## Abstract

We propose a reduced-complexity space-demultiplexing algorithm based on higher-order Poincaré spheres (HoPs) which is modulation format agnostic, free of training sequences and robust to the local oscillator phase fluctuations and frequency offsets. The signal tributaries are space-demultiplexed by calculating and realigning the best fit plane in the HoPs, with the inverse channel matrix being iteratively constructed by sequentially space-demultiplexing all pairs of tributaries. When compared with the previous proposed HoPs-based space-demultiplexing algorithm, results show a complexity reduction gain of 99% along with an improvement of 97% in terms of convergence speed.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Space-division multiplexing (SDM) has been proposed to increase the capacity of optical transmission systems using multiple cores in multicore fibers (MCFs) or orthogonal modes of few-mode fibers (FMFs) [1]. Although both approaches may suffer from crosstalk, only FMF based transmission systems present high values of differential mode group delay or modal dispersion [1, 2]. Both, crosstalk and modal dispersion, can be successfully equalized in the coherent receiver by using digital signal processing [2, 3]. Usually, multiple-input and multiple-output (MIMO) algorithms with large memory and aided by training sequences are employed to compensate both effects and can be implemented in the time [3, 4] or in the frequency domain [5–8]. However, it was demonstrated that modal dispersion can be substantially reduced by using FMF with modal interleaving [9] or coupled-core (CC)-MCF [10]. For such links with negligible modal dispersion, the crosstalk still remain as a performance limiting factor and it needs to be compensated by suitable digital signal processing subsystems without memory. In [11], it was proposed a digital space-demultiplexing algorithm based on higher-order Poincaré spheres (HoPs), which compared with other compensation techniques, has the following advantages: it is free of training sequences, it is modulation format agnostic and it tends to be more robust to phase fluctuations and frequency offsets. Nonetheless, this algorithm still presenting a high computational complexity, which makes difficult envisioning its implementation in real transmission systems. The development of more efficient HoPs-based space-demultiplexing algorithms is therefore an open issue.

Polarization-division multiplexing over single-mode fiber can be considered as a basic form of SDM [12]. In polarization-division multiplexing transmission systems, the two independent tributaries appear mixed at the coherent receiver due to the random polarization rotations [13]. However, polarization-demultiplexing (PolDemux) algorithms allow to compensate for such rotations [14]. PolDemux algorithms based on Stokes space were proposed and experimentally validated, showing promising improvements in the convergence speed [15–17]. Moreover, these kind of algorithms are modulation format agnostic and tend to be robust against phase fluctuations and frequency offsets; which means that they have higher laser linewidth resilience [8,14–17]. Therefore, the coherent receiver and transmitter can use low-cost lasers, potentially decreasing the overall cost of the optical transceiver. In Stokes space based PolDemux, the demultiplexing matrix is defined using the spatial orientation of the best fit plane to the received samples in the Stokes space [14–18]. Such digital filters do not have “temporal memory”; hence, they are only able to compensate impairments without delayed time response, which is the case of crosstalk between polarization-multiplexed signals. However, it was demonstrated that time delay effects, such as polarization-mode dispersion, can be previously compensated by suitable digital signal processing subsystems before PolDemux [19], which in principle can be also extended to SDM systems with non-negligible values of modal dispersion.

HoPs are defined as a full set of geometric representations of a given vector space where a pair of orthogonal states is represented in a single HoPs, likewise the Bloch or the Poincaré spheres [20]. Initially, it was introduced to represent light carrying orbital angular momentum (OAM) by incorporating in the basis, both, polarization and OAM states [21,22]. In [21], the Jones circular polarization basis is rewritten by replacing the set of two orthogonal polarization states by two orthogonal vector modes. In [22], it is proposed a hybrid spatio-polarization description to represent simultaneously the polarization and space into two spheres [22]. Using a semi-simple representation of the generalized Stokes space, it was proposed a hybrid spatio-polarization description of the optical signal based on ${g}_{s}=\left(\begin{array}{c}2n\\ 2\end{array}\right)$ HoPs, with *n* denoting the number of distinguishable modes or data pathways supported by the SDM transmission system [11, 23]. In [11], it is also shown that two carries supporting an arbitrary complex-modulation format can be symmetrically represented around the best fit plane in a given HoPs, which was exploited to develop a space-demultiplexing algorithm. HoPs-based space-demultiplexing algorithms also show high robustness to laser linewidth allowing for the use of low-cost lasers in coherent optical communications systems based on SDM [8, 11]. Like in the single-mode case, the crosstalk between two arbitrary tributaries can be compensated by realigning the best fit plane in the respective HoPs. In this space-demultiplexing algorithm, the sequence of demultiplexing matrices is found by analyzing the least squares calculations employed to trace the best fit plane, which includes the computation of the sum of the absolute value of the residuals (arising from the calculations of the best fit plane) for all the possible combinations of pairs of tributaries. After, the inverse channel matrix is calculated by progressively demultiplexing all pairs of tributaries. However, the computational effort of this algorithm evolves as ${n}_{t}^{2}!$, being *n _{t}* the number of tributary signals considered in the SDM system. With the current technology, the computational effort required by this algorithm can scale for impracticable values.

