## 3 Traditional KRLS Algorithm
### Traditional KRLS Algorithm
\right], \tag{23}\]

and \(e(i)=y_{i}-h(i)^{T}\alpha(i-1)\) is the prediction error in the \(ith\) time slot. ### Case II: The Size of \(D(i)\) Is Unchanged

In the case when the size of the dictionary did not change, the information of the discarded sample \(u_{j^{*}}\) in \(Q(i-1)\) needed to be deleted. The information of \(u_{j^{*}}\) lay in the \(j^{*}th\) column and \(j^{*}th\) row of \(Q(i-1)\). In order to not influence the update of the matrix, we moved the \(j^{*}th\) column and \(j^{*}th\) row of \(Q(i-1)\) into the first row and the first column, and obtained \(\tilde{Q}(i-1)\). Correspondingly, we applied the same transformation to \(K_{i-1}\) and obtained \(\tilde{K}(i-1)\). For \(Q(i-1)=[\lambda B(i-1)+B(i-1)K_{i-1}]^{-1}\); after the movement, the \(jth\) column that meets \(j<j^{*}\) should be multiplied by \(\beta\). Suppose that the matrix after removing the first column and first row of \(\tilde{Q}(i-1)\) and \(\tilde{K}(i-1)\) is \(\tilde{Q}(i-1)\) and \(\tilde{K}(i-1)\), respectively. We can obtain its inversion \(\tilde{Q}(i-1)^{-1}\) according to Appendix B,

\[\tilde{Q}(i-1)=\left[ \begin{array}{cc}e&\mathrm{f}^{T}\\ l&G\end{array} \right] \tag{24}\]

\[\tilde{Q}(i-1)^{-1}=G-l\mathrm{f}^{T}/e. \tag{25}\]New matrix \(Q(i)\) can be formulated into a partitioned matrix:

\[Q(i)=\left[ \begin{array}{cc}\beta\tilde{Q}(i-1)^{-1}&\beta B(i-1)K_{i}\\ k(i)^{T}&\kappa(u_{i},u_{i})+\lambda\end{array} \right]^{-1}=\left[ \begin{array}[ ]{cc}A&b\\ p^{T}&\kappa(u_{i},u_{i})+\lambda\end{array} \right]^{-1}. \tag{26}\]

Then, \(Q(i)\) can be obtained using the partitioned matrix:

\[Q(i)=\left[\lambda\beta^{i}I(i)+B(i)K_{i}\right]^{-1}=\left[ \begin{array}{ cc}A^{-1}(I+bp^{T}A^{-1}g)&-A^{-1}bg\\ -p^{T}A^{-1}g&g\end{array} \right], \tag{27}\]

where \(A^{-1}=\beta^{-1}\tilde{K}(i-1)^{-1}B(i-1)^{-1}\), \(g=(\kappa(u_{i},u_{i})+\lambda-p^{T}A^{-1}b)^{-1}\). Then, the weight coefficient is updated:

\[\alpha(i)=Q(i)\tilde{y}_{i}=Q(i)B(i)y_{i}, \tag{28}\]

where \(y_{i}\) is composed of the output value of the samples \(y_{i}=[y_{1},y_{2},\ldots,y_{i}]^{T}\). Lastly, we can obtain the prediction value for the next time slot as \(\hat{y}_{i+1}=\alpha(i)k(i)^{T}\). ## 5 Performance Evaluation

Based on the analysis above, we present algorithmic steps as Algorithm 2 and in this section we show the simulation results of our proposed SSW-KRLS algorithmcompare its performance to that of the ALD-KRLS algorithm as a baseline. The basic simulation parameters are listed in Table 2. We adopted a 3D urban macro scenario, and considered a typical setting of 3 kHz with 30 kHz of subcarrier spacing. We considered 20 MHz of bandwidth that contained 51 resource partitions. Our adopted channel model was a CDL-A channel model, and the DL precoder was RZF. ```
1:Initialize \(Q(1)=(\lambda+\kappa(u_{1},u_{1}))^{-1}\), \(B(1)=[1]\), \(\alpha(1)=Q(1)y_{1}\), \(D(1)=[u_{1}]\). 2:Step 1: Iterate for \(i>1\): judging the number of samples in \(D(i)\), which is \(L\). If \(L<M\), perform Step 2; otherwise, perform Step 3. 3:Step 2: For each  
Figure 5 shows the normalized mean squared error (NMSE) performance of different algorithms. We show the performance of the ALD-KRLS and SSW-KRLS algorithms with the 60 and 120 km/h velocity levels for all UEs. When the UE speed was 60 km/h, the prediction algorithms was more accurate than with a UE speed of 120 km/h. Regarding the performance of the SSW-KRLS algorithm, we set different sample budgets: \(M=30,90,150\). The algorithm performed better with a higher sample budget. The SSW-KRLS algorithm with sample budget \(M=30\) performed better than the ALD-KRLS algorithm with \(v=0.001\), and the SSW-KRLS algorithm with sample budget \(M=90\) greatly outperformed the ALD-KRLS algorithm. However, the performance was insignificantly improved when changing the sample budget from 90 to 150. Figure 6 shows the kernel dictionary size with 400 iterations. The kernel dictionary size for all algorithms first grew and then slowed down; the size for SSW-KRLS remained unchanged.