## Abstract

In this paper, we demonstrate channel equalization for two-modefiber based MIMO system with the size of receiver larger than that of transmitter. The proposed configuration is verified via experiments of 102-Gb/s CO-OFDM mode-division multiplexed systems. We compare performances of three channel equalization algorithms including zero-forcing (ZF), minimum-mean-square-error (MMSE), and successive-interference-cancellation (SIC). Our results show that the 4x6 MIMO system can improve the receiver sensitivity by 1.8, 2.9, and 4.9 dB for ZF, MMSE, and SIC, respectively.

©2012 Optical Society of America

## 1. Introduction

The Shannon capacity limit of single-mode fiber (SMF) systems in the presence of fiber nonlinearity has been extensively studied in the last decade [1–5]. This capacity has been quickly approached by the recent demonstration within some practical engineering margin [6]. It is therefore impossible to enjoy the same dramatic capacity improvement in the future as in the past two decades, if we continue to stay with the SMF platform. Therefore, the spatial-division multiplexing (SDM) has been explored recently to overcome the capacity barrier [7–13]. There are a few techniques available at the moment for SDM, including multi-core fiber (MCF) [7–9] and few-mode fiber (FMF) [10–13]. Particularly, in the FMF based systems, there exist a variety of mechanisms inducing mode dependent loss (MDL), for instance, due to non-ideal mode multiplexing and de-multiplexing modules, and MDL of the splices, couplers, and optical amplifiers. Excessive MDL results in significant system penalty [14, 15]. Various methods are proposed to reduce MDL. For instance, MDL can be reduced by coupling and decoupling modes with novel system components (low-loss mode coupler), or using carefully aligned mode multiplexers and demultiplexers [14–16]. There was report on performance improvement using delayed signals from additional receivers [17]. In this paper, we introduce a systematic and detailed discussion on digital signal processing to improve receiver sensitivity by using larger receiver sets. In particular, we demonstrate signal processing of 4x6 MIMO systems using 3 different channel equalization algorithms: zero-forcing (ZF), minimum-mean-square-error (MMSE), and successive interference cancellation (SIC). The results show that the receiver sensitivity can be improved respectively by 1.8, 2.9, and 4.9 dB for ZF, MMSE, and SIC equalization in a 102-Gb/s 4x6 MIMO-OFDM systems.

## 2. Algorithms for TMF-based MIMO channel equalization

We now introduce the principle of the three equalization methods. We consider the MIMO channel with *N* transmitters and *M* receivers. The received signal can be expressed as

*H*is the

_{$M\times N$}channel matrix,

*y*is the

_{$M\times 1$}received signal,

*x*is the

_{$N\times 1$}transmitted signal, and

*n*is the additive white Gaussian noise. The superscript

*k*denotes the

*k-th*OFDM subcarrier, subscript

*i*denotes

*i-th*receiver, and

*j*denotes the

*j-th*transmitter. For the reminder of the paper, we shall omit the subcarrier index for brevity, which implies that the signal processing is done on the subcarrier basis.

There are a myriad of available techniques for channel equalization and estimation [18, 19]. In this paper, we apply three different algorithms in our FMF optical communication systems. They are ZF equalization, MMSE equalization, and SIC equalization.

#### 2.1 Zero-forcing (ZF)

The ZF equalizer applies the inverse of the channel frequency response to the received signal, to remove the signal distortion. As the name suggests, ZF would completely remove the inter-modal-interference (IMI) if there were no noise. The ZF equalization utilizes general method of Pseudo-inverse of the _{$M\times N$} channel matrix, which is defined as

*H*denotes Hermitian conjugation of a matrix.

*W*is the channel equalization matrix, and the superscript ‘-1’ represents the inverse of matrix. The estimated transmitted symbol, $\widehat{x}$is therefore given by

#### 2.2 Minimum mean square error (MMSE)

The ZF equalizer removes all linear distortion, but amplifies noise greatly where the channel response has small amplitude (fading). An improved equalizer for a noisy channel is the MMSE equalizer, which does not usually eliminate interference completely but instead minimizes the total power of the noise and the interference components. Formally, it is an approach that tries to find an equalization matrix *W* minimizing the criterion

*E*’ stands for ensemble average. Minimizing Eq. (4) yieldswhere ${\delta}_{n}^{2}$and ${\delta}_{s}^{2}$are respectively the variance for the noise and transmitted signal, and

*I*represents identity matrix.