We propose a different approach to perform the space-demultiplexing based on the HoPs representation with the inverse channel matrix being calculated in a iterative way. This new approach allows to substantially decrease the computational effort of the previous HoPs-based space-demultiplexing algorithm. When compared with the one proposed in [11], the reduced-complexity algorithm achieves a complexity reduction gain of 99% and an improvement of 97% in the convergence speed for a SDM transmission system with four tributary signals. Furthermore, the algorithm remains modulation format agnostic, free of training sequences and robust to the local oscillator phase fluctuations and frequency offsets.

This paper is organized in four sections. In section 2, the signal representation in HoPs is briefly presented, and the reduced-complexity space-demultiplexing algorithms deeply described and discussed. The convergence speed and the complexity of the proposed space-demultiplexing algorithm are investigated in section 3. Finally, the main conclusions are presented in section 4.

## 2. Reduced-complexity space-demultiplexing algorithm based on HoPs

#### 2.1. Higher-order Poincaré spheres

Throughout this paper, we will make use of the Dirac notation to denote the optical signal in the Jones space. We start by writing the electric field of a signal transmitted in a SDM transmission system as

*j*= 1, 2, ...

*n*distinguishable data pathway,

*i.e*., core or mode, with the

*l*=

*x*,

*y*polarization state and the superscript

*T*denoting the transpose operator. Equation (1) can be rewritten as

*f*= 1, 2..., 2

*n*− 1 and

*g*=

*f*+ 1, ..., 2

*n*represent the

*υ*and

_{f}*υ*tributary signals, respectively, selected from (2). In the case of HoPs, the

_{g}*Pauli spin vector*is given by

*i*= 1, 2 and 3, can be comprehensively written as [11]:

- the first matrix can be written by considering the elements ${\mathrm{\Lambda}}_{1}^{(f,g)}(f,f)$ and ${\mathrm{\Lambda}}_{1}^{(f,g)}(g,g)$ set to $\sqrt{n}$ and to $-\sqrt{n}$, respectively, in the intramode case or $\kappa \sqrt{n}$ and to $-\kappa \sqrt{n}$, respectively, in the intermode case, while the remaining elements are set to zero;
- the second matrix is written by considering the elements outside of the main diagonal, ${\mathrm{\Lambda}}_{2}^{(f,g)}(f,g)$ and ${\mathrm{\Lambda}}_{2}^{(f,g)}(g,f)$, set to $\sqrt{n}$ and to $-\sqrt{n}$, respectively, and the remaining elements are set to zero;
- the third matrix is written by considering the elements ${\mathrm{\Lambda}}_{3}^{(f,g)}(f,g)$ set to $i\sqrt{n}$, with $i=\sqrt{-1}$, and the symmetric element to $-i\sqrt{n}$. The remaining elements are set to zero.

*g*residuals are affected in the same way. Notice that the factor

_{s}*κ*was introduced in [11] to describe the mathematical relationship between the generalized Stokes space and the HoPs. Such factor depends on the number of spatial channels considered in the Jones space and, it can be calculated from the Pauli matrices for the generalized Stokes space and the Pauli matrices for the HoPs [11]. In the space-demultiplexing algorithm previously proposed, the factor

*κ*is not considered in the calculations of the Stokes vectors enabling the use of the residual as a reliable criterion to choose a suitable sequence of HoPs,

*i.e*., the sequence with better least squares calculations. In the following sections, the factor

*κ*will also be neglected in order to make the residual a metric of the quality of the least square calculations and, additionally, to simplify the computation of the Pauli spin vector and following calculations.

