## Abstract

In this paper, we propose a constant modulus algorithm (CMA) for mode division multiplexing (MDM) systems with improved convergence performance. In order to adapt to sparse channels with large differential mode group delay (DMGD) in MDM systems, the CMA adopts a variable step size similar to the improved proportionate normalized least-mean-square (IPNLMS) algorithm. In additional to that, when a singularity problem is encountered or the tap values fail to converge, it reinitializes the tap coefficients according to the tap vectors of the successfully de-multiplexed data tributaries. The proposed initialization approach is based on the fact that the channel matrix is unitary in the frequency domain in the absence of mode dependent loss (MDL), which means the channel coefficient vectors for each data tributary should be orthogonal to each other. By limiting the initial values of the taps within the null space of the complex conjugate vectors of the successfully de-multiplexed channels, singularity can be effectively avoided and the convergence of the taps is guaranteed. When the number of modes is two, the proposed algorithm becomes the constrained CMA, which has been commonly implemented in polarization division multiplexing (PDM) systems. Although the algorithm has been derived under zero MDL assumption, it is found that the proposed CMA can be quite resilient to MDL. No singularity/tap convergence failure problem occurs when the MDL is below 4 dB at both the input and the output ports.

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

## 1. Introduction

Mode division multiplexing (MDM) [1,2] is one of the promising technologies to further increase the system capacity of the optical communication systems, which has been under investigation by the research community and several leading companies [1].

In MDM systems, the channels are highly coupled to each other especially for the modes with relatively close propagation constants, which are usually referred to as mode groups [3]. It is possible to directly de-multiplex the modes within different mode groups with the mode de-multiplexers [3]. However, it is rather difficult to de-multiplex the modes within the same mode group without any signal processing techniques, because those modes share almost the same propagation constants and the mode coupling among them becomes significantly strong. The coupling can be caused by the index change induced by the random temperature and stress variations. The interconnections between the optical links without the mode MUXs/DEMUXs guaranteeing the mode purity further randomize the channel matrix. The coupling effect is unpredictable and varies in the time scale of milliseconds. Henceforth, de-multiplexing of the modes within the same mode group will be highly reliant on the digital signal processing (DSP) technique. Particularly in the long haul MDM transmission systems, the coupling of the modes within different mode groups should be significant to suppress group delay (GD) spread [4], and the de-multiplexing process is more dependent on the DSP technology.

There have been literatures on the DSP algorithms to de-multiplex the MDM signals. Training sequence (TS) is one way to monitor the MIMO channel matrix [5]. However, it requires additional overheads and reduces the spectral efficiency. Therefore, the blind channel estimation algorithms have gained significant attentions. For example, L. Zhao et al. [6] and Z. He et. al. [7] proposed to de-multiplex the MDM system modes by the independent components analysis (ICA). Among the blind channel estimation algorithms, the constant modulus algorithm (CMA) [8–15] is much simpler and more efficient to be implemented in the MDM systems [8–10].

In comparison with the successful implementation in the polarization division multiplexing (PDM) systems, CMA encounters two major problems while being applied to the MDM systems. First of all, the MDM systems usually have very large differential mode group delay (DMGD). This results in a sparse yet long channel impulse response [16]. The traditional CMA becomes difficult to converge in this case. The problem can be partially resolved by introducing the improved proportionate normalized least-mean-square (IPNLMS) algorithm [16-17], which adopts variable step size according to tap weights. However, sometimes, failure to converge still occurs. The second problem which degrades the CMA performance is the singularity issue, i.e. several de-multiplexed signals converge to the same data tributary. In the PDM systems, several modified CMAs have been proposed to mitigate the singularity problem [11–15], such as the tap coefficient constrained CMA [13] (which resets the tap initial value of the second tributary after the first tributary is successfully de-multiplexed), the two stage CMA [14] and the MU-CMA [15]. However, the former two techniques are difficult to be implemented in the MDM systems because the number of data tributaries can be larger than two and the principles to derive the algorithms [13,14] are no longer valid. The multi-user (MU) CMA [15] can be extended to the MDM systems, however, it needs quite significant computational effort because it requires to evaluate the cross-correlation function between the de-multiplexed signals in real time. Hence, there are no suitable singularity avoidable CMAs for MDM systems without any prior channel knowledge. The singularity problem becomes even more significant in the presence of mode dependent loss (MDL), which usually arise from the mode multiplexer (MUX) or de-multiplexers (DE-MUX).

