# Novel Equalization Techniques for Space Division Multiplexing Based on Stokes Space Update Rule
## 4 Frequency Domain Equalization  
Due to signal impairments such as DMGD, SDM systems may much have longer impulse responses, resulting in an increment of the number of taps of the equalizer [25]. In addition, multi-mode equalization systems with M modes need \(M^{2}\) times as many filters compared to single-mode equalization. Hence, the computational complexity is increased dramatically and efficient implementations of the MIMO equalization are required to overcome the increase of the number of taps and filters in multi-mode equalization. A well-known approach to decrease the computational complexity is the frequency domain (FD) equalization [26]. The FD equalization is rooted in the equivalence of time domain (TD) convolutions and multiplications in FD, allowing to effectively reduce the number of arithmetic operations of the equalizer [7]. As multiplications in FD correspond to cyclic but not linear convolutions in TD, different methods have been introduced to resolve this issue: most common ones are the overlap-save and the overlap-add techniques [27]. In this paper, we focus on the overlap-save, but similar results are expected from the overlap-add technique. Figure 2 illustrates the overlap-save technique, where two blocks are concatenated together. The light blue color is used to differentiate the last block from the initial one. For simplicity, same length is assumed for both blocks. The concatenated signals are Fourier-transformed to FD, where they are multiplied by the filter coefficients also expressed in FD. Once the multiplications are performed, the resultant signals are converted back to TD. The initial block is discarded, while the last block is the result of the desired linear convolution, and hence saved. The single- and multi-mode equalizations previously explained can be written in FD as:  
\[\overline{\mathbf{x}}_{v}^{(n)} =\sum_{j=1}^{M}\mathbf{X}_{j}^{(n)}*\mathbf{H}_{x_{v}j}^{(n)}+ \mathbf{Y}_{j}^{(n)}*\mathbf{H}_{x_{v}j}^{(n)}, \tag{18}\] \[\overline{\mathbf{Y}}_{v}^{(n)} =\sum_{j=1}^{M}\mathbf{X}_{j}^{(n)}*\mathbf{H}_{y_{v}j}^{(n)}+ \mathbf{Y}_{j}^{(n)}*\mathbf{H}_{y_{v}j}^{(n)},\]  
where \(\mathbf{X}_{v}^{(n)}\), \(\mathbf{Y}_{v}^{(n)}\) are the input signals of the received mode \(v\), \(\mathbf{H}_{x_{v}j}^{(n)}\), \(\mathbf{H}_{x_{v}j}^{(n)}\), \(\mathbf{H}_{y_{v}x_{j}}^{(n)}\), \(\mathbf{H}_{y_{v}y_{j}}^{(n)}\) are the filter coefficients and \(\overline{\mathbf{X}}_{v}^{(n)}\), \(\overline{\mathbf{Y}}_{v}^{(n)}\) the result of the equalization. The notation \({}^{\prime}{}_{s^{\prime}}\) is defined as the element wise multiplication of the vectors: \(A*B=[a_{1}b_{1},\dots,a_{n}b_{n}]\). As defined by the overlap-save technique, the input signal vectors are formed by two blocks concatenated and transformed into the Fourier domain:  
\[\mathbf{X}_{v}^{(n)}=FFT\Big{\{}\Big{[}\mathbf{x}_{v}^{(n-1)},\mathbf{x}_{v}^{ (n)}\Big{]}\Big{\}};\mathbf{Y}_{v}^{(n)}=FFT\Big{\{}\Big{[}\mathbf{y}_{v}^{(n- 1)},\mathbf{y}_{v}^{(n)}\Big{]}\Big{\}}. \tag{19}\]The '\([\mathbf{a},\mathbf{b}]\)' operation represents the concatenation of vectors \(\mathbf{a}\) and \(\mathbf{b}\); and '\(FFT[t]\)' is the Fourier transform operator. The response of the frequency domain filter is limited by a constraint \(\mathbf{K}\)[27], which limits the non-zero filters coefficients to a certain range in order to speed up the convergence:  
\[\mathbf{H}_{l_{d}m_{g}}^{(n+1)}=FFT\Big{\{}\mathbf{K}*\ \ IFFT\Big{\{}\mathbf{H}_{l_{d}m_{g}}^{(n )}\Big{\}}\Big{\}};l,m\in\{x,y\};d,g\in[1,M]. \tag{20}\]  
Finally the equalized signal in TD (\(\overline{\mathbf{x}}_{v}^{(n)}\), \(\overline{\mathbf{y}}_{v}^{(n)}\)) is obtained by IFFT of the FD signal as:  
\[