## 3 The Gradient-Based Iterative Algorithm  
This section derives the gradient-based iterative identification algorithm for the Wiener nonlinear OEMA model. Define the stacked output vector \(Y(N)\), the stacked information vector \(\Psi(N)\) and the white noise vector \(V(N)\) as  
\[Y(N) := [y(N),y(N-1),\cdots,y(1)]^{\mathrm{ T}}\in\mathbb{R}^{N} \tag{9}\] \[\Psi(N) := [\phi(N),\phi(N-1),\cdots,\phi(1)]^{\mathrm{ T}}\in\mathbb{R}^{N\times n_{0}}\]  
Using the negative gradient search for the optimization problem in Equation (13), we obtain the iterative algorithm of computing \(\hat{\vartheta}_{k}\) as follows:  
\[\hat{\vartheta}_{k} = \hat{\vartheta}_{k-1}-\frac{1}{r_{k}}\mathrm{grad}[J(\hat{ \vartheta}_{k-1})] \tag{14}\] \[= \hat{\vartheta}_{k-1}+\frac{1}{r_{k}}\Psi^{\intercal}(N)[Y(N)- \Psi(N)\hat{\vartheta}_{k-1}]\] \[r_{k} = r_{k-1}+\|\Psi^{\intercal}(N)\Psi(N)\|^{2} \tag{15}\]  
However, \(\Psi(N)\) in Equations (14) and (15) containing unknown inner variable \(m(t)\) and the unmeasurable noise term \(v(t)\) lead to a difficulty that the iterative solution \(\hat{\vartheta}_{k}\) of \(\vartheta\) is impossible to be computed. In order to solve this difficulty, the approach here is based on the auxiliary model idea. Let \(\hat{m}_{k}(t)\) and \(\hat{v}_{k}(t)\) be the estimate of \(m(t)\) and \(v(t)\) at iteration \(k\), respectively, and define  
\[\hat{\phi}_{k}(t) := \begin{bmatrix}\hat{\phi}_{s,k}(t)\\ \hat{\phi}_{\gamma,k}(t)\\ \hat{\phi}_{n,k}(t)\end{bmatrix}  \tag{16}\]  
where  
\[\hat{\boldsymbol{\phi}}_{s,k}(t) = [-\hat{m}_{k-1}(t-1),-\hat{m}_{k-1}(t-2),\cdots,-\hat{m}_{k-1}(t- n_{a}),u(t-1),u(t-2),\cdots,u(t-n_{b})]^{\intercal}\] \[\hat{\boldsymbol{\phi}}_{\gamma,k}(t) = [\hat{m}_{k-1}^{2}(t),\hat{m}_{k-1}^{3}(t),\cdots,\hat{m}_{k-1}^ {n_{\gamma}}(t)]^{\intercal}\] \[\hat{\boldsymbol{\phi}}_{n,k}(t) = [\hat{v}_{k-1}(t-1),\hat{v}_{k-1}(t-2),\cdots,\hat{v}_{k-1}(t-n_ {d})]^{\intercal}\]  
Replacing \(\phi_{s}(t)\) and \(\vartheta_{s}\) in Equation (7) with \(\hat{\phi}_{s,k}(t)\) and \(\hat{\vartheta}_{s,k}\), respectively, the iterative estimate \(\hat{m}_{k}(t)\) can be obtained by the following auxiliary model:  
\[\hat{m}_{k}(t)=\hat{\phi}_{s,k}^{\intercal}(t)\hat{\vartheta}_{s,k},\;t=1,2, \cdots,N \tag{17}\]  
Similarly, from Equation (8), the estimate \(v_{k}(t)\) can be computed by  
\[\hat{v}_{k}(t)=y(t)-\hat{\phi}_{k}^{\intercal}(t)\hat{\vartheta}_{k} \tag{18}\]  
Define  
\[\hat{\Psi}_{k}(N) := \begin{bmatrix}\hat{\phi}_{k}^{\intercal}(N)\\ \hat{\phi}_{k}^{\intercal}(N-1)\\ \vdots\\ \hat{\phi}_{k}^{\intercal}(1)\end{bmatrix} \in\mathbb{R}^{N\times n_{0}} \tag{19}\]  
Let \(\hat{\vartheta}_{k}= \begin{bmatrix}\hat{\vartheta}_{s,k}\\ \hat{\vartheta}_{\gamma,k}\\ \hat{\vartheta}_{n,k}\end{bmatrix} \) be the estimate of \(\vartheta= \begin{bmatrix}\vartheta_{s}\\ \vartheta_{\gamma}\\ \vartheta_{n}\end{bmatrix} \) at iteration \(k\). Using \(\hat{\Psi}_{k}(N)\) in place of \(\Psi(N)\) in Equations (14) and (15), we have  
\[\hat{\vartheta}_{k} = \hat{\vartheta}_{k-1}+\frac{1}{r_{k}}\hat{\Psi}_{k}^{\intercal}( N)[Y(N)-\hat{\Psi}_{k}(N)\hat{\vartheta}_{k-1}] \tag{20}\]\[r_{k} = r_{k-1}+\|\hat{\Psi}_{k}^{\mathrm{ T}}(N)\hat{\Psi}_{k}(N)\|^{2} \tag{21}\]  
Equations (14)-(21) form the gradient-based iterative (GI) identification algorithm for the Wiener nonlinear OEMA model, which can be summarized as follows:  
\[\hat{\vartheta}_{k} = \hat{\vartheta}_{k-1}+\frac{1}{r_{k}}\hat{\Psi}_{k}^{\mathrm{ T}}(N)[Y(N)-\hat{\Psi}_{k}(N)\hat{\vartheta}_{k-1}] \tag{22}\] \[r_{k} = r_{k-1}+\|\hat{\Psi}_{k}^{\mathrm{ T}}(N)\hat{\Psi}_{k}(N)\|^{2}\] (23) \[\hat{\Psi}_{k}(N) = \left[ \begin{matrix}\hat{\phi}_{k}^{\mathrm{ T}}(N)\\ \hat{\phi}_{k}^{\mathrm{ T}}(N-1)\\ \vdots\\ \hat{\phi}_{k}^{\mathrm{ T}}(1)\end{matrix}