In this paper, we propose a novel CMA for MDM systems. A variable step size similar to the IPNLMS algorithm is used to adapt to the sparse MDM channel. A tap initialization technique is introduced afterwards. It initializes the tap coefficients based on the null space formed by the complex conjugate tap channel vectors of the successfully de-multiplexed signals. The derivation of the null space basis vector initialization is based on the unitary property of the channel matrix without the MDL effect. When the number of modes used in the MDM system is two, the algorithm reduced the tap coefficient constrained CMA in [13]. It was found that the algorithm works well in the relatively large MDL regimes, although it was derived based on the unitary channel matrix assumption.

## 2. Theory

#### 2.1 Channel model for the MDM systems

In the MDM system without MDL and random mode coupling, the input and output mode vectors can be related to each other by the following

when**a**

_{in}and

**a**

_{out}are the input and the output mode vectors and

**T**a diagonal unitary matrix in the frequency domain

^{th}mode. When the MDL and the random mode coupling effect are in presence, the transfer matrix

**T**becomes non-unitary in the frequency domain. According to the theory in [18], it is possible to decompose

**T**into two matriceswhere

**A**is a positive definite matrix and

**U**an unitary matrix.

**A**stands for the MDL and

**U**stands for the DMGD and the random mode rotation within the link. Generally

**A**and

**U**are frequency dependent and random. Proper FIR filter matrix should be chosen to equalize the channel matrix

**T**. The MDL of the MDM system is defined as

_{T}stands for the eigen value of matrix

**T**.

#### 2.2 Variable step size for CMA to resolve sparse channel

It has been shown in [16,17] that the channel in multimode fiber is sparse. Hence only a few taps have large coefficients while other taps have their coefficients close to zero. In order to accelerate the convergence speed of CMA to resolve sparse channel, variable step size is assumed like the IPNLMS algorithm in [16,17].

^{th}tap, and

*l*and

*L*denote the tap number and length of the FIR filter. $\alpha $is a constant varying between −1 to 1. $\epsilon $is a small positive constant used to avoid the division by zero.

#### 2.3 Tap initialization based on the null space technique

With the introduction of the IPNLMS-based variable step size, the convergence of the CMA is greatly improved. However, the singularity problem cannot be avoided by changing the step size, because the conventional CMA adjusts the taps for each tributary independently and the converging results depend on the initial values for the taps [13]. Hence, we propose the tap re-initialization algorithm to resolve the problem. The algorithm is firstly derived by assuming the DMGD and the MDL are not present in the link and only the coupling between the modes is under consideration. Therefore the channel matrix **T** = **U** is unitary and frequency independent. The more general algorithm for the non zero DMGD case is going to be presented in the next section. Although the algorithm is derived under zero MDL assumption, it can efficiently avoid singularity in the presence of MDL.

The modified CMA starts with the standard CMA just like the tap coefficient constrained CMA for PDM signals [12]. It is discussed and shown in [19,20] that at least one of the tributaries will be successfully de-multiplexed by the standard CMA. Since the channel is a constant unitary matrix, only one tap matrix is required to equalize the channel. Assuming there are totally *N* signals, and denoting the tap coefficients for the *k* successfully de-multiplexed signals as [**w**_{1} **w**_{2}…**w**_{k}] ^{T}, where **w**_{i} stands for the corresponding tap vector for the i^{th} successfully de-multiplexed signal. Since the channel matrix **T** is unitary, its inverse (the tap matrix) is also a unitary matrix and we have

