# Application of Generalized S-Transform in the Measurement of Dynamic Elastic Modulus
## 2 The Principle of Generalized S-Transform  
As a time-frequency analysis tool, S-transform is developed from short-time Fourier transform and continuous wavelet transform (CWT). In S-transform, the basic wavelet is composed of the product of simple harmonic wave and Gaussian wave function. The simple harmonic wave only performs a scaling transform in the time domain, while the Gaussian function does scaling and translation. In the transformation, the mother wavelet is fixed, which limits its application [28; 29; 30]. The S-transform can be considered the phase correction of CWT. \[S(\tau,f)=\text{CWT}(\tau,d)\,\text{esp}(i2\pi f\tau) \tag{1}\]  
The equation of CWT is expressed as:  
\[\text{CWT}(\tau,d)=\int_{-\infty}^{\infty}x(t)w(t-\tau,d)\text{d}t \tag{2}\]  
where \(S(\tau,f)\) is the S-transform result of the original signal \(x(t)\), \(x(t)\in L^{2}(R)\) ( \(L^{2}(R)\) is a function space with limited energy), \(f\) is the frequency, \(t\) and \(\tau\) are the time, \(w(t,d)\) is the wavelet function in the S-transform, where the wavelet function is defined as:  
\[w(t,f)=\frac{|f|}{\sqrt{2\pi}}\,\text{esp}\bigg{(}-\frac{t^{2}f^{2}}{2}\bigg{)} \,\text{esp}(-i2\pi ft) \tag{3}\]  
where the stretching factor \(d\) and frequency \(f\) are reciprocals of each other. Then, the equation of S-transform can be written as:  
\[S(\tau,f)=\int_{-\infty}^{\infty}x(t)\frac{|f|}{\sqrt{2\pi}}\,\text{esp} \bigg{(}-\frac{(\tau-t)^{2}f^{2}}{2}\bigg{)}\,\text{esp}(-i2\pi ft)\text{d}t \tag{4}\]  
The frequency f controls the size of the Gaussian window \(\frac{|f|}{\sqrt{2\pi}}e^{-\frac{t^{2}f^{2}}{2}}\), and the time variable \(\tau\) controls the movement of the center of the Gaussian window on the time \(t\) axis. It can be seen that the sliding Gaussian window adaptively adjusts the window size in inverse proportion to the frequency \(f\)[31]. In the calculation of the actual S-transform coefficients, \(S(\tau,f)\) is calculated by FFT:  
\[S(\tau,f)=\int_{-\infty}^{\infty}X(\alpha+f)\,\text{esp}\bigg{(}-\frac{2\pi^ {2}\alpha^{2}}{f^{2}}\bigg{)}\,\text{esp}(i2\pi\alpha\tau)\text{d}\alpha \tag{5}\]  
where \(\alpha\) is the frequency variable, \(X(\alpha+f)\) represents the distribution of the Fourier spectrum of the signal \(x(t)\) after translation in the frequency domain, and the S-transform spectrum is the weighted filtering result of the translated Fourier spectrum of the signal \(x(t)\). A simulated vibration signal \(x(t)\) is processed to verify the validity of the S-transform for dynamic elastic modulus measurement, which is composed of three parts: the frequency \(f_{s}\) of sweep signal \(x_{1}\) increases continuously from 1 Hz to 450 Hz, and a sine wave \(x_{2}\) with frequency \(f_{r1}\) of 300 Hz is superimposed between time points 200\(\sim\)250; a sine wave \(x_{3}\) with frequency \(f_{r2}\) of 100 Hz is superimposed between time points 700\(\sim\)800 as follows:\[\left\{ \begin{array}{ll}x1(t)=\sin[2\pi f_{s}t(1:1000)]&(f_{s}=1\sim 450\;{\rm Hz },f_{s}\;{\rm is\;proportional\;to\;}t)\\ x2(200:250)=\sin[2\pi f_{s}t(200:250)]&(f_{s}1=300\;{\rm Hz})\\ x3(700:800)=\sin[2\pi f_{s}2t(700:800)]&(f_{s}2=100\;{\rm Hz})\\ x(t)=x1(t)+x2(200:250)+x3(700:800)&\end{array} \right. \tag{6}\]  
The original simulated signal \(x(t)\) is shown in Figure 1a, and its corresponding spectrum is shown in Figure 1b. From the frequency spectrum, the frequency range of the sweep signal \(x_{1}\) can be obviously seen (0-450 Hz), and we can also accurately obtain the peak frequency of signal \(x_{3}\) is 100 Hz, but the measurement accuracy of signal \(x_{2}\) at 300 Hz is low. Figure 1c is the S-domain time-frequency spectrum, the vertical axis is the frequency range of the signal, and the horizontal axis corresponds to the occurrence time of instantaneous frequency of the sweep signal.