Recent Advances in DSP Techniques for Mode Division Multiplexing Optical Networks with MIMO Equalization: a Review

applied sciences

Review Recent Advances in DSP Techniques for Mode Division Multiplexing Optical Networks with MIMO Equalization: A Review

Yi Weng 1, Junyi Wang 2 and Zhongqi Pan 1,*

1 Department of Electrical and Computer Engineering, University of Louisiana at Lafayette, Lafayette, LA 70504, USA; [email protected] 2 Qualcomm Technologies Inc., San Diego, CA 95110, USA; [email protected] * Correspondence: [email protected]; Tel.: +1-337-482-5899

Received: 12 February 2019; Accepted: 7 March 2019; Published: 20 March 2019

Abstract: This paper provides a technical review regarding the latest progress on multi-input multi-output (MIMO) digital signal processing (DSP) equalization techniques for high-capacity fiber-optic communication networks. Space division multiplexing (SDM) technology was initially developed to improve the demanding capacity of optic-interconnect links through mode-division multiplexing (MDM) using few-mode fibers (FMF), or core-multiplexing exploiting multicore fibers (MCF). Primarily, adaptive MIMO filtering techniques were proposed to de-multiplex the signals upon different modes or cores, and to dynamically compensate for the differential mode group delays (DMGD) plus mode-dependent loss (MDL) via DSP. Particularly, the frequency-domain equalization (FDE) techniques suggestively lessen the algorithmic complexity, compared with time-domain equalization (TDE), while holding comparable performance, amongst which the least mean squares (LMS) and recursive least squares (RLS) algorithms are most ubiquitous and, hence, extensively premeditated. In this paper, we not only enclose the state of the art of MIMO equalizers, predominantly focusing on the advantage of implementing the space–time block-coding (STBC)-assisted MIMO technique, but we also cover the performance evaluation for different MIMO-FDE schemes of DMGD and MDL for adaptive coherent receivers. Moreover, the hardware complexity optimization for MIMO-DSP is discussed, and a joint-compensation scheme is deliberated for chromatic dispersion (CD) and DMGD, along with a number of recent experimental demonstrations using MIMO-DSP.

Keywords: digital signal processing; multi-input multi-output; mode-division multiplexing; least mean squares; frequency-domain equalization; recursive least squares; space–time block-coding; mode-dependent loss

1. Introduction The space division multiplexing (SDM) systems were proposed to improve the capacity of fiber-optic transmission links, either by means of core multiplexing exploiting multicore fibers (MCF), or with mode-division multiplexing (MDM) using few-mode fibers (FMF) [1–3]. Device integration is the most likely approach to accomplish power consumption and cost reduction, through a variety of proposals for realizing parallel spatial channels, including integrating parallel transmitters, receivers, and amplifiers in the same device, as well as utilizing various multimode or multicore fibers, as shown in Figure1[4]. Amongst these proposals, the simplest solution is to solely apply fiber bundles to have the same behavior as N single-mode fibers, which can consist of a fiber ribbon or a multi-element fiber as

Appl. Sci. 2019, 9, 1178; doi:10.3390/app9061178 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 1178 2 of 28

Appl. Sci. 2019, 9, x FOR PEER REVIEW 2 of 29 displayed in Figure1a. Such a multi-element fiber provides a smooth upgrade path where all existing fiber bundles. In contrast, MCFs epitomize the next step in parallel fiber integration, where the components can be reused, and the power consumption and cost go up with the scale of the fiber adjacent fiber cores are enclosed in the same glass cladding, and it allows novel system architectures bundles. In contrast, MCFs epitomize the next step in parallel fiber integration, where the adjacent such as local oscillator sharing with less temperature-dependent fluctuation, as revealed in Figure fiber cores are enclosed in the same glass cladding, and it allows novel system architectures such as 1b. It is worth mentioning that MCFs are usually designed with low inter-core cross-talk; thus, local oscillator sharing with less temperature-dependent fluctuation, as revealed in Figure1b. It is multiple-input multiple-output (MIMO) digital signal processing (DSP) is not required at the worth mentioning that MCFs are usually designed with low inter-core cross-talk; thus, multiple-input receiver. On the other hand, mode coupling is difficult to avoid in FMFs and, hence, necessitates the multiple-output (MIMO) digital signal processing (DSP) is not required at the receiver. On the other usage of MIMO DSP in the SDM systems, where orthogonal spatial modes within the same fiber core hand, mode coupling is difficult to avoid in FMFs and, hence, necessitates the usage of MIMO DSP form the parallel channels, as exposed in Figure 1c. Furthermore, strongly coupled MCFs that in the SDM systems, where orthogonal spatial modes within the same fiber core form the parallel behave like FMFs were also proposed with certain combined advantages, which shows there is great channels, as exposed in Figure1c. Furthermore, strongly coupled MCFs that behave like FMFs were potential to combine MCF with few-mode cores in next-generation optical-fiber communication also proposed with certain combined advantages, which shows there is great potential to combine systems [5,6]. MCF with few-mode cores in next-generation optical-fiber communication systems [5,6].

Figure 1. The typical examples of optical fibers for space division multiplexing (SDM) technology: (a) multi-element fiber; (b) multicore fiber; (c) multi-mode fiber [4]. Figure 1. The typical examples of optical fibers for space division multiplexing (SDM) technology: (a) multi-elementPrincipally, thefiber; adaptive (b) multicore multi-input fiber; (c multi-output) multi-mode fiber (MIMO) [4]. frequency-domain equalization (FDE) techniques were recognized to de-multiplex the signals upon diverse modes, as well as compensate for the differentialPrincipally, modethe adaptive group delay multi-input (DMGD) dynamicallymulti-output through (MIMO) digital frequency-domain signal processing equalization (DSP) [7–9 ]. (FDE)TheFDE techniques techniques were can recognized reduce the to algorithmicde-multiplex complexity the signals evocatively, upon diverse in comparison modes, as withwell theas compensatetime-domain for equalization the differential (TDE), mode while holdinggroup dela equivalenty (DMGD) performance, dynamically amongst through which digital the least signal mean processingsquares (LMS) (DSP) and [7–9]. recursive The FDE least techniques squares (RLS) can algorithmsreduce the arealgorithmic most widely complexity held and, evocatively, hence, broadly in comparisonstudied [10 –with12]. Although the time-domain the LMS approachequalization has lower(TDE), complexity, while holding it suffers equivalent from severe performance, performance amongstdeprivation which and the exhibits least mean slower squares convergence, (LMS) and while recursive encountering least squares a large (RLS) number algorithms of parameters are most to widelybe simultaneously held and, hence, adapted broadly in futurestudied SDM [10–12]. channels Although [13]. the Alternatively, LMS approach faster has convergence lower complexity, can be itaccomplished suffers from severe by applying performance RLS with deprivation a moresophisticated and exhibits slower algorithmic converge conformationnce, while encountering at the price of ahigher large complexitynumber of [ 14parameters,15]. Hereafter, to itbe is simultaneously significant to develop adapted a MIMO-DSP in future FDESDM algorithm channels with [13]. the Alternatively,intention of achieving faster convergence an RLS-level can performance be accomplished at an by LMS-adapted applying RLS complexity with a more for ansophisticated SDM-based algorithmicoptical communication conformation system. at the price of higher complexity [14,15]. Hereafter, it is significant to developAnother a MIMO-DSP bewildering FDE algorithm deficiency with of FMF-based the intention optical of achieving fiber systems an RLS-level is the mode-dependent performance at lossan LMS-adapted(MDL), which complexity soars from for inline an SDM-based component optical defectiveness communication and disorders system. the modal orthogonality, consequentlyAnother bewildering degrading thedeficiency overall of capacity FMF-based of MIMO optical channels fiber systems [16]. With is the the mode-dependent aim of lessening loss the (MDL),impact which of MDL, soars old-fashioned from inline tacticscomponent take accountdefectiveness of mode and scrambler disorders and the specialtymodal orthogonality, fiber designs. consequentlyThese approaches degrading were the primarily overall disadvantagedcapacity of MIMO with channels high cost, [16]. yet With they the cannot aim of utterly lessening eradicate the impact of MDL, old-fashioned tactics take account of mode scrambler and specialty fiber designs. the accumulated MDL in optical links [17]. Furthermore, the space–time trellis codes (STTC) were These approaches were primarily disadvantaged with high cost, yet they cannot utterly eradicate the proposed to reduce the MDL, but they suffered from great complexity as well [18]. Henceforward, accumulated MDL in optical links [17]. Furthermore, the space–time trellis codes (STTC) were MDL is categorically another key challenge that needs to be triumphed for next-generation optical proposed to reduce the MDL, but they suffered from great complexity as well [18]. Henceforward, transmission systems. MDL is categorically another key challenge that needs to be triumphed for next-generation optical The spatial multiplicity versus the transmission distance accomplished in recent SDM- wavelength transmission systems. division multiplexing (WDM) transmission experiments is plotted in Figure2[ 19], where the region The spatial multiplicity versus the transmission distance accomplished in recent SDM- with a spatial multiplicity over 30 is designated as the “DSDM region”. The dots in Figure2 represent wavelength division multiplexing (WDM) transmission experiments is plotted in Figure 2 [19], the cases of multimode fiber (MMF), MCF, and coupled-core fibers (only three-core and six-core), where the region with a spatial multiplicity over 30 is designated as the “DSDM region”. The dots in as well as the latest multicore multimode fiber experiments [20]. Figure 2 represent the cases of multimode fiber (MMF), MCF, and coupled-core fibers (only three-core and six-core), as well as the latest multicore multimode fiber experiments [20]. Appl. Sci. 2019,, 9,, 1178x FOR PEER REVIEW 33 of of 2829

Figure 2. The recent records of spatial multiplicity vs. the transmission distance in the state-of-the-art SDM-WDMFigure 2. The experiments recent records [19 ].of spatial multiplicity vs. the transmission distance in the state-of-the-art SDM-WDM experiments [19]. The outline of this review article is sketched as follows: Section2 describes the principles of MIMO equalizers, the LMS and RLS techniques, and the space–time block-coding (STBC) assisted The outline of this review article is sketched as follows: Section 2 describes the principles of RLS algorithms, while Section3 introduces the conﬁguration of a typical multimode coherent MIMO equalizers, the LMS and RLS techniques, and the space–time block-coding (STBC) assisted transmission system transmission system with FMFs for high-order modulation formats. In Section4, RLS algorithms, while Section 3 introduces the configuration of a typical multimode coherent the performance analysis of various adaptive ﬁltering schemes is presented for quadrature phase shift transmission system transmission system with FMFs for high-order modulation formats. In Section keying (QPSK), 16 quadrature amplitude modulation (16QAM), and 64QAM, correspondingly. Then, 4, the performance analysis of various adaptive filtering schemes is presented for quadrature phase Section5 explores the STBC-based enhancement amongst different modulation formats. In Section6, shift keying (QPSK), 16 quadrature amplitude modulation (16QAM), and 64QAM, correspondingly. we further investigate the bit error rate (BER) versus optical signal-to-noise ratio (OSNR) curves Then, Section 5 explores the STBC-based enhancement amongst different modulation formats. In of mode-dependent loss equalization. Last but not least, Section7 demonstrates the complexity Section 6, we further investigate the bit error rate (BER) versus optical signal-to-noise ratio (OSNR) optimization using the proposed single-stage architecture. Finally, Section8 provides a summary curves of mode-dependent loss equalization. Last but not least, Section 7 demonstrates the of recent expressive experiments of SDM transmission using MIMO DSP for short-, medium- and complexity optimization using the proposed single-stage architecture. Finally, Section 8 provides a long-haul distances. summary of recent expressive experiments of SDM transmission using MIMO DSP for short-, 2.medium- Principle and long-haul distances.

2. PrincipleThis section introduces the theoretical background of MIMO equalizers, including LMS, RLS, and the space–time block-coding (STBC) algorithm for SDM optical transmission. Moreover, we This section introduces the theoretical background of MIMO equalizers, including LMS, RLS, also cover the aspects of polarization-related dynamic equalization, orthogonal frequency-division and the space–time block-coding (STBC) algorithm for SDM optical transmission. Moreover, we also multiplexing (OFDM) FDE requirements, and the MIMO hardware requirements in this section. cover the aspects of polarization-related dynamic equalization, orthogonal frequency-division Although hypothetically any modal cross-talk can be electronically managed by DSP, the algorithmic multiplexing (OFDM) FDE requirements, and the MIMO hardware requirements in this section. complexity of the MDM coherent receiver remains one of the main limiting factors to the MIMO Although hypothetically any modal cross-talk can be electronically managed by DSP, the equalization scheme. In general, the total complexity of the DSP algorithm can be measured by the algorithmic complexity of the MDM coherent receiver remains one of the main limiting factors to the number of complex multiplications per symbol per mode, for the reason that the complex multipliers MIMO equalization scheme. In general, the total complexity of the DSP algorithm can be measured are usually the most resource-consuming arithmetic logics in terms of power consumption, area, and by the number of complex multiplications per symbol per mode, for the reason that the complex cost in application-speciﬁc integrated circuit (ASIC) design [21]. Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 29

Appl.multipliers Sci. 2019 , 9are, 1178 usually the most resource-consuming arithmetic logics in terms of power4 of 28 consumption, area, and cost in application-specific integrated circuit (ASIC) design [21].

2.1. MIMO Equalizers: State of the Art Although the SDM conception was present for quite a long time,time, the realreal SDMSDM transmissiontransmission systems were not pursued in industry until recent yearsyears [[22].22]. This is because, in MDM systems, it is assumed that signals could be immaculately conveyedconveyed under a particular mode without interfering with otherother spatialspatial channelschannels within within an an ideal ideal FMF; FMF; however, however, in in reality, reality, that that might might not not be thebe the case, case, for thefor signalsthe signals in numerous in numerous modes modes are often are cross-coupledoften cross-coupled to each to other each dueother to bending,due to bending, twisting, twisting, and/or fiberand/or fabrication fiber fabrication imperfections; imperfections; henceforward, hencefor theward, orthogonality the orthogonality of modes might of modes only be might preserved only forbe apreserved very short for distance a very short in industrial distance SDM in industrial applications. SDM applications. In orderorder to to resolve resolve such such issue, issue, MIMO MIMO signal signal processing processing was developed was developed to undo channelto undo cross-talk channel andcross-talk recover and the transmittedrecover the data transmitted quality. The data coherent quality. receiver The structure coherent for areceiver single-carrier structure system for with a asingle-carrier 6 × 6 MIMO system FDE is exemplifiedwith a 6 × 6 in MIMO Figure 3FDE, where is exemplified the coefficient in adaptationFigure 3, where can be accomplishedthe coefficient withadaptation LMS or can RLS be algorithms accomplished along with with LMS data-aided or RLS algorithms training. along with data-aided training.

