Impact and mitigation of XT-induced time-synchronization errors in MDM transmission systems through a minimized residual inter-block-interference (MRI) algorithm

Yu Tian; Juhao Li; Yingchao Xin; Zhongying Wu; Paikun Zhu; Yongqi He; Zhangyuan Chen

doi:10.1364/OE.25.028794

1. Introduction

Recently, mode-division-multiplexed (MDM) transmission using few-mode fiber (FMF) instead of single-mode fiber (SMF) has been widely investigated to further enhance the capacity of optical transmission systems and networks [1,2]. In MDM systems, all orthogonal spatial and polarization modes in FMF are exploited as independent data channels, which potentially boosts capacity by a factor of the number of modes. At the receiver, coherent detection and multiple-input-multiple-output (MIMO) equalization are typically required to demultiplex the signals. Both time domain equalization (TDE) [3,4] and frequency domain equalization (FDE) [5–7] have been investigated for MIMO-MDM transmission. Compared with TDE, FDE techniques such as orthogonal frequency division multiplexing (OFDM) and single-carrier (SC) transmission with FDE (SC-FDE) are more computation efficient in MIMO-MDM transmission.

As we know, timing synchronization is critical for proper operation of MIMO equalization. Currently, most of the synchronization methods are based on the maximum-correlation (MC) criterion [8,9]. In these methods, sliding correlation of received signal is performed to recognize a known symbol pattern, and synchronization is accomplished by finding the peak of the correlation trajectory. In MDM systems, typically correlation trajectory on the fastest mode is used for synchronization of all modes. These MC based timing synchronization methods work well for additive white Gaussian noise (AWGN) channels. However, in MIMO-MDM systems, perturbations of FMF will give rise to mode crosstalk and the FMF will behave like a multipath channel [10–12], which will output multiple delayed signal replicas after transmission. When MC algorithms are used in MDM systems, the correlation trajectory can be severely deteriorated by mode crosstalk and the peak value may no longer indicate the desired symbol arriving time [13–15].

For FDE approaches, the resulted synchronization error will lead to misplacing of DFT (Discrete Fourier transform) window and cause the inter-block interference (IBI) leak out of cyclic prefix (CP) [16,17]. To mitigate the synchronization error induced penalty, extra CP has to be allocated to each data block [15], so that the leaked IBI can be completely isolated. The length of extra required CP is proportional to modal dispersion. Because of the significant accumulated modal dispersion in MDM systems, the excessive CP overhead will seriously degrade the transmission and power efficiency especially when the number of multiplexed modes is very large.

For TDE approaches, in the presence of synchronization error, delayed signal replicas can no longer be completely collected by equalizer taps, and the leaked signal replicas will seriously degrade the equalization performance. Redundant equalizer taps have to be used to compensate the synchronization error, which will significantly increase the computation complexity.

In this paper, for the first time, we study the impact of random mode crosstalk on system timing synchronization. On this basis, a robust synchronization method against random mode crosstalk is proposed for MIMO-MDM transmission. The proposed synchronization method is designed based on the minimum residual IBI (MRI) criterion. With negligible computation complexity and no sacrifice of transmission efficiency, the MRI synchronization can effectively compensate the synchronization error induced by random mode crosstalk. The effectiveness of proposed MRI synchronization is numerically validated in 100-km 12 × 12 MDM FDE transmission system. Q²-factor improvement up to 8.7-dB has been observed and the system outage probability has been substantially reduced from 0.3 to 5e⁻⁴.

2. Impact of mode crosstalk on timing synchronization and the MRI algorithm

2.1 Impact of mode crosstalk on timing synchronization

In MIMO-MDM systems, all mode channels are jointly transmitted, received and processed. So timing synchronization of all mode channels is typically performed jointly. For MC method, at the transmitter a known synchronization symbol pattern is transmitted as preamble on a certain mode (e.g. the fastest mode). At the receiver, the target symbol pattern is searched on that mode and timing synchronization is acquired based on MC criterion for all modes.

In MIMO-MDM systems, signal may couple from original mode into the other modes due to mode crosstalk, and further scatter after transmission over some random distances. As a result, different portions of a specific signal may go through different modes and have different propagation paths [11,12]. Since modes in FMF propagate with different velocities, different paths will possess different propagation delays and multiple delayed signal replicas will be received after transmission.

