# Recent Progress on Novel DSP Techniques for Mode Division Multiplexing Systems: A Review
## 3 RLS MIMO Algorithms and Its Application on Equalization of TV Effects  
The procedure of an overlap-based FD-RLS algorithm is shown schematically in Figure 4 for a MIMO MDM system with \(m\) degrees of freedom. Here, \(y_{i}(n)\) and \(x_{i}(n)\) represent the time-domain input signal vectors and output signal vectors of mode \(i\) with \(1\leq i\leq m\). After parallelization from serial sequences, two consecutive blocks of TD input vectors are concatenated into FFT with an overlap rate of 0.5 to obtain the FD input signal vector \(\mathbf{Y}(k)\) and the FD output signal vector \(\mathbf{X}(k)\) would be equalized in frequency domain through the overlap-save method. The posses of this algorithm is similar to that of the FD-LMS algorithm shown in Figure 1[37] except the correlation matrix updating process contained in the dotted box. The RLS algorithm is intended to minimize its exponentially weighted cost function with iterations in frequency domain, which utilizes the assumption \begin{table}
\begin{tabular}{c c c} \hline \hline
**Operation** & \multicolumn{2}{c}{**Number of Complex Multiplications**} \\ \hline  & **LMS** & **RLS** \\ \hline Compute output & \(4m\) & \(4m\) \\ Update filter coef. & \(4m\) & \(4m\) \\ FFT/IFFT & \((4+4m)\log_{2}N_{f}\) & \((4+4m)\log_{2}N_{f}\) \\ Update Kalman vector & - & \(8+4m\) \\ Update correlation matrix & - & \(12m\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Complexity comparison between the FD-RLS and FD-LMS. Figure 3: Complexity ratio of FD-RLS and FD-LMS. of mutual independence between different frequency components. The iterative matrix updating can be formulated mathematically as Equations (5)-(7):  
\[\mathbf{K}(k)\leftarrow\frac{\lambda^{-1}\mathbf{R}(k)\mathbf{Y}(k)}{1+\lambda^{ -1}\mathbf{Y}(k)^{H}\mathbf{R}(k)\mathbf{Y}(k)} \tag{5}\]  
\[\mathbf{W}(k)\leftarrow\mathbf{W}(k)+\mathbf{K}(k)\mathbf{E}(k)^{H} \tag{6}\]  
\[\mathbf{R}(k)\leftarrow\lambda^{-1}\mathbf{R}(k)-\lambda^{-1}\mathbf{K}(k) \mathbf{Y}(k)^{H}\mathbf{R}(k) \tag{7}\]  
In the Equations (5)- (7), \(\mathbf{R}(k)\) is an \(m\times m\) inverse matrix of the correlation of input signals at the \(k\)-th frequency component, which can be initialized as a unit matrix multiplied by a constant inversely proportional to the signal to noise ratio (SNR) of the input data; \(\lambda\) is a forgetting factor satisfying \(0\ll\lambda<1\) and the superscript H denotes the conjugate transpose operator. Kalman vectors, \(\mathbf{K}\), which is the key parameter to update the filter coefficients, always is computed based on the frequency domain signal input vector \(\mathbf{Y}(k)\) and the inverse correlation matrix \(\mathbf{R}(k)\) of the previous block. Besides, the even and odd tributaries should be updated separately because they use different input vectors and naturally different correlation matrix. Generally, RLS could exploit more information from the input signals by the Kalman vector compared to typical LMS algorithm based on stochastic gradient descent [43]. It is noteworthy that variable \(\lambda\) could be time-varying to further improve the performance of RLS. Since a fixed \(\lambda\) leads to a trade-off between a faster convergence speed and a lower misadjustment error, this compromise could be balanced well with a variable \(\lambda\), which algorithm is referred to VFF-RLS. Lots of work around VFF-RLS has been done for different signal processing applications [44; 45; 46]. However, calculating the specific value of \(\lambda\) is always operated by division or extraction of a root, which leads to rather higher computational complexity compared that with fixed forgetting factor. To avoid higher complexity while still holding the advantages of VFF, we proposed a FD-RLS schemes with an exponential VFF in Equation (8) [47]:  
\[\lambda(n)=\lambda_{\text{max}}-\gamma\times 10^{-n\tau} \tag{8}\]  
where \(n\) represents the FD-block index. \(\lambda\) decreases with the increasing of \(n\).