Figure 3. The typical coherent receiver configuration for a single-carrier transmission system with a 6Figure× 6 multi-input 3. The typical multi-output coherent receiver (MIMO) configuration frequency-domain for a single-carrier equalization transmission (FDE) [7]. system with a 6 × 6 multi-input multi-output (MIMO) frequency-domain equalization (FDE) [7]. The conventional time-domain equalization (TDE) uses a finite impulse response (FIR) filter matrixThe adapted conventional by the LMStime-domain algorithm. equa Thelization complexity (TDE) for uses the TDEa finite CTDE impulseis given response by [23] (FIR) filter matrix adapted by the LMS algorithm. The complexity for the TDE C is given by [23] CTDE = 3·m·∆β1·L·RS, (1) 𝐶 =3∙𝑚∙∆𝛽 ∙𝐿∙𝑅, (1) where ∆β indicates the DMGD of the fiber, m signifies the number of modes within the fiber, where ∆𝛽1 indicates the DMGD of the fiber, 𝑚 signifies the number of modes within the fiber, 𝐿 L represents the length of the fiber, and R denotes the symbol rate. Subsequently, the complexity represents the length of the fiber, and 𝑅 S denotes the symbol rate. Subsequently, the complexity of ofTDE TDE scales scales linearly linearly with with the the mode mode group group delay delay of of the the optical optical link, link, the the number number of of modes, and the symbol rate. Meanwhile, the complexity for FDE C can be written in the form of symbol rate. Meanwhile, the complexity for FDE CFDE can be written in the form of C =4∙(𝑚+1) ∙log (2∙∆𝛽 ∙𝐿∙𝑅 ) +8∙𝑚. (2) CFDE = 4·(m + 1)· log2(2·∆β1·L·RS) + 8·m. (2) It can be resolved from the equations above that FDE might substantially reduce the complexity It can be resolved from the equations above that FDE might substantially reduce the complexity compared with TDE. Characteristically, a block length equal to twice the equalizer length is applied compared with TDE. Characteristically, a block length equal to twice the equalizer length is applied for the adaptive FDE. By dint of even and odd sub-equalizers, an equivalent half-symbol period for the adaptive FDE. By dint of even and odd sub-equalizers, an equivalent half-symbol period delay delay spacing finite impulse response (FIR) filter can be realized in the frequency domain. The spacing finite impulse response (FIR) filter can be realized in the frequency domain. The required required MIMO equalizer duration to compensate for DMGD in the weak-coupling regime can be MIMO equalizer duration to compensate for DMGD in the weak-coupling regime can be transcribed as transcribed as

NNDMGD = ⌈∆∆𝛽β1·∙𝐿∙𝑅L·ROS·R∙𝑅S,⌉, (3)

where ⌈… ⌉ means the ceiling function, Δβ signifies the DMGD in the fiber, and R represents where ... means the ceiling function, ∆β1 signiﬁes the DMGD in the ﬁber, and ROS represents the oversamplingthe oversampling rate. rate. Furthermore, Furthermore, the MDLthe MDL can can be expressed be expressed as [24 as] [24] Appl. Sci. 2019, 9, 1178 5 of 28

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2 2 MDL = max λk / min λk , (4) k=1∼N k=1∼N MDL = max 𝜆 min,𝜆 ~ ~ (4) where λk stands for singular values of the coupler transfer matrix, which can be calculated by employing thewhere singular λ stands value for decomposition singular values (SVD). of the coupler transfer matrix, which can be calculated by employingTo achieve the singular the MIMO value equalization, decomposition the (SVD). impulse-response matrix measurement needs to be performedTo achieve ﬁrst. the An MIMO example equalization, of the squared the magnitudeimpulse-response of the polarization-divisionmatrix measurement multiplexingneeds to be (PDM)performed SDM first. 6 × An6 impulse example responses of the squared for 96 kmmagnitude of a six-mode of the FMFpolarization-division is depicted in Figure multiplexing4, where the(PDM) columns SDM correspond6 × 6 impulse to responses the transmitted for 96 portskm of and a six-mode the rows FMF correspond is depicted to the in Figure received 4, where ports [ 25the]. Incolumns Figure correspond4, the sharp to peaks the transmitted in the subplots ports identify and the the rows main correspond coupling to points the received evidently, ports which [25]. are In Figure 4, the sharp peaks in the subplots identify the main coupling points evidently, which are divided into four regions designated with A (the coupling between the polarization modes LP01x divided into four regions designated with A (the coupling between the polarization modes LP01x and and LP01y), B (the coupling between the spatial and polarization modes LP11ax, LP11ay, LP11bx, and LP01y), B (the coupling between the spatial and polarization modes LP11ax, LP11ay, LP11bx, and LP11by), LP11by), and C and D (the cross-talk between LP01 and LP11 modes). Such channel estimation provides aand clear C and picture D (the of thecross-talk cross-talk between introduced LP01 and by theLP11 mode modes). multiplexer Such channel and the estimation propagation provides through a clear the six-modepicture of FMF, the whichcross-talk allows introduced a better understandingby the mode ofmultiplexer the MIMO and DSP the performance. propagation through the six-mode FMF, which allows a better understanding of the MIMO DSP performance.

Figure 4. The squared magnitude of the 6 × 6 impulse responses for MIMO equalization [25]. Figure 4. The squared magnitude of the 6 × 6 impulse responses for MIMO equalization [25]. Furthermore, a typical eye diagram is shown below of two de-correlated 12.5-Gbps NRZ 7 Furthermore, a typical eye diagram is shown below of two de-correlated 12.5-Gbps NRZ 27-1 2 -1 pseudorandom binary sequences (PRBSs) launched into one fundamental mode LP01 and a pseudorandom binary sequences (PRBSs) launched into one fundamental mode LP01 and a higher-order LP12 mode, with power adjusted such that both channels received the same power of −12 dBm at the receiver, with decent isolation, low los,s and minimal modal mixing [26]. Consequently, we can observe any substantial coupling between the modal groups within the FMF Appl. Sci. 2019, 9, 1178 6 of 28

Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 29 higher-order LP12 mode, with power adjusted such that both channels received the same power of −12 dBm at the receiver, with decent isolation, low los,s and minimal modal mixing [26]. Consequently, which would easily close the eye at the receiver. The eye diagram with heavy modal dispersion can we can observe any substantial coupling between the modal groups within the FMF which would easily be seen in Figure 5b, while the eyes of multiplexing independent channels LP01 and LP12 are close the eye at the receiver. The eye diagram with heavy modal dispersion can be seen in Figure5b, displayed in Figures 5c,d. while the eyes of multiplexing independent channels LP01 and LP12 are displayed in Figure5c,d.

Figure 5. Eye diagrams of a typical mode-division multiplexing (MDM) transmission system for (a)

Figureback-to-back; 5. Eye diagrams (b) over-fill of launcha typical without mode-division coupler; (multiplexingc) LP01 channel; (MDM) (d) LP transmission12 channel [26 system]. for (a) back-to-back; (b) over-fill launch without coupler; (c) LP01 channel; (d) LP12 channel [26]. 2.2. LMS and RLS Techniques 2.2. LMSBoth and LMS RLS and Techniques RLS algorithms are intended to iteratively minimalize the squared error at all discreteBoth frequencies. LMS and RLS The algorithms LMS algorithm are intended stands as to a it stochasticeratively gradientminimalize descent the squared minimization error at using all discreteinstantaneous frequencies. error estimates,The LMS algorithm with the corresponding stands as a stochastic weight updategradient as descent [27] minimization using instantaneous error estimates, with the corresponding weight update as [27] W(k + 1) = W(k) + µ[U(k) − W(k)V(k)], (5) 𝑊(𝑘+1) =𝑊(𝑘) +𝜇𝑈(𝑘) −𝑊(𝑘)𝑉(𝑘), (5) where W(k) signifies the MIMO channel matrix vector at frequency k, U(k) indicates fast Fourier where W(𝑘) signifies the MIMO channel matrix vector at frequency 𝑘, U(𝑘) indicates fast Fourier transform (FFT) of detected data blocks, µ denotes the step size, and V(k) represents the FFT of transform (FFT) of detected data blocks, 𝜇 denotes the step size, and V(𝑘) represents the FFT of sampled received signals. sampled received signals. In conventional LMS, the convergence rate and equalization performance depend on the same In conventional LMS, the convergence rate and equalization performance depend on the same value of the scalar step size µ. The convergence speed can be enlarged by choosing an adaptive step value of the scalar step size μ. The convergence speed can be enlarged by choosing an adaptive step size based on the power spectral density (PSD) of input signal, which can be conveyed as [28] size based on the power spectral density (PSD) of input signal, which can be conveyed as [28] αα µ𝜇((k𝑘))== ,, (6) (6) SS((k𝑘)) wherewhere S(k) S(k) signifies signifies the the PSD PSD of of a a posterior posterior error error block, block, and and αα symbolizessymbolizes the the adaptation adaptation rate, rate, which which determinesdetermines both thethe convergenceconvergence speed speed and and equalization equalization performance performance of the of noisethe noise PSD directedPSD directed (NPD) (NPD)LMS algorithm. LMS algorithm. The basic The schematic basic schematic of an MDM of an transmission MDM transmission system using system the LMSusing algorithm the LMS is algorithmshown in Figureis shown6. in Figure 6. In contrast, the RLS algorithm depends on the iterative minimization of an exponentially weighted cost function, whose convergence speed is not strongly dependent on the input statistics, with the corresponding weight update as h i W(k + 1) = W(k) + [U(k) − W(k)V(k)]·V(k)H· R(k)·β−1 , (7) Appl. Sci. 2019, 9, 1178 7 of 28 where R(k) symbolizes a tracked inverse time-averaged weighted correlation matrix, the superscript H signifies the Hermitian conjugate, and β denotes a forgetting factor, which gives an exponentially lower weight to older error samples. Consequently, the main difference between the conventional LMS and RLS algorithms is that RLS has a growing memory due to R(k) and β, and, thus, might achieve an even lower OSNR and faster adaptation. Specifically, the tracked inverse time-averaged weighted correlation matrix can be denoted as [29] ( ) h i R(k)·β−1·V(k)·V(k)H·R(k)·β−1 R(k + 1) = R(k)·β−1 − , (8) 1 + V(k)H·[R(k)·β−1]·V(k) which assigns a different step size to each adjustable RLS equalizer’s coefficient at their updates, rendering all frequency bins with a uniform convergence speed, thus cultivating the total algorithmic performance.

Figure 6. Schematic of an MDM system utilizing the MIMO FDE [31]. Figure 6. Schematic of an MDM system utilizing the MIMO FDE [31]. The space and time allocation for the STBC-aided RLS algorithm can be briefly represented In contrast, the RLS algorithm depends on the iterative minimization of an exponentially as shown in Figure7, as a matrix of spatial channels and time slots, whereby each element is the weighted cost function, whose convergence speed is not strongly dependent on the input statistics, modulated symbol to be transmitted in mode i at time slot j. We may choose different combinations, with the corresponding weight update as and we can apply one or more of received copies to properly decode the received signal and, therefore, achieve better performance.W(𝑘+1) =W(𝑘) + U(𝑘) −W(𝑘)V(𝑘) ∙V(𝑘) ∙ R(𝑘) ∙𝛽, (7) The architecture of the adaptive STBC-RLS equalizer in an m-mode FMF transmission system is where R (k) symbolizes a tracked inverse time-averaged weighted correlation matrix, the illuminated in Figure8. At coherent detection, the serial-to-parallel (S/P) converters split each data superscript H signifies the Hermitian conjugate, and β denotes a forgetting factor, which gives an sequence y , . . . y into even/odd sequences, while two consecutive blocks are concatenated. After the exponentially1 lowerm weight to older error samples. Consequently, the main difference between the FFT, the samples of the k-th block are overlapped with the (k − 1)th block at frequency k, which are conventional LMS and RLS algorithms is that RLS has a growing memory due to R(k) and β, and, further alienated into sequential blocks yP(k), where superscript P specifies an even or odd sequence. thus, might achieve an even lower OSNRq and faster adaptation. Specifically, the tracked inverse P Eachtime-averaged block is then weighted converted correlation to the matrix frequency can domainbe denoted with as FFT[29] as Yq(k), which goes through an inversed channel filter for the data matrix production with the even or odd tributaries, while HP (k) 𝑅(𝑘) ∙𝛽 ∙V(𝑘) ∙V(𝑘) ∙ 𝑅(𝑘) ∙𝛽 q,j stands for the inversed𝑅(𝑘+1 channel) = 𝑅 filter(𝑘) ∙𝛽 in Jones-vector − notation, while both q and j denote, mode indices(8) 1+V(𝑘) ∙ 𝑅(𝑘) ∙𝛽 ∙V(𝑘) between 1 and m. After an adaptive MIMO equalization, the carrier recovery is applied to mitigate which assigns a different step size to each adjustable RLS equalizer’s coefficient at their updates, rendering all frequency bins with a uniform convergence speed, thus cultivating the total algorithmic performance.

2.3. STBC-Assisted RLS Algorithms One of the primary issues regarding the conventional RLS implementation might be its comparatively sophisticated computational complexity, attributable to the recursive updating with a growing memory. Henceforth, in order to accomplish a quasi-RLS performance at a quasi-LMS cost, the STBC technique can be applied to aid the RLS approach, which is defined as follows: assuming multiple copies of data stream are transmitted across a number of spatial modes in an FMF transmission, some of the received copies are better than the others under the same scattering, reflection, or thermal noise; thus, the various received versions of the data can be exploited to increase the overall system reliability [30]. The space and time allocation for the STBC-aided RLS algorithm can be briefly represented as shown in Figure 7, as a matrix of spatial channels and time slots, whereby each element is the modulated symbol to be transmitted in mode 𝑖 at time slot 𝑗. We may choose different combinations, and we can apply one or more of received copies to properly decode the received signal and, therefore, achieve better performance.

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the laser phase noise, the outputs of which are transformed back to the time domain using inverse FFT (IFFT),Figure 7. compared The space toand the time desired allocation response for the to space–time generate block-coding an error vector, (STBC)-aided and to update recursive the least receiver coefﬁcients in a training mode untilsquares they converge. (RLS) algorithm Then, [31]. it switches to a decision-directed mode, whereby reconstructed data are transformed back using FFT through the zero padding to update the Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 29 coefﬁcientsThe architecture [31]. of the adaptive STBC-RLS equalizer in an m-mode FMF transmission system is illuminated in Figure 8. At coherent detection, the serial-to-parallel (S/P) converters split each data

sequence y,…y into even/odd sequences, while two consecutive blocks are concatenated. After the FFT, the samples of the k-th block are overlapped with the (k − 1)th block at frequency k, which are further alienated into sequential blocks y (k), where superscript P specifies an even or odd sequence. Each block is then converted to the frequency domain with FFT as Y (k), which goes through an inversed channel filter for the data matrix production with the even or odd tributaries, while H,(k) stands for the inversed channel filter in Jones-vector notation, while both q and j denote mode indices between 1 and m. After an adaptive MIMO equalization, the carrier recovery is applied to mitigate the laser phase noise, the outputs of which are transformed back to the time domain using inverse FFT (IFFT), compared to the desired response to generate an error vector, and to update the receiver coefficients in a training mode until they converge. Then, it switches to a Figure 7. The space and time allocation for the space–time block-coding (STBC)-aided recursive least decision-directed mode, whereby reconstructed data are transformed back using FFT through the Figuresquares 7. (RLS)The space algorithm and time [31]. allocation for the space–time block-coding (STBC)-aided recursive least zero padding to update the coefficients [31]. squares (RLS) algorithm [31].