When MC algorithm is used in MIMO-MDM systems, the target symbol pattern in every delayed replica is searched and the calculated correlation trajectory may contain multiple peaks. The peak value characterizes the amplitude and the position indicates the arriving time of a certain replica. Then only the highest peak of correlation trajectory is detected, which indicates the arriving of the strongest signal replica (SSR) and may be different with the first-arriving signal replica (FSR) [15]. In fact, FSR is not the same as SSR in most cases [12,18,19] and the arriving time difference between FSR and SSR varies stochastically due to the random nature of mode crosstalk. For time-domain equalizers, if the starting point of frame is selected at the SSR, the signal replica ahead of it will not be collected by equalizer taps. For FDE approaches, the IBI caused by previous signal replicas will not be isolated by CP. As a result, this discrepancy fundamentally leads to synchronization error and degrades system performance.

The effect of synchronization error on channel estimation (CE) is analyzed as follow. We assume M modes are multiplexed in the considered MDM system. The modal dispersion lasts L samples, and the lengths of data block and CP are N and L. For ideal timing synchronization, FSR of the beginning symbol in each block is detected and its arriving time is taken as the starting point of that block. Assume that the estimated frequency domain channel response between the n-th input mode and m-th output mode is $H_{m n} (k)$ , and the transformed time-domain CE is $h_{m n} (l) = IDFT [H_{m n} (k)]$ . Ideal synchronization ensures that all the non-zero channel taps are distributed among the first L (L < N) taps of $h_{m n} (l)$ , in other words,

h_{m n} (l) = 0, l \in [L, N - 1] .

Assume that the transmitted training sequence (TS) on the n-th mode is $x_{n} (l)$ . Then the received TS on the m-th mode is

\begin{matrix} y_{m} (l) = \sum_{n = 1}^{M} h_{m n} (l) \otimes x_{n} (l) \\ = \sum_{n = 1}^{M} \sum_{k = 0}^{N - 1} h_{m n} (k) \cdot x_{n} [(l - k) \mod N], l \in [0, N - 1], \end{matrix}

where notation

\otimes

denotes the operation of cyclic convolution, and symbol x and y respectively represent the transmitted and received TS, and the subscript _m and _n are used to indicate the index of mode channel. The reason why cyclic convolution is used is that TS is prepended with cyclic prefix and therefore is periodic.

In the presence of random mode crosstalk, MC timing synchronization undergoes synchronization ambiguity. When the timing synchronization deviates $e r r$ samples, by substituting Eq. (2) and adopting the commutative property of cyclic convolution [20], we have

\begin{matrix} y_{m} (l - e r r) = \sum_{n = 1}^{M} \sum_{k = 0}^{N-1} h_{m n} (k) \cdot x_{n} [(l - e r r - k) \mod N] \\ = \sum_{n = 1}^{M} \sum_{k = 0}^{N-1} h_{m n} [(l - e r r - k) \mod N] \cdot x_{n} (k) \\ = \sum_{n = 1}^{M} \sum_{k = 0}^{N-1} h_{m n} {[(l - e r r) \mod N - k] \mod N} \cdot x_{n} (k) \\ = \sum_{n = 1}^{M} \sum_{k = 0}^{N-1} {\bar{h}}_{m n} (k) \cdot x_{n} [(l - k) \mod N] \\ = \sum_{n = 1}^{M} {\bar{h}}_{m n} (l) \otimes x_{n} (l), l \in [0, N - 1], \end{matrix}

where

{\bar{h}}_{m n} (l)

represents the CE in the presence of synchronization error,

{\bar{h}}_{m n} (l) = h_{m n} [(l - e r r) \mod N], l \in [0, N - 1] .

It can be seen in Eq. (4) that synchronization error will result in cyclic shift of the time-domain CE with period of N, which is illustrated in Fig. 1.

Fig. 1 Illustration of the impact of timing synchronization error on CE. (a) CE under ideal timing synchronization. (b) CE when err = 20. (c) CE when err = −20. The length of data block is 256. Modal dispersion lasts 50 samples.

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Non-zero taps in ${{\bar{h}}_{m n} (l), l \in [L, N - 1]}$ essentially characterize the leaked IBI caused by synchronization error. Recalling the definition of inverse discrete Fourier transform (IDFT), ${\bar{h}}_{m n} (l)$ represents the principal-value sequence of the periodically extended time-domain CE. The more intuitive time-domain CE ${\bar{h}}_{m n} (l)$ defined in $[- N / 2, N / 2 - 1]$ is given by

{\bar{h}}_{m n} (l) = {\begin{matrix} {\bar{h}}_{m n} (l), \\ {\bar{h}}_{m n} (l + N), \end{matrix} \begin{matrix} l \in [0, N / 2 - 1] \\ l \in [- N / 2, - 1] \end{matrix} .