_{ij}is the Kronecker delta,

*H*denote Hermitian transpose. For

**w**

_{k + 1}to

**w**

_{N}, they should be orthogonal to the vectors

**w**

_{1}to

**w**

_{k}according to the orthognality between the row/column vectors of the unitary matrix, and hence, we have

**w**

_{k + 1}

*to*

^{H}**w**

_{N}

*belong to the null space of the matrix [*

^{H}**w**

_{1}

**w**

_{2}…

**w**

_{k}]

^{T}. The null space has the dimension of

*N-k*, and we may find

*N-k*basis vectors of it and use their complex conjugate vectors to initialize

**w**

_{k + 1}to

**w**

_{N}. In this way,

**w**

_{k + 1}to

**w**

_{N}will never converge to

**w**

_{1}to

**w**

_{k}. During the second step, we implement standard CMA for the unsuccessfully de-multiplexed data tributaries with the above initial tap values for

**w**

_{k + 1}to

**w**

_{N}, we may expect data tributary

*k*+ 1 to data tributary

*N*will have at least one data tributary successfully de-multiplexed and it is different from the data tributaries 1 to k. Adding the tap coefficients of the newly de-multiplexed signals to the set [

**w**

_{1}

**w**

_{2}…

**w**

_{k}] and redo the null space initialization for the rest taps. The iteration process stops until every data tributary is de-multiplexed. In this way, one may efficiently avoid the singularity problem.

When only two modes are used in the MDM system, if the first data tributary has its tap coefficients as [*w*_{11} *w*_{12}], the second data tributary should have its initial value as [-*w*_{12}* *w*_{11}*], which is the same as the tap coefficient constrained CMA proposed for the PDM systems [12].

#### 2.4 Extension of the proposed CMA in the presence of DMGD

When we introduce DMGD, multiple taps should be used instead of a single tap. In the absence of MDL, the matrix **T** remains unitary in the frequency domain, so should its inverse matrix. Therefore, the tap matrix should be unitary after Fourier transform, and the null space initialization CMA can be modified to be applicable in the presence of DMGD. The procedure is as follows (shown in Fig. 1):

- 1. Run the standard CMA and summarize the taps coefficient vectors of the successfully de-multiplexed signals, i.e. [
**w**_{1}(1)**w**_{2}(1)…**w**_{k}(1)]^{T}, [**w**_{1}(2)**w**_{2}(2)…**w**_{k}(2)]^{T}, ….[**w**_{1}(*L*)**w**_{2}(*L*)…**w**_{k}(*L*)]^{T}, with*k*being the number of the successfully de-multiplexed data tributaries, and*L*being the number of taps. If*k*≠*N*, go to step 2. - 2. Take fast Fourier transform (FFT) of above tap values and find the corresponding vectors in the frequency domain. Initialize the tap vectors for the rest of signals according to the null space of the matrices in the frequency domain.
- 3. Take the inverse Fourier transform (IFFT) to find initial values of the tap vectors in the time domain.
- 4. Run the standard CMA with the time domain initial tap values for data tributaries
*k*+ 1 to*N*. - 5. Add the tap values of the successfully de-multiplexed signals to tap set described in step 1 and start the iteration again.

When only two modes are used in MDM system, step 1-5 indicate that if the first data tributary has the tap vectors as [*w*_{11}(1) *w*_{12}(1)], [*w*_{11}(2) *w*_{12}(2)]…., [*w*_{11}(*L*) *w*_{12}(*L*)], the second data tributary should have the initial tap values as [-*w*_{12}*(*L*) *w*_{11}*(*L*)], [-*w*_{12}*(*L*-1) *w*_{11}*(L-1)], …. [-*w*_{12}*(1) *w*_{11}*(1)], which is still exactly the same as the tap coefficient constrained CMA designed for the PDM signals [12].

It can be seen from the above descriptions that the algorithm is derived based on the assumption that the channel matrix is unitary in the frequency domain. It can, however, also be applied to avoid singularity in the presence of MDL. It is also worth mentioning that the proposed null space tap initialization can work independently if the channel is not sparse, and it is verified that in the non-sparse MIMO channels, the proposed singularity avoidance technique works well.