The architecture of the adaptive STBC-RLS equalizer in an m-mode FMF transmission system is illuminated in Figure 8. At coherent detection, the serial-to-parallel (S/P) converters split each data sequence y,…y into even/odd sequences, while two consecutive blocks are concatenated. After the FFT, the samples of the k-th block are overlapped with the (k − 1)th block at frequency k, which are further alienated into sequential blocks y (k), where superscript P specifies an even or odd sequence. Each block is then converted to the frequency domain with FFT as Y (k), which goes through an inversed channel filter for the data matrix production with the even or odd tributaries, while H,(k) Figurestands 8. forThe the architecture inversed of channel an STBC-aided filter RLSin Jones-vector equalizer for MDMnotation, receivers while [29 ].both q and j Figure 8. The architecture of an STBC-aided RLS equalizer for MDM receivers [29]. denote mode indices between 1 and m. After an adaptive MIMO equalization, the carrier recovery is appliedThe to weight mitigate update the laser of thephase STBC-aided noise, the RLS outputs algorithm of which follows are transformed the rules below, back whereasto the time the The weight update of the STBC-aided RLS algorithm follows the rules below, whereas the domaincorresponding using inverse equalizer FFT coefﬁcients (IFFT), compared are updated to the every desired two response blocks (k +to 2generate) instead an of error each blockvector, (k and+ 1 ) corresponding equalizer coefficients are updated every two blocks (k+2) instead of each block (k+ toin update the recursion. the receiver coefficients in a training mode until they converge. Then, it switches to a 1) in the recursion. decision-directed mode, whereby reconstructed data! are transformed back using FFT through the PkP+2 00 ∗ ∗ zero padding to updateW(kW+( 𝑘+2the2) = coefficients)W=W(k)(+𝑘) + [31]. ((UUk+)2)∙ ·[DDk+2−U− Uk+2∙W·W((𝑘k))],, (9) 00P Pk+2 where U denotes the Alamouti matrix containing the received symbols from block k+2 and where U denotes the Alamouti matrix containing the received symbols from block k + 2 and k + 3, k+3, andk+ 2D represents the desired response vector for training and decision-directed tracking and D represents the desired response vector for training and decision-directed tracking [32], while [32], whilek+2 the superscript * indicates the complex conjugation. Meanwhile, the diagonal term P the superscript * indicates the complex conjugation. Meanwhile, the diagonal term P is given by is given by k+2

−1 h −1 i PPk+2 ==𝛽β · ∙PPk−−𝛽β ·P∙Pk·Ω∙Ωk+2·Pk∙P,, (10)

where 𝛽 is the forgetting factor, and the term Ω is given by where β is the forgetting factor, and the term Ωk+2 is given by Figure 8. The architecture of an STBC-aided RLS equalizer for MDM receivers−1 [29]. 2 2 −1 −1 Ωk+2 = {Diagonal[|V(k)| + |V(k + 1)| ] + β ·Pk} , (11) The weight update of the STBC-aided RLS algorithm follows the rules below, whereas the where V(k) stands for the FFT of sampled received signals. By utilizing the STBC scheme, we may corresponding equalizer coefficients are updated every two blocks (k+2) instead of each block (k+ choose different combinations, and apply one or more of received copies to correctly decode the 1) in the recursion. received signal, and, thus, attain a better performance. Meanwhile, the STBC is able to accomplish an P 0 RLS-level performanceW(𝑘+2 with) a=W much(𝑘) lower+ computational (U complexity,)∗ ∙ D −U because∙W( no𝑘) matrix, inversion is 0P (9) essential during the recursion. Instead of adding a training sequence to each data block, only a few where U denotes the Alamouti matrix containing the received symbols from block k+2 and k+3, and D represents the desired response vector for training and decision-directed tracking [32], while the superscript * indicates the complex conjugation. Meanwhile, the diagonal term P is given by

P =𝛽 ∙ P −𝛽 ∙P ∙Ω ∙P, (10) where 𝛽 is the forgetting factor, and the term Ω is given by Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 29

Ω = 𝐃𝐢𝐚𝐠𝐨𝐧𝐚𝐥|V(𝑘)| + |V(𝑘+1)| +𝛽 ∙P , (11) where V(k) stands for the FFT of sampled received signals. By utilizing the STBC scheme, we may choose different combinations, and apply one or more of received copies to correctly decode the receivedAppl. Sci. 2019 signal,, 9, 1178 and, thus, attain a better performance. Meanwhile, the STBC is able to accomplish9 of an 28 RLS-level performance with a much lower computational complexity, because no matrix inversion is essential during the recursion. Instead of adding a training sequence to each data block, only a few training blocks are added in the beginning, and the channel variations are tracked at the decision-directed training blocks are added in the beginning, and the channel variations are tracked at the stage in an attempt to reduce the overall system overhead [33]. decision-directed stage in an attempt to reduce the overall system overhead [33]. 2.4. Polarization-Related Dynamic Equalization 2.4. Polarization-Related Dynamic Equalization Polarization characterizes a fundamental multiplexing technique, which allows doubling the Polarization characterizes a fundamental multiplexing technique, which allows doubling the spectral efficiency of fiber optical transmission systems [34]. To accommodate the usage of two spectral efficiency of fiber optical transmission systems [34]. To accommodate the usage of two polarization tributaries, polarization de-multiplexing (PolDemux) algorithms represent a crucial part polarization tributaries, polarization de-multiplexing (PolDemux) algorithms represent a crucial of digital coherent receivers, which makes it conceivable to efficiently compensate for time-varying part of digital coherent receivers, which makes it conceivable to efficiently compensate for state of polarization (SOP). The PolDemux approaches include MIMO algorithms such as LMS and time-varying state of polarization (SOP). The PolDemux approaches include MIMO algorithms such RLS, and the constant-modulus algorithms (CMA). as LMS and RLS, and the constant-modulus algorithms (CMA). In particular, Stokes space-based DSP techniques improve the polarization de-multiplexing, In particular, Stokes space-based DSP techniques improve the polarization de-multiplexing, polarization dependent loss (PDL) compensation, and other polarization-related propagation polarization dependent loss (PDL) compensation, and other polarization-related propagation impairments in coherent receivers, particularly in terms of convergence speed and transparency impairments in coherent receivers, particularly in terms of convergence speed and transparency to to higher-level M-ary modulated signals. The block diagram of the DSP subsystems of a coherent higher-level M-ary modulated signals. The block diagram of the DSP subsystems of a coherent transceiver for adaptive equalization of both polarization and phase diversity is presented in Figure9, transceiver for adaptive equalization of both polarization and phase diversity is presented in Figure which includes the front-end compensation, static equalization, adaptive equalization, carrier phase 9, which includes the front-end compensation, static equalization, adaptive equalization, carrier recovery, and the symbol decision [35]. phase recovery, and the symbol decision [35].

Figure 9. Block diagram of the digital signal processing (DSP) subsystems of a coherent transceiver Figure 9. Block diagram of the digital signal processing (DSP) subsystems of a coherent transceiver using a Stokes space-based polarization de-multiplexing (PolDemux) algorithm [35]. using a Stokes space-based polarization de-multiplexing (PolDemux) algorithm [35]. Furthermore, the Stokes space-based PolDemux method has speciﬁc advantages including the higherFurthermore, convergence the ratio, Stokes the space-based improved robustnessPolDemux method against phasehas specific noise, advantages and the transparency including the to higherhigher-level convergence M-ary modulated ratio, the signals.improved robustness against phase noise, and the transparency to higher-level M-ary modulated signals. 2.5. OFDM FDE Requirements The high-speed orthogonal frequency-division multiplexing (OFDM) in SDM optical long-haul transmission systems serves as another important aspect of MIMO equalization techniques [36]. The complex multiplications per bit for various equalizer types in terms of modal dispersion for a 2000-km transmission distance and 10 × 10 MIMO is shown in Figure 10, where an FMF with three Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 29

2.5. OFDM FDE Requirements The high-speed orthogonal frequency-division multiplexing (OFDM) in SDM optical long-haul transmissionAppl. Sci. 2019 systems, 9, 1178 serves as another important aspect of MIMO equalization techniques [36]. 10The of 28 complex multiplications per bit for various equalizer types in terms of modal dispersion for a 2000-km transmission distance and 10 × 10 MIMO is shown in Figure 10, where an FMF with three modes with two polarizations (LP01, LP11a, LP11b, LP21a, LP21b) results in the number of tributaries modes with two polarizations (LP01, LP11a, LP11b, LP21a, LP21b) results in the number of tributaries being 10. being 10.

Figure 10. The orthogonal frequency-division multiplexing (OFDM) FDE equalizer complexity requirements Figure 10. The orthogonal frequency-division multiplexing (OFDM) FDE equalizer complexity compared with other MIMO approaches [37]. requirements compared with other MIMO approaches [37]. In principle, the complexity is calculated in a two-dimensional matrix, where one variable is the samplingIn principle, rate and the thecomplexity other is theis calculated size of the in FFT. a two-dimensional For each conﬁguration, matrix, thewhere optimum one variable sampling is the rate samplingresulting rate in and minimum the other complexity is the size satisfying of the FFT. the For overhead each configuration, constraint isthe considered. optimum sampling As we can rate see, resultingOFDM in requires minimum the lowestcomplexity equalizer satisfying complexity the overhead for cross-talk constraint compensation is considered. in anAs MDM we can receiver, see, OFDMfor most requires of the the multiplications lowest equalizer required complexity for FDE/TDE for cross-talk are caused compensation by time-domain in an operations. MDM receiver, However, forOFDM most cannotof the toleratemultiplications a modal required dispersion for of moreFDE/TDE than 5.9are ps/kmcaused modal by time-domain dispersion dueoperations. to the 10% However,overhead OFDM constraint cannot for tolerate FMF-based a modal optical dispersion transmission of more [37]. than 5.9 ps/km modal dispersion due to the 10% overhead constraint for FMF-based optical transmission [37]. 2.6. MIMO Hardware Requirements 2.6. MIMO Hardware Requirements Last but not least, this subsection discusses the design and architecture of spatial multiplexing MIMOLast decodersbut not least, for field-programmable this subsection discusses gate arrays the (FPGAs).design and Conventionally, architecture of the spatial optimal multiplexing hard-decision MIMOdetection decoders for MIMO for systemsfield-programmable is the maximum gate likelihood arrays (FPGAs). (ML) detector, Conventionally, which is well the known optimal due to hard-decisionits superior performancedetection for MIMO in terms systems of BER, is but the its maximum direct implementation likelihood (ML) grows detector, exponentially which is well with a knownhigher-level due to modulationits superior scheme, performance causing in its terms FPGA of implementation BER, but its direct to be relatively implementation infeasible grows [38]. exponentiallyOn the with other a hand, higher-level soft-output modulation generation scheme, for a spherecausing detector its FPGA can implementation solve the ML detection to be relativelyissue in infeasible a computationally [38]. efﬁcient manner, for a real-time implementation on a DSP processor at high-performanceOn the other hand, parallel soft-output computing generation platforms. for a Thesphere scalable detector architecture can solve ofthe soft-output ML detection generation issue in fora computationally a sphere detector isefficient presented manner, in Figure for 11 a, whichreal-time requires implementation further micro-architecture on a DSP processor optimizations at high-performanceand trade-offs [39 parallel]. computing platforms. The scalable architecture of soft-output generation for a sphere detector is presented in Figure 11, which requires further micro-architecture optimizations and trade-offs [39]. Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 29 Appl. Sci. 2019Appl., 9 Sci., x FOR2019 ,PEER9, 1178 REVIEW 11 of 29 11 of 28

Figure 11. A scalable architecture of spatial multiplexing MIMO decoders for field-programmable gate Figure 11.Figure A scalable 11. A architecturescalable architecture of spatial of multiple spatial xingmultiple MIMOxing decoders MIMO decoders for field-programmable for field-programmable arrays (FPGAs) [39]. gate arraysgate (FPGAs) arrays (FPGAs)[39]. [39]. 3. System Configuration 3. System3. SystemConfiguration Configuration The system configuration of a six-mode coherent transmission system is illustrated in Figure 12, The system configuration of a six-mode coherent transmission system is illustrated in Figure 12, Thewith system a 223 configuration− 1 pseudorandom of a six-mode binary coherent sequence transmission (PRBS) sequence system modulated is illustrated on in every Figure mode 12, over a with23 a 223 − 1 pseudorandom binary sequence (PRBS) sequence modulated on every mode over a with a 230-km− 1 pseudorandom FMF. The OSNR binary setting sequ blockence indicates (PRBS) that sequence the additive modulated white Gaussian on every noise mode (AWGN) over a noise is 30-km FMF. The OSNR setting block indicates that the additive white Gaussian noise (AWGN) noise 30-km FMF.added The after OSNR the FMFsetting transmission, block indicates while that the the optical additive filtering white block Gaussian specifies noise that (AWGN) a Gaussian noise filter with is added after the FMF transmission, while the optical filtering block specifies that a Gaussian filter is addeda after 33-GHz the bandwidthFMF transmission, is usedto while suppress the optical the out-of-band filtering block noise specifies before the that coherent a Gaussian detection. filter with a 33-GHzwith a 33-GHzbandwidth bandwidth is used tois usedsuppress to suppress the out-of-band the out-of-band noise before noise the before coherent the coherent detection. detection.