Thus, for a certain data block i, taps

{{\bar{h}}_{m n} (l), l \in [0,L]} = {{\bar{h}}_{m n} (l), l \in [0,L]}

represent the IBI isolated by CP. Taps

{{\bar{h}}_{m n} (l), l \in [L, N / 2 - 1]} = {{\bar{h}}_{m n} (l), l \in [L, N / 2 - 1]}

can be interpreted as the IBI leaked into block i + 1, and taps

{{\bar{h}}_{m n} (l), l \in [N / 2, N - 1]} = {{\bar{h}}_{m n} (l), l \in [-N / 2, - 1]}

are the IBI leaked into block i-1. It is the task of the proposed MRI algorithm to minimize

{{| {\bar{h}}_{m n} (l) |}^{2}, l \in [L, N - 1]}

by finely tuning the timing point given by MC synchronization.

2.2 Implementation of MRI timing synchronization

Being aware of the impact of synchronization error on channel estimation, MRI algorithm infers the synchronization error by identifying the cyclic shifts in channel estimation. Compared with conventional MIMO-MDM receiver, additional MRI timing synchronization is added in MRI-based receiver as shown in Figs. 2(a) and 2(b). Here we assume FDE approach is used. Application of MRI with TDE approach is similar, where equalizer in Fig. 2(b) should be replaced with TDE.

Fig. 2 Receiver DSP for (a) conventional FDE and (b) MRI-FDE. (c) Principle of MRI timing synchronization. TS: training sequence. CE: channel estimation.

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The schematic of MRI algorithm is shown in Fig. 2(c). To implement MRI, the frequency-domain CE ${\bar{H}}_{m n} (k)$ is firstly transformed into time-domain ${\bar{h}}_{m n} (l)$ . Then ${\bar{h}}_{m n} (l), m, n \in [1, M]$ are cyclically shifted together tap by tap, and the following normalized timing metric is calculated

f (u) = \sum_{n = 1}^{M} \sum_{m = 1}^{M} \sum_{l = L}^{N - 1} {| {\bar{h}}_{m n} [(l + u) \mod N] |}^{2} / \sum_{n = 1}^{M} \sum_{m = 1}^{M} \sum_{l = 0}^{N - 1} {| {\bar{h}}_{m n} (l) |}^{2}, u \in [- S,S],

where S defines the scanning range. According to Eqs. (1) and (4),

f (u)

reaches its minimum value 0 when

u = e r r

. So the cyclic shift length (synchronization error) can be estimated as

\bar{\bar{e r r}} = \underset{u}{\arg \max} [1 - f (u)] .

Considering the interpretation of

{{\bar{h}}_{m n} (l), l \in [L, N - 1]}

in previous subsection and Eq. (4),

f (u)

can be interpreted as power of residual IBI when certain correction

u

is applied to compensate synchronization error. Therefore Eq. (7) expresses the minimum residual IBI (MRI) criterion. At last, the synchronization error in received MDM signal is corrected and CE is updated

{\bar{\bar{R}}}_{1} (l) = {\bar{R}}_{1} (l + \bar{\bar{e r r}}),

{\bar{\bar{H}}}_{M \times M} (k) = {\bar{H}}_{M \times M} (k) \cdot \exp [j \cdot \frac{2 π}{N} \cdot k \cdot \bar{\bar{e r r}}] .

2.3 Complexity analysis

The complexity of FDE with and without MRI synchronization are compared in this subsection. Assessed in terms of complex multiplications per symbol, the complexity of conventional FDE is given by [6]

C_{w/o . MRI} = \log_{2}^{N} + M + \frac{T_{b}}{T_{c}} \cdot (\frac{\log_{2}^{N}}{2} {+M+M}^{2}) .

The last term in Eq. (10) represents the complexity introduced by frequency domain channel estimation.

T_{c}

is channel coherent time, which measures the time scale of mode crosstalk dynamics.

T_{b} =N/ R_{B}

represents the duration of one data block, where

R_{B}

is baud rate. For FDE with MRI, extra complex multiplications are introduced by IDFT, synchronization error estimation, and channel estimation update as shown in Fig. 2(c). The complexity is given by

C_{w . MRI} = C_{w/o . MRI} + \frac{T_{b}}{T_{c}} \cdot (\frac{M}{2} \log_{2}^{N} +2M) .

