# Application of Generalized S-Transform in the Measurement of Dynamic Elastic Modulus
## 4 Measurement Based on Generalized S-Transform  
In the processing program of vibration signal, the generalized S-transform algorithm is modified based on C28x Floating-Point Unit Library, mainly referring to its fast Fourier algorithm. Based on the above sampling sequence of vibration signals, the generalized S-transform is computed relatively fast due to the use of the convolution theorem and FFT. In practical applications, the S-transform is calculated according to the Algorithm 1. ```
1: The signal \(x(t)\) is sampled with \(T\) as the period, and the total number of samples is \(N\). Finally, the discrete signal sequence \(x(NT)\) is obtained, and then calculate its FFT spectrum \(X(\frac{m}{NT})\);
2: By shifting its frequency to the \(n\)-th frequency point, the frequency shift spectrum \(X(\frac{n+m}{NT})\) is obtained;
3: For this frequency point, the Gauss local window function \(w(m,n)\) with parameters is calculated;
4: For \(X(\frac{n+m}{NT})\), the Gauss local window function \(w(m,n)\) is used for windowing calculation, and the inverse Fourier transform is performed on the dot product \(X(\frac{n+m}{NT})w(m,n)\) to obtain the S-transform spectrum for that frequency point;
5: Repeat step 2 to step 4 until all frequency points of the signal spectrum are calculated, and the final result of the generalized S-transform of \(x(t)\) is \(\text{GST}_{x}(iT,\frac{n}{NT})\). Figure 3: The structural diagram of the dynamic elastic measurement system. After the repeat (RPT) pipeline of the DSP is started, the hardware multiplier performs multiplication and addition in a single instruction cycle, to complete convolution calculation quickly. In order to improve the operation speed, the parameters \(\lambda\) and \(p\) of the generalized S-transform are stored in the RAM space in the form of dynamic data. To verify the effectiveness of the algorithm in measuring dynamic elasticity modulus, the excitation frequency output by the ePWM module increases continuously from 50 Hz to 5 kHz. The pulse width modulation is set at 30% to reduce the temperature drift of the electromagnetic coil in the impulser, and the average output power of the impulser is \(P_{T}=5\) W. To measure the dynamic elastic modulus in batches, machine-produced concrete blocks are chosen as specimens to ensure uniform size and consistent frequency. The size of the blocks is 400 mm \(\times\) 100 mm \(\times\) 100 mm, and the experimental setup is shown in Figure 4.

For the rectangular cement sample weighing 9.263 kg, and the correction factor \(T_{1}\) is 1.38, after removing the 50 Hz noise by digital filtering, the resonant frequency was measured to be 2.315 kHz, and the dynamic modulus of elasticity of the sample was calculated to be 41.499 GPa using Equation (10). The final time-frequency spectrum is shown in Figure 5.

To further evaluate the precision of the GST algorithm in measuring the dynamic elastic modulus, in addition to concrete bricks, we also conducted repeated measurements on structurally stable and compact aluminum samples [40; 41]. The FFT and GST algorithms were applied to various samples with an equal number of sampling points. Additionally, a swept-frequency measurement based on a sweep signal generator and spectrum analyzer was chosen for comparison. Finally, Table 1 shows the standard deviation of 10 repeated measurements. Figure 4: Experimental setup. (**a**) Placement of probes and the concrete sample. (**b**) Rods of impulser and transducer. Figure 5: (**a**) Time-frequency spectrum of concrete. (**b**) Three-dimensional display of the Time-frequency spectrum. According to the table, for the resonance frequency measurement of the same concrete