Figure 12. Conﬁguration of a six-mode coherent transmission system [29]. Figure 12.Figure Configuration 12. Configuration of a six-mode of a six-mode coherent coherent transmission transmission system [29]. system [29]. Appl.Appl. Sci. Sci.2019 2019, 9,, 9 1178, x FOR PEER REVIEW 1212 of of 28 29

After mode de-multiplexer (Mod DEMUX), parallel signals are launched into the receiver, and areAfter converted mode to de-multiplexer baseband in the (Mod optical DEMUX), front parallelend by internally signals are mixing launched with into the the local receiver, oscillator and are(LO) convertedat a linewidth to baseband of 100 kHz. in the The optical electrical front signal end by in-phase internally and mixing quadrature with the (I/Q) local components oscillator (LO)are then at are-sampled linewidth of at 100 two kHz. samples The electricalper symbol signal by the in-phase analog-to-digital and quadrature converter (I/Q) (ADC). components After the are MIMO then re-sampledprocessing, at a twoBER samples is estimated per symbolfor each by mode the analog-to-digital[40]. converter (ADC). After the MIMO processing, a BER is estimated for each mode [40]. 4. Performance Evaluation for Different Adaptive Filtering Schemes 4. Performance Evaluation for Different Adaptive Filtering Schemes In this section, the convergence speed comparison amongst the conventional LMS algorithm, theIn noise this section,PSD directed the convergence (NPD) LMS speed algorithm, comparison the RLS amongst algorithm, the conventional and the proposed LMS algorithm, STBC-RLS thealgorithm noise PSD for directedPDM-QPSK (NPD) signal LMS is exemplified algorithm, thein Figure RLS algorithm, 13, whereas and the the normalized proposed mean STBC-RLS square algorithmerror (NMSE) for PDM-QPSK represents signal the average is exempliﬁed square in errors Figure of 13 each, whereas block theafter normalized the MIMO mean equalization. square error The (NMSE)adaptation represents rate and the averagestep size square of four errors algorithms of each blockwere afterset to the their MIMO optimal equalization. values; the The block adaptation length ratewas and 8192 step samples size of fourusing algorithms 50% overlap were with set toOSNR their set optimal at 18 values;dB. In the the meantime, block length the was inset 8192 on samples the right usingdisplays 50% the overlap constellation with OSNR diagram set at 18before dB. Inthe the ST meantime,BC compensation, the inset as on the the received right displays signal thewas constellationseverely distorted, diagram attributable before the to STBC mode compensation, coupling and as delay, the received while the signal inset was on the severely left indicates distorted, the attributableconstellation to modediagram coupling after andapplying delay, the while STBC-RLS the inset compensation, on the left indicates as the the modal constellation distortions diagram were afterremarkably applying compensated the STBC-RLS for compensation, by the MIMOas equalizers the modal over distortions a 30-km wereFMF. remarkably compensated for by the MIMO equalizers over a 30-km FMF.

Figure 13. The normalized mean square error (NMSE) vs. the number of fast Fourier transform (FFT)Figure blocks 13. The for variousnormalized algorithms mean square for a polarization-division error (NMSE) vs. the multiplexing number of fast (PDM) Fourier quadrature transform phase (FFT) shiftblocks keying for various (QPSK) algorithms signal; inset: for constellation a polarization-division diagrams before multiplexing (right) and (PDM) after quadrature (left) the STBC-RLS phase shift compensationkeying (QPSK) [27 ].signal; inset: constellation diagrams before (right) and after (left) the STBC-RLS compensation [27]. As for the computational complexity per data symbol for the MIMO FDE, which is mainly determinedAs for by the the computational number of complex complexity multiplications, per data thesymbol RLS andfor the LMS MIMO algorithms FDE, wouldwhich consumeis mainly 148determined and 302 complexby the number multiplications, of complex respectively, multiplications, while the the RLS proposed and LMS STBC-RLSalgorithms wouldwould requireconsume 172148 complex and 302 multiplications, complex multiplications, which is roughly respectively, a 16.2% while increase the proposed in hardware STBC-RLS complexity would compare require to172 LMS,complex but 75.6% multiplications, less than that which of the is RLS roughly approach. a 16.2% increase in hardware complexity compare to LMS,To but further 75.6% upsurge less than the that spectral of the efﬁciency RLS approach. of the next-generation SDM transmission systems, the convergenceTo further speed upsurge comparisons the spectral between efficiency different of th algorithmse next-generation for PDM-16QAM SDM transmission and PDM-64QAM systems, the signalsconvergence are presented speed comparisons in Figures 14 between and 15, respectively,different algorithms whereas for the PDM-16QAM comparison between and PDM-64QAM different modulationsignals are formatspresented is summarized in Figures 14 in and Table 15,1. respectively, whereas the comparison between different modulationFrom the formats plot above, is summarized we can see that in Table RLS converges 3. faster to a lower asymptotic NMSE than LMS, because it has a growing memory due to the forgetting factor. The NPD-LMS algorithm can achieve a faster convergence than the traditional LMS, because it adopts variable bin-wise step size to render posterior error of every frequency bin convergent to the background noise of the AWGN channels. Still, the NMSE convergence of the proposed STBC-RLS algorithm seems a bit inferior to that of the conventional RLS in the beginning, for a smaller block size is used for training than the data block size to reduce the system overhead at the expense of a minor loss of performance. Subsequently, to Appl. Sci. 2019, 9, x FOR PEER REVIEW 13 of 29

Appl. Sci. 2019, 9, 1178 13 of 28 achieve −10 dB steady NMSE, which is equal to a 9.8-dB Q-value, the LMS and NPD-LMS schemes needAppl. 55 Sci. and 2019 47, 9, FFTx FOR blocks, PEER REVIEW while the RLS and STBC-RLS approaches require roughly 27 and 31 blocks,13 of 29 respectively;Figure henceforward,14. The NMSE vs. the the convergence number of FFT rate blocks would for be various enhanced algorithms by 50.9% for anda PDM 43.6%, 16 quadrature respectively. amplitude modulation (16QAM) signal; inset: constellation diagrams before (right) and after (left) the STBC-RLS compensation [27].

From the plot above, we can see that RLS converges faster to a lower asymptotic NMSE than LMS, because it has a growing memory due to the forgetting factor. The NPD-LMS algorithm can achieve a faster convergence than the traditional LMS, because it adopts variable bin-wise step size to render posterior error of every frequency bin convergent to the background noise of the AWGN channels. Still, the NMSE convergence of the proposed STBC-RLS algorithm seems a bit inferior to that of the conventional RLS in the beginning, for a smaller block size is used for training than the data block size to reduce the system overhead at the expense of a minor loss of performance. Subsequently, to achieve −10 dB steady NMSE, which is equal to a 9.8-dB Q-value, the LMS and NPD-LMS schemes need 55 and 47 FFT blocks, while the RLS and STBC-RLS approaches require roughly 27 and 31 blocks, respectively; henceforward, the convergence rate would be enhanced by 50.9%Figure and 14. 43.6%,The NMSE respectively. vs. the number of FFT blocks for various algorithms for a PDM 16 quadrature amplitudeFigure 14. modulation The NMSE (16QAM) vs. the number signal; inset:of FFT constellation blocks for various diagrams algorithms before (right for a) PDM and after 16 quadrature (left) the STBC-RLSamplitude compensation modulation (16QAM) [27]. signal; inset: constellation diagrams before (right) and after (left) the STBC-RLS compensation [27].

Figure 15. The NMSE vs. the number of FFT blocks for various algorithms for a PDM-64QAM signal; inset:Figure constellation 15. The NMSE diagrams vs. the before number (right of )FFT and blocks after ( leftfor )various the STBC-RLS algorithms compensation for a PDM-64QAM [27]. signal; Tableinset: 1. constellationComparison diagrams of multiple before modulation (right) and formats after (left [21].) the QPSK—quadrature STBC-RLS compensation phase shift [27]. keying; QAM—quadrature amplitude modulation. Table 1. Comparison of multiple modulation formats [21]. QPSK—quadrature phase shift keying; Modulation # of SymbolsQAM—quadratureLevels per amplitude# of Bits modulation. per Symbol Corresponding Format Transmitted Carrier Contained Data Rate Modulation # of Symbols Levels per # of Bits per Symbol Corresponding QPSK 4 = 22 2 2 112 Gb/s 16QAM 16 = 24 4 4 224 Gb/s 64QAM 64 = 26 8 6 336 Gb/s

From the plots, we can see that with higher-order modulation formats, the advantage of RLS convergence rate over that of the LMS becomes even larger, owing to a growing memory, while the difference in NMSE between the proposed STBC-RLS algorithm and the conventional RLS shrinks, which indicates that such proposed adaptive receivers could lower system overhead requirements for higher-orderFigure 15. modulation The NMSE formats. vs. the number of FFT blocks for various algorithms for a PDM-64QAM signal; inset: constellation diagrams before (right) and after (left) the STBC-RLS compensation [27].

Table 1. Comparison of multiple modulation formats [21]. QPSK—quadrature phase shift keying; QAM—quadrature amplitude modulation.

Modulation # of Symbols Levels per # of Bits per Symbol Corresponding Appl. Sci. 2019, 9, x FOR PEER REVIEW 14 of 29

Format Transmitted Carrier Contained Data Rate QPSK 4=2 2 2 112 Gb/s 16QAM 16 = 2 4 4 224 Gb/s 64QAM 64 = 2 8 6 336 Gb/s

From the plots, we can see that with higher-order modulation formats, the advantage of RLS convergence rate over that of the LMS becomes even larger, owing to a growing memory, while the difference in NMSE between the proposed STBC-RLS algorithm and the conventional RLS shrinks, which indicates that such proposed adaptive receivers could lower system overhead requirements for higher-order modulation formats. Appl. Sci. 2019, 9, 1178 14 of 28 5. STBC-Assisted Improvement between Various Modulation Formats

5. STBC-AssistedTo irradiate Improvementthe benefits of betweenusing the Various STBC scheme Modulation to mitigate Formats the MDL impairment in the SDM transmission systems, the performance results of the space–time coded FMF transmission with or To irradiate the beneﬁts of using the STBC scheme to mitigate the MDL impairment in the SDM without using the STBC-RLS scheme are presented in Figure 16, from which we can achieve a transmission systems, the performance results of the space–time coded FMF transmission with or roughly 3-dB OSNR improvement for the PDM-QPSK signal at a 10−3 BER, whereas the condition for without using the STBC-RLS scheme are presented in Figure 16, from which we can achieve a roughly the dotted line marked as Gaussian indicates the performance over a perfect MDL-free Gaussian 3-dB OSNR improvement for the PDM-QPSK signal at a 10−3 BER, whereas the condition for the channel. dotted line marked as Gaussian indicates the performance over a perfect MDL-free Gaussian channel.

Figure 16. The bit error rate (BER) vs. optical signal-to-noise ratio (OSNR) comparison for PDM-QPSK Figure 16. The bit error rate (BER) vs. optical signal-to-noise ratio (OSNR) comparison for signal in LP11a mode with or without using the STBC-RLS scheme [31]. PDM-QPSK signal in LP11a mode with or without using the STBC-RLS scheme [31]. Additionally, for the higher-order modulation formats, the performance comparison plots for Appl. Sci. 2019, 9, x FOR PEER REVIEW 15 of 29 PDM-16QAMAdditionally, and PDM-64QAM for the higher-order signals withmodulation or without formats, using the the performance purposed STBC-RLS comparison scheme plots are for furtherPDM-16QAM shown in and Figures PDM-64QAM 17 and 18, signals respectively. with or without using the purposed STBC-RLS scheme are further shown in Figures 17 and 18, respectively.

Figure 17. The BER vs. OSNR comparison for PDM-16QAM signal in LP11a mode with or without usingFigure the STBC-RLS17. The BER scheme vs. OSNR [31]. comparison for PDM-16QAM signal in LP11a mode with or without using the STBC-RLS scheme [31]. In summary, the performance comparison for various modulation formats with or without using the STBC-RLSIn summary, scheme the is performance summarized comparison in Figure 19 for, whereas various the modulation scheme for formats not using with STBC-RLS or without designatesusing the the STBC-RLS conventional scheme RLS scheme,is summarized from which in Figure we can 19, observe whereas that, the as morescheme bits for per not symbol using areSTBC-RLS transmitted, designates a larger the OSNR conventional tolerance RLS improvement scheme, from could which be accomplished we can observe by usingthat, as STBC more for bits higher-orderper symbol modulation are transmitted, formats. a larger The overall OSNR OSNRtolerance tolerance improvement can be improved could be by accomplished means of the by STBC using approachSTBC for by higher-order approximately modulation 3.1, 4.9, and formats. 7.8 dB The for QPSK,overall 16QAM, OSNR tolerance and 64QAM can be with improved respect toby BER. means of the STBC approach by approximately 3.1, 4.9, and 7.8 dB for QPSK, 16QAM, and 64QAM with respect to BER.

Figure 18. The BER vs. OSNR comparison for PDM-64QAM signal in LP11a mode with or without using the STBC-RLS scheme [31]. Appl. Sci. 2019, 9, x FOR PEER REVIEW 15 of 29

Figure 17. The BER vs. OSNR comparison for PDM-16QAM signal in LP11a mode with or without using the STBC-RLS scheme [31].

In summary, the performance comparison for various modulation formats with or without using the STBC-RLS scheme is summarized in Figure 19, whereas the scheme for not using STBC-RLS designates the conventional RLS scheme, from which we can observe that, as more bits per symbol are transmitted, a larger OSNR tolerance improvement could be accomplished by using STBC for higher-order modulation formats. The overall OSNR tolerance can be improved by means of the STBC approach by approximately 3.1, 4.9, and 7.8 dB for QPSK, 16QAM, and 64QAM with Appl.respect Sci. 2019 to, 9 BER., 1178 15 of 28

Appl. Sci. 2019, 9, x FOR PEER REVIEW 16 of 29 Figure 18. The BER vs. OSNR comparison for PDM-64QAM signal in LP11a mode with or without Figure 18. The BER vs. OSNR comparison for PDM-64QAM signal in LP11a mode with or without using the STBC-RLS scheme [31]. using the STBC-RLS scheme [31].

Figure 19. The summary of BER vs. OSNR for various modulation formats with or without using STBC-RLS compensation [29]. Figure 19. The summary of BER vs. OSNR for various modulation formats with or without using 6. MIMOSTBC-RLS Equalization compensation for [29]. Mode-Dependent Loss

6. MIMOThe Equalization mode-dependent for Mode-Dependent loss (MDL) arises Loss from imperfections in optical components such as optical amplifiers, couplers, and multiplexers, as well as from non-unitary cross-talk in the fiber and atThe fiber mode-dependent splices and connectors, loss (MDL) where arises the from modes im experienceperfections differentialin optical components losses or gains such when as opticalpropagating amplifiers, through couplers, the optical and multiplexers, link, which leadsas well to as signal-to-noise from non-unitary ratio cross-talk disparities in along the fiber with and a loss at offiber orthogonality splices and between connectors, modes. where The the MDL modes is a capacity-limiting experience differential effect reducing losses or the gains multiplexing when propagatingbenefit of SDMthrough communication the optical link, systems, which which leads cannotto signal-to-noise be completely ratio removed disparities at along the receiver with a by losssimply of orthogonality inserting mode between scramblers. modes. The probabilityThe MDL distributionis a capacity-limiting functions (PDF) effect of thereducing accumulated the multiplexingMDL for the benefit different of SDM coupling communication levels, with andsystems, without which mode cannot scrambling, be completely are shown removed in Figure at the20 as receiveran example by simply [41]. inserting mode scramblers. The probability distribution functions (PDF) of the accumulatedFurthermore, MDL for as the there different are up tocoupling three spatial levels modes, with and being without transmitted mode in scrambling, our modeling, are Figureshown 21 in illustratesFigure 20 as the an equalization example [41]. performance of the BER curves at different OSNRs by means of the STBC-RLS algorithm for a PDM-QPSK signal, from which we can see that the BERs of all spatial modes fall after MIMO equalization.