In comparison to the MU-CMA in [15], which requires the evaluation of the cross correlation function in real time, the algorithm proposed here is more suitable for real time signal processing. Because the proposed algorithm does not add any “real time” computational effort as the FFT and IFFT only take place in the initialization stage, and the initialization is required when there is a sudden channel variation. When updating the tap coefficients, no additional computation is required. Meanwhile the MU-CMA needs to evaluate the cross-correlation when updating the coefficients, and significant real time computational effort is required. Since the MU-CMA is extremely time-consuming especially in the large DMGD regime and the singularity avoidance performance of the tap re-initialization CMA and the MU-CMA have been proved to be comparable in the two-mode case [12], we focus on the proposed tap re-initialization CMA based on the null space concept in this work.

As for the IPNLMS algorithm, it is used to increase the convergence rate for the sparse channel and it is not related to singularity avoidance. As mentioned above, the proposed null space tap initialization does not necessarily require to be used in combination with the IPNLMS algorithm if the channel is not sparse. Hence, the corresponding IPNLMS algorithm complexity should not be taken into account when the algorithm complexity for the singularity avoidance techniques is evaluated.

## 3. Results and discussions

#### 3.1 Singularity occurrence with respect to the orientation of the principal modes in the MDM systems

First of all, the algorithm is verified by scanning the principal modes orientations in the MDM systems like the case in the PDM systems. The computational effort to scan the principal modes orientations is highly related to the size of the unitary matrix. In order to save the computational effort, the transfer matrix is limited to the size of 3X3. Similar to the PDM case, the transfer matrix **T** can be decomposed into [11,12,21]

**B**represents an arbitrary unitary matrix with the size of 3X3, and matrix

**R**relates the three principal modes in the MDL matrix

**D**to the three orientations, and MDL is the mode dependent loss. According to the formula in Eq. (9), the parameter a, b, c, d, e, and f can be written as

_{1}, δ

_{1}, κ

_{1}, θ

_{2}, δ

_{2}, κ

_{2}, θ

_{3}, δ

_{3}, and κ

_{3}, can be used to determine an arbitrary unitary matrix, which is in accordance with the theoretical dimension of an arbitrary unitary 3X3 matrix. Since the input signal phase rotation can be compensated by the Viterbi-Viterbi algorithm, the phases of the elements in the first row of

**B**can be assumed to be zero. Hence, we have δ

_{1}= 0, δ

_{2}= -κ

_{1}, and κ

_{2}= -κ

_{1}. The essential parameters are reduced to θ

_{1}, θ

_{2}, θ

_{3}, κ

_{1}, δ

_{3}, and κ

_{3}.

Similar to the representation of **B**, matrix **R** can be characterized by the 6 parameters as well. A full scan of the nine parameters in **B** and **R** (totally 12 parameters) can provide a thorough examination of the impact of the principal modes orientations on the singularity of the algorithms. In our case, only matrix **B** is scanned thoroughly to save the computational effort, i.e., θ_{1}, θ_{2}, θ_{3}, κ_{1}, δ_{3}, and κ_{3} are divided into 16 × 16 × 16 × 5 × 5 × 5 sections. They are fixed in each section with the principal modes in the matrix **R** varying uniformly in the matrix space U(3). 500 different matrix **R**s are chosen in the simulation. The OSNR is fixed as 18dB, the step size ${\mu}_{0}$for the CMA is 1/400, $\alpha $is −1 (no FIR filter is required in the case) and symbol length 1 × 10^{4}.

