## 4 Numerical Simulation Results  
In this part, we test the filtering performance of the proposed algorithms in system identification. In this work, all simulation results have been averaged over 50 independent trials. The signal-to-noise ratio (SNR) was set to 30 dB, and background noise has been added to \(d(n)\). The impulsive noise \(v(n)\) is modeled by \(b(n)G(n)\), where \(b(n)\) is a Bernoulli process with a probability of success of \(P[b(n)=1]=p_{r}\), and \(G(n)\) denotes zero-mean Gaussian noise with the variance \(\sigma_{G}^{2}=1000\sigma_{d}^{2}\), where \(\sigma_{d}^{2}\) is the average power of the desired output \(d(n)=\mathbf{u}^{T}(n)\mathbf{w}^{o}\). We set \(\xi_{1}=\xi_{2}=0.001\), \(K=2\), \(p_{r}=0.1\), \(\mu_{1}=0.01\), and \(\mu_{2}=0.0001\) in all simulations. Following normalized MSD,  
\[\text{NMSD}=10\log_{10}\left(\frac{\|\mathbf{w}^{o}-\mathbf{w}(n)\|_{2}}{\| \mathbf{w}^{o}\|_{2}}\right) \tag{42}\]  
was used to measure the performance of the algorithms. The color input datum were generated by filtering the white and zero-mean Gaussian signals with a unit power through a first-order system, as  
\[H_{1}=\frac{1}{1-0.7z^{-1}}. \tag{43}\]The unknown weight \(\mathbf{w}^{o}\) was randomly generated with \(M=32\) taps, and turned into \(-\mathbf{w}^{o}\) at the middle of the iterations. First of all, we investigated the influence of the parameter \(C\) on the performance of CSS-APSA. We chose \(C\in\{1.1,2,3,4,5,6,7\}\), \(\mu_{s}=1\). Figure 2 shows the corresponding NMSD curves. From this figure, we know that CSS-APSA with \(C=5\) and \(C=6\) can realize better filtering performance than that of other \(C\) values. In other words, proper \(C\) values can enhance the filtering performance of CSS-APSA. This result demonstrates the effectiveness of constraint on \(s(n)\in[-0.3365,0.3365]\). **Remark 1**. : _For CSS-type algorithms, when \(C\) is in a certain interval (such as \(C\in(1,6]\)), the performance of CSS-APSA can be improved by increasing the value of \(C\). In addition, we can have following observations for \(C\in(1,6]\):_  
* If the value of \(C\) is relatively small (such as \(C\leq 2\)), the change rate of \(\lambda(n)\) from 0 to 1 will be reduced. In this case, the combined algorithm has slower convergence speed. Figures 1 and 2 illustrate such behavior. * If the value of \(C\) is too large (such as \(C=7\)), the change rate of \(\lambda(n)\) from 0 to 1 is increased in the process of iteration. In this case, \(\lambda(n)\) tends to 1 too quickly, resulting in \(\mathbf{w}(n)\) dominated by \(\mathbf{w}_{2}(n)\), which is related to the lower steady-state error. Hence, the combined algorithm will still realize a slower convergence speed. Secondly, we investigate the influence of \(\mu_{s}\) on CSS-APSA with \(C\in\{5,6\}\), since the \(\mu_{s}\) as a step-size factor affects the update rate of \(s(n)\) and indirectly influences the change rate of \(\lambda(n)\). In this part, we set \(\mu_{s}\in\{0.1,1,10\}\). Figure 3 plots the corresponding NMSD curves. From this figure, we know that \(\mu_{s}=1\) is a better choice than other values. Hence, except where mentioned otherwise, we set \(\mu_{s}=1\) in the following experiments. Furthermore, we compare the filtering performance between CSS-APSA and SCSS-APSA. Figure 4 plots the NMSD curves of the corresponding algorithms. Such results reveal that:  
* CSS-APSA with \(C=6\) realizes faster convergence than \(C=5\) after an abrupt change;
* SCSS-APSA with \(C=6\) outperforms CSS-APSA in terms of convergence rate in the whole filtering process. Moreover, Table 2 compares the execution times of CSS-APSA, SCSS-APSA, MCSS-APSA, and SMCSS-APSA at each iteration (computation platform: Intel(R) Core(TM) i7-4770K, CPU @ 3.50 GHz). One can clearly see that SCSS-APSA is more computationally efficient than CSS-APSA. Figure 2: The NMSD curves of APSA and CSS-APSA with different \(C\) values. Thirdly, we investigate the effect of parameter \(\beta\) on the performance of MCSS-APSA. We set \(C=6\) and \(\beta\in\{1,5,10,20\}\).