Figure 20. Mode-dependent loss (MDL) probability distribution functions under different coupling levels [41].

Furthermore, as there are up to three spatial modes being transmitted in our modeling, Figure 21 illustrates the equalization performance of the BER curves at different OSNRs by means of the STBC-RLS algorithm for a PDM-QPSK signal, from which we can see that the BERs of all spatial modes fall after MIMO equalization. Appl. Sci. 2019, 9, x FOR PEER REVIEW 16 of 29

Figure 19. The summary of BER vs. OSNR for various modulation formats with or without using STBC-RLS compensation [29].

6. MIMO Equalization for Mode-Dependent Loss The mode-dependent loss (MDL) arises from imperfections in optical components such as optical amplifiers, couplers, and multiplexers, as well as from non-unitary cross-talk in the fiber and at fiber splices and connectors, where the modes experience differential losses or gains when propagating through the optical link, which leads to signal-to-noise ratio disparities along with a loss of orthogonality between modes. The MDL is a capacity-limiting effect reducing the multiplexing benefit of SDM communication systems, which cannot be completely removed at the receiver by simply inserting mode scramblers. The probability distribution functions (PDF) of the Appl.accumulated Sci. 2019, 9, 1178MDL for the different coupling levels, with and without mode scrambling, are 16shown of 28 in Figure 20 as an example [41].

Figure 20. Mode-dependent loss (MDL) probability distribution functions under different coupling Appl. Sci. 2019, 9, x FOR PEER REVIEW 17 of 29 levelsFigure [41 20.]. Mode-dependent loss (MDL) probability distribution functions under different coupling levels [41].

Figure 21. The BER vs. OSNR for three different modes for a PDM-QPSK signal using the STBC-RLS Figure 21. The BER vs. OSNR for three different modes for a PDM-QPSK signal using the STBC-RLS scheme [27]. scheme [27]. −3 The LP01 mode is able to realize a lower required OSNR for a 10 BER than the LP11 mode The LP01 mode is able to realize a lower required OSNR for a 10−3 BER than the LP11 mode groups, predominantly because its more centralized mode pattern makes the LP01 mode suffer a lesser groups, predominantly because its more centralized mode pattern makes the LP01 mode suffer a level of impact from mode coupling and cross-talk. Al though the LP11a and LP11b modes share an lesser level of impact from mode coupling and cross-talk. Al though the LP11a and LP11b modes share identical mode pattern as a pair of degenerate modes, their BER performance appears a bit dissimilar an identical mode pattern as a pair of degenerate modes, their BER performance appears a bit triggered by spatial mismatching and rotation. Since we found that the transmitted signals on both x- dissimilar triggered by spatial mismatching and rotation. Since we found that the transmitted and y-polarizations provide a comparable performance, here, in these plots, the LP mode is adopted signals on both x- and y-polarizations provide a comparable performance, here,11b in these plots, the as the average of two orthogonal polarizations for an easier analysis. LP11b mode is adopted as the average of two orthogonal polarizations for an easier analysis. Additionally, the BER versus OSNR relationships based on the STBC-RLS algorithm for Additionally, the BER versus OSNR relationships based on the STBC-RLS algorithm for PDM-16QAM and PDM-64QAM signal compensation for LP01, LP11a, and LP11b modes are presented PDM-16QAM and PDM-64QAM signal compensation for−3 LP01, LP11a, and LP11b modes are presented in Figures 22 and 23, respectively. To reach a BER below 10 , the LP01 mode requires a 16.5-dB OSNR, in Figures 22 and 23, respectively. To reach a BER below 10−3, the LP01 mode requires a 16.5-dB while the LP11b and LP11a modes need about 17.7-dB and 18.1-dB OSNRs for the PDM-16QAM signal, OSNR, while the LP11b and LP11a modes need about 17.7-dB and 18.1-dB OSNRs for the PDM-16QAM while, for the PDM-64QAM signal, the LP01 mode requires a 21.3-dB OSNR, while the LP11b and LP11a signal, while, for the PDM-64QAM signal, the LP01 mode requires a 21.3-dB OSNR, while the LP11b modes need about 22.8-dB and 23.6-dB OSNRs, respectively. and LP11a modes need about 22.8-dB and 23.6-dB OSNRs, respectively.

Figure 22. The BER vs. OSNR for three different modes for a PDM-16QAM signal using the STBC-RLS scheme [27]. Appl. Sci. 2019, 9, x FOR PEER REVIEW 17 of 29

Figure 21. The BER vs. OSNR for three different modes for a PDM-QPSK signal using the STBC-RLS scheme [27].

The LP01 mode is able to realize a lower required OSNR for a 10−3 BER than the LP11 mode groups, predominantly because its more centralized mode pattern makes the LP01 mode suffer a lesser level of impact from mode coupling and cross-talk. Al though the LP11a and LP11b modes share an identical mode pattern as a pair of degenerate modes, their BER performance appears a bit dissimilar triggered by spatial mismatching and rotation. Since we found that the transmitted signals on both x- and y-polarizations provide a comparable performance, here, in these plots, the LP11b mode is adopted as the average of two orthogonal polarizations for an easier analysis. Additionally, the BER versus OSNR relationships based on the STBC-RLS algorithm for PDM-16QAM and PDM-64QAM signal compensation for LP01, LP11a, and LP11b modes are presented in Figures 22 and 23, respectively. To reach a BER below 10−3, the LP01 mode requires a 16.5-dB OSNR, while the LP11b and LP11a modes need about 17.7-dB and 18.1-dB OSNRs for the PDM-16QAM signal, while, for the PDM-64QAM signal, the LP01 mode requires a 21.3-dB OSNR, while the LP11b Appl. Sci. 2019, 9, 1178 17 of 28 and LP11a modes need about 22.8-dB and 23.6-dB OSNRs, respectively.

Appl. Sci. 2019, 9, x FOR PEER REVIEW 18 of 29 Figure 22. The BER vs. OSNR for three different modes for a PDM-16QAM signal using the STBC-RLS Figure 22. The BER vs. OSNR for three different modes for a PDM-16QAM signal using the scheme [27]. STBC-RLS scheme [27].

Figure 23. The BER vs. OSNR for three different modes for a PDM-64QAM signal using the STBC-RLS Figure 23. The BER vs. OSNR for three different modes for a PDM-64QAM signal using the scheme [27]. STBC-RLS scheme [27]. 7. Hardware Complexity Optimization for MIMO 7. Hardware Complexity Optimization for MIMO This section focuses on the hardware complexity for MIMO equalization, which includes the FIR filter tapThis number section for focuses MIMO, on single-stage the hardware architecture, complexity and for joint MIMO DSP equalization, for an MCF-based which SDM.includes For the a largerFIR filter mode tap number numberm, frequency-domainfor MIMO, single-stage (FD)-RLS architecture, might provide and ajoint better DSP performance for an MCF-based pertaining SDM. to theFor conventional a larger mode FD-LMS number algorithm, 𝑚, frequency-domain for the convergence (FD)-RLS speed ofmi FD-RLSght provide is not a strongly better performance dependent onpertaining the input statistics.to the conventional Until now, the FD-LMS main issue algorith of them, adaptive for the FD-RLS convergence implementation speed of isFD-RLS its relatively is not higherstrongly computational dependent complexity,on the input due statistics. to the recursiveUntil now, updating the main with issue a growing of the memory.adaptive InFD-RLS this section,implementation a joint chromatic is its relatively dispersion higher (CD) and computat DMGDional optimization complexity, technique due to is the proposed recursive for aupdating further reductionwith a growing in the algorithmic memory. In complexity this section, of a the joint FD-RLS chromatic approach. dispersion (CD) and DMGD optimization technique is proposed for a further reduction in the algorithmic complexity of the FD-RLS approach. 7.1. Filter Tap Number for MIMO 7.1. Filter Tap Number for MIMO The structure of an MIMO equalizer as an extension of the standard CMA is depicted in Figure 24, whichThe is based structure on multiple of an MIMO FIR filtersequalizer in aas butterfly an extens structure.ion of the Thestandard hardware CMAcomplexity is depicted ofin Figure such an24, architecture which is based per transmitted on multiple bit FIR is filters only doubledin a butterfly for the structure. 4 × 4 MIMO The hardware configuration complexity compared of such to standardan architecture single-mode per transmitted operation, forbit sixteenis only adaptivedoubled filtersfor the are 4 × prerequisite 4 MIMO configuration to process 2 ×compared100 Gb/s to withstandard a 4 × 4 single-mode MIMO compared operation, to four for filters sixteen for 1adaptive× 100 Gb/s filters with are a prerequisite 2 × 2 MIMO to as process a reference. 2 × 100 On theGb/s with a 4 × 4 MIMO compared to four filters for 1 × 100 Gb/s with a 2 × 2 MIMO as a reference. On the right graph of Q2-factor versus the FIR filter tap number, which is used to estimate the required length of the MIMO equalizer allowing the detection of all the polarization and mode tributaries, an almost flat factor from nine taps to 15 taps can be observed for both 100 Gb/s PDM-QPSK and 2 × 100 Gb/s MDM after a 40-km transmission [42]. Appl. Sci. 2019, 9, 1178 18 of 28

right graph of Q2-factor versus the FIR ﬁlter tap number, which is used to estimate the required length

Appl.of theSci. 2019 MIMO, 9, x equalizerFOR PEER REVIEW allowing the detection of all the polarization and mode tributaries, an19 almost of 29 ﬂatAppl. factor Sci. 2019 from, 9, x nine FOR tapsPEER to REVIEW 15 taps can be observed for both 100 Gb/s PDM-QPSK and 2 × 10019 Gb/s of 29 MDM after a 40-km transmission [42].

2 FigureFigure 24. 24. TheThe finite finite impulse impulse response response (FIR) (FIR) filter filter illustration, illustration, along along with with Q Q2-factor-factor vs. vs. tap tap number number for for Figure 24. The finite impulse response (FIR) filter illustration, along with Q2-factor vs. tap number for bothboth 2 2× ×2 and2 and 4 × 4 4× MIMO4 MIMO cases cases [42]. [42 ]. both 2 × 2 and 4 × 4 MIMO cases [42]. TheThe required required numbernumber ofof taps taps per per carrier carrier used used as a functionas a function of distance of indistance recent MDMin recent transmission MDM The required number of taps per carrier used as a function of distance in recent MDM transmissionexperiments experiments is illuminated is in illuminated Figure 25, where in Figure the DMGD 25, where increases the DMGD linearly withincreases distance linearly in the with weak transmission experiments is illuminated in Figure 25, where the DMGD increases linearly with distancecoupling in regime the weak [19]. Fibercoupling management regime [19]. is expedient Fiber management in decreasing is the expedient maximum in DMGD decreasing in an FMF.the distance in the weak coupling regime [19]. Fiber management is expedient in decreasing the maximumIn addition, DMGD as different in an FMF. forms In ofaddition, signals, as fibers, different and spatialforms of channels signals, arefibers, likely and to spatial cohabit channels in future maximum DMGD in an FMF. In addition, as different forms of signals, fibers, and spatial channels areSDM likely transport to cohabit networks, in future it is SDM also significanttransport thatnetw MIMOorks, it DSP is also has significant sufficient processing that MIMO capability DSP has to are likely to cohabit in future SDM transport networks, it is also significant that MIMO DSP has sufficienthandle all processing received signals.capability to handle all received signals. sufficient processing capability to handle all received signals.

Figure 25. The number of taps per carrier for MIMO DSP vs. distance in recent few-mode fiber (FMF) Figuretransmission 25. The number experiments of taps [19 per]. carrier for MIMO DSP vs. distance in recent few-mode fiber (FMF) Figure 25. The number of taps per carrier for MIMO DSP vs. distance in recent few-mode fiber (FMF) transmission experiments [19]. 7.2. Single-Stagetransmission Architecture experiments [19]. 7.2. Single-StageIn this subsection, Architecture we discuss the complexity optimization using the single-stage architecture 7.2. Single-Stage Architecture of CDIn this and subsection, DMGD joint we discuss compensation. the complexity The equalization optimization of using CD the and single-stage DMGD can architecture be performed of In this subsection, we discuss the complexity optimization using the single-stage architecture of CDtogether and DMGD using joint FFT forcompensation. efficient implementation The equalization of convolution, of CD and DMGD as a favorable can be performed structure fortogether digital CD and DMGD joint compensation. The equalization of CD and DMGD can be performed together usingcoherent FFT for receivers, efficient for implementation sparing the CD of compensation convolution, as modules a favorable and, structure hence, reducing for digital the coherent total DSP using FFT for efficient implementation of convolution, as a favorable structure for digital coherent receivers,implementation for sparing complexity the CD [43]. compensation modules and, hence, reducing the total DSP receivers, for sparing the CD compensation modules and, hence, reducing the total DSP implementationThe schematic complexity diagram [43]. of a coherent receiver for the compensation of the linear channel effects implementation complexity [43]. suchThe as schematic CD and DMGD diagram is of exemplified a coherent inreceiver Figure for 26 the. To compensation attain a lower of computationalthe linear channel complexity, effects The schematic diagram of a coherent receiver for the compensation of the linear channel effects suchthe as equalization CD and DMGD of CD is and exemplified DMGD can in Figure be implemented 26. To attain together, a lower compared computational with thecomplexity, conventional the such as CD and DMGD is exemplified in Figure 26. To attain a lower computational complexity, the equalization of CD and DMGD can be implemented together, compared with the conventional equalization of CD and DMGD can be implemented together, compared with the conventional two-stage FDE structure, using FFT for an efficient implementation of convolution; thus, 𝑚 two-stage FDE structure, using FFT for an efficient implementation of convolution; thus, 𝑚 independent static CD compensation modules would be spared. Subsequently, a joint MIMO FDE of independent static CD compensation modules would be spared. Subsequently, a joint MIMO FDE of the static CD along with the dynamic DMGD would be an auspicious configuration for digital the static CD along with the dynamic DMGD would be an auspicious configuration for digital Appl. Sci. 2019, 9, x FOR PEER REVIEW 20 of 29

coherent receivers, because it would save a separate CD compensation module, thereby decreasing the total DSP implementation complexity.

Appl. Sci. 2019, 9, 1178 19 of 28

two-stage FDE structure, using FFT for an efﬁcient implementation of convolution; thus, m independent

Appl.static Sci. CD 2019 compensation, 9, x FOR PEER REVIEW modules would be spared. Subsequently, a joint MIMO FDE of the20 staticof 29 CD along with the dynamic DMGD would be an auspicious conﬁguration for digital coherent coherentreceivers, receivers, because itbecause would it save would a separate save a sepa CD compensationrate CD compensation module, module, thereby decreasingthereby decreasing the total theDSP total implementation DSP implementation complexity. complexity.

Figure 26. The representation of a coherent receiver with the joint compensation of chromatic dispersion (CD) plus differential mode group delay (DMGD) through a single-stage MIMO architecture [10].