It is found that when mdl = 0dB, the traditional CMA already falls into the singularity problem while our proposed algorithm is singularity free. The singularity free performance maintains until mdl = 6dB. When mdl> = 7dB, the proposed algorithm can no long guarantee the singularity free operation. To demonstrate this, we fix four of the six parameters as 0 and only vary θ_{1} and θ_{2}, and the singularity occurrence is plotted in Fig. 2. Figure 2(a), Fig. 2(c) and Fig. 2(e) are for the conventional CMA and Fig. 2(b), Fig. 2(d) and Fig. 2(f) are for the proposed CMA with null space tap initialization. The singularity occurrence is indicated by the white color while the black color indicates that the regime is singularity free. The MDLs are set as 0dB, 6dB and 7dB. Thereafter, the parameter θ_{3} is changed to 48 degree while others remain unchanged. The corresponding singularity occurrence has been plotted in Fig. 3. It can be seen from the Figs. 2 and 3 that the conventional CMA experiences the singularity problem even when MDL = 0dBm, while the proposed CMA with null space initialization remains singularity free until MDL = 6dB. When MDL = 7dB, the proposed CMA can no longer avoid the singularity problem although very few chances to have singularity are observed (Fig. 3(f)).

The OSNR has little impact on the singularity avoidance performance of the proposed algorithm. When the OSNR decreases from 18dB to 10dB, the maximum tolerable MDL for singularity free operation decreases from 6dB to 5.5dB, which demonstrates the robustness of the proposed algorithm with respect to the OSNR variation.

#### 3.2 System level verification

We have conducted the numerical simulations on an MDM system to further verify the proposed CMA. The MDM system uses the multimode fibers with four degenerated modes [5], namely the x and the y polarized LP11a/LP11b modes [5]. In the simulation, the multimode fiber transmission system is divided into multiple subsections. Each subsection has a constant DMGD. Random coupling matrix between the degenerated modes is assumed between the subsections as the random rotation / index perturbation of the fiber at the interconnections impacts the coupling effect significantly if mode MUXs/DEMUXs are not used to ensure the mode purity. Amplified spontaneous noise (ASE) is added and the OSNR is tuned by varying the ASE power. Such a configuration will maintain the sparsity of the channel [16,17] while incorporating the random coupling effect between the modes. The proposed channel model is also in accordance with the one in [22]:

**U**and

_{k}**V**are two random coupling matrices,

_{k}**D**the diagonal matrix standing for the DMGD and the mode dependent loss (MDL) of the kth multimode fiber section, g

_{k}_{i}and τ

_{i}the MDL and the mode group delay of the ith mode. On the right and left sides of matrix

**T**, we may further multiply

**D**

_{0}and

**D**

_{K + 1}, which are the DMGD and MDL matrices of the input and output mode multiplexers. When MDL = 0, g

_{i}= 0.

It is worth noting that the nonlinear coupling effects [23] are not considered here due to the significant computational effort. Its contribution can be taken into account by incorporating the ASE noise [24] in the simulation setup.

The signal wavelength is 1550nm and QPSK modulation is assumed for the four data tributaries with the Baud rate of 20G/s each. Chromatic dispersion is assumed to be perfectly compensated by the DSP and will not be considered during the CMA de-multiplexing process.

The basic optical schematic and the signal processing unit [25] are illustrated in Fig. 4.

First of all, the mean DMGD of the each multimode fiber section is assumed to be 3.5ns (assuming four modes are with the DMGD of 0 0 3.5 3.5ns, such an assumption is reasonable as two polarizations of the LP11a of LP11b modes should have similar DMGDs) with the length of 10km, which matches the measured DMGD value for the LP11 mode group in [4] (a 26km length fiber with 10ns channel delay spread). The fiber transmission system is 100km, which is composed of 10 sections of the above fiber. A random 4 times 4 coupling matrix is inserted between sections. No MDL is added at the input or output. The OSNR is 18dB. Totally 1501 taps are used in the CMA to combat the DMGD, which is a randomly varying value [26]. The initial tap values are assumed to be the identity matrix for the center tap and the zero matrices for the other taps. The number of symbols used in the simulation is 1 × 10^{6}, which is set to be long enough to ensure the convergence. Mont Carlo simulation is conducted for 3000 times with the proposed method and the conventional CMA with the IPLMS algorithm and the results are summarized in Table 1. When multiple de-multiplexed signals converge to the same data tributary, it is regarded that only one signal is successfully de-multiplexed. It can be seen that no singularity/failure of convergence has been observed for the proposed CMA reinitialized by the null space technique while the conventional CMA has over 90% probability to fall into the singularity problem or fail to converge to the four different tributaries.