#### 2.2. Algorithm description

In the proposed HoPs-based space-demultiplexing algorithm, the received signal is space-demultiplexed by iteratively reversing the crosstalk between all the pairs of tributaries. This means that, for an arbitrary pair of tributaries, the samples are represented in the respective HoPs and a best fit plane is calculated by means of a least square regression. Afterwards, the crosstalk between these two tributary signals is compensated by realigning the best fit plane with the respective Ψ_{1} = 0 plane, see the schematic representation in Fig. 1(a). Henceforth, this process is called the space-demultiplexing (SpDemux) step, see appendix A for a detailed explanation. It is worth noting that the sequence of SpDemux steps is extremely important to achieve a perfect space-demultiplexing because SpDemux steps (which are rotation matrices of dimension 2*n*, with *n* > 1) do not commute [24].

In order to find the first pair of tributaries for space-demultiplexing, the *g _{s}* SpDemux steps are applied to the received signal. After that, the pair of tributaries with the better fit is chosen for space-demultiplexing. In this sense, the “quality” of the fit can be assessed by the residual provided by the least square calculations. Then, a similar process is used to choose the second pair of tributaries for space-demultiplexing, and so on; until all the

*g*SpDemux steps are properly applied to the received signal. Notice that the pair of tributaries previously considered is not taken into account in the following calculations. In summary, a comparison among the several residuals (provided by the least square regression employed to calculate the best fit planes) can be used to guide the algorithm in the search for an optimum sequence of SpDemux steps.

_{s}For an arbitrary HoPs, however, the best fit plane can be described by two distinct spatial orientations of the normal, *i.e*., the normal can point to opposite sides of the best fit plane. Although both spatial orientations can be successfully used to calculate the demultiplexing matrix, an unsuitable orientation induces a signal permutation between both tributaries considered. For instance, if the space-demultiplexing is performed by considering an improper spatial orientation of the normal, the SDM signal |*ϕ*〉 = *(υ*_{1}, ...*υ _{f}*, ...

*υ*, ...

_{g}*υ*

_{2n})

*gives rise to |*

^{T}*ϕ*〉 = (

*υ*

_{1}, ...

*υ*, ...

_{g}*υ*, ...

_{f}*υ*

_{2n})

*. It should be highlighted, once again, that after applied a given SpDemux step this one is no longer considered in the following calculations. Therefore, the change in the sequence of tributaries may lead to a double space-demultiplexing of a given pair of tributaries, while another pair of tributaries is not space-demultiplexed; this can jeopardize the accurate space-demultiplexing of the SDM signal. In order to avoid this effect, one considers all possible combinations of two consecutive pairs of tributaries, instead of a single pair, and their normals. The sequence with minimum residual at the second SpDemux step is chosen. Then, the SDM signal is space-demultiplexed considering the first SpDemux step and their normal. Using this heuristic, the two possible spatial orientations of the normal can be considered in the residual minimization, or in other words, in the search for a suitable sequence of SpDemux steps. The above process is consecutively repeated until all the tributaries are space-demultiplexed. The set of operations described above are called space-demultiplexing (SpDemux) filter, see the schematic representation in Fig. 1(b). For a detailed explanation, see appendix B.*

^{T}The aforementioned simplification tremendously reduces the complexity of the space-demultiplexing algorithm at the expense of a certain signal-to-noise ratio (SNR) penalty. Such impairments in the space-demultiplexing are due to a residual misalignment between the best fit plane and the Ψ_{1} = 0 plane in one or more HoPs. This misalignment can be compensated by again applying the SpDemux filter to the SDM signal. With this idea in mind, we introduce the parameter,

*r*| denotes the residual for the least squared regression at the

_{j}*j*

^{th}SpDemux step considered in the SpDemux filter. As the iterative process goes forward, the crosstalk is progressively compensated and the parameter

*ζ*tends asymptotically to a given value; which depends on the number of tributaries supported by the transmission system and in the SNR of the received signal. Of course, the difference between consecutive values of

*ζ*tends to decrease with the number of SpDemux filters because the next filter only takes into account the remaining misalignments between the best fit plane and the respective Ψ

_{1}= 0 plane. In that way, the relative difference between two consecutive parameters

*ζ*can be used to assess the performance of the space-demultiplexing. Using the residuals as a metric of the space-demultiplexing performance, we define the parameter with

*ζ*defined as in (5) and the superscript denoting the

^{n}*n*SpDemux filter. When this relative variation,

^{th}*ε*, is below than a given threshold,

*ε*

_{ST}, it is assumed that the algorithm achieved the steady state. Hence, the iterative process is stopped. The iterative process of the reduced-complexity algorithm is schematically represented in Fig. 1(c). Both space-demultiplexing performance and computational complexity mainly depend on the number of SpDemux filters required in the iterative process and, consequently, in