Above and beyond, Figure 27 provides the comparison of the required complex multiplications per output symbol at different transmission distances for the two-stage and single-stage FD-RLS algorithms. For a transmission distance rising from 100 km up to 1000 km, the single-stage method continuously consumes fewer complex multiplications than the conventional two-stage method, becauseFigure both 26. Themethods representation use the ofsame a coherent FFT receiversize for with DMGD the joint compensation compensation for of chromaticthe same dispersion transmission (CD) plus differential mode group delay (DMGD) through a single-stage MIMO architecture [10]. distance,Figure whereas 26. The representationthe FFT size ofof a staticcoherent FDE receiver in the with two-stage the joint methodcompensation is determined of chromatic by the FMF-induceddispersion CD.(CD) Forplus instance, differential the mode two-stage group FD-RLSdelay (DMGD) method through consumes a single-stage 323 and 351 MIMO complex Above and beyond, Figure 27 provides the comparison of the required complex multiplications per multiplicationsarchitecture [10].per output symbol at 400-km and 800-km transmissions, whereas our proposed output symbol at different transmission distances for the two-stage and single-stage FD-RLS algorithms. single-stage method takes only 297 and 328 complex multiplications per output symbol at the same For a transmission distance rising from 100 km up to 1000 km, the single-stage method continuously transmissionAbove and distances, beyond, respectively. Figure 27 provides the comparison of the required complex multiplications consumes fewer complex multiplications than the conventional two-stage method, because both per outputFurthermore, symbol Figureat different 28 irradiatestransmission the distQ-valueances forversus the two-stagethe step sizeand ofsingle-stage adaptive FD-RLSFD-RLS methods use the same FFT size for DMGD compensation for the same transmission distance, whereas algorithms.algorithms Forin both a transmission two-stage anddistance single-stage rising from methods 100 km for up compensating to 1000 km, thefor single-stageDMGD and method CD. To the FFT size of static FDE in the two-stage method is determined by the FMF-induced CD. For instance, continuouslyattain a <0.5-dB consumes Q-penalty fewer at 400-, complex 700-, and multiplications 1000-km transmissions, than the conventional the maximum two-stage step sizes method, are 4.5 × the two-stage FD-RLS method consumes 323 and 351 complex multiplications per output symbol at because10−5, 3 ×both 10− 5methods, and 1.5 use× 10 the−5, samerespectively. FFT size In for addition, DMGD compensationfor the same fortransmission the same transmissiondistance, the 400-km and 800-km transmissions, whereas our proposed single-stage method takes only 297 and distance,maximum whereas step sizes the required FFT size to achieveof static the FDE same in Q-valuethe two-stage after equalization method is aredetermined identical forby boththe 328 complex multiplications per output symbol at the same transmission distances, respectively. FMF-inducedapproaches. CD. For instance, the two-stage FD-RLS method consumes 323 and 351 complex multiplications per output symbol at 400-km and 800-km transmissions, whereas our proposed single-stage method takes only 297 and 328 complex multiplications per output symbol at the same transmission distances, respectively. Furthermore, Figure 28 irradiates the Q-value versus the step size of adaptive FD-RLS algorithms in both two-stage and single-stage methods for compensating for DMGD and CD. To attain a <0.5-dB Q-penalty at 400-, 700-, and 1000-km transmissions, the maximum step sizes are 4.5 × 10−5, 3 × 10−5, and 1.5 × 10−5, respectively. In addition, for the same transmission distance, the maximum step sizes required to achieve the same Q-value after equalization are identical for both approaches.

Figure 27. The required complex multiplications per output symbol of the two-stage and single-stage Figure 27. The required complex multiplications per output symbol of the two-stage and single-stage frequency-domain (FD)-RLS approaches as a function of transmission distance in an FMF-based frequency-domain (FD)-RLS approaches as a function of transmission distance in an FMF-based transmission system [10]. transmission system [10]. Furthermore, Figure 28 irradiates the Q-value versus the step size of adaptive FD-RLS algorithms in both two-stage and single-stage methods for compensating for DMGD and CD. To attain a <0.5-dB Q-penalty at 400-, 700-, and 1000-km transmissions, the maximum step sizes are 4.5 × 10−5, 3 × 10−5, and 1.5 × 10−5, respectively. In addition, for the same transmission distance, the maximum step sizes required to achieve the same Q-value after equalization are identical for both approaches.

Figure 27. The required complex multiplications per output symbol of the two-stage and single-stage frequency-domain (FD)-RLS approaches as a function of transmission distance in an FMF-based transmission system [10]. Appl. Sci. 2019, 9, 1178 20 of 28 Appl. Sci. 2019, 9, x FOR PEER REVIEW 21 of 29

FigureFigure 28.28. TheThe Q-valueQ-value vs.vs. step size usingusing bothboth two-stagetwo-stage andand single-stage FD-RLSFD-RLS algorithms as aa functionfunction ofof transmissiontransmission distancedistance inin anan FMF-basedFMF-based transmissiontransmission systemsystem [[15].15].

LastLast butbut not not least, least, Figure Figure 29 exhibits 29 exhibits the convergence the convergence speed comparison speed comparison among three among algorithms three atalgorithms different OSNRat different levels, OSNR from levels, which from we learnt which the we time learnt required the time for required convergence for convergence grows more slowlygrows formore FD-RLS slowly than for FD-RLS FD-LMS than algorithms FD-LMS as algorithms the OSNR as increases, the OSNR equivalent increases, to equivalent plummeting to plummeting the overall trainingthe overall sequence training overhead sequence by overhead approximately by approximately 30%. 30%.

Figure 29. The convergence speed comparison of two FD-LMS algorithms and single-stage FD-RLS Figure 29. The convergence speed comparison of two FD-LMS algorithms and single-stage FD-RLS algorithms at different OSNRs [7]. algorithms at different OSNRs [7]. 7.3. Joint DSP for MCF-Based SDM 7.3. Joint DSP for MCF-Based SDM Last but not least, this subsection focuses on a joint DSP scheme for the MCF-based SDM transmissionLast but system not least, to reduce this subsection the overall costfocuses and poweron a joint consumption DSP scheme of integrated for the receivers.MCF-based There SDM is atransmission strong correlation system between to reduce the phasethe overall ﬂuctuations cost and of the power different consum sub-channels,ption of integrated and the master–slave receivers. phaseThere recoveryis a strong causes correlation no BER degradation.between the Thephase block fluctuations diagram ofof a the joint different DSP method sub-channels, for an MCF-based and the SDMmaster–slave transmission phase system recovery with causes master–slave no BER degradation. phase recovery The is block presented diagram in Figure of a joint30, which DSP method utilizes thefor phasean MCF-based recovered SDM from transmission a single “master” system sub-channel with master–slave to eradicate phase phase recovery recovery is blockspresented in the in “slave”Figure 30, sub-channels, which utilizes thus the decreasing phase recovered the overall from DSP a single burden “master” at the receiver sub-channel side [ 44to]. eradicate phase recovery blocks in the “slave” sub-channels, thus decreasing the overall DSP burden at the receiver side [44]. Appl. Sci. 2019, 9, 1178 21 of 28 Appl.Appl. Sci. Sci. 2019 2019, 9, ,9 x, xFOR FOR PEER PEER REVIEW REVIEW 2222 of of 29 29

FigureFigure 30. 30. A A Ajoint joint DSP DSP approach approach for for multicore multicore fiber fiber ﬁber (MCF)-based (MCF)-based SDM SDM transmission transmission system system with with master–slavemaster–slave phase phase recovery recovery [44]. [[44].44 ].

InInIn addition, addition, the thethe BER BERBER versus versus OSNROSNR OSNR observedobserved observed for fo for master–slave rmaster–slave master–slave phase phase phase recovery recovery recovery using using using the the jointthe joint joint DSP DSPDSPscheme scheme scheme for MCF-basedfor for MCF-based MCF-based SDM SDM SDM transmission transmission transmission system system system is presented is is presented presented in Figure in in Figure Figure 31, which 31, 31, which which is equivalent is is equivalent equivalent to that totofound that that found withfound independentwith with independent independent phase phase recoveryphase recovery recovery [45]. [45]. [45].

Figure 31. BER vs. OSNR for MCF-based SDM transmission system using the joint DSP scheme [45]. Figure 31. BER vs. OSNR for MCF-based SDM transmission system using the joint DSP scheme [45]. 8. RecentFigure Experiments 31. BER vs. OSNR of SDM for MCF-based Transmission SDM Usingtransmission FMFs system using the joint DSP scheme [45].

8.8. Recent RecentLast Experiments butExperiments not least, of aof SDM couple SDM Transmission ofTransmission topical experimental using using FMFs FMFs verifications are assessed facilitating a higher transmission capacity for short, medium, and long distances in this section, utilizing FMFs and/or Last but not least, a couple of topical experimental verifications are assessed facilitating a higher MCFs,Last along but not with least, the progressivea couple of topical maturity experiment of the SDMal verifications amplifier, spatial are assessed mode coupler,facilitating and a higher digital transmission capacity for short, medium, and long distances in this section, utilizing FMFs and/or transmissionMIMO equalizers capacity to safeguard for short, higher medium, capacity and long and spectraldistances efficiency. in this section, utilizing FMFs and/or MCFs, along with the progressive maturity of the SDM amplifier, spatial mode coupler, and digital MCFs,For along instance, with the Ryf progressive et al. conducted maturity an of experiment the SDM amplifier, in 2018 for spatial a MIMO-based mode coupler, 36-mode and digital (two MIMO equalizers to safeguard higher capacity and spectral efficiency. MIMOpolarization equalizers modes) to transmissionsafeguard higher over capacity a 2-km-long and 50-spectralµm graded-index efficiency. multimode fiber (MMF) with For instance, Ryf et al. conducted an experiment in 2018 for a MIMO-based 36-mode (two a spectralFor instance, efficiency Ryf of 72et bit/s/Hz,al. conducted as denoted an experiment in Figure 32 in along 2018 withfor a the MIMO-based close-ups of the36-mode heterodyne (two polarizationpolarization modes) modes) transmis transmissionsion over over a a2-km-long 2-km-long 50- 50-μμmm graded-index graded-index multimode multimode fiber fiber (MMF) (MMF) withwith a aspectral spectral efficiency efficiency of of 72 72 bit/s/Hz, bit/s/Hz, as as deno denotedted in in Figure Figure 32 32 along along with with the the close-ups close-ups of of the the Appl. Sci. 2019, 9, 1178 22 of 28

Appl. Sci. 2019, 9, x FOR PEER REVIEW 23 of 29 receiver arrangement and multimode acousto-optic modulator, as well as the spectrum of the test signalheterodyne and location receiver of arrangement the heterodyne and ﬁlters multimode and local acou oscillatorsto-optic [46 modulator,]. The results as well were as attained the spectrum using aof 72 the× 72test coherent signal and MIMO-DSP, location withof the ﬁve heterodyne 15-Gbaud filters dual-carrier and local PDM-QPSK oscillator signals[46]. The with results a channel were spacingattained of using 50 GHz a 72 transmitted. × 72 coherent MIMO-DSP, with five 15-Gbaud dual-carrier PDM-QPSK signals with a channel spacing of 50 GHz transmitted.

FigureFigure 32.32. ((aa)) SetupSetup of a 72 72 ×× 7272 MIMO-based MIMO-based transmission; transmission; (b ()b heterodyne) heterodyne receiver receiver arrangement; arrangement; (c) (cspectrum) spectrum of ofthe the test test signal signal and and location location of of the the heterodyne heterodyne filters ﬁlters and local oscillator;oscillator; ((dd)) multimodemultimode acousto-opticacousto-optic modulatormodulator [46[46].].

FigureFigure 33 33 denotes denotes the the experimental experimental set-up set-up of an of MIMO-based an MIMO-based 45-mode 45-mode (two polarization (two polarization modes) µ MDMmodes) transmission MDM transmission over a 26.5-km-long over a 26.5-km-long 50- m graded-index 50-μm graded-index MMF over MMF 20 wavelength over 20 wavelength channels resultingchannels inresulting a total capacityin a total of capacity 101 Tb/s of and 101 a Tb/s spectral and efficiencya spectral of efficiency 202/bit/s/Hz. of 202/bit/s/Hz. The received The × 90-degreereceived 90-degree complex amplitude complex signalsamplitude were signals processed were by processed a 90 90 MIMO-FDE by a 90 × 90 with MIMO-FDE 300 symbol-spaced with 300 taps,symbol-spaced whereas the taps, initial whereas convergence the initial of the converge equalizernce was of obtainedthe equalizer using was the data-aidedobtained using FD-LMS the algorithmdata-aided with FD-LMS CMA algorithm used after with that. CMA Then, used the afte carrier-phaser that. Then, the recovery carrier-phase and BER recovery counting and were BER performed,counting were while performed, the Q-factors while were the computedQ-factors were by evaluating computed an by inverse evaluating Q function an inverse of the Q BER function [47]. Theof the corresponding BER [47]. The plots corresponding also include plots the also Q-factors include of the 90 Q-factors spatial tributaries of 90 spatial for tributaries QPSK and for 16QAM QPSK transmissionand 16QAM transmission in line with thein line mode with groups, the mode sorted grou byps, the sorted performance by the performance within the within mode groupthe mode in Figuregroup 33in b,Figure with 33b, the intensity-averaged with the intensity-averaged impulse response impulse attainedresponse from attained channel from estimation channel estimation over the MMFover the in Figure MMF 33inc, Figure and the 33c, intensity and the transfer intensity matrix transfer showing matrix the couplingshowing betweenthe coupling the mode between groups the inmode Figure groups 33d, alongin Figure with 33d, the along average with Q-factor the average for QPSK Q-factor and 16QAMfor QPSK as and a function 16QAM of as wavelength a function inof Figurewavelength 33e. The in errorFigure bars 33e. within The error the plots bars signify within the the best plots and signify the worst thespatial best and tributaries. the worst spatial tributaries.In another example, a 138 Tbit/s coherent transmission over six spatial modes plus two polarizations,In another and example, 120 wavelength a 138 Tbit/s channels coherent carrying transmission 120 Gbit/s over mapped six spatial 16QAM modes signals plus over two a 65polarizations,km FMF12 fiberand are120 shownwavelength in Figure channels 34, where carrying low 120 insertion Gbit/s and mapped MDL 16QAM mode-selective signals photonicover a 65 lanternskm FMF12 are fiber used are along shown with in a coherentFigure 34, 12 where× 12 MIMOlow insertion equalization and MDL using mode-selective a data-aided FD-LMSphotonic algorithmlanterns are for used resolving along the with modal a coherent mixing [ 4812]. × The 12 FMF12MIMO fiberequalization is a special using FMF a designed data-aided to have FD-LMS low attenuationalgorithm for for resolving the first 12 the spatial modal channels, mixing while[48]. The having FMF12 sufficiently fiber is a high special losses FMF for designed the higher-order to have modeslow attenuation to guarantee for an the effective first 12 cut-off. spatial At channels, the receiver while side, having optical sufficiently front-end impairment, high losses CD, forand the frequency-offsethigher-order modes compensation to guarantee is performed, an effective while cu thet-off. carrier At the phase receiver recovery side, is performedoptical front-end before calculatingimpairment, performance CD, and frequency-offset metrics over 8 million compensation bits per spatialis performed, channel. while Moreover, the carrier the Q-factors phase recovery versus theis performed launch power before averaged calculating for 15 WDMperformance channels, metrics as well over as for8 million 120 WDM bits channels per spatial after channel. 650-km transmissionMoreover, the are Q-factors shown in versus Figure the 35 ,launch where power the gradient averaged of the for shaded 15 WDM areas channels, epitomizes as well the variationas for 120 WDM channels after 650-km transmission are shown in Figure 35, where the gradient of the shaded areas epitomizes the variation in WDM channel performance [48]. In addition, the larger Appl. Sci. 2019, 9, x FOR PEER REVIEW 24 of 29 Appl. Sci. 2019, 9, 1178 23 of 28 Appl. Sci. 2019, 9, x FOR PEER REVIEW 24 of 29 performance variations on the edges of the spectrum, and the singular values for each WDM channel inperformanceAppl.averaged, WDM Sci. 2019 channel corresponding, 9 variations, x FOR performance PEER on REVIEW to the the [edges48 measured]. In of addition, the MDL,spectr the areum, larger depicted and performancethe singularin Figures values variations 35c,d, for respectively. each on the WDM edges channel24 of of the 29 spectrum,averaged, andcorresponding the singular to valuesthe measured for each MDL, WDM are channel depicted averaged, in Figures corresponding 35c,d, respectively. to the measured performance variations on the edges of the spectrum, and the singular values for each WDM channel MDL, are depicted in Figure 35c,d, respectively. averaged, corresponding to the measured MDL, are depicted in Figures 35c,d, respectively.