In synchronization error estimation module, before timing metric Eq. (6) is evaluated, all

\sum_{n = 1}^{M} \sum_{m = 1}^{M} {| {\bar{h}}_{m n} (l) |}^{2}, l = 1, \dots, N

are calculated and restored. This increases the complexity by

T_{b} / T_{c} \cdot M

, which is included in the last term in Eq. (11). During the search phase, different

\sum_{n = 1}^{M} \sum_{m = 1}^{M} {| {\bar{h}}_{m n} (l) |}^{2}

are retrieved and combined to evaluate Eq. (6). In this process no additional complex multiplication is introduced.

The complexity of FDE with and without MRI synchronization are shown in Fig. 3 under different baud rates. The multiplexed mode number of 6, 12, 20 and 30 and data block size of 128, 256, 512 and 1024 are considered here. $T_{c}$ is set to be 25-us [7]. As can be seen, extra complexity introduced by MRI synchronization is negligible. This is because that the MDM channel is slow-varying over time (T_b / T_c << 1) and MRI synchronization only have to be performed one time after channel estimation. The complexities of both approaches decrease when baud rate increases. The reason is that for a given channel coherence time, high baud rate system can transmit more data blocks before next channel estimation is performed, which leads to lower computation overhead.

Fig. 3 Complexity of conventional FDE and FDE with MRI. Markers are for MRI-FDE. Lines are for conventional FDE.

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3. Numerical simulation

Numerical Monte Carlo simulations are carried out based on VPItransmissionMaker 9.3 and Matlab. The simulation setup is shown in Fig. 4. The MDM system supports 12 spatial and polarization modes (LP₀₁, LP_11a, LP_11b, LP_21a LP_21b, LP₀₂ and two polarizations). Independent 10-GBaud quadrature phase shift keying (QPSK) signal streams are modulated on all mode channels. Each signal stream consists of 65536 symbols. Data block size N is chosen as 256. The length of FMF link is set to be 100-km. The maximum differential mode group delay (DMGD) is set to be 85-ps/km [21]. DMGD of different mode groups are assumed to be uniformly distributed between 0 and the maximum DMGD. To simulate random mode crosstalk, FMF is divided into uncoupled sections. At the end of each section, a random unitary matrix is used to scatter all supported modes. The section lengths are drawn from a Gaussian distribution with mean section length of 25-km and standard deviation of 5-km [22]. The random matrices are assumed to be uniformly distributed and are generated according to [23]. At the receiver, conventional MC synchronization is firstly carried out based on cross-correlation, whose timing metric is provided in Appendix for reference. Then the Q²-factors of conventional FDE, MRI-FDE are evaluated. We carry out Monte Carlo simulations with 10000 runs. In different runs, different random number seeds are used to generate transmitted data, FMF section lengths and mode crosstalk matrices.

Fig. 4 Simulation setup of the 240-Gbit/s 12-mode transmission system. PDM: polarization division multiplexing. CRX: coherent receiver.

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MC timing metrics under different crosstalk conditions are investigated in Fig. 5. To generate different crosstalk conditions, different random number seeds are used to generate the crosstalk matrices in FMF. As can be seen, the location of SSR varies randomly. For conventional FDE, MC synchronization can only estimate the arriving time of SSR, and it is difficult to determine the position of FSR and the exact starting point of DFT window. As a result, block partition in this case cannot ensure that IBI is completely isolated by CP. Residual IBI will lead to serious performance penalty. Without loss of generosity, assumption that SSR locates in the middle of timing metric is adopted during block partition.

Fig. 5 MC timing metric and Q²-factors under different OSNR for three typical conditions of mode crosstalk. (a), (b) and (c) are MC timing metrics. (d), (e) and (f) are the corresponding Q²-factors. The CP length is chosen to be 40.

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Reception performance of conventional FDE and MRI-FDE are compared in Fig. 6. Here the OSNR is set to be 25-dB, and the CP length is chosen to be 40. For conventional FDE without MRI, synchronization error leads to IBI leak, and Q²-factor fluctuates over a large range. By contrast, MRI-FDE effectively mitigates the performance fluctuations. In Fig. 7, we show the probabilistic histogram of Q²-factors under different CP lengths. To measure the Q²-factor free of IBI, we choose CP length L = 80, and the mean IBI-free Q²-factor is measured to be 21.9-dB. For FDE without MRI, worst Q²-factor of 13.2-dB is observed at L = 46, showing a performance penalty of 8.7-dB. By contrast, the performance of MRI-FDE is much more stable and is converged at L = 46.

