# 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
With regard to algorithm schemes, the procedure of ED-VFF-RLS is similar with the above conventional FD-RLS algorithm except for the forgetting factor updating process. Figure 6 shows schematically the procedure of the ED-VFF-RLS algorithm of an \(m\times m\) MDM system. The forgetting factor \(\lambda\) is decided by the estimation of the power of noise signal and priori error signal. Specifically, \(\lambda\) would be recalculated at each frequency-domain block as Equation (9) [44]:  
\[\lambda(n)= \begin{cases}\lambda_{\text{max}}&\text{if }\partial_{\text{e}}(n) \leq\gamma\partial_{v}(n)\\ \min\Big{\{}\frac{\partial_{\text{e}}(n)\partial_{v}(n)}{\xi+|\partial_{\text {e}}(n)-\partial_{v}(n)|},\lambda_{\text{max}}\Big{\}}&\text{if }\partial_{\text{e}}(n)\geq\gamma\partial_{v}(n)\end{cases}  \tag{9}\]  
where \(\lambda_{\text{max}}\cong 1\), \(1\leq\gamma\leq 2\), \(\xi\) is a small positive number preventing division by zero and \(\mathbf{q}(n)=\mathbf{Y}_{\text{ev,od}}^{\text{H}}(n)\mathbf{P}_{\text{ev,od} }^{\text{H}}(n)\mathbf{Y}_{\text{ev,od}}(n)\). In addition, \(\sigma_{q}^{2}(n)=E\big{[}\mathbf{q}^{2}(n)\big{]}\), while \(\sigma_{\text{e}}^{2}(n)\) and \(\sigma_{v}^{2}(n)\) is the mean power of the priori error signal and the noise signal, respectively. For convenience of calculation, an exponential window [50] was adopted to computed the three mean powers in Equation (9) with \(\hat{\sigma}_{\text{e}}^{2}(n)=\alpha\hat{\sigma}_{\text{e}}^{2}(n-1)+(1- \alpha)E\big{[}\mathbf{e}^{2}(n)\big{]}\), \(\hat{\sigma}_{q}^{2}(n)=\alpha\hat{\sigma}_{q}^{2}(n-1)+(1-\alpha)E\big{[} \mathbf{q}^{2}(n)\big{]}\) and \(\hat{\sigma}_{v}^{2}(n)=\hat{\beta}\hat{\sigma}_{v}^{2}(n-1)+(1-\beta)E\big{[} \mathbf{e}^{2}(n)\big{]}\), where the weight factors \(\alpha\) and \(\beta\) meet the condition of \(\alpha<\beta<1\). We use two sets of parameters to represent the different trade-off between misalignment error and tracking capability, respectively. And Figure 7a,b have compared the tracking capability of ED-VFF-RLS with that of conventional LMS under the two sets of parameters. First of all, Figure 7a shows that the proposed ED-VFF-RLS has the nearly same performance as LMS under the first set of parameters when the angular frequency of rotation matrix is low. When angular frequency is further increased to over 100 krad/s, the Q\({}^{2}\) factors of the ED-VFF-RLS algorithm have more than 2.2-dB improvement, in comparison with LMS algorithm adopting a smaller unnormalized step-size. It can be observed that the consistency of FD algorithms agrees well with TD algorithms. While in Figure 7b, the ED-VFF-RLS algorithm with smaller \(\alpha\) factor displays nearly identical tracking capability as the conventional LMS algorithm using a larger unnormalized step-size, but the former's Q\({}^{2}\) factor is improved around 0.5-dB as compared to LMS. It is worth noting that at the cost of a minor Q\({}^{2}\) penalty in static environment, the performance of equalizers could be significantly promoted in fast time-varying situations. Figure 6: Block diagram of the FD-VFF-RLS algorithm.