## 4 Numerical Simulation
### Timing Analysis  
Measuring the required time to encrypt/decrypt a voice signal depends on the operating system, programming language, and CPU frequency. Although measuring the encryption/decryption time does not directly impact the security of the system, it reflects the execution efficiency of the encryption algorithm. Timing analysis is crucial for identifying the optimal encryption system, as it helps achieve a higher level of security in less time. Table 3 shows the results of encryption/decryption time for each voice sample of the original voice signal using different values for N-Levels. The PC specs are Intel(R) Core(TM) i7-8565U CPU @ 1.80 GHz hp LAPTOP-UFTCGPRC, 8 GB RAM, Windows10. The results show that increasing the number of levels N creates more time delay in the process of encryption/decryption as expected. Table 3 shows the performance metrics of N-level cascaded coupled chaotic systems using 12 different values of N. The results are obtained using the fourth-order Runge-Kutta integrator with a fixed step size (0.001). The parameter values for all the cascaded hyperchaotic systems in the transmitter and receiver are \(a_{1}=45\), \(a_{2}=3\), \(a_{3}=10\) and \(a_{4}=13\) and the initial conditions are \(x(0)=[1,1,1,1]^{T}\), \(y(0)=[1,1,1,1]^{T}\). Based on the security metrics presented in Table 3, we observe that as the number of levels (N) increases, SNR, PSNR, and PRD values also increase, indicating improved security. Conversely, the decreasing values of the correlation coefficient (CC) suggest enhanced security as well. However, increasing the number of cascaded levels also leads to longer encryption and decryption times, as the algorithm becomes more complex. representing the system's overall quality. This unified performance measure allows for a more comprehensive analysis of how increasing the number of cascaded levels affects the general performance of chaotic systems. In this section, we introduce a new measure called Value-Based Performance Metrics (VBPM) to evaluate the studied systems based on both security metrics and resource utilization, as discussed in Section 4. The returned values of VBPM can assist in the comparison since higher values of VBPM reflects higher quality. The mathematical description of VBPM is shown in Equation (8):  
\[VBPM=\frac{\frac{1}{i}(|SNR|+PSNR+PRD+F_{max})}{\frac{1}{j}(CC+t_{e}+t_{d}+LUT)} \tag{8}\]  
where i indicates the number of direct proportion metrics and j represents inverse proportion ones. In our study, \(i=j=4\). However, there is an issue when the features are on drastically different scales. For example, considering data in Table 3, the values of \(|SNR|\) and PSNR are in the ranges [60.1, 92.02] and [7.44, 12.56], respectively, while the values of PRD are in the range [\(1.0124\times 10^{+5}\), \(3.9904\times 10^{+6}\)]. The VBPM should represent the relationship between the values. However, substituting those values in Equation (8) on a large scale, i.e., PRD will completely dominate the others. The same issue exist for the values of the correlation coefficient, encryption time, and decryption time, where the correlation coefficient values are on the larger scale and dominate the others. Based on the previous discussion, since the ranges of metrics in Table 3 are diverged, we need to adjust the values of metrics on different scales to a notionally common scale by using normalization. The goal of normalization is to make every data point have the same scale, so each metric is equally important. Table 4 presents the normalized values for the performance metrics used in the comparison, along with the corresponding VBPM values for all the studied systems. The values in Tables 2 and 3 using min-max normalization, which is one of the most common ways to normalize data. For each column in Tables 2 and 3, the min-max normalization equation is as follows:  
\[NormalizedValue=\frac{value-min}{max-min} \tag{9}\]  
where min and max indicate the minimum and maximum values for each studied metric.