*ε*

_{ST}. If

*ε*

_{ST}is too high, the space-demultiplexing can be jeopardized. On the other hand, if

*ε*

_{ST}is too small, the performance of the algorithm will be substantially enhanced, resulting into a successful space-demultiplexing at the expensive of a higher computation complexity. As aforementioned, the calculated space-demultiplexing matrix tends to converge to the inverse channel matrix as the number of SpDemux filters employed by the algorithm increases. Due to the iterative process, the proposed algorithm allows an accurate calculation of the inverse channel even for a small number of samples.

#### 2.3. Algorithm assessment

In this subsection, the proposed algorithm is assessed in terms of operation principle. In that way, the cornerstone of the reduced-complexity space-demultiplexing algorithm, *i.e.*, the SpDemux filter, is schematically represented for a transmission system with four tributaries in Fig. 2(a). Furthermore, it is also schematically represented in Fig. 2(b) the algorithm proposed in [11]. In that way, the operation principle of both aforementioned algorithms can be easily compared. In the case of the SpDemux filter, the six pairs of tributaries are space-demultiplexed giving rise to twelve output signals; afterwards, each one of these signals is again space-demultiplexed. In the second set of SpDemux steps, the SpDemux step previously applied to the signal is not taken into account, see Fig. 2(a). For instance, the tributaries 1 and 2 are initially space-demultiplexed; subsequently the resultant SDM signal is again space-demultiplexed taking into account only the remaining combinations of tributaries. By choosing the sequence with the lower value of residual for the second set of SpDemux steps, the suitable SpDemux step and the spatial orientation of the normal are found. In Fig. 2(a) this operation is denoted by min{*r*^{(2)}}, with the superscript index pointed out the SpDemux step considered at the SpDemux stage. Such sequence comprises two concatenated SpDemux steps and the respective signs for the normals, the first SpDemux step and its normal are chosen. In addition, the SpDemux step also provides the first residual, *r*_{1}, considered in the calculation of *ζ*. In the second SpDemux stage, a similar operation is performed in which the space-demultiplexing step with the Pauli spin vector Λ⃗^{(1,3)} is not considered in the space-demultiplexing stage because it was already chosen as the first suitable SpDemux step. In the algorithm proposed in [11], all the possible sequences of SpDemux steps and normals are evaluated in order to calculate the inverse channel matrix. Thereby, it is considered the permutations of the six pairs of tributaries taking into account for each SpDemux step both possible spatial orientations of the normal, *n⃗ _{p}* and −

*n⃗*. Due to the two possible spatial orientations of the normal, each sequence of SpDemux steps has 64 additional combinations giving rise to a total of 46080 possible solutions. For each sequence, the parameter

_{p}*ζ*is defined as: $\zeta ={\sum}_{i=1}^{6}\left|{r}_{i}\right|$, with

*r*being the residual for the

_{i}*i*SpDemux step. Lastly, the sequence with the lower value of

^{th}*ζ*is chosen, see Fig. 2(b).

## 3. Results and discussion

In this section, the performance of the reduced-complexity space-demultiplexing algorithm is analyzed for several scenarios. For this purpose, we assume a pseudo-random binary sequence of 2^{18} (2^{19}) bits mapped into the I and Q components of a quadrature phase-shift keying (QPSK) (16 quadrature amplitude modulation (QAM) and 64QAM) with an averaged SNR of 17 dB (23 and 30 dB, respectively) and a standard deviation of 0.1 dB. It should be noted that the results presented in this section are obtained by averaging over 100 trials, with the exception of subsection 3.4.3 in which are averaging over 10 trials.

#### 3.1. SDM link modeling

The multi-section model is used to emulate the signal propagation in a SDM link under the strong coupling regime [25]. Both approaches, FMF and MCF can be properly described by such model considering a section with a length slightly longer than the correlation length and the dispersive properties for each spatial channel. The channel impulse response can be written as

where |*ψ*〉

_{out}and |

*ψ*〉

_{in}represent the output and the input signals, respectively. The channel impulse response is given by

*L*the total length of the link,

*β̄*

_{2}the averaged chromatic dispersion per unit length and ${\text{M}}_{\text{MD}}^{k}(\omega )$ represents the linear effects in the

*k*span; such as differential loss and crosstalk. Assuming that the differential loss can be compensated by in-line amplifiers, ${\text{M}}_{\text{MD}}^{k}(\omega )$ can be expressed as

*H*denoting the Hermitian conjugate operator. The V

*and U*

_{kl}*matrices denote the frequency-independent random unitary matrices and ${g}_{i}^{k}$, with*

_{kl}*i*= 1, 2 ... 2

*n*, represents the gain/loss for each tributary signal. Matrix Θ(Ω) accounts for the dispersion in a single section,

*τ*the delay for the

_{i}*i*tributary signal.

