint probability of the maximum joint probability of receiving the observation sequence $[{{\boldsymbol x}_1},…,{{\boldsymbol x}_{{T_{eq}}}}]$ and the hidden state sequence of the constellation class $[{y_1},{y_2},…,{y_t} = {c_i}],t = 1,2,…,{T_{eq}},i = 1,…,{M_j}$ [34]:
The second parameter ${\omega _t}(i)$ records the chosen hidden state of the constellation class and functions as a back pointer as follows:
The Viterbi algorithm is then applied as follows is as shown in Fig. 5(c).
- (2) Recursion. For each timestep $t = 2\textrm{ to }{T_{eq}} - 1$, ${\rho _t}(i)$ and ${\omega _t}(i)$ are calculated for each constellation class at timestep $t$ based on the result at the previous timestep $t - 1$.$$\begin{array}{l} {\rho _t}(i) = \mathop {\max }\limits_{l \in \{ 1,\ldots ,{M_j}\} } [{\rho _{t - 1}}(l){a_{l,i}}] \cdot {b_i}({{\boldsymbol x}_t}),i = 1,\ldots ,{M_n},\\ {\omega _t}(i) = \mathop {argmax}\limits_{l \in \{ 1,\ldots ,{M_j}\} } [{\rho _{t - 1}}(l){a_{l,i}}],i = 1,\ldots ,{M_n}. \end{array}$$
- (3) Termination. The maximum joint probability is calculated at the last timestep ${T_{eq}}$ as:
- (4) Backtracking. Trace back through the back pointers ${\omega _{end}}(i)\textrm{ and }{\omega _t}(i),\textrm{ }t = 1,2,…,{T_{eq}} - 1 {\omega _{end}}(i)\textrm{ and }{\omega _t}(i),\textrm{ }t = 1,2,…,{T_{eq}} - 1$ to obtain the hidden state sequence of the constellation classes with the highest a posteriori probability, i.e., $[c_1^*,c_2^*,…,c_{{T_{eq}}}^*]$ as:
In OAM-MDM RCF communications systems with NMCAP modulation, heavy IBI combined with inter-MG XT can cause severe signal distortion. The proposed ExtraTrees-HMM equalizer uses the statistical characteristics of the received distorted signals to model the nonlinear channel of the system, thus enabling the probabilities of initialization, transition, and emission to be acquired, finally, it decodes the distorted signals into constellation classes by a dynamic programming-based Viterbi algorithm, hence completing the equalization of NMCAP signal in OAM-MDM RCF communication systems.
3. Experimental setup
To verify the effectiveness of the proposed method, an experimental platform was built, as shown in Fig. 6, in which two OAM modes (l = + 2, + 3) were multiplexed. Each mode was loaded with a six-sub-band NMCAP signal with 4.5 GBaud 16-QAM modulation in each sub-band. The total band ${B_{CAP}}$ was 22.5 GHz.
Fig. 6. Experimental setup. DSP: digital signal processing; AWG: arbitrary waveform generator; MZM: Mach-Zehnder modulator; PC: polarization controller; ECL: external cavity laser; EDFA: erbium-doped fiber amplifier; SMF: single-mode fiber; Col.: collimator; LP: linear polarizer; SLM: spatial light modulator; HWP: half-wave plate; QWP: quarter-wave plate; PBC: polarization beam combiner; BS: beam splitter; MR: mirror; VPP: vortex phase plate; VOA: variable optical attenuator; OC: optical coupler; PD: photodetector; OSC: oscilloscope; Intensity profiles of two OAM modes (i) l = + 3 and (ii) l = + 2 after transmission; (iii) cross-sectional structure of the RCF.
