# Behavioral Modeling of Memristor-Based Rectifier Bridge
## 3 Forming the Sets of Input and Output Signals of the Memristor-Based Rectifier Bridge
### Smoothing Output Signals of the Memristor-Based Rectifier Bridge  
To construct polynomial models, the set of the test input signals of the rectifier and the corresponding set of the output signals are formed. The harmonic input signal is written  
\[v(t)=A\cdot\sin(\omega_{0}\cdot t), \tag{8}\]  
where \(A\) is an amplitude; \(\omega_{0}=2\pi\cdot f_{0}\) is an angular frequency; \(f_{0}\) is a frequency; \(t\ \in\ [0,\ T)\) is time interval on a signal period; \(T\) is a signal period. The set of the test input signals is generated on specifying the following vectors of amplitude and frequency:  
\[\mathbf{A}=[12;13;14;15;16]\ \text{V}, \tag{9}\]  
\[\mathbf{F}_{0}=[1;2;3;4;5]\ \text{Hz}, \tag{10}\]  
where \(\mathbf{A}\) and \(\mathbf{F}_{0}\) are the amplitude and frequency vectors of the harmonic test exciters. The set of the corresponding output signals results from simulating the circuit shown in Figure 4. To build the mathematical models (a split polynomial), the number \(N_{\text{p}}=2001\cdot 5=10,005\) of the input-output pairs is used. We use high amplitudes (9) of the input signal (8), since the rectifier bridge model based on the Yakopcic model of a memristor is severely nonlinear and the Yakopcic model requires high nonlinearity. With the given set of parameters this nonlinearity is achieved at high input amplitudes. One of our targets is to compare the split multi-dimensional polynomial of the rectifier bridge with the model based on the Yakopcic memristor model. It is well known that the memristor resistance can change for smaller voltages, leading to the so-called voltage-time dilemma and the resistance may degrade even at zero voltage [60]. It should be pointed out that the investigated \"black box\" approach is independent of the mentioned problem, since a behavioral model does not take the memristor model into account. The input and output signals are sampled and normalized according to equation  
\[\omega_{0}t_{n}=2\pi f_{0}t_{n}=2\pi t_{n}/T=\tilde{t}_{n},\]  
where \(t_{n}\), \(t_{n}\ \in\ [0,\ T)\) is a time sample; \(N=2001\) is the number of the time samples on the signal period; \(t_{n}=(n-1)T/(N-1)\), \(n=1\), \(2\),..., \(N\); \(\tilde{t}_{n}\), \(\tilde{t}_{n}\ \in\ [0,\ 2\pi)\) is a time-normalized sample; \(\tilde{t}_{n}=2\pi\cdot(n-1)/(N-1)\). Thus, when the frequency of the input signal (8) changes according to the vector \(\text{F}_{0}\) from expression (10), we always have the time-normalized range \(\tilde{t}_{n}\in[0,\,2\pi)\). The rectifier outputs \(v_{\text{out}}(t)\) are also converted to the time-normalized signals \(v_{\text{out}}\big{(}\tilde{t}_{n}\big{)}\). The time-normalized output signals of the rectifier at the excitation (8), in which the amplitude and frequency are assigned by the combination of elements from vectors (9) and (10), are shown in Figure 4. As seen from Figure 4, the output signals of the rectifier simulated with the Yakopcic model of memristors are not smooth functions. Every output signal has a point similar to discontinuity. This discontinuity is known to be generated by high-frequency harmonic components. To smooth signals, a low-pass filter, connected at  
Figure 4: For the test input signals, the load voltages obtained sequentially after simulated in LTspice (a), after smoothing by the low-pass filter (b), after normalizing (c). the output of the rectifier, is used. Let's apply an ideal low-pass filter. We simulate the filter action as follows. Each output signal of the rectifier is processed with the discrete Fourier transform of length \(N=2001\). We keep the first 15 harmonics and zero the remaining ones. Then, we carry out the inverse discrete Fourier transform. As a result, the rectifier outputs are smoothed out. Figure 4b shows signals smoothed in comparison with those which are depicted in Figure 4a. It should be pointed out that the spectrum of the rectifier output signals contains even harmonics, since the nonlinear operator of a full-wave rectifier is even.