#### 3.2. Convergence speed

We assume a coupled-core (CC-) MCF based transmission system in which all linear and nonlinear transmission impairments are fully compensated. It is assumed that the optical losses are compensated by in-line amplifiers, and that the chromatic and modal dispersions can be fully compensated by suitable digital signal processing subsystems before the space-demultiplexing. After space-demultiplexing, a Viterbi-Viterbi algorithm for carrier phase estimation was employed [13]. Against this backdrop, the signal quality is only degraded due to optical noise and due to crosstalk. The performance of the algorithm can be assessed in terms of the SNR penalty,

where the SNR_{out}and SNR

_{in}are the SNR in the electrical domain for the demultiplexed signal and for the signal launched in the fiber link, respectively, and Δ is the remaining penalty. Then, the error vector magnitude (EVM) and the SNR of each post-processed tributary signal is calculated. Note that, the SNR is calculated from the tributary signal with higher EVM, i.e., the worst case. After a successful space-demultiplexing, the tributaries may be recovered at a distinct spacial channel. However, such ambiguity may be easily solved at the symbol identification subsystems.

We start by analyzing the convergence speed for a SDM transmission system based on a 3-core CC-MCF, supporting each core a polarization-multiplexed (PM)-QPSK signal. In Fig. 3(a), we show the average value for Δ after the *i ^{th}* SpDemux filter, with

*i*= 1, 2 and 3, and the reduced-complexity algorithm with

*ε*

_{ST}= 10% and 1% as function of the number of samples. After a single SpDemux filter, the average value of Δ takes values between 7 and 10 dB. Therefore, the demultiplexed signal still presents a considerable amount of crosstalk. When a second SpDemux filter is applied to the space-demultiplexed signal, the remaining penalty decreases,

*i.e*., Δ ≈ 3 dB. For a third SpDemux filter, the penalty decreases again for values lower than 1 dB. However, the remaining penalty still exceeds the value of 0.1 dB. Such value is assumed as the minimum acceptable value to consider that space-demultiplexing was successfully achieved. For the reduced-complexity algorithm with

*ε*

_{ST}= 10%, the Δ target is achieved for ≈ 850 samples. The average value of Δ substantially decreases for the first ≈ 200 samples; whereas, for a larger number of samples > 200, Δ decreases slowly. As the SpDemux filter only takes into account two consecutive SpDemux steps in the demultiplexing process, the crosstalk cannot be fully compensated by a single filter. However, the remaining crosstalk, or the misalignments between the best fit plane and the Ψ

_{1}= 0 plane, can be successfully compensated by the progressive application of several SpDemux filters. In that way, the algorithm produces an enhanced space-demultiplexing matrix by each SpDemux filter considered. For instance, in the case of

*ε*

_{ST}= 1%, the average number of SpDemux filter increase for ≈ 5 and the average value of Δ decreases from a maximum of ≈ 0.02 dB, see Fig. 3. These results show that the SNR penalty of the post-processed signal can be substantially enhanced by considering lower

*ε*, which tends to increases the numbers of SpDemux filters employed by the space-demultiplexing algorithm. In that way, the SNR improvements are due to the progressive use of SpDemux filters and, therefore, the performance of the algorithm is not mainly limited by the number of samples employed in the calculation of the inverse channel matrix. Such behavior can be observed in Fig. 3(a), where the improvements in the SNR penalty produced by the progressive application of the SpDemux filters are evident. In Fig. 3(b), it is shown the average number of SpDemux filters,

_{ST}*N*, required by the reduced-complexity algorithm as a function of the number of samples for both cases aforementioned. Inset show the constellations for the received signal and the space-demultiplexed signal at the end of the first, second, third SpDemux filters and the reduced-complexity algorithm with

_{f}*ε*

_{ST}= 10%; assuming 1000 samples in the calculations of the inverse channel matrix. Like the average SNR penalty, the average number of SpDemux filters required by the algorithm tends to substantially decrease from the first ≈ 200 samples; whereas, after ≈ 200 samples, the average number of SpDemux tends to stabilize for both cases considered, see Fig. 3(b). For

*ε*

_{ST}= 10%, the average number of SpDemux filters tends to 4.1. For

*ε*

_{ST}= 1%, the algorithm requires on average 5.25 SpDemux filters. Notice that although the number of SpDemux filters considered in the iterative process is an integer, the aforementioned non-integer values result of the averaging over 100 trials.