#### 2.3 Successive interference cancellation (SIC)

The performance of the channel equalization can be further improved by having some prior knowledge of channel information, for instance, the strength of each mode. The interference from stronger modes to the weaker ones can be cancelled by recovering them earlier. Such an equalizer estimates and cancels the interference of transmitted signal one by one, until all the transmitted signals have been processed. The estimation of transmitted waveform (the interference) is done by either ZF or MMSE equalization. In this paper, we choose MMSE approach for its better performance compared with ZF equalization. Usually, the signal with higher SNR (e.g._{}_{${\widehat{x}}_{j}$}) will be equalized and processed earlier using Eq. (5). The signal after cancelling the interference of the first (*j-1*) estimated modes:

_{${H}_{1:M,1:j-1}$}is the first (

*j-1*) columns corresponding the first (

*j-1*) recovered modes

_{${\widehat{x}}_{1:j-1}$}. At the end of each iteration, the channel matrix shrinks to

_{$M\times (N-p)$}for the

*p-th*iteration.

_{${y}^{\prime}$}will be applied to Eq. (5) to recover the next mode.

The SIC channel equalization procedure is as follows [18], the first step of the procedure is performing channel estimation and demodulating the signal. Then the columns of the matrix are re-ordered according to the intensity of transmitted signals. More specifically, the channel matrix *H* will be re-ordered according to amplitude of each column. The transmitted signal with strongest power is placed to the first column and the transmitted signal with weakest power is placed to the last column. By applying such re-ordering, we will be able to identify the strongest mode in each iteration and remove its interference from the other modes using Eq. (6). Then the column that represents the strongest mode is dropped. The iteration repeats until all the transmitted signals are processed.

## 3. Experimental results and discussion

We conduct 4x6 back-to-back measurement for a 102 Gb/s TMF CO-OFDM system to verify the performance of the above mentioned three algorithms. As shown in Fig. 1
, the signal is created by combining 3 optical tones spaced at 6.5185 GHz. The 3 tones are fed into optical IQ modulator, which is driven by OFDM signal from arbitrary waveform generator (AWG). The polarization multiplexing is emulated by splitting the CO-OFDM signal from IQ modulator into two branches, which are delayed with each other by 1 OFDM symbol (500 ns), and recombined with a polarization beam combiner (PBC). We then emulate the signals for two orthogonal LP_{11} modes by splitting the signal and delaying one branch by 2 OFDM symbols (1 μs). Two pairs of long-period fiber grating (LPFG)-based mode converters (MC) and mode strippers (MS) are used to convert LP_{01} mode to LP_{11} mode. Free space mode combining and splitting is achieved with collimators and prism beam splitters (BS). The mode-division multiplexed signal is then sent to the receiver by a 7-meter TMF fiber. We employ two synchronized sampling scopes for coherent heterodyne detection. We use a narrow band pass filter (BPF) to remove the unwanted bands and keep an intermediate frequency gap between the LO and signal to avoid image folding. For 4x6 MIMO configuration, at the transmitter side, there are 4 transmitted signal (2 degenerate LP_{11} modes with each being polarization-division multiplexed), and at the receiver we implement 6 receivers to collect not only the 4 LP_{11} signals but also the 2 LP_{01} signals which is excited by the non-ideal coupling. In contrast, for 4x4 MIMO configuration, only the 4 LP_{11} signals are processed at the receiver. The OFDM parameters are: OFDM symbol length of 2560 points; cyclic prefix of 452 points; 4 OFDM training symbols employed to represent alternate launch of 4 combinations of polarizations and modes, which is used for 4x6 (or 4x4) channel matrix by means of intra-symbol averaging [20]; unique word length of 32 used at each end of OFDM symbol to seed phase estimation for data symbols [21]. The phase estimation of training symbols is done by using RF pilot subcarrier [22]. The data symbols use DFT-spread (DFT-S) OFDM signal, and the phase estimation of data symbol is done using decision feedback seeded by using unique word [21]. The CO-OFDM single channel occupies 19.5 GHz bandwidth, carrying a net data rate of 102 Gb/s.

