# Recent Progress on Novel DSP Techniques for Mode Division Multiplexing Systems: A Review
## 2 Linear DSP MIMO Technique for MDM Systems  
An optical MIMO system with \(N_{T}\) transmitting channels and \(N_{R}\) receiving channels can be linearly modeled as Equation (1). \[\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n} \tag{1}\]  
where \(\mathbf{x}\), \(\mathbf{y}\), \(\mathbf{n}\), and \(\mathbf{H}\) denote the \(N_{T}\ \times\ 1\) transmitted symbol vector, the \(N_{R}\ \times\ 1\) received symbol vector, the \(N_{R}\ \times\ 1\) zero-mean complex noise vector, and the \(N_{R}\ \times\ N_{T}\) channel transmission matrix, respectively. To reconstruct the signal \(\mathbf{\hat{x}}\) from the received symbols \(\mathbf{y}\), MIMO processing must be used to achieve the inverse transmission matrix which can be expressed as Equation (2):  
\[\mathbf{\hat{x}}=\mathbf{W}\mathbf{y} \tag{2}\]  
where \(\mathbf{W}\) is the weight matrix of MIMO equalizer. Zero-forcing and minimum mean square error method are usually utilized to achieve \(\mathbf{W}\) if transmission matrix \(\mathbf{H}\) is already known [33; 34] Generally, there is always no priori information of MIMO channel, so adaptive filters are widely used to solve out the weight matrix in iteration process. So far, two main linear equalization schemes have been investigated in detail: adaptive TDE [35; 36] and FDE [37; 38]. Taking LMS algorithm as an example, LMS-FDE has significant advantage on the computational complexity compared with LMS-TDE. The computational complexity of MIMO equalizers increases with the number of finite impulse response (FIR) filters length. The transmitted signal suffers a large amount of DMGD constantly accumulating with the transmission distance, which finally leads to a large time spread or delay. Thus, a longer FIR filter will be required, which means a corresponding higher computational complexity. In order to relax the complexity of the equalizers, FDE is well adopted by utilizing fast Fourier transform (FFT) to achieve complex convolutions in a block-by-block way [37] as shown in Figure 1[10; 39]. Figure 1: Block diagram of the proposed FDE for one mode channel. The computational complexity for the equalization algorithm could be measured by the number of complex multiplications per mode per symbol. The complexity of TDE increases linearly with the filter length, while the FDE scales logarithmically with the taps number. The complexity except for carrier recovery for these two types of algorithms can be expressed as Equations (3) and (4) [40]:  
\[C_{TDE}=3m\Delta\tau LR_{s} \tag{3}\]  
\[C_{FDE}=(4+4m)\log_{2}(2\Delta\tau LR_{s})+8m \tag{4}\]  
where \(\Delta\tau\) is the fiber DMGD, \(L\) is the fiber link length, \(R_{s}\) is symbol rate, \(m\) is the number of channels for transmission. The computational complexities as a function of number of taps are shown in Figure 2, in which we assumed \(m\) equals 10. We could see that FDE reduce the complexity dramatically compared with TDE. For long MDM transmission link, the advantage of FDE will be more significant because the required taps number will be higher. As the complexity of LMS-TDE and LMS-FDE has been compared above, we could also discuss complexity of the RLS algorithm by the way to prove that the complexity of RLS-FDE is not so sophisticated as it seems to be for per symbol per mode. RLS algorithm is also a main receiver-side equalization candidate for SDM systems, which utilizes iterative updating to minimize the exponentially weighted cost function, whose convergence speed is nearly independent on the input vectors. In the simulation of [9], the RLS algorithm was demonstrated through a cyclic prefix (CP) for each data block in MDM systems. The CP could reduce the complexity per symbol at the cost of throughput spectrum efficiency. We also previously proposed an overlap-save based FD-RLS algorithm for MDM systems with a fixed forgetting factor. It proved that RLS has the ability to achieve higher convergence speed, faster adaptation, better performance at the cost of moderately higher computational complexity compared with LMS algorithm [41].