Figure 33. (a) Setup of a 90 × 90 MIMO-based transmission; (b) Q-factors for QPSK and 16QAM; (c) Figureintensity-averaged 33. (a) Setup impulse of a 90 ×response 90 MIMO-based attained; transmission; (d) intensity transfer(b) Q-factors matrix; for ( QPSKe) average and 16QAM;Q-factor for(c) Figureintensity-averagedQPSK and 33. (16QAMa) Setup impulse[47]. of a 90 response× 90 MIMO-based attained; (d transmission;) intensity transfer (b) Q-factors matrix; ( fore) average QPSK and Q-factor 16QAM; for (QPSKFigurec) intensity-averaged and 33. 16QAM(a) Setup [47]. of impulse a 90 × 90 response MIMO-based attained; transmission; (d) intensity ( transferb) Q-factors matrix; for (eQPSK) average and Q-factor16QAM; for(c) QPSKintensity-averaged and 16QAM [impulse47]. response attained; (d) intensity transfer matrix; (e) average Q-factor for QPSK and 16QAM [47].

Figure 34. (a) Set-up of a 12 × 12 MIMO-based transmission; (b) impulse response after 65-, 195-, and × Figure650-km 34.34. transmission ((a)) Set-upSet-up ofof [48]. aa 1212 × 1212 MIMO-based MIMO-based transmission; transmission; ( b) impulse response after 65-, 195-, andand 650-km transmissiontransmission [[48].48]. Figure 34. (a) Set-up of a 12 × 12 MIMO-based transmission; (b) impulse response after 65-, 195-, and 650-km transmission [48].

FigureFigure 35.35.( a()a Q-factors) Q-factors vs. vs. power power for 15for WDM 15 WDM channels; channels; (b) Q-factors (b) Q-factors vs. WDM vs. channels; WDM channels; (c) singular (c) values after 650-km transmission; (d) average MDL vs. frequency per WDM channel [48]. Figuresingular 35. values (a) Q-factors after 650-km vs. transmission;power for 15 ( dWDM) average channels; MDL vs. (b )frequency Q-factors per vs. WDM WDM channel channels; [48]. ( c) singular values after 650-km transmission; (d) average MDL vs. frequency per WDM channel [48]. Figure 35. (a) Q-factors vs. power for 15 WDM channels; (b) Q-factors vs. WDM channels; (c) singular values after 650-km transmission; (d) average MDL vs. frequency per WDM channel [48]. Appl. Sci. 2019, 9, x FOR PEER REVIEW 25 of 29

Appl. Sci. 2019, 9, 1178 24 of 28 Appl. Sci.Last 2019 but, 9, notx FOR least, PEER theREVIEW set-up of another 6 × 6 MIMO-based SDM/WDM transmission system25 of 29 using a 70-km DGD-compensated three-mode fiber is shown in Figure 36, whereas the received signalsLast are but off-line notnot least,least, processed thethe set-upset-up by firstly ofof anotheranother re-sampling 66× × 66 the MIMO-basedMIMO-based signals to two SDM/WDMSDM/WDM samples per transmission signal, along system with CDusing and aa 70-km 70-kmfrequency-offset DGD-compensated DGD-compensated compensation, three-mode three-mode which fiber fiberis isthen shown is shownfollowed in Figure in byFigure 36 a ,6 whereas ×36, 6 MIMOwhereas the received FDE the withreceived signals 600 symbol-spacedaresignals off-line are off-line processed taps processed using by firstly a data-aided by re-sampling firstly re-sampling LMS the algori signals thmthe tosignals along two samples withto two carrier-phase samples per signal, per along recoverysignal, with along and CD withBER and countingfrequency-offsetCD and frequency-offsetfor Q-factor compensation, calculations compensation, which [49]. is The then which Q-factors followed is then byof followedthe a 6 ×center6 MIMO by channel a FDE6 × at6 with MIMO1550 600 nm symbol-spacedFDE for with64QAM, 600 16QAM,tapssymbol-spaced using and a data-aided QPSKtaps using signals LMS a data-aided algorithmas a function LMS along algori of with thmdistance carrier-phase along for with full recoverycarrier-phase space-division and recovery BER multiplexed, counting and BER for quasi-single-mode,Q-factorcounting calculationsfor Q-factor and [calculations49 ].single-mode The Q-factors [49]. fiber The of thetransmisQ-factors centersion, channelof the respectively, center at 1550 channel nm are for presentedat 64QAM, 1550 nm in 16QAM, for Figure 64QAM, and37, withQPSK16QAM, the signals markersand asQPSK a indicating function signals ofthe distanceas measured a function for fulldistance space-divisionof sdistance [50]. A summaryfor multiplexed, full space-divisionof these quasi-single-mode, recent experimentalmultiplexed, and validationssingle-modequasi-single-mode, using fiber transmission,MIMO and single-mode DSP is respectively, presented fiber transmisin are Table presentedsion, 4, while respectively, in Figurean updated 37 ,are with graphpresented the markersof mode in Figure indicating numbers 37, versusthewith measured the the markers transmission distances indicating [ 50distance]. Athe summary measured is presented of thesedistance in recent Figures [50]. experimental 38, A summarywhich summarizes validations of these recent usingthe state-of-the-art MIMOexperimental DSP is SDM-WDMpresentedvalidations in usingexperiments Table MIMO2 while [51-53].DSP an updated is Furtpresentedher graph improvement in of Table mode 4, numbers inwhile the nearan versus updated future the will graph transmission most of likely mode distancecome numbers from is thepresentedversus soft-output the in transmission Figure maximum-likelihood 38, which distance summarizes is presented algorithms, the state-of-the-art in Figureincluding 38, SDM-WDM whichrepeated summarizes tree experiments search the and [51state-of-the-art– 53single]. Further tree search,improvementSDM-WDM to further experiments in theimprove near [51-53]. futurethe efficiency Furt willher most of improvement the likely MIMO come DSP in from the [54]. near the soft-outputfuture will most maximum-likelihood likely come from algorithms,the soft-output including maximum-likelihood repeated tree search algorithms, and single including tree search, repeated to further tree improve search and the efficiencysingle tree of thesearch, MIMO to further DSP [54 improve]. the efficiency of the MIMO DSP [54].

Figure 36. Setup of a 6 × 6 MIMO-based transmission [49]. Figure 36. Setup of a 6 × 6 MIMO-based transmission [49]. Figure 36. Setup of a 6 × 6 MIMO-based transmission [49].

Figure 37. Q-factors for 64QAM, 16QAM, and QPSK [50]. Figure 37. Q-factors for 64QAM, 16QAM, and QPSK [50].

Figure 37. Q-factors for 64QAM, 16QAM, and QPSK [50].

Appl. Sci. 2019, 9, x FOR PEER REVIEW 26 of 29

Appl. Sci.Table2019, 2.9, 1178Summary table of recent experimental validations using multi-input multi-output (MIMO)25 of 28 digital signal processing (DSP). MMF—multimode fiber; FMF—few-mode fiber; MCF—multicore fiber. Table 2. Summary table of recent experimental validations using multi-input multi-output (MIMO) Transmissiondigital signal processingSpectral (DSP). MMF—multimodeDSP fiber; FMF—few-modeNumber fiber; MCF—multicore of fiber. Fiber Type Year Reference Distance Efficiency Approach Modes Transmission Spectral Number of DSP Approach Fiber Type Year Reference Distance2 km 72Efficiency bit/s/Hz 72 × 72 MIMO 50 μm MMF Modes 36 × 2 2018 [46] 26.52 kmkm 72202 bit/s/Hz bit/s/Hz 9072 ×× 7290 MIMOMIMO 50 50µ μmm MMF MMF 36 × 452 × 2 2018 2018 [46] [47] 6526.5 km km 34.91 202 bit/s/Hz bit/s/Hz 1290 ×× 9012 MIMOMIMO 50 µ FMF12m MMF 45 × 62 × 2 2018 2017 [47] [48] 7065 km km 34.91 9 bit/s/Hz bit/s/Hz 12 6 ×× 126 MIMO MIMO FMF12 FMF 6 × 2 3 × 2 2017 2017 [48] [49] 70 km 9 bit/s/Hz 6 × 6 MIMO FMF 3 × 2 2017 [49] 9.89.8 km km 947 947 bit/s/Hz bit/s/Hz 6 ×× 6 MIMOMIMO FMF FMF + MCF+ MCF 3 × 2 3 × 2 2016 2016 [51] [51] 23.823.8 km km 43.63 43.63 bit/s/Hz bit/s/Hz 30 ×× 30 MIMOMIMO FMF9 FMF9 15 × 152 × 2 2015 2015 [52] [52] 53.753.7 km km 217.6 217.6 bit/s/Hz bit/s/Hz 6 ×× 6 MIMOMIMO MCF MCF 3 × 2 3 × 2 2016 2016 [53] [53]

Figure 38. Updated graph of mode numbers vs. the transmission distance in the state-of-the-art SDM-WDMFigure 38. experiments.Updated graph of mode numbers vs. the transmission distance in the state-of-the-art SDM-WDM experiments. 9. Conclusions 9. ConclusionsIn this review paper, we presented an overview regarding the state-of-the-art adaptive MIMO equalizationIn this forreview high-speed paper, MDMwe presented coherent an receivers, overview including regarding the the LMS state-of-the-art and RLS techniques, adaptive aswell MIMO as theequalization STBC-assisted for RLShigh-speed algorithms. MDM The coherent performance receivers, analysis including of various the adaptive LMS and filtering RLS techniques, schemes was as presentedwell as the for STBC-assisted QPSK, 16QAM, RLS and algorithms. 64QAM, along The with performance the STBC-based analysis enhancement of various amongstadaptive different filtering modulationschemes was formats. presented Last for but QPSK, not least, 16QAM, the complexity and 64QAM, optimization along with using the the STBC-based single-stage enhancement architecture ofamongst CD and different DMGD jointmodulation compensation formats. was Last discussed, but not least, for reducing the complexity the total optimization DSP implementation using the complexity.single-stage Additionally, architecture aof number CD and of DMGD recent experimentaljoint compensation validations was discussed, were assessed, for reducing enabling the higher total transmissionDSP implementation capacity andcomplexity. spectral efficiencyAdditionally, for different a number transmission of recent experimental distances by meansvalidations of various were MIMOassessed, DSP enabling approaches. higher transmission capacity and spectral efficiency for different transmission distances by means of various MIMO DSP approaches. Author Contributions: Y.W. leads the development of algorithms and results used in the paper. J.W. contributes toAuthor the development Contributions: of simulation Author (1) mode leads and the results. development Z.P. supervises of algorith overallms and project. results used in the paper. Author Funding:(2) contributesThis research to the development received no externalof simulation funding mode and results. Author (3) supervises overall project. Acknowledgments: We would like to thank the unknown reviewers that helped improve this manuscript with theirFunding: valuable This feedback. research received no external funding ConflictsAcknowledgments: of Interest: TheWe would authors like declare to thank no conflicts the unknown of interest. reviewers that helped improve this manuscript with their valuable feedback.

