## 3 Proposed Digital Canceler
### Multi-Layered Generalized HP-Based Adaptive Algorithm
Furthermore, their average energy follows the conditions \(\text{ISR}=E(\|w\|^{2}/\|h\|^{2})\) and \(\text{SNR}=E(\|s(n)\|^{2}/\sigma_{a}^{2})\). The received signal \(y(n)\) is subject to corruption from two sources: Gaussian white noise \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Symbol** \\ \hline \(p\)-th order basis functions & \(\phi_{p}(x[n];c_{p})\) \\ Coefficients of the \(p\)-th basis function & \(c_{p}\) \\ Nonlinear order of PA & \(p\) \\ \(p\)-th SI channel response & \(\mathbf{w}_{p}\) \\ \(p\)-th RT channel response & \(\mathbf{h}_{p}\) \\ Joint channel response & \(\mathbf{h}_{joint}\) \\ \(p\)-th even-order partial moments & \(\mathbf{M}_{p}\) \\ Number of layers & \(M\) \\ Input of \(m\)-th layers & \(y_{(m)}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Table of notations. Figure 4: Average NMSD learning curve with different impulsive noise probabilities. (**a**) \(P_{i}=0\); (**b**) \(P_{i}=0.01\). \(v_{a}\) and random impulsive noise \(v_{i}\). The occurrence probability of the impulsive noise is denoted as \(P_{i}\). Additionally, the local reference signal \(i(n)\) is influenced by Gaussian noise \(v_{b}\), where the condition \(\sigma_{a}^{2}=\sigma_{b}^{2}\) holds true. In the context of this simulation, we employ the m-MTLS algorithm with a total of number of \(M=2\) layers, which is mainly a compromise considering the additional computational cost brought by multi-layer filters. As \(M\) increases, the effectiveness of SIC initially improves and then tends to stabilize or even decline. This is due to the propagation of estimation errors across different layers. To assess the efficacy of the proposed multi-layered joint estimator, a comparative analysis is conducted against the minimum mean square error (MMSE) estimator and the single-layer MTLS estimator, focusing on their performance in estimating the RT channel. The physical meaning of the parameters involved in this simulation is summarized in Table 3.

Figure 5 illustrates the performance curves of several algorithms in terms of NMSD as the SNR varies across different noise environments, with \(\text{ISR}=20\) dB. Compared to the MMSE estimator, the proposed method exhibits higher robustness. The MMSE estimator shows a noticeable decline in performance at certain sampling points when affected by impulsive noise pollution. However, thanks to the M-estimate function, the m-MTLS estimator remains virtually unaffected. Furthermore, as the SNR increases, the NMSD of the m-MTLS estimator gradually decreases, whereas the MMSE estimator experiences limited gain due to convergence being disrupted by interference from the interest signal. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Length of SI channel & \(N_{1}\) & 4 \\ Length of RT channel & \(N_{2}\) & 10 \\ Layers & \(M\) & 2 \\ Input noise & \(v_{in}\) & \(v_{b}\) \\ Output noise & \(v_{out}\) & \(v_{a}+v_{i}\) \\ Gaussian noise & \(v_{a},v_{b}\) & \(\sigma_{a}^{2}=\sigma_{b}^{2}\) \\ Impulsive noise & \(v_{i}\) & \(\sigma_{i}^{2}=10\) \\ ISR & - & 20 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Table of parameters. Figure 5: The performance of different estimators under the impulse noise probability. (**a**) \(P_{i}=0.01\); (**b**) \(P_{i}=0.05\). ### Nonlinear SI Cancel

In this section, We use the [PERSON]'s PA model [27] to simulate the nonlinear distortion, and the SI signal received after ADC can be described as follows:

\[f(x[n])=\sum_{l=0}^{L-1}h[l]\frac{\gamma x[n-l]}{1+\beta|x[n-l]|^{2}}, \tag{32}\]

where \(\gamma=3\) and \(\beta=0.03\) in this simulation, and \(h[n]\) is the overall memory length, set to \(L=5\).