# Behavioral Modeling of Memristor-Based Rectifier Bridge
## 4 Results of the Rectifier Modeling Based on Multi-Dimensional Polynomials  
The rectifier polynomial describes the mapping between the input signals set and the set of the smoothed rectifier output signals. The rectifier models are two-dimensional polynomials of different powers. Note there are the polynomials of the even-only degree, since the full-wave rectifier is described by an even operator. The two-dimensional polynomial of the 6 th degree (the number of parameters is 15) is used as an example. The polynomial is of the form  
\[\overline{y}(\overline{t}_{n})=\sum_{m=2}^{L}\sum_{i_{1}=0}^{I_{1}}\sum_{i_{2} =i_{1}}^{I_{2}}\cdots\sum_{i_{m}=i_{m-1}}^{I_{m}}C_{i_{1}i_{2}...i_{m}} \overline{v}(\overline{t}_{n}-\overline{t}_{i_{1}})\overline{v}(\overline{t}_ {n}-\overline{t}_{i_{2}})\,\ldots\,\overline{v}(\overline{t}_{n}-\overline{t} _{i_{m}}), \tag{11}\]  
where \(C_{i_{1}i_{2}...i_{m}}\) are the model parameters; \(L\) is the even power of the polynomial; \(m\) is an even degree; \(I_{1}=I_{2}=\cdots=I_{m}=1\) are constants that determine the dimension of the polynomial in the corresponding coordinates. As the analysis of expression (11) shows, the presented polynomial is two-dimensional for any degree, its basis functions are the samples of the input signal at the current and previous time instants. The model parameters result from solving the optimization problem:  
\[\|\overline{v}_{\text{out,sm}}(\overline{t}_{n})-\overline{y}(\overline{t}_{n })\|\to\min_{\mathbf{C}},\]  
where \(\overline{v}_{\text{out,sm}}(\overline{t}_{n})\) is discrete amplitude and time-normalized smoothed rectifier output signal, \(\mathbf{C}\) is a vector containing the parameters of the model (8). To estimate the accuracy of the constructed polynomials, three test input signals (\(k=1,\,2,\,3\)) were used, specified by expression (8) at amplitudes and frequencies from Table 1, and we calculated the following:  
* the uniform error \[\overline{\Delta}_{k}(\overline{t}_{n})=\overline{v}_{\text{out,sm,k}}( \overline{t}_{n})-\overline{y}_{k}(\overline{t}_{n}),\,\,n=1,2,\ldots,2001,\,k =1,2,3;\] (12)
* the maximum absolute error \[\max\bigl{(}|\overline{\Delta}(\overline{t}_{n})|\bigr{)}=\max_{k=1,\,2,\,3} \bigl{(}\max\bigl{(}|\overline{\Delta}_{k}(\overline{t}_{n})|\bigr{)}\bigr{)},\]\[\max\bigl{(}\bigl{|}\overline{\Delta}_{k}(\vec{t}_{n})\bigr{|}\bigr{)}=\max_{\vec{t} _{n}\in(0,2\pi)}\bigl{(}\bigl{|}\overline{v}_{\text{out,sm,}k}(\vec{t}_{n})- \overline{y}_{k}(\vec{t}_{n})\bigr{|}\bigr{)},\ n=1,2,\ldots,2001;\]
* the root-mean-square error \[\varepsilon_{k}=\sqrt{\frac{1}{N-1}\sum_{q=1}^{N-1}\bigl{(}\,\overline{v}_{ \text{out,sm}}(\vec{t}_{q})-\overline{y}_{k}(\vec{t}_{q})\,\bigr{)}^{2}},\ N=2001,\ k=1,2,3.\]  
The maximum absolute and root-mean-square errors of modeling are represented in Table 1.  
For the 3 rd trial input signal, as shown in Figure 5a, the blue curve is the response \(\overline{y}_{3}(\vec{t}_{n})\) of polynomial (11) of the 6 th power, the red curve is the smoothed output signal \(\overline{v}_{\text{out,sm,}3}(\vec{t}_{n})\), the black curve is the uniform error \(\overline{\Delta}_{3}(\vec{t}_{n})\) from Equation (12) in the range \(\vec{t}_{n}\ \in\ [0,2\pi)\). Figure 5b shows correspondingly the response of the split polynomial, the non-smoothed load voltage \(\overline{v}_{\text{out,}3}(\vec{t}_{n})\), the uniform error obtained without smoothing by the low-pass filter. By comparing Figure 5a with Figure 5b, it can be seen that the non-smoothed output signal is described with a significantly greater error (black curve) than the smoothed output signal.