ReferencesConflicts of Interest: The authors declare no conflicts of interest. 1. Winzer, P.J. Making spatial multiplexing a reality. Nat. Photonics 2014, 8, 345–348. [CrossRef] 2.ReferencesLi, A.; Chen, X.; Amin, A.A.; Ye, J.; Shieh, W. Space-division multiplexed high-speed superchannel 1. transmissionWinzer, P.J. Making over few-mode spatial multiplexing ﬁber. J. Lightw. a reality. Technol. Nat.2012 Photonics, 30, 3953–3964. 2014, 8, [ CrossRef345–348. ] 3. Richardson, D.J.; Fini, J.M.; Nelson, L.E. Space-division multiplexing in optical ﬁbres. Nat. Photonics 2013, 7, 354–362. [CrossRef] Appl. Sci. 2019, 9, 1178 26 of 28

4. Ip, E. Recent advances in mode-division multiplexed transmission using few-mode fibers. In Proceedings of the Twenty-Third National Conference on Communications (NCC), Chennai, India, 2–4 March 2017. 5. Zhang, Z.; Guo, C.; Cui, L.; Mo, Q.; Zhao, N.; Du, C.; Li, X.; Li, G. 21 spatial mode erbium-doped fiber amplifier for mode division multiplexing transmission. Opt. Lett. 2018, 43, 1550–1553. [CrossRef][PubMed] 6. Jain, S.; Rancaño, V.J.F.; May-Smith, T.C.; Petropoulos, P.; Sahu, J.K.; Richardson, D.J. Multi-element fiber technology for space-division multiplexing applications. Opt. Express 2014, 22, 3787–3796. [CrossRef] [PubMed] 7. Pan, Z.; Weng, Y.; He, X.; Wang, J. Adaptive frequency-domain equalization and MIMO signal processing in mode division multiplexing systems using few-mode fibers. In Proceedings of the Signal Processing in Photonic Communications 2016 (SPPCom)—SpW2G.1, Vancouver, BC, Canada, 18–20 July 2016. 8. Arık, S.Ö.; Askarov, D.; Kahn, J.M. Adaptive frequency-domain equalization in mode-division multiplexing systems. Opt. Express 2014, 32, 1841–1852. [CrossRef] 9. Faruk, M.S.; Kikuchi, K. Adaptive frequency-domain equalization in digital coherent optical receivers. Opt. Express 2011, 19, 12789–12798. [CrossRef][PubMed] 10. Weng, Y.; He, X.; Pan, Z. Performance analysis of low-complexity adaptive frequency-domain equalization and MIMO signal processing for compensation of differential mode group delay in mode-division multiplexing communication systems using few-mode fibers. In Proceedings of the SPIE 9774, Next-Generation Optical Communication: Components, Sub-Systems, and Systems V, 97740B, San Francisco, CA, USA, 13 February 2016. 11. Zhu, C.; Tran, A.V.; Do, C.C.; Chen, S.; Anderson, T.; Skafidas, E. Digital signal processing for training-aided coherent optical single-carrier frequency-domain equalization systems. J. Lightw. Technol. 2014, 32, 4110–4120. 12. Pittalà, F.; Slim, I.; Mezghani, A.; Nossek, J.A. Training-aided frequency-domain channel estimation and equalization for single-carrier coherent optical transmission systems. J. Lightw. Technol. 2014, 32, 4247–4261. [CrossRef] 13. He, X.; Weng, Y.; Wang, J.; Pan, Z. Noise power directed adaptive frequency domain least mean square algorithm with fast convergence for DMGD compensation in few-mode fiber transmission systems. In Proceedings of the Optical Fiber Communication Conference (OFC)—Th3E.1, San Francisco, CA, USA, 9–13 March 2014. 14. Puttnam, B.J.; Sakaguchi, J.; Mendinueta, J.M.D.; Klaus, W.; Awaji, Y.; Wada, N.; Kanno, A.; Kawanishi, T. Investigating self-homodyne coherent detection in a 19 channel space-division-multiplexed transmission link. Opt. Express 2013, 21, 1561–1566. [CrossRef] 15. Weng, Y.; Wang, T.; Pan, Z. Fast-convergent adaptive frequency-domain recursive least-squares algorithm with reduced complexity for MDM transmission systems using optical few-mode fibers. In Proceedings of the Conference on Lasers and Electro-Optics (CLEO)—SW4F.6, San Jose, CA, USA, 5–10 June 2016. 16. Ryf, R.; Weerdenburg, J.V.; Alvarez-Aguirre, R.A.; Fontaine, N.K.; Essiambre, R.-J.; Chen, H.; Alvarado-Zacarias, J.C.; Amezcua-Correa, R.; Koonen, T.; Okonkwo, C. White Gaussian noise based capacity estimate and characterization of fiber-optic links. In Proceedings of the Optical Fiber Communication Conference (OFC)—W1G.2, San Diego, CA, USA, 11–15 March 2018. 17. Choutagunta, K.; Arik, S.Ö.; Ho, K.-P.; Kahn, J.M. Characterizing mode-dependent loss and gain in multimode components. J. Lightw. Technol. 2018, 36, 3815–3823. [CrossRef] 18. Andrusier, A.; Meron, E.; Feder, M.; Shtaif, M. Optical implementation of a space-time-trellis code for enhancing the tolerance of systems to polarization-dependent loss. Opt. Lett. 2013, 38, 118–120. [CrossRef] 19. Mizuno, T.; Takara, H.; Shibahara, K.; Sano, A.; Miyamoto, Y. Dense space division multiplexed transmission over multicore and multimode fiber for long-haul transport systems. J. Lightw. Technol. 2016, 34, 1484–1493. [CrossRef] 20. Ip, E.; Milione, G.; Li, M.-J.; Cvijetic, N.; Kanonakis, K.; Stone, J.; Peng, G.; Prieto, X.; Montero, C.; Moreno, V.; Liñares, J. SDM transmission of real-time 10GbE traffic using commercial SFP + transceivers over 0.5 km elliptical-core few-mode fiber. Opt. Express 2015, 23, 17120–17126. [CrossRef][PubMed] 21. Pan, Z.; He, X.; Weng, Y. Hardware efficient frequency domain equalization in few-mode fiber coherent transmission systems. In Proceedings of the SPIE 9009, Next-Generation Optical Communication: Components, Sub-Systems, and Systems III, San Francisco, CA, USA, 1 February 2014. 22. Ferreira, F.; Jansen, S.; Monteiro, P.; Silva, H. Nonlinear semi-analytical model for simulation of few-mode fiber transmission. IEEE Photonics Technol. Lett. 2012, 24, 240–242. [CrossRef] Appl. Sci. 2019, 9, 1178 27 of 28

23. Poletti, F.; Horak, P. Description of ultrashort pulse propagation in multimode optical fibers. J. Opt. Soc. Am. B 2008, 25, 1645–1654. [CrossRef] 24. Mumtaz, S.; Essiambre, R.J.; Agrawal, G.P. Nonlinear propagation in multimode and multicore fibers: Generalization of the manakov equations. J. Lightw. Technol. 2013, 31, 398–406. [CrossRef] 25. Ryf, R.; Randel, S.; Gnauck, A.H.; Bolle, C.; Sierra, A.; Mumtaz, S.; Esmaeelpour, M.; Burrows, E.C.; Essiambre, R.-J.; Winzer, P.J.; et al. Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 × 6 MIMO processing. J. Lightw. Technol. 2012, 30, 521–531. [CrossRef] 26. Carpenter, J.; Wilkinson, T.D. Mode Division Multiplexing over 2 km of OM2 fibre using rotationally optimized mode excitation with fibre coupler demultiplexer. In Proceedings of the IEEE Photonics Society Summer Topical Meeting Series, Seattle, WA, USA, 9–11 July 2012. 27. Weng, Y.; He, X.; Yao, W.; Pacheco, M.; Wang, J.; Pan, Z. Investigation of adaptive filtering and MDL mitigation based on space-time block-coding for spatial division multiplexed coherent receivers. Opt. Fiber Technol. 2017, 36, 231–236. [CrossRef] 28. Winzer, P.J.; Foschini, G.J. MIMO capacities and outage probabilities in spatially multiplexed optical transport systems. Opt. Express 2011, 19, 16680–16696. [CrossRef][PubMed] 29. Weng, Y.; He, X.; Yao, W.; Pacheco, M.C.; Wang, J.; Pan, Z. Mode-dependent loss mitigation scheme for PDM-64QAM few-mode fiber space-division-multiplexing systems via STBC-MIMO equalizer. In Proceedings of the Conference on Lasers and Electro-Optics (CLEO), San Jose, CA, USA, 14–19 May 2017. 30. Raptis, N.; Grivas, E.; Pikasis, E.; Syvridis, D. Space-time block code based MIMO encoding for large core step index plastic optical fiber transmission systems. Opt. Express 2011, 19, 10336–10350. [CrossRef][PubMed] 31. Weng, Y.; He, X.; Wang, J.; Pan, Z. Rigorous study of low-complexity adaptive space-time block-coded MIMO receivers in high-speed mode multiplexed fiber-optic transmission links using few-mode fibers. In Proceedings of the SPIE 10130, Next-Generation Optical Communication: Components, Sub-Systems, and Systems VI, San Francisco, CA, USA, 28 January 2017. 32. Mizusawa, Y.; Kitamura, I.; Kumamoto, K.; Zhou, H. Proposal analysis and basic experimental evaluation of MIMO radio over fiber relay system. In Proceedings of the SPIE 10559, Broadband Access Communication Technologies XII, San Francisco, CA, USA, 29 January 2018. 33. He, X.; Weng, Y.; Pan, Z. Fast convergent frequency-domain MIMO equalizer for few-mode fiber communication systems. Opt. Comm. 2018, 409, 131–136. [CrossRef] 34. Li, G. Recent advances in coherent optical communication. Adv. Opt. Photonics 2009, 1, 279–307. [CrossRef] 35. Muga, N.J.; Fernandes, G.M.; Ziaie, S.; Ferreira, R.M.; Shahpari, A.; Teixeira, A.L.; Pinto, A.N. Advanced digital signal processing techniques based on Stokes space analysis for high-capacity coherent optical systems. In Proceedings of the 19th International Conference on Transparent Optical Networks (ICTON 2017), Girona, Spain, 2–6 July 2017. 36. Lin, C.; Djordjevic, I.B.; Zou, D. Achievable information rates calculation for optical OFDM few-mode fiber long-haul transmission systems. Opt. Express 2015, 23, 16846–16856. [CrossRef][PubMed] 37. Inan, B.; Jansen, S.L.; Spinnler, B.; Ferreira, F.; Borne, D.V.D.; Kuschnerov, M.; Lobato, A.; Adhikari, S.; Sleiffer, V.A.; Hanik, N. DSP requirements for MIMO spatial multiplexed receivers. In Proceedings of the IEEE Photonics Society Summer Topical Meeting Series, Seattle, WA, USA, 9–11 July 2012. 38. Pan, Z.; He, X.; Weng, Y. Frequency domain equalizer in few-mode fiber space-division-multiplexing systems. In Proceedings of the 23rd Wireless and Optical Communication Conference (WOCC), Newark, NJ, USA, 9–10 May 2014. 39. Dick, C.; Amiri, K.; Cavallaro, J.R.; Rao, R. Design and architecture of spatial multiplexing MIMO decoders for FPGAs. In Proceedings of the 42nd Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, USA, 26–29 October 2008. 40. Cvijetic, N.; Ip, E.; Prasad, N.; Li, M.J.; Wang, T. Experimental time and frequency domain MIMO channel matrix characterization versus distance for 6 × 28 Gbaud QPSK transmission over 40 × 25 km few mode fiber. In Proceedings of the Optical Fiber Communication Conference (OFC), San Francisco, CA, USA, 9–13 March 2014. 41. Awwad, E.; Othman, G.R.; Jaouën, Y. Space-time coding and optimal scrambling for mode multiplexed optical fiber systems. In Proceedings of the IEEE International Conference on Communications (ICC), London, UK, 8–12 June 2015. Appl. Sci. 2019, 9, 1178 28 of 28

42. Salsi, M.; Koebele, C.; Sperti, D.; Tran, P.; Mardoyan, H.; Brindel, P.; Bigo, S.; Boutin, A.; Verluise, F.; Sillard, P.; et al. Mode-Division Multiplexing of 2 × 100 Gb/s Channels Using an LCOS-Based Spatial Modulator. J. Lightw. Technol. 2012, 30, 618–623. [CrossRef] 43. Weng, Y.; He, X.; Pan, Z. Space division multiplexing optical communication using few-mode fibers. Opt. Fiber Technol. 2017, 36, 155–180. [CrossRef] 44. Feuer, M.D.; Nelson, L.E.; Zhou, X.; Woodward, S.L.; Isaac, R.; Zhu, B.; Taunay, T.F.; Fishteyn, M.; Fini, J.F.; Yan, M.F. Demonstration of joint DSP receivers for spatial superchannels. In Proceedings of the IEEE Photonics Society Summer Topical Meeting Series, Seattle, WA, USA, 9–11 July 2012. 45. Feuer, M.D.; Nelson, L.E.; Zhou, X.; Woodward, S.L.; Isaac, R.; Zhu, B.; Taunay, T.F.; Fishteyn, M.; Fini, J.F.; Yan, M.F. Joint digital signal processing receivers for spatial superchannels. IEEE Photonics Technol. Lett. 2012, 24, 1957–1960. [CrossRef] 46. Ryf, R.; Fontaine, N.K.; Chen, H.; Wittek, S.; Li, J.; Alvarado-Zacarias, J.C.; Amezcua-Correa, R.; Antonio-Lopez, J.E.; Capuzzo, M.; Kopf, R.; et al. Mode-multiplexed transmission over 36 spatial modes of a graded-index multimode fiber. In Proceedings of the European Conference on Optical Communication (ECOC), Rome, Italy, 23–27 September 2018. 47. Ryf, R.; Fontaine, N.K.; Wittek, S.; Choutagunta, K.; Mazur, M.; Chen, H.; Alvarado-Zacarias, J.C.; Amezcua-Correa, R.; Capuzzo, M.; Kopf, R.; et al. High-spectral-efficiency mode-multiplexed transmission over graded-index multimode fiber. In Proceedings of the European Conference on Optical Communication (ECOC), Rome, Italy, 23–27 September 2018. 48. Weerdenburg, J.V.; Ryf, R.; Alvarado-Zacarias, J.C.; Alvarez-Aguirre, R.A.; Fontaine, N.K.; Chen, H.; Amezcua-Correa, R.; Koonen, T.; Okonkwo, C. 138 Tbit/s transmission over 650 km graded-index 6-mode fiber. In Proceedings of the European Conference on Optical Communication (ECOC), Gothenburg, Sweden, 17–21 September 2017. 49. Rademacher, G.; Ryf, R.; Fontaine, N.K.; Chen, H.; Essiambre, R.-J.; Puttnam, B.J.; Luis, R.S.; Awaji, Y.; Wada, N.; Gross, S.; et al. 3500-km mode-multiplexed transmission through a three-mode graded-index few-mode fiber link. In Proceedings of the European Conference on Optical Communication (ECOC), Gothenburg, Sweden, 17–21 September 2017. 50. Rademacher, G.; Ryf, R.; Fontaine, N.K.; Chen, H.; Essiambre, R.-J.; Puttnam, B.J.; Luis, R.S.; Awaji, Y.; Wada, N.; Gross, S.; et al. Long-Haul Transmission Over Few-Mode Fibers with Space-Division Multiplexing. J. Lightw. Technol. 2018, 36, 1382–1388. [CrossRef] 51. Igarashi, K.; Soma, D.; Wakayama, Y.; Takeshima, K.; Kawaguchi, Y.; Yoshikane, N.; Tsuritani, T.; Morita, I.; Suzuki, M. Ultra-dense spatial-division-multiplexed optical fiber transmission over 6-mode 19-core fibers. Opt. Express 2016, 24, 10213–10231. [CrossRef] 52. Fontaine, N.K.; Ryf, R.; Chen, H.; Velazquez Benitez, A.; Antonio Lopez, J.E.; Amezcua Correa, R.; Guan, B.; Ercan, B.; Scott, R.P.; Yoo, S.J.B.; et al. 30 × 30 MIMO Transmission over 15 Spatial Modes. In Proceedings of the Optical Fiber Communication Conference (OFC), Los Angeles, CA, USA, 22–26 March 2015. 53. Luís, R.S.; Rademacher, G.; Puttnam, B.J.; Awaji, Y.; Wada, N. Long distance crosstalk-supported transmission using homogeneous multicore fibers and SDM-MIMO demultiplexing. Opt. Express 2018, 26, 24044–24053. [CrossRef] 54. Garcia-Moll, V.M.; Simarro, M.A.; Martínez-Zaldívar, F.J.; González, A.; Vidala, A.M. Maximum likelihood soft-output detection through Sphere Decoding combined with box optimization. Signal Process. 2016, 125, 249–260. [CrossRef]

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