# Application of Generalized S-Transform in the Measurement of Dynamic Elastic Modulus
## 2 The Principle of Generalized S-Transform
Among them, the low-frequency signal \(x_{3}\) (at time points 700\(\sim\)800) has higher accuracy in the frequency direction than the high-frequency signal \(x_{2}\) (at time points 200\(\sim\)250) and, correspondingly, the high-frequency signal can obtain higher time resolution, such as the starting and ending time of signal \(x_{2}\) at 300 Hz is clearer. For the time-frequency coefficient in Figure 1c, we integrate it along the time axis, and the frequency domain result can be obtained, as shown in Figure 1d, which is consistent with the FFT result. In the standard S-transform, the Gaussian window function (kernel function) \(w(t,f)\) cannot adjust the time width or frequency resolution, while the width of the Gaussian window changes inversely with the frequency of the measured signal. Therefore, the time window is wider at low frequencies, and a higher frequency resolution can be obtained, but the time resolution is reduced. In the process of dynamic elastic modulus measurement, due to the sample material, defects, geometry and other reasons, the resonance frequency bandwidth formed by the excitation signal through the transfer function may have a large distribution range, and the primary resonance frequency is easily aliased with the secondary resonance signal. While  
Figure 1: Transformation results of simulated vibration signal. **(a)** Original vibration signal. **(b)** Frequency spectrum calculated by FFT. **(c)** Time-frequency spectrum calculated by S-transform. **(d)** Frequency spectrum obtained from time-frequency coefficients. calculating the frequency-time variation relationship of the primary vibration signal caused by the excitation signal, it requires higher time and frequency resolution. And, if the window function of S-transform is a hyperbolic function or other asymmetric function, it will increase the computational complexity, which is not conducive to complete the calculation in a processor with low-performance such as embedded devices. Therefore, the Gaussian window function can be modified to a generalized Gaussian window function, thus realizing a generalized S-transform to improve the resolution of the high frequency band [32]. In the generalized S-transform, the Gaussian window is improved by adding two adjustable parameters \(\lambda\) and \(p\) to adjust the time width and attenuation trend of wavelet basis function, the window function is modified as:  
\[w(t,f)=\frac{\lambda|f|^{p}}{\sqrt{2\pi}}\exp\left(-\frac{\lambda^{2}t^{2}f^{ 2p}}{2}\right) \tag{7}\]  
By analyzing the time-frequency characteristic of the measured non-stationary signal, the parameters \(\lambda\) and \(p\) are modified so that the window width of the generalized Gaussian window function can be adjusted by the trend of the signal frequency. In this way, the Gaussian window function can be better applied to the analysis and processing of non-stationary signals. After substituting Equation (7) into the standard S-transform, the generalized S-transform can be expressed as:  
\[\mathrm{GST}_{x}(\tau,f)=\int_{-\infty}^{\infty}x(t)\frac{\lambda|f|^{p}}{ \sqrt{2\pi}}\exp\left(-\frac{\lambda^{2}(\tau-t)^{2}f^{2p}}{2}\right)\exp(-i2 \pi ft)\mathrm{d}t \tag{8}\]  
Similarly, the original simulated signal \(x(t)\) is simulated by generalized S-transform with different parameters \(\lambda\) and \(p\). The simulation results are shown as in Figure 2.  
Through the analysis of the above figures, if the value \(p\) is fixed, when \(\lambda>1\), the width of the window function changes inversely with frequency faster; when \(\lambda<1\), it slows down. When \(\lambda\) is fixed, the width increases exponentially as \(p\) varies from large to small. Figure 2: The generalized S-transform simulation with different parameters.