## 3 Results
### Example of Symbolic Transfer Function Complexity Calculation  
This chapter contains an example of symbolic expression complexity calculation based on a transfer function obtained in symbolic form using the FSM. Specifically, it shows the extraction of feature O1.2, \"Complexity of transfer function\". The complexity value obtained is an integer number, which could be used in a similar way to extract the features with IDs I2.4, M1.2, O1.2, and O1.3 described above. For this example, the one-circuit parametric amplifier shown in Figure 6 has been selected. A parametric circuit with a single varying coefficient, shown in Figure 6, has been analyzed. The time-dependent parameter \(c(t)\) of the system is the capacitance, which is represented by a function:  
\[c(t)=C_{0}\{1+m_{c}\text{cos}(\Omega t)\},\ \ C_{0}=10\ \text{pF}. \tag{13}\]  
The parameters of the LPTV circuit shown in Figure 6 are as follows: a constant angular frequency \(\Omega\) for the parametric function \(c(t)\): \(\Omega=2\pi f\), where \(f=200\) MHz, and a modulation coefficient \(m_{c}=0.01\). The inductance is \(L=253.3\) nH, and the conductance values are \(Y_{1}=0.25\) S and \(Y_{2}=0.4\) mS. A monoharmonic input signal is a current \(i(t)\) with a time waveform:  
\[i(t)=I_{m}\text{cos}\left(\omega t+\varphi_{0}\right),\ I_{m}=0.1\ \text{mA},\ \varphi_{0}=45^{\text{\tiny\textdegree}},\ \omega=2\pi f_{m},\ f_{m}=100\ \text{MHz}. \tag{14}\]  
Three parameters remain in symbolic form: time \(t\), constant angular frequency of the parametric function \(\Omega\), and \(m_{c}\)--modulation coefficient of the time-varying capacitance \(c\). Using FSM, the transfer function was obtained for three cases: 1, 2, and 3 harmonics used to approximate the transfer function. It can be noted that the FSM makes it possible to obtain the approximation of the transfer function using Fourier series with a predetermined number of harmonics. Figure 7 (magnitude) and Figure 8 (phase) present plots of the  
Figure 6: Model of a single parametric-variable circuit. transfer function (with 1 harmonic approximation) for presented circuit. It can be observed that there is an influence of variable capacitance at frequency \(f=200\) MHz. The following formula is proposed to calculate the complexity \(C\) of an expression (transfer function):  
\[C=\sum_{k=1}^{K}w_{i}\cdot N_{i}, \tag{15}\]  
where:  
\(N_{i}\) is number of particular operations (for \(i\)-th mathematical operand),  
\(w_{i}\) is complexity weight of the \(i\)-th operand,  
\(K\) is total number of distinguished operands (depending on the case). Figure 8: Phase plot of complex transfer function (for 1 harmonic). Figure 7: Magnitude of complex transfer function (for 1 harmonic). There following operands are distinguished (with abbreviations): addition (add), subtraction (sub), multiplication (mul), division (div), and exponentiation (exp). Negative numbers are not counted as subtraction (they have assumed particular hardware representation). Deciding on the weights of each operation is a debatable topic, as it depends very much on the tools, algorithms, hardware implementation, etc. Therefore, the following assumptions have been made for this example, which may not apply to other environments:  
1. Every number in the expression in the generic case is a complex number and it serves as our reference. 2. The addition or subtraction of two complex numbers has the same complexity as two additions/subtractions of real numbers and is considered as a unit weight with value \(w_{add}=w_{sub}=1\). 3. Multiplication of two real numbers typically uses hardware acceleration, resulting in a weight of 1. 4. Complex multiplication involves two multiplications and one addition, and based on the assumptions above, its weight is \(w_{mul}=3\). 5. The division of two complex numbers uses multiplication by the conjugate, and its weight is \(w_{div}=7\). 6. Binary exponentiation is calculated with complexity \(O(2\cdot\log_{2}n)\).