Fig. 6 Q²-factor of (a) conventional FDE and (b) MRI-FDE under different conditions of mode crosstalk. The OSNR is 25-dB. The CP length is 40.

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Fig. 7 Probabilistic histogram of Q²-factors for (a) conventional FDE and (b) MRI-FDE. The OSNR is 25-dB. The CP length is 40.

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When the performance fluctuation exceeds the acceptable threshold, it will lead to system outage, and the required communication capacity and quality cannot be guaranteed. Outage probability is defined as the probability that the synchronization error induced performance penalty exceeds certain threshold. The outage probabilities of conventional FDE and MRI-FDE under different CP lengths are compared in Fig. 8. As can be seen, MRI synchronization substantially reduces the outage probability when same length of CP is used. For given a CP length of 46 and threshold of 1.5-dB, the outage probability is reduce from 0.3 to 5e⁻⁴. The proposed method can substantially improve the performance stability of MDM-FDE transmission systems without introducing extra transmission overhead. According to Fig. 3(b), when MRI method is used, complex multiplications per symbol only increase from 20.16 to 20.24 and the increased complexity is less than 0.5%. Outage probability for different mode numbers and transmission distances are investigated in Fig. 9. The DMGD values are chosen with the reference to FMFs introduced in [21] [24]. For each point, 10000 different crosstalk variations are considered. For all cases, outage probability reductions of more than two orders of magnitude have been observed.

Fig. 8 Outage possibility under different CP lengths. The OSNR is set to be 25-dB.

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Fig. 9 Outage possibility for different mode numbers and transmission distances. The OSNR is set to be 25-dB.

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4. Conclusion

In this paper, for the first time, we discuss the impact of mode crosstalk on timing synchronization in MIMO-MDM transmission. We put forward that traditional maximum-correlation (MC) based synchronization algorithms are vulnerable to random mode crosstalk, which ultimately degrades the transmission performance. On this basis, a crosstalk robust synchronization method is proposed for FDE MDM systems. The proposed synchronization method is designed based on minimum residual IBI (MRI) criterion. With negligible computation complexity and no sacrifice of transmission efficiency, the MRI synchronization can effectively compensate the synchronization error induced by mode crosstalk. The effectiveness of proposed MRI synchronization is numerically validated in 100-km 12 × 12 MDM FDE transmission system. Q²-factor improvement up to 8.7-dB has been observed and the system outage probability has been substantially reduced from 0.3 to 5e⁻⁴. In practical MIMO-MDM systems, mode dependent loss (MDL) is inevitably introduced by all kinds of inline components. MDL may randomly change the signal power of the mode carrying the synchronization sequence, and potentially impact the performance of MC method. By contrast, MDL does not change the effect of synchronization error on channel estimation (Eq. (4)), and therefore MDL has little effect on MRI criterion. The proposed MRI timing synchronization will be beneficial for the design of practical MIMO-MDM systems.

5 Appendix

For cross-correlation based MC timing synchronization, the following timing metric is calculated [9]

ρ (u) = \frac{\sum_{k = 1}^{K} s^{*} (k) r (u + k - 1)}{\sqrt{\sum_{k = 1}^{K} {| s (k) |}^{2}} \sqrt{\sum_{k = 1}^{K} {| r (u + k - 1) |}^{2}}},

where

s (k), k = 1, \dots, K

is the target synchronization sequence, K represents the length of synchronization sequence, and

r (k)

is the received signal. Synchronization is accomplished by finding the highest peak of the correlation trajectory.

Funding

973 Program (No. 2014CB340105 and 2014CB340101); National Natural Science Foundation of China (NSFC) (61377072, 61627814, 61690194 and 61505002); Fundamental Research Project of Shenzhen Science and Technology Foundation (JCYJ) (20170412153729436, 20170307172513653).

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Impact and mitigation of XT-induced time-synchronization errors in MDM transmission systems through a minimized residual inter-block-interference (MRI) algorithm

Abstract

1. Introduction

2. Impact of mode crosstalk on timing synchronization and the MRI algorithm

2.1 Impact of mode crosstalk on timing synchronization

2.2 Implementation of MRI timing synchronization

2.3 Complexity analysis

3. Numerical simulation

4. Conclusion

5 Appendix

Funding

References and links

Cited By

Figures (9)

Equations (12)

Optics Express