## 2 The Derivation of the Wiener OEMA Model  
Let us firstly introduce some notations. The superscript \(\mathrm{T}\) denotes the matrix transpose; \(1_{n}\) represents an \(n\)-dimensional column vector whose elements are 1; the norm of a matrix \(X\) is defined by \(\|X\|^{2}=\mathrm{tr}[XX^{\mathrm{T}}]\).  
Consider a Wiener nonlinear OEMA system shown in Figure 1:  
\[m(t)=\frac{B(z)}{A(z)}u(t) \tag{1}\]  
\[y(t)=f(m(t))+D(z)v(t) \tag{2}\]  
where \(u(t)\) is the system input, \(y(t)\) is the system output and \(v(t)\) is an additive noise with zero mean; the inner variable \(m(t)\)(namely, the output of the linear block) is unmeasurable; \(A(z)\), \(B(z)\) and \(D(z)\) are polynomials in the shift operator \(z^{-1}[z^{-1}y(t)=y(t-1)]\) with  
\[A(z) := 1+a_{1}z^{-1}+a_{2}z^{-2}+\cdots+a_{n_{a}}z^{-n_{a}}\] \[B(z) := b_{1}z^{-1}+b_{2}z^{-2}+\cdots+b_{n_{b}}z^{-n_{b}}\] \[D(z) := 1+d_{1}z^{-1}+d_{2}z^{-2}+\cdots+d_{n_{d}}z^{-n_{d}}\]  
The nonlinear part \(f(\cdot)\) in the Wiener system is a polynomial of a known order as follows:  
\[f(m(t))=\gamma_{1}m(t)+\gamma_{2}m^{2}(t)+\cdots+\gamma_{n_{\gamma}}m^{n_{ \gamma}}(t) \tag{3}\]  
where \(n_{\gamma}\) is the polynomial orders.  
Equation (1) can be rewritten to  
\[m(t)=[1-A(z)]m(t)+B(z)u(t) \tag{4}\]  
In order to get unique parameter estimates, we introduce the key term separation technique presented in [7, 29] and let the first coefficient of the nonlinear part be unity, _i.e._, \(\gamma_{1}=1\). Then, we have  
\[y(t)=m(t)+\sum_{i=2}^{n_{\gamma}}\gamma_{i}m^{i}(t)+D(z)v(t) \tag{5}\]  
Here \(m(t)\) in Equation (5) is called as the key term. Substituting Equation (4) into Equation (5) gives  
\[y(t)=[1-A(z)]m(t)+B(z)u(t)+\sum_{i=2}^{n_{\gamma}}\gamma_{i}m^{i}(t)+D(z)v(t) \tag{6}\]  
Define the information vectors and the parameter vectors  
\[\phi_{s}(t) := [-m(t-1),-m(t-2),\cdots,-m(t-n_{a}),u(t-1),u(t-2),\cdots,u(t-n_{b} )]^{\mathrm{ T}}\in\mathbb{R}^{n_{a}+n_{b}}\] \[\phi_{\gamma}(t) := [m^{2}(t),m^{3}(t),\cdots,m^{n_{\gamma}}(t)]^{\mathrm{ T}}\in\mathbb{R}^{n_{\gamma}-1}\] \[\phi_{n}(t) := [v(t-1),v(t-2),\cdots,v(t-n_{d})]^{\mathrm{ T}}\in\mathbb{R}^{n_{d}}\]  
Figure 1: The Wiener nonlinear OEMA system.  
\[\phi(t) := \begin{bmatrix}\phi_{s}(t)\\ \phi_{\gamma}(t)\\ \phi_{n}(t)\end{bmatrix}\in\mathbb{R}^{n_{a}+n_{b}+n_{\gamma}+n_{d}-1}\] \[\vartheta_{s} := [a_{1},a_{2},\cdots,a_{n_{a}},b_{1},b_{2},\cdots,b_{n_{b}}]^{ \mathrm{ T}}\in\mathbb{R}^{n_{a}+n_{b}}\] \[\vartheta_{\gamma} := [\gamma_{2},\gamma_{3},\cdots,\gamma_{n_{\gamma}}]^{\mathrm{ T}}\in\mathbb{R}^{n_{\gamma}-1}\] \[\vartheta_{n} := [d_{1},d_{2},\cdots,d_{n_{d}}]^{\mathrm{ T}}\in\mathbb{R}^{n_{d}}\] \[\vartheta := \begin{bmatrix}\vartheta_{s}\\ \vartheta_{\gamma}\\ \vartheta_{n}\end{bmatrix}\in\mathbb{R}^{n_{a}+n_{b}+n_{\gamma}+n_{d}-1}\]  
Thus, Equation (4) can be written in a vector form  
\[m(t)=\phi_{s}^{\mathrm{ T}}(t)\vartheta_{s} \tag{7}\]  
Combining Equations (6) and (7), we can obtain the following identification model:  
\[y(t) = \phi_{s}^{\mathrm{ T}}(t)\vartheta_{s}+\phi_{\gamma}^{ \mathrm{ T}}(t)\vartheta_{\gamma}+\phi_{n}^{\mathrm{ T}}(t)\vartheta_{n}+v(t) \tag{8}\] \[= \phi^{\mathrm{ T}}(t)\vartheta+v(t)\]  
The objective of this paper is to present a gradient based iterative identification algorithm to estimate the parameters \(a_{i}\), \(b_{i}\), \(\gamma_{i}\) and \(d_{i}\) for the Wiener nonlinear OEMA model using the auxiliary model identification idea in [23].