In Fig. 4, it is shown the average value for Δ and the *N _{f}* required by the space-demultiplexing algorithm as function of

*ε*

_{ST}for the previous SDM system assuming several number of samples in the calculations of the inverse channel matrix. Figure 4(a) shows the results in terms of convergence speed for the case of a PM-QPSK signal. For all the cases considered, the reduced-complexity algorithm allows for a successful space-demultiplexing of the received signal. The average number of SpDemux filters is approximately the same for all the cases considered and its value tends to increase for smaller values of

*ε*

_{ST}. Figure 4(b) shows the results for the case of PM-16QAM signals. When compared with the QPSK, the space-demultiplexing of the 16QAM requires a smaller

*ε*

_{ST}. Regarding the number of SpDemux filters, results show that its value tends to increases as

*ε*

_{ST}decreases. In Fig. 4(c), it is shown the average value of Δ as a function of the

*ε*

_{ST}for a PM-64QAM signal. As in the previous cases, the received signal is successfully space-demultiplexed. However, the values of

*ε*

_{ST}required to keep Δ below than 0.1 dB is even smaller. The remaining penalty apparently tends for Δ ≈ 0.03 dB, with the number of SpDemux filters required by the algorithm varying between ≈ 40 and ≈ 60. This set of results show that the convergence speed tends to be faster for larger sets of samples and the number of SpDemux filters required by the algorithm tends to increase with the density of the modulation format. The increment in the number of SpDemux filters makes the algorithm more complex in terms of computational effort and, additionally, increases it’s latency because the filters are consecutively applied to the received signal. However, the algorithm has a degree of freedom in

*ε*

_{ST}, which can be used to adjust the remaining penalty and, therefore, to control the number of SpDemux filters applied by the algorithm.

In order to analyse the scalability of the reduced-complexity algorithm, we have performed simulations considering a SDM system supporting a large number of tributaries. In Fig. 5, it is shown the average value of Δ and *N _{f}* as function of

*ε*

_{ST}for a SDM system based on

*n*= 2, 3, 4, 5 and 7-cores CC-MCF, where each core carries a PM-QPSK signal. Such results are obtained considering 100 samples in the calculations of the inverse channel matrix. For all the cases considered, the algorithm is able to successfully space-demultiplex the received signal. It should be noted that the successful space-demultiplexing is first achieved for SDM transmission systems supporting a lower number of cores or tributaries. As the number of tributaries increase, the algorithm tends to require more SpDemux filters because it needs to take into account more HoPs; hence, increasing the computational complexity of the space-demultiplexing algorithm. Such increment in the number of HoPs tends to increase the number of iterations needed to successfully realign the best fit plane with the Ψ

_{1}= 0 plane in all the HoPs considered.

#### 3.3. Signal-to-noise ratio tolerance

In this subsection, we analyze the role of the SNR in the convergence speed of the reduced-complexity space-demultiplexing algorithm. For this purpose, we assume a PM-QPSK signal with SNR_{in} equal to 10, 20 and 30 dB. The average value of Δ as function of *ε*_{ST} is shown in Fig. 6, for the three aforementioned SNR and assuming 100 samples in the calculations of the inverse channel matrix. The algorithm is able to successfully space-demultiplex the received signals in the three cases analyzed. In addition, we note that the algorithm shows a faster convergence speed for signals with lower SNR_{in}. Although not shown in Fig. 6, the number of SpDemux filters tends to increase as the parameters *ε*_{ST} decrease, like in Fig. 4. For the three cases considered, the algorithm require approximately the same number of SpDemux filters to achieve the steady state. Nevertheless, the number of SpDemux filters marginally increases for signals with higher SNR_{in}.

#### 3.4. Comparison between the HoPs-based and NLMS algorithms

Here, we compare the convergence speed and complexity of the two HoPs-based algorithms and a 4 × 4 time-domain equalizer based on data-aided NLMS algorithm with 1 tap [3]. In order to make the text more clear, along this subsection the term “static” is used to designate the space-demultiplexing algorithm proposed in [11]. Furthermore, both HoPs-based space-demultiplexing algorithms are compared in terms of SNR tolerance. Without loss of generality, we assume a SDM transmission system based on a 2-core CC-MCF in which a single core carries a PM-QPSK signal with a SNR of 17 dB.