In the digital signal processing at the transmitter (TXDSP), two sets of six independent PRBS in NRZ format with a length of ${2^{17}} - 1$ were generated. Each PRBS was mapped onto 16-QAM modulation constellation symbols followed by a pulse shaping filter, with a roll-off factor of 0.2 and a compression factor ranging from 0 to 0.2 with a spacing of 0.05 in different settings (corresponding to ${\eta _{SE}}$ 3.33, 3.51, 3.70, 3.92 and 4.17, respectively). The outputs of the shaping filter pairs were summed to obtain an NMCAP signal. Digital-to-analogue conversion was realized using an arbitrary waveform generator (AWG, Keysight 8194A) with a sampling rate of 108 GSa/s. The analog signals were then amplified by two electrical amplifiers (EA) and fed into two MZMs (XBlue). The Gaussian beams were generated using two external cavity lasers (ECLs) with a 100 kHz linewidth. The two modulated Gaussian beams were then amplified using two erbium-doped fiber amplifiers (EDFAs). After collimation and linear polarization, the two Gaussian beams were converted into OAM beams with l = + 2 and l = + 3 using two SLMs separately. The two OAM beams with orthogonal linear polarization were then multiplexed with a half-wave plate (HWP) and a PBC. The multiplexed OAM beams were converted into circular polarization through a quarter-wave plate (QWP) and then coupled into a 2.3 km RCF. The intensity profiles of the two transmitted OAM modes l = + 2, + 3 are shown in Figs. 6. (I) and (II), respectively. The cross-sectional structure of the RCF is shown in Fig. 6 (III). The characteristics of the employed RCF are shown in Table 1. The values of the effective refractive index ${n_{eff}}$ of all guided modes in the employed RCF are shown in Fig. 7(a). The employed RCF is designed to minimize the effective refractive index difference $\Delta {n_{eff}}$ between the intra-MG modes to around $1 \times {10^{ - 5}}$ to suppress the frequency-selective fading of each MG channel and supports four azimuthal MGs from |l|=0 to |l|=3. Moreover, the measured DMDs of all supported modes of the employed RCF at 1550 nm are shown in Fig. 7(b). Relatively large DMD between high-order MGs means weak coupling among high-order MGs, while the intra-MG DMD is almost constant for all MGs, indicating intra-MG modes with similar characteristics can be considered as one data channel. The crosstalk of -22.3 dB between |l|=2 and |l|=3 was measured by using a vector network analyzer (VNA) in the 2.3 km RCF based system. Because the relatively large $\Delta {n_{eff}}$, two modes of |l|=2 and |l|=3 were chosen for transmission experiment due to the low crosstalk.
Fig. 7. (a) The calculated ${n_{eff}}$ of all supported modes of the employed RCF at 1550 nm; (b) the measured DMDs of all supported modes of the employed RCF at 1550 nm.
Table 1. Characteristics of the employed RCF
At the receiver side, the multiplexed OAM beams were split into two beams via a BS, and two QWPs were used to convert them into linearly polarized beams. The two beams were then converted into Gaussian beams through two opposite VPPs. By changing the order of the VPPs and recording the power using an SMF-pigtailed power meter, the mode purities were measured and shown to be above 97% for both modes (l = + 2, + 3). The total insertion loss of the mode multiplexer was ∼8 dB, which contains the coupling loss into the RCF, and the insertion loss of mode demultiplexer was ∼6 dB. The two Gaussian beams were coupled to SMFs via collimators and converted into electrical signals via two PDs. The ROPs were measured with a power meter and adjusted using a VOA. The electrical waveforms were captured and recorded with a real-time OSC with a sampling rate of 256 GSa/s.
Receiver’s DSP included low-pass filtering (LPF), resampling, clock recovery, matched filtering, equalization for all sub-bands, demapping, and BER calculation. In addition, all the equalizers in this manuscript are operated with one sample per symbol.
4. Experimental results and analysis
This section is organized as follows. First, electrical spectra for the received OAM-MDM NMCAP signals are given and analyzed. Then, constellation diagrams of each sub-band and their IBI from adjacent sub-bands are illustrated. Third, an equalization example of probabilistic distributions of input feature vectors belonging to each constellation class are given. Forth, the BER performance on OAM-MDM NMCAP signals with the proposed ExtraTrees-HMM equalizer is evaluated and compared with VNE. The BER performance of each sub-band is then given and the impact of the compression factor on BER performance is evaluated. Moreover, the effects of key parameters in the proposed equalizer on the equalization performance have been evaluated. Finally, the computational complexity of the ExtraTrees-HMM equalizer is analyzed and compared with that of VNE. In the experiments $Dept{h_{\max }}$ was set to 18 and $W$ to 100 for both OAM modes. Besides, in the feature vectors construction, $L$ was set to three and in the subset selection process, the subset size was set to 16.
