## 3 Identification Algorithm
 \tag{24}\]Since \(\mathbf{Y}^{T}[\hat{\mathbf{\Phi}}_{k,n},-\hat{\mathbf{\Phi}}_{k,n}]\mathbf{z}_{n}\) is a scalar, it follows that  
\[\mathbf{Y}^{T}[\hat{\mathbf{\Phi}}_{k,n},-\hat{\mathbf{\Phi}}_{k,n}]\mathbf{z}_{n}=(\mathbf{Y}^{ T}[\hat{\mathbf{\Phi}}_{k,n},-\hat{\mathbf{\Phi}}_{k,n}]\mathbf{z}_{n})^{T}=\mathbf{z}_{n}^ {T}[\hat{\mathbf{\Phi}}_{k,n},-\hat{\mathbf{\Phi}}_{k,n}]^{T}\mathbf{Y}=[\mathbf{Y}^{T}\hat{\mathbf{ \Phi}}_{k,n},-\mathbf{Y}^{T}\hat{\mathbf{\Phi}}_{k,n}]\mathbf{z}_{n}.\]  
Let  
\[b:=\hat{\mathbf{\Phi}}_{k,n}^{T}\mathbf{Y}\in\mathbb{R}^{n}. \tag{25}\]  
Equation (24) can be further written as  
\[\|\mathbf{Y}-\hat{\mathbf{\Phi}}_{k,n}\mathbf{\theta}_{n}\|_{2}^{2} =\mathbf{Y}^{T}\mathbf{Y}-2[b^{T},-b^{T}]\mathbf{z}_{n}+\mathbf{z}_{n}^{T} [\hat{\mathbf{\Phi}}_{k,n},-\hat{\mathbf{\Phi}}_{k,n}]^{T}[\hat{\mathbf{\Phi}}_{k,n},-\hat {\mathbf{\Phi}}_{k,n}]\mathbf{z}_{n} \tag{26}\] \[=\mathbf{Y}^{T}\mathbf{Y}-2[b^{T},-b^{T}]\mathbf{z}_{n}+\mathbf{z}_{n}^{T }\left[ \begin{array}{cc}\hat{\mathbf{\Phi}}_{k,n}^{T}\hat{\mathbf{\Phi}}_{k,n}&-\hat {\mathbf{\Phi}}_{k,n}^{T}\hat{\mathbf{\Phi}}_{k,n}\\ -\hat{\mathbf{\Phi}}_{k,n}^{T}\hat{\mathbf{\Phi}}_{k,n}&\hat{\mathbf{\Phi}}_{k,n}^{T}\hat{ \mathbf{\Phi}}_{k,n}\end{array} \right]\mathbf{z}_{n}\] \[=\mathbf{Y}^{T}\mathbf{Y}-2[b^{T},-b^{T}]\mathbf{z}_{n}+\mathbf{z}_{n}^{T }b\mathbf{z}_{n},\]  
where  
\[b:=\left[ \begin{array}{cc}\hat{\mathbf{\Phi}}_{k,n}^{T}\hat{\mathbf{\Phi}}_{k,n}&- \hat{\mathbf{\Phi}}_{k,n}^{T}\hat{\mathbf{\Phi}}_{k,n}\\ -\hat{\mathbf{\Phi}}_{k,n}^{T}\hat{\mathbf{\Phi}}_{k,n}&\hat{\mathbf{\Phi}}_{k,n}^{T}\hat{ \mathbf{\Phi}}_{k,n}\end{array} \right]\in\mathbb{R}^{2n\times 2n}. \tag{27}\]  
From Equations (22) and (26), we have  
\[\min_{\theta_{n}}\frac{1}{2}\|\mathbf{Y}-\mathbf{\Phi}_{k,n}\theta_{n}\|_ {2}^{2}+\lambda\|\theta_{n}\|_{1} \tag{28}\] \[=\min_{\mathbf{z}_{n}}\frac{1}{2}\mathbf{Y}^{T}\mathbf{Y}-[b^{T},-b^{T}] \mathbf{z}_{n}+\frac{1}{2}\mathbf{z}_{n}^{T}b\mathbf{z}_{n}+\lambda\mathbf{1}_ {2n}^{T}\mathbf{z}_{n}\] \[=\min_{\mathbf{z}_{n}}\frac{1}{2}\mathbf{Y}^{T}\mathbf{Y}+\mathbf{C}^{T} \mathbf{z}_{n}+\frac{1}{2}\mathbf{z}_{n}^{T}b\mathbf{z}_{n},\]  
where  
\[\mathbf{C}:=\lambda\mathbf{1}_{2n}+[-b^{T},b^{T}]^{T}=\sigma\sqrt{2 \log(n)}\mathbf{1}_{2n}+[-b^{T},b^{T}]^{T}\in\mathbb{R}^{2n}. \tag{29}\]  
Since \(\frac{1}{2}\mathbf{Y}^{T}\mathbf{Y}\) is a constant, Equation (28) can be constructed in a standard QP framework,  
\[\min_{\mathbf{z}_{n}}\mathbf{C}^{T}\mathbf{z}_{n}+\frac{1}{2}\mathbf{ z}_{n}^{T}b\mathbf{z}_{n},\ \ \text{s.t.}\ \ z_{nj}\geq 0,\ \ j=1,2,\cdots,n. \tag{30}\]  
For the inequality constrained QP problem in Equation (30), the common solution is the active set method [33]. While for the sake of simplicity, the QP problem can be directly solved by calling the relevant function from the standard scientific software toolbox. For example, the MATLAB toolbox provides a function \"quadprog\". Then, the parameter vector \(\hat{\mathbf{\theta}}_{n}\) can be obtained from the optimum solution \(\hat{\mathbf{z}}_{n}\),  
\[\hat{\mathbf{\theta}}_{n}=\hat{\mathbf{z}}_{n}(1:n)-\hat{\mathbf{z}}_{ n}(n+1:2n), \tag{31}\]  
where \(\hat{\mathbf{z}}_{n}(1:n)\) represents a vector formed by the first \(n\) elements of \(\hat{\mathbf{z}}_{n}\), and \(\hat{\mathbf{z}}_{n}(n+1:2n)\) a vector formed by the last \(n\) elements of \(\hat{\mathbf{z}}_{n}\). Considering that the system in Equation (8) is contaminated with noise, the parameter estimation error can be large. To further reduce the estimation error, a small threshold TH\(=\epsilon\) can be set to filter the elements close to zero in \(\hat{\mathbf{\theta}}_{n}\). Let \(\hat{\mathbf{\theta}}_{nj}=0\) if \(\hat{\mathbf{\theta}}_{nj}<\epsilon\) and denote the filtered parameter vector as \(\hat{\mathbf{\theta}}_{n,e}\).