# A Kernel Least Mean Square Algorithm Based on Randomized Feature Networks
## 5 Simulations and Results
### Nonlinear Channel Equalization
#### 5.2.2 Abruptly Changed Channel Equalization  
With several practical nonlinear channel models, the channel impulse response of the linear part can be described by  
\[h(k)=\frac{1}{2}\left[1+cos(\frac{2*pi}{\Lambda})(k-2)\right]for\,k=1,2,3 \tag{41}\]  
where the parameter \(\Lambda\) determines the eigenvalue ratio (EVR) of the channel and denotes the difficulty of equalizing a channel [34; 35]. Here, \(\Lambda\) is chosen as 2.9 in the first 3000 samples and increases to 3.1 with abrupt change rate for the next 3000 samples and returns to 2.9 for the last 3000 samples. The corresponding normalized channel impulse responses in the z-transform domain are given by  
\[Channel(\Lambda=2.9) :H_{1}(z)=0.209+0.995z^{-1}+0.209z^{-2} \tag{42}\] \[Channel(\Lambda=3.1) :H_{2}(z)=0.260+0.930z^{-1}+0.260z^{-2}\]  
The output of the nonlinear channel is defined by \(r(n)=x(n)+0.2x^{2}(n)-0.1x^{3}(n)+v(n)\), where \(v(n)\) represents 20-dB SNR white Gaussian noise. A series of binary signal \(s(n)\in\{+1,-1\}\) is used as the desired channel input. In this simulation, the kernel size of the KLMS-RFN, QKLMS, and QKLMS-FB are set to \(\sigma=1\), and all step sizes are set to \(\mu=1\). The parameter \(\gamma\) of the QKLMS and QKLMS-FB are set to be the same with 0.7. The parameter \(\beta\) and fixed-budget size of the QKLMS-FB are chosen as 0.96 and 50, respectively. Two learning curves of the KLMS-RFN with dimensions of 50 and 100 are used to demonstrate the performance of the proposed algorithm. The learning and network size growth curves over 100 Monte Carlo runs are shown in Figures 12 and 13. As indicated in the nonstationary channel equalization with an abrupt change in condition, the following remarks can be made. For the performance of convergence, the proposed method can effectively track the non-stationary change of channel, as shown in Figure 12. On comparing two learning curves of the KLMS-RFN with different dimensions, increasing the number of dimensions did not accelerate the convergence speed of the KLMS-RFN algorithm. This result is reasonable because these features are independent of each other in the process of convergence and because the updating of their weight coefficients is controlled by the step size, which influences the convergence speed. An analysis of the details is indicated in Figure 12. For the first segment, the QKLMS-FB can achieve the same MSE of \(-17.13\) dB with the QKLMS but with lower computational complexity. After the channel deterioration in segment 2, the QKLMS-FB achieves convergence first and then suffers from performance degradation, with a final MSE of \(-15.12\) at i  
Figure 11: The run time curves of the QKLMS, QKLMS-FB, and KLMS-RFN algorithms in time-varying channel equalization. limitation of the preset budget (see the separation point at almost 4150 in Figure 13). In segment 3 (iteration from 8001 to 12,000), the channel model returns to the initial condition. Compared with the performance in segment 1, the steady-state MSE of the QKLMS-FB algorithm is \(-17.18\) dB, whereas the KLMS-RFN can rapidly achieve MSE values of \(-16.53\) dB (\(M\) = 50) and \(-19.15\) dB (\(M\) = 100) in the steady state. In segment 3, the QKLMS-FB achieves an MSE value of \(-16.83\) dB, which is worse than the MSE value of \(-17.18\) dB obtained in segment 1. This difference may be due to the limitation in the budget size and the defect in pruning criteria. When the channel model changes, the QKLMS-FB will update the dictionary by pruning some historical centers and adding new centers. During that process, some relatively unimportant new feature samples are retained, and some significant centers cannot be retrieved based on the significance metrics criteria. However, this does not affect KLMS-RFN because the explicit mapping-based algorithm does not rely on historical samples. In the steady state performance, the final MSEs of the QKLMS, QKLMS-FB, KLMS-RFN (\(M\) = 50), and KLMS-RFN (\(M\) = 100) are \(-18.28\) dB, \(-16.83\) dB, \(-17.52\) dB, and \(-20.55\) dB, respectively.