Figures 8(a) and (b) show the electrical spectra of the received OAM-MDM NMCAP signals for l = + 2 and l = + 3, respectively. When $\alpha = 0$, each sub-band is independent of the others, with no overlapping. With a decrease in the compression factor $\alpha$, each sub-band begins to overlap to the adjacent sub-bands, resulting in IBI to the signal. However, it can be seen that the signal bandwidth gradually decreases with the increasing the compression factor $\alpha$. Compared with conventional MultiCAP, NMCAP with $\alpha = 0.2$ reduces the bandwidth from 32.4 GHz to 25.92 GHz, thus saving 6.48 GHz of spectral resources.
Fig. 8. Electrical spectra for the received OAM-MDM NMCAP signals (a) mode l = + 2; (b) mode l = + 3.
Figures 9(a) to (p) show diagrams for each sub-band and the IBI from adjacent sub-bands at an ROP of 8 dBm for mode l = + 3. It can be seen that the first and last sub-bands (sub-band 1 and 6) are less distorted, as there is only one source of IBI, while the constellation diagrams of the middle sub-bands (sub-band 2 to 5) are severely affected, with two adjacent IBIs.
Fig. 9. Constellation diagrams of (a) sub-band 1; (b) IBI from sub-band 2 to sub-band 1; (c) IBI from sub-band 1to sub-band 2; (d) sub-band 2; (e) IBI from sub-band 3 to sub-band 2; (f) IBI from sub-band 2 to sub-band 3; (g) sub-band 3; (h) IBI from sub-band 4 to sub-band 3; (i) IBI from sub-band 3 to sub-band 4; (j) sub-band 4; (k) IBI from sub-band 5 to sub-band 4 (l) IBI from sub-band 4 to sub-band 5; (m) sub-band 5; (n) IBI from sub-band 6 to sub-band 5; (o) IBI from sub-band 5 to sub-band 6; (p) sub-band 6 at an ROP of 8 dBm for mode l = + 3. (q) Probabilistic distributions of input feature vectors ${{\boldsymbol x}_1}$ to ${{\boldsymbol x}_{10}}$ belonging to each constellation class.
Figure 9(q) shows the probabilistic distributions of ten input feature vectors, i.e., ${{\boldsymbol x}_1}$ to ${{\boldsymbol x}_{10}}$, assigning to each constellation class with the output of all ExtraTrees according to Eq. (11). Taking ${{\boldsymbol x}_1}$ for example, it can be seen that all outputs of ExtraTrees are considered when using ExtraTrees to calculate emission probabilities for the HMM according to Eqs. (11) and (12), and the classification abilities of each ExtraTree are fully utilized.
The BER performance of OAM-MDM NMCAP signals with six sub-bands processed with the ExtraTrees-HMM equalizer, VNE, and without equalization were experimentally evaluated, as shown in Figs. 10(a) and (b). Due to the nonlinear impairments of the OAM-MDM system and the IBI induced by the NMCAP modulation, the BERs without equalization did not fall below the 7% HD-FEC threshold of 3.8 × 10−3 in the ROP ranges of 6-9.3 dBm for l = + 2 and 6-9.6 dBm for l = + 3. With the ExtraTrees-HMM equalizer, the BERs for both OAM modes was below the 7% FEC threshold when the ROP exceeded 7.2 dBm for l = + 2 and 7.8 dBm for l = + 3, thus demonstrating the effectiveness of the proposed equalizer. Compared with VNE, the receiver sensitivity was improved by 1 dB for l = + 2 and 0.6 dB for l = + 3 at the 7% HD-FEC threshold with the ExtraTrees-HMM equalizer.
Fig. 10. BER performance versus ROPs for (a) l = + 2 and (b) l = + 3 with different equalizers when $\alpha = 0.2$.
