# Behavioral Modeling of Memristor-Based Rectifier Bridge
## 5 Results of the Rectifier Modeling Based on Piecewise Multi-Dimensional Polynomials  
To construct a piecewise polynomial model of the rectifier, we divide the range of the normalized modulus of the input signal amplitude into several sub-ranges. The thresholds of the sub-ranges we denote as \(\lambda_{j-1}\); \(j=1,2,\ldots,j\); \(0<\lambda_{j-1}<1\); \(\lambda_{0}=0\); \(\lambda_{J}=1\). The input sub-signals correspond to the mentioned amplitude sub-ranges, and every input signal is described by the sum  
\[\overline{v}(\overline{t}_{n})=\sum_{j=1}^{J}\overline{v}^{(j)}(\overline{t}_ {n})\]  
where \(\overline{v}^{(j)}(\overline{t}_{n})\) is the input sub-signal with the normalized amplitude and time variables,  
\[\overline{v}^{(j)}(\overline{t}_{n})=\left\{ \begin{array}{ll}0&\left| \overline{v}^{(j)}(\overline{t}_{n})\right|\leq\lambda_{j-1}\\ \left|\overline{v}(\overline{t}_{n})\right|&\lambda_{j-1}<\left|\overline{v}^{ (j)}(\overline{t}_{n})\right|\leq\lambda_{j}\\ 0&\left|\overline{v}^{(j)}(\overline{t}_{n})\right|>\lambda_{j}\end{array} \right.\]  
As a result, we form the sub-sets from the input sub-signals \(\overline{v}^{(j)}(\overline{t}_{n}),\overline{t}_{n}\in[0,\,2\pi)\), \(j=1,2,\ldots,J\). The sub-sets of the output sub-signals \(\overline{v}^{(j)}_{\text{out,sm}}(\overline{t}_{n})\), \(\overline{t}_{n}\in[0,\,2\pi)\), \(j=1,2,\ldots,J\) correspond to the sub-sets of the input sub-signals. The block-scheme of dividing the input signal into sub-signals according to the amplitudes and the formation of the response of the split piecewise-polynomial model is shown in Figure 6.  
Figure 6: The block-scheme of dividing the input signal into sub-signals and the formation of the response of the split piecewise-polynomial model. In the root-mean-square norm, using the sub-sets of the input sub-signals \(\overline{v}^{(j)}(\overline{t}_{n})\), \(\overline{t}_{n}\in[0,\,2\pi),\,\,j\,\,=\,\,1,\,2,\,\ldots,\,\,J\). and the output sub-signals \(\overline{v}^{(j)}_{\text{out,sm}}(\overline{t}_{n})\), we build models:  
\[\overline{y}^{(j)}_{\text{pw}}(\overline{t}_{n})=\sum_{m=2}^{L}\sum_{i_{1}=0}^ {L}\sum_{i_{2}=i_{1}}^{I_{1}}\ldots\sum_{i_{m}=i_{m-1}}^{I_{m}}C^{(j)}_{i_{1}i_{ 2}\ldots i_{m}}\overline{v}^{(j)}(\overline{t}_{n}-\overline{t}_{i_{1}}) \overline{v}^{(j)}(\overline{t}_{n}-\overline{t}_{i_{2}})\,\ldots\,\overline {v}^{(j)}(\overline{t}_{n}-\overline{t}_{i_{m}}) \tag{13}\]  
where \(\overline{y}^{(j)}_{\text{pw}}(\overline{t}_{n})\) and \(C^{(j)}_{i_{1}i_{2}\ldots i_{m}}\); \(j\,\,=\,\,1,\,2,\,\ldots,\,\,J\) are the output sub-signals and parameters of the piecewise model on the sub-sets of the input signal respectively; \(L\) is the even power of the polynomial; \(m\) is an even degree. The output signal of the piecewise model results from the following summation  
\[\overline{y}_{\text{pw}}(\overline{t}_{n})=\sum_{j=1}^{J}\overline{y}^{(j)}_{ \text{pw}}(\overline{t}_{n}) \tag{14}\]  
where \(\overline{y}^{(j)}_{\text{pw}}(\overline{t}_{n})\) is described in (13). Table 2 demonstrates the errors of modeling the memristive rectifier with the split piecewise-polynomial of the 6 th power at three trial excitations. We used three sub-ranges of the normalized amplitude of the input signal, the thresholds of sub-ranges are \(\lambda_{0}=\,\,0,\,\lambda_{1}=\,\,0.8,\,\,\lambda_{2}=\,\,0.9,\,\,\lambda_{3 }=\,\,1\). The total number of the model parameters is 45. As shown in Figure 7, the blue curve is the response of the polynomial (14) of the 6 th power, the red curve is the smoothed output signal \(\overline{v}_{\text{out,sm,3}}(\overline{t}_{n})\) of the circuit depicted in Figure 2, the black curve is the uniform error \(\overline{\Delta}_{3}(\overline{t}_{n})\). Comparison of the errors noted in Tables 1 and 2, as well as in Figures 5a and 7 results in the fact, that the piecewise-polynomial model ensures lower errors than a general polynomial model does.