### 3.4.1. Convergence speed

The average value of the SNR penalty as function of the number of samples for the static, the reduced-complexity and the NLMS algorithms is shown in Fig. 7. For a small amount of samples, both HoPs-based space-demultiplexing algorithms achieve a better performance than the NLMS. In terms of SNR penalty, the NLMS requires ≈64 and ≈201 samples, for a step size of *μ* = 0.1 and 0.01, respectively, to achieve a penalty as low as Δ = 0.1 dB, see Fig. 7(b). While, the static algorithm requires 375 samples to achieve the same penalty. If we assume *ε*_{ST} = 90% in the reduced-complexity algorithm, the space-demultiplexing can be carried out with an average SNR penalty of ≈0.7 dB. For a *ε*_{ST} = 50%, the algorithm needs 1250 samples to achieve a SNR penalty of 0.1 dB, despite the SNR penalty does not exceed the value of 0.2 dB for all the sets of samples considered. For *ε*_{ST} = 5%, the value of the SNR penalty is lower than 0.1 dB for all the sets of samples considered in the calculations of the inverse channel matrix. Despite Δ may present small fluctuations when assumed a lower number of samples, these results show that the performance of the reduced-complexity algorithm mainly depends on the choice of the parameter *ε*_{ST}. While, in the static algorithm, the performance mainly depends on the number of samples considered in the calculation of the inverse channel matrix. In order to accurately estimate the best fit plane, we require 10 points in the HoPs as the minimum amount of samples, see Fig. 7(b). Assuming *ε*_{ST} = 5% and 10 samples in the calculations of the inverse channel matrix, the reduced-complexity space-demultiplexing algorithm shows an improvement of the convergence speed of 84.38% over the NLMS with faster convergence speed. It should also be noted that the NLMS algorithm is working on data-aided mode to estimate the inverse channel matrix, while both HoPs-based space-demultiplexers are free of training symbols. When comparing both HoPs space-demultiplexing algorithms, the results show an improvement in the convergence speed of 97.33% for the reduced-complexity algorithm.

### 3.4.2. Computational complexity

Since both HoPs-based space-demultiplexing algorithms trace the best fit plane by means of a linear least square regression, the computational complexity depends on the number of best fit planes evaluated and on the number of samples considered in the calculations of the best fit plane. The number of HoPs required by the static algorithm is given by

and, therefore, the complexity evolves as $\approx O\left({n}_{t}^{2}!\right)$. For the reduced-complexity algorithm, the number of HoPs required by a single SpDemux filter can be expressed as*N*

_{m}, and additions,

*N*

_{s}, required by the least-mean square algorithm to calculate the best fit plane is given by [26] and respectively, with

*n*denoting the number of samples. Thereby, the complexity reduction gain,

_{s}*G*, of the reduced-complexity approach over the static approach can be written as

_{CR}*N*= 4 for successful space-demultiplexing the received signal, thus leading to a complexity reduction gain of

_{f}*G*= 50.36 dB or 99.99%.

_{CR}The NLMS based space-demultiplexing algorithm needs ${n}_{t}^{2}$ finite impulse response (FIR) filters with 1 tap, each filter requiring *N*_{m} = 20*n _{s}* and

*N*

_{s}= 4

*n*. When the reduced-complexity algorithm is compared with the NLMS in terms of computational complexity, one obtains a complexity reduction gain of

_{s}*G*= −19.41 dB, which means that the reduced complexity approach is 98.86% more complex than the NLMS. Note that, such values were obtained assuming

_{CR}*n*= 64.

_{s}### 3.4.3. Signal to noise ratio tolerance

Both HoPs-based space-demultiplexing algorithms are compared in terms of SNR tolerance. Thus, the SNR penalty after the space-demultiplexing considering SDM signals with distinct SNR is compared. In Fig. 8, it is shown the SNR penalty after space-demultiplexing considering the static, with *n _{s}* = 2000, and the reduced-complexity algorithm with

*ε*

_{ST}= 5% and

*n*= 700. For the static algorithm, Δ tends to slightly increase with the SNR. However, such penalty can be reduced by increasing the number of samples considered in the calculations of the inverse channels matrix. The SNR penalty provided by the reduced-complexity algorithm is substantially lower than the static approach, in the entire range of SNRs considered in the analyses. Furthermore, the SNR penalty also tends to slightly increase with the SNR.