The average BER performance for varying values of the compression factor $\alpha $ was also investigated, as shown in Fig. 11. It was observed that the proposed ExtraTrees-HMM equalizer outperformed VNE at each compression factor. According to the BER curves and the 7% HD-FEC threshold used as a reference, the maximal compression factor achieved with the proposed equalizer was 0.2. Compared with conventional Multiband modulation, SE of the NMCAP at the maximal compression factor was improved from 3.33 to 4.17. Consequently, the proposed equalizer could support higher spectral compression and with a significant SE improvement.
The effects on the equalization performance of the two key parameters, the number of trees $W$ and the maximal depth $Dept{h_{\max }}$ were also evaluated. Figures 12(a) and(b) show the BER contour plots for two OAM modes with different values of $W$ and $Dept{h_{\max }}$, for 6 dBm ROP. It can be seen that the BER performance improved as the values of $W$ and $Dept{h_{\max }}$ increased. With an increase in $W$ and $Dept{h_{\max }}$, the number of ExtraTrees and splitting nodes increased, resulting in a higher accuracy of classification. The performance of the ExtraTrees-HMM equalizer therefore significantly improves. However, large values of $W$ and $Dept{h_{\max }}$ also increase the computational complexity of the ExtraTrees algorithm. Hence, in this experiment $Dept{h_{\max }}$ was set to 18 and $W$ to 100 for both OAM modes, to find a balance between the BER performance and computational complexity.
Fig. 12. log(BER) contour plots for (a) l = + 2 and (b) l = + 3 with the proposed equalizer for OAM-MDM NMCAP in performance optimization.
The number of real multiplications can be used as an indicator for the analysis and comparison of the computational complexity of different equalizers. To simplify presentation, $M = 16$ is used here to denote the modulation order for all sub-bands. Noted that there are many comparison operations in the proposed ExtraTrees-HMM equalizer. According to Refs. [36] and [37], the computational complexity of a comparison operation is 0.75 times that of a real multiplication operation.
The number of real multiplications for the proposed ExtraTrees-HMM equalizer can be expressed as:
The corresponding number of real multiplications for the VNE can be expressed as
The configurations and computational complexities for ExtraTrees-HMM and VNE equalizer for an ROP of 6 dBm with l = + 2 are shown in Table 2.. It can be seen that, the computational complexity of the proposed equalizer is 43.94% lower than for VNE, which evidences its lower consumption of computational resources.
Table 2. Configurations and computational complexities for VNE and ExtraTrees-HMM
In addition, the proposed equalizer is expected to have good performance in other MDM systems since it compensates for the nonlinear impairments through robust ensemble learning and temporal modeling.
5. Conclusion
In this manuscript, an ExtraTrees-HMM equalizer for OAM-MDM RCF communication systems with NMCAP modulation has been proposed. Although the use of NMCAP modulation could improve the spectral efficiency, severe IBI is introduced due to the overlapping of sub-bands. To counteract this effect, IBI components are extracted, and an ExtraTrees-HMM is used with the statistical characteristics of the received distorted signals to model the nonlinear channel. Our experimental results show that the proposed equalizer can mitigate the nonlinear distortions and alleviate the IBI effect for each sub-band effectively. Compared with the conventional VNE, the ExtraTrees-HMM equalizer improved the receiver sensitivity by 1 dB for OAM mode l = + 2 and 0.6 dB for OAM mode l = + 3. In addition, the computational complexity of the proposed equalizer is reduced by 43.94% compared with VNE. Moreover, it showcases robust generalization performance through the effective utilization of ensemble learning (ExtraTrees), temporal modeling (HMM), and rigorous evaluation techniques (cross-validation and testing). In brief, the ExtraTrees-HMM is a promising candidate equalization candidate for use in the ultra-high- capacity inter-data- center interconnections.
Funding
National Key Research and Development Program of China (2023YFB2905601); Beijing Municipal Natural Science Foundation (L232049); National Natural Science Foundation of China (61835002, 62022016, 62305027); Science Fund for Creative Research Groups of China (62021005); Open Fund of State Key Laboratory of Information Photonics and Optical Communications (IPOC) (BUPT) (IPOC2022A04); Beijing Institute of Technology (BIT) Research and Innovation Promoting Project (2023YCXY028).
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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