As an example, one of the realizations of the Mont Carlo simulation is demonstrated below. The taps values by the conventional CMA (Fig. 5) and the proposed CMA with null space initialization (Fig. 6) are plotted. It can be seen from Fig. 5 that the tap weights **w**_{11} to **w**_{14} and **w**_{21} to **w**_{24} are almost the same, which indicates the convergence to an identical data tributary. Hence, singularity has occurred. The fact that the tap weights **w**_{41} to **w**_{44} have quite noisy behavior indicates failure to converge. The corresponding de-multiplexed signals by the taps in Fig. 5 are shown in Fig. 7 which demonstrates that the last data tributary has failed to converge (Fig. 7d). In Fig. 6, it can be seen that the tap coefficient distributions are quite different for the four de-multiplexed modes hence indicating no singularity has occurred. Also no noisy behavior is observed in Fig. 6. The successfully de-multiplexed data tributaries by the taps in Fig. 6 are plotted in Fig. 8. It is worth mentioning that since the conventional CMA has implemented the IPNLMS algorithm as well, the successfully de-multiplexed data tributaries for the conventional CMA should have the identical BER as the one by the proposed method.

The proposed CMA with null space initialization is quite robust with respect to different DMGD, OSNR and MDL. We first vary the total DMGD from 35ns to 0ns, and 100% probability maintains to successfully de-multiplex the four different data tributaries. Afterwards, the proposed algorithm and the standard CMA are checked under different OSNRs and MDLs with the results shown in Fig. 9. Other parameters, such as the DMGD and the random coupling between the sections, remain unaltered as the ones for the first simulation. Without loss of generality, we assume equal MDLs at the input and the output ports, which vary from 0dB to 4dB. The average link MDL changes from 0dB to about 5.6dB (the link MDL varies from one realization to another due to the different random coupling matrix). The maximum possible link PDL is from 0dB to 8dB. Meanwhile the OSNR changes from 18dB to 15dB and 11dB. It is found that the proposed algorithm can always successfully de-multiplex the four different signals without the singularity problem encountered, while the standard CMA experiences a very high probability of singularity. Further increase of the MDL (in/out MDL = 5dB and average link MDL = 7dB) will inevitably bring the singularity problem for the proposed algorithm. Henceforth, the proposed singularity avoidance technique is quite resilient to MDL and is very robust under different OSNRs.

Finally, we investigate how steady state BER changes Vs OSNR. In the simulation, the theoretical curve has been obtained by adding the noise on the ideal QPSK signals with the corresponding OSNRs. The BTB curve has been obtained by sending the signals throughout the transmitter/receiver without the fiber link but with the CMA implemented. For the same system setup, the CMAs with different initialization techniques should have the identical steady state BER if singularity does not occur. Therefore, only the curves for the proposed CMA are shown with different MDLs and OSNRs. From Fig. 10, one may infer that the algorithm works well under different MDLs and OSNRs and particularly, it functions well around the FEC limit.

## 4. Summary

The main contribution of this work is to propose a new blind CMA with the novel initialization procedures. This resolves the problem for CMA, which is currently not applicable for MDM systems with over two data tributaries if the training signals are not used. The modified CMA uses the null space technique to initialize of the tap values when singularity/convergence failure is encountered. In comparison with the training signal based algorithms [17], it saves significant transmission overhead. While compared with the other blind equalization algorithms [6-7], it implements the well-known CMA which is already a mature technique in PDM systems and higher computational efficiency and more compact DSP structure can be expected. The proposed blind algorithm is very robust with respect to different OSNR, DMGD and MDL, and it can be useful for the future MDM systems.

## Funding

National Natural Science Foundation of China (NSFC) (61775168); Shanghai Natural Science Foundation (16ZR1438600).

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