To study the effectiveness of the 4x6 MIMO equalization, and to verify how the algorithms can adapt the changing channel, we measured the CO-OFDM system under two different channel conditions: (i) the two LP_{11} modes are launched in non-orthogonal orientation to emulate non-ideal mode coupling, which is called system I, (ii) the two LP_{11} modes are launched in orthogonal orientation, which is called system II. Fig. 2
shows the BER as a function of OSNR in system I, where the system suffers large penalty due to the non-ideal mode coupling. It can be seen that compared with 4x4 configuration, the system sensitivity is improved by 1.8 dB, 2.9 dB, and 4.9 dB using equalization of 4x6 ZF, MMSE, and SIC respectively. All the three bands of the 102 Gb/s OFDM signal are measured and the average BER is plotted. There are 5,836,800 bits measured in total for each OSNR value. We consider OSNR penalty at FEC BER threshold of 1.8x10^{-2} for 19.5% overhead [23].

Figure 3(a) shows BER performance for comparison of systems I and II but with measurement of only 1 band of 34-Gb/s data. We point out a few instructive observations from Fig. 3(a). When the channel condition is ‘good’ in system II, there is no obvious difference between the three algorithms. While the channel becomes ‘bad’ in system I, i.e., the channel matrix is badly skewed implying excessive MDL, there are significant disparity among the three methods. We conclude that although SIC provides the best performance, for the channel without severe fading, ZF is sufficient. Moreover, channel equalization using 4x6 to recover the non-ideal coupling (system I) is extremely effective, especially for SIC algorithm, the sensitivity difference between system I and system II are reduced from 2.8 dB to merely 0.2 dB after SIC equalization.

To further confirm the sensitivity improvement from larger receiver sets, we also apply ZF, MMSE, and SIC to 4x4 CO-OFDM signals of system I. Both 4x4 and 4x6 MIMO results are plotted in Fig. 3(b). The 4x4 ZF is the conventional 4x4 MIMO system which serves as the baseline of the comparison. By applying SIC to 4x4 signal, the sensitivity can be improved by 1.1 dB. Increasing number of receivers from 4 to 6 improves the sensitivity by 3.8 dB for SIC equalization. In practice, this performance improvement should be traded off against the increased cost of higher receiver complexity.

We further analyze the MDL in our system by applying singular value decomposition (SVD) to the channel matrix [14, 24]. We plot the power ratio between the maximum singular value and minimum singular value, which represents the MDL in our system. In other words, we define MDL = 20*log_{10}(Max(Eigen value)/ Min(Eigen value)). The ratios are plotted per each subcarrier for both 4x4 and 4x6 system under system I and system II. The MDL is shown in Fig. 4
. As illustrated in the figure, when comparing 4x4 and 4x6 configuration, the 4x6 scheme evidently reduces MDL (from 19.6 dB to 14.3 dB for system I, and from 19.1 dB to 12.2 dB for system II). Moreover, besides the reduction of the average MDL, the variance of MDL over all the frequencies in 4x6 system is significantly smaller as well. More specifically, the variance of MDL over all frequencies reduces from 6.8 to 2.2 for system I, and from 3.3 to 1.2 for system II. The above observations suggest that 4x6 configuration is an effective approach to combating MDL in a MDM system.

## 4. Conclusions

We have demonstrated three-channel equalization methods for FMF-based CO-OFDM systems by using more receivers than transmitters. The results show that the 4x6 MIMO system can improve the receiver sensitivity by 1.8, 2.9, and 4.9 dB for ZF, MMSE, and SIC, respectively. In the case of small MDL, ZF is sufficient without resorting to computationally intensive MMSE and SIC. However, in the case of large MDL, SIC is much more effective in alleviating the system from large sensitivity penalty.

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