# Advanced DSP for Coherent Optical Fiber Communication
## 4 SDM and Frequency Domain Equalization (FDE)  
SDM can increase the system capacity and exceed the nonlinear Shannon limit per fiber. Many efforts have been made in this area and important results have been presented. The maximum number of multiplexing has reached 72 and the capacity has reached beyond 10 Pbit/s. The authors of [43] reported a 10 \(\times\) 112 Gb/s PDM-QPSK transmission in FMF over 5032 km as quasi-signal mode (QSM) application, after which the reach is extended to 7326 km [44]. [PERSON] et al. deduced the theoretical analysis result about the minimum number of spans required for a 3000 km QSM system [45]. More demonstrations such as transmission distance reach 14,350 km by using 12 core fiber were also reported [46]. [PERSON] et al. demonstrated a WDM and MDM transmission with 146 channels and 3 linear polarized (LP) modes over 500 km. The experimental setup is shown in Figure 13[47]. Another experiment by [PERSON] et al. demonstrated a \(12\times 12\) MIMO transmission over 130 km FMF with 8 wavelengths [48]. [PERSON] et al. demonstrated a \(30\times 30\) MIMO transmission over 15 spatial modes [49] and [PERSON] et al. demonstrated a \(72\times 72\) MIMO DSP experimental result which is the current record of multiplexing numbers [50]. The SDM system capacity reached 1 Pbit/s in 2013 [51], and exceeded 2 Pbit/s two years later [52]. In 2017, an over 10 Pbit/s SDM transmission was demonstrated by [PERSON] et al. which made a new milestone to fiber transmission field [53]. The mode coupling and core coupling cause serious signal distortions so the MIMO DSP also needed to be applied to recover the information on each mode or fiber core. In the SDM system, the inverse transmission matrix is extended from \(2\times 2\) to \(D\times D\) dimension where \(D\) is the overall coupling degree of freedom in the SDM system. The computational complexity of FIR filters increases with the number of delay taps. The transmitted signal suffers a large amount of DMGD and will continuously accumulate with the transmission distance. Large DMGD requires the digital filter with a greater number of taps, so the computational complexity will also increase dramatically. To reduce the complexity of the equalization algorithm, FDE is applied by employing discrete Fourier transform (DFT) and complex multiplications in a block-by-block way instead of convolutions in the time-domain filters [54; 55]. Theoretically, the change rate of channel parameters should be much slower than the packet rate to ensure the equalization performance. Optical fiber communication systems are generally slow-varying so this condition can be easily satisfied. The evaluation criterion for the equalization algorithm is the number of complex multiplications per symbol per mode. The computational cost of TDE scales linearly with the number of taps, while the FDE scales logarithmically with the filter length. The complexity excluding carrier recovery for these two types of algorithms can be expressed as:  
\[C_{TDE}=3m\Delta\tau LR_{s} \tag{21}\]  
Figure 13: Experiment setup for a few-mode fiber (FMF) loop experiment. \[C_{FDE}=(4+4m)\log_{2}(2\Delta\tau LR_{s})+8m \tag{22}\]  
where \(\Delta\tau\) is the DMGD of the fiber, \(L\) is the link distance, \(R_{s}\) is the symbol rate, \(m\) can be considered as the number of degree of freedoms utilized for transmission. The algorithmic complexities of both FDE and TDE as a function of filter length are shown in Figure 14. More detailed explanations can be found in Ref. [56]. As mentioned above, the computational complexity is determined by the degree of freedom and the differential mode group delay (DMGD). It is necessary to study and reduce the computational complexity of SDM applications. Several works have been reported to address this issue [57; 58; 59; 60]. Compared with the TDE method, the FDE method can significantly reduce the computational complexity while maintaining the same performance. Some experimental results have been presented for SDM transmission, achieved by FDE.