_{s}## 4. Conclusions

A reduced-complexity space-demultiplexing algorithm based on HoPs is proposed. The performance of the proposed algorithm is assessed for signals with distinct modulation formats, different SNR values and even for SDM systems with distinct degrees of spatial diversity. For the particular case of four tributaries, the complexity-reduced algorithm presents improvements of 97% in terms of convergence speed and 99% in terms of computational complexity over the previously proposed HoPs-based space-demultiplexing algorithm. The proposed algorithm is also compared with the NLMS, showing an improvement of 84% in the convergence speed, despite it has a higher computational complexity than the NLMS. It should be noted that HoPs-based space-demultiplexing algorithms do not require training symbols, in contrast to the NLMS, and is modulation format agnostic and robust to phase fluctuations and frequency offsets. The reduced-complexity algorithm proposed here can therefore contribute to pave the way for the development of a low complexity modulation format agnostic space-demultiplexing algorithm more robust to the phase fluctuations and frequency offsets.

## Appendix A: SpDemux step

The crosstalk between an arbitrary pair of tributaries can be compensated by using the information of the best fit plane to the received samples represented in the respective HoPs [11]. The Stokes parameters are calculated using (3) and the best fit plane is estimated by means of a linear least square regression. Then, the two tributaries are space-demultiplexed by applying the demultiplexing matrix to the received signal in the Jones space. The demultiplexing matrix F^{(f,g)} is defined as [11]

*g*and

*f*denotes both tributaries. The parameters $p=\text{arctan}\left(a,\sqrt{{b}^{2}+{c}^{2}}\right)$ and

*q*= arctan (

*b*,

*c*) are the angles between the normal to the best fit plane

*n⃗*= (

_{p}*a*,

*b*,

*c*)

*and the vector (1, 0, 0)*

^{T}*. However, the two spatial orientations of the normal,*

^{T}*n⃗*and −

_{p}*n⃗*, must be considered in the space-demultiplexing process because an unsuitable direction of

_{p}*n⃗*produces a permutation between both tributaries. Such ambiguity may seem as a degeneracy in the spatial orientation of

_{p}*n⃗*.

_{p}## Appendix B: SpDemux filter

In order to find a suitable sequence of SpDemux steps, we assume that the crosstalk only changes the spatial orientation of the symmetry plane without breaking the symmetric distribution of the samples. Thereby, the impulse response channel can be progressively constructed by choosing the pairs of tributaries with better least square regression, *i.e*., the space-demultiplexing steps with lower residual. However, a second SpDemux step is required to lift the degeneracy in the spatial orientation of the normals. Using these heuristics, the sequence of the *g _{s}* SpDemux steps can be tracked as follows. Firstly, the

*g*SpDemux steps are applied to the signal obtaining 2

_{s}*g*output signals. Then, each one of these output signals are again space-demultiplexed. It should be noted that the SpDemux step previously applied to the signal is not taken into account in the second set of SpDemux steps. Thus, a suitable SpDemux step (and the spatial orientation of the normal) is found by choosing the first SpDemux step (and the respective normal) from the sequence with lower residual at the end of the second set of SpDemux steps. Although the residuals analyzed are provided by the second set of space-demultiplexing steps, the two tributaries space-demultiplexed and the spatial orientation of the normal chosen are the ones considered in the first space-demultiplexing steps. This process is designated as space-demultiplexing (SpDemux) stage, see Fig. 1(b). Subsequently, the output signal of the first SpDemux stage is launched in the second SpDemux stage and the process is repeated for the remaining (

_{s}*g*− 3) stages up to space-demultiplexed all the pairs of tributaries, see Fig. 1(b). It must be again emphasized that the SpDemux steps previously applied to the signal are not considered in the following SpDemux stages.

_{s}## Funding

Fundação para a Ciência e a Tecnologia (FCT).

## Acknowledgments

This work was supported in part by Fundação para a Ciência e a Tecnologia (FCT) through national funds, and when applicable co-funded by FEDER-PT2020 partnership agreement, under the project UID/EEA/50008/2013 (actions SoftTransceiver, COHERENTINUOUS, and OPTICAL-5G). The work of G. M. Fernandes and N. J. Muga, was supported by the doctoral/postdoctoral research under the grants SFRH/BD/102631/2014 and SFRH/BPD/77286/2011, respectively, by FCT.

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