## 3 Proposed Digital Canceler
### Robust \(m\)-MTLS Adaptive Algorithm  
We consider a linear model where both the independent and dependent variables are subject to measurement errors, which can be illustrated by  
\[(y_{n}+v_{n})=(\mathbf{x}_{n}+\mathbf{u}_{n})^{T}\mathbf{w}, \tag{9}\]  
where \(\mathbf{w}\in\mathbb{R}^{L\times 1}\) denotes the system vector to be estimated, with \(\mathbf{x}_{n}\in\mathbb{R}^{L\times 1}\) as the input vector and \(y_{n}\in\mathbb{R}\) as the output signal at time \(n\). The noise vectors \(\mathbf{u}_{n}\), distributed as \(N(\mathbf{0},\sigma_{i}^{2}\mathbf{1})\in\mathbb{R}^{L\times 1}\), represent the input noise, and \(v_{n}\), distributed as \(N(0,\sigma_{o}^{2})\in\mathbb{R}\), represent the output noise, where \(\sigma_{i}^{2}\) and \(\sigma_{o}^{2}\) are the variances in the input and output noise, respectively. Here, \(\mathbf{0}\) indicates the zero vector and \(\mathbf{1}\) the identity matrix. Total least squares (TLS) considers errors in all variables, making it particularly useful for more accurate modeling and estimation when errors are present in both predictors and outcomes [25]. Its cost function can be formulated as  
\[\min_{\mathbf{w}}J(\mathbf{w})=\frac{1}{N}\sum_{n=1}^{N}\frac{\left(\hat{y}_{n}-\mathbf{w} ^{T}\mathbf{\hat{x}}_{n}\right)^{2}}{\|\mathbf{w}\|^{2}+\gamma}, \tag{10}\]  
where \(\gamma\overset{\text{def}}{=}\sigma_{o}^{2}/\sigma_{i}^{2}\) is a parameter to normalize the noise variances, \(\hat{y}_{n}=y_{n}+v_{n}\) and \(\mathbf{\hat{x}}_{n}=\mathbf{x}_{n}+\mathbf{u}_{n}\). Similar to other adaptive algorithms, the iterative formula for TLS based on stochastic gradient descent can be expressed as  
\[ \begin{split}\mathbf{\hat{w}}_{n+1}&=\mathbf{\hat{w}}_{n}- \mu\mathbf{\hat{g}}(\mathbf{\hat{w}}_{n})\\ &=\mathbf{\hat{w}}_{n}+\mu\mathbf{a}_{n}(\mathbf{\hat{x}}_{n}+\mathbf{a}_{n}\mathbf{ \hat{w}}_{n}),\end{split}  \tag{11}\]  
where \(\mu\) is the step-size parameter, and \(\mathbf{\hat{g}}\) is the instantaneous gradient of \(J(\mathbf{w})\), which can be represented as  
\[ \begin{split}\mathbf{\hat{g}}(\mathbf{w})&=\alpha_{n}(\mathbf{ \hat{x}}_{n}+\mathbf{a}_{n}\mathbf{w}_{n}),\\ \alpha_{n}&\overset{\text{def}}{=}\frac{\hat{y}_{n}- \mathbf{\hat{x}}_{n}^{T}\mathbf{w}_{n}}{\|\mathbf{w}_{n}\|^{2}+\gamma}.\end{split}  \tag{12}\]  
To enhance the robustness of the algorithm when the received signal is subjected to impulsive noise, an M-estimate function can be utilized to improve TLS. This improved algorithm, referred to as the M-estimate total least mean square (MTLS) adaptive algorithm, has its cost function defined as  
\[\min_{\mathbf{w}}J(\mathbf{w})=E\bigg{[}\frac{\rho(e_{n}^{2})}{\|\mathbf{w}\|^{2}+\gamma} \bigg{]}, \tag{13}\]where \(\rho(\cdot)\) is the M-estimate function, which is a real-valued even function, given by  
\[\rho(e_{n})= \begin{cases}e_{n}^{2}/2,&|e_{n}|<\xi\\ \xi^{2}/2,&|e_{n}|\geq\xi,\end{cases}  \tag{14}\]  
where \(\xi=c_{1}\sigma_{e}\) represents the threshold parameter, with \(c_{1}\) set to 2.576, and \(\sigma_{e}\) can be calculated by  
\[\hat{\sigma}_{e_{n}}^{2}=\lambda_{e}\hat{\sigma}_{e_{n-1}}^{2}+c_{2}(1-\lambda _{e})\text{med}(A_{e_{n}}), \tag{15}\]  
where \(A_{e}[n]=\left[e^{2}[n],e^{2}[n-1],\ldots,e^{2}[n-N_{w}+1]\right]\), \(\text{med}(\cdot)\) denotes the median operation, and the parameter \(\lambda_{e}\), indicative of the weighting factor, is typically selected within the range of 0.98 to 0.99. The constant \(c_{2}=1.483(1+5/(N_{w}-1))\), and \(N_{w}\) represents the length of the window over which the estimation is performed. The iterative formula (11) can be rewritten as  
\[\hat{\mathbf{g}}(\mathbf{w})= \begin{cases}-\frac{\left(\|\mathbf{w}\|^{2}+\gamma\right)e_ {n}\mathbf{x}_{n}+e_{n}^{2}\mathbf{w}}{\left(\|\mathbf{w}\|^{2}+\gamma\right)^{2}},&|e_{n} |<\xi\\ 0,&|e_{n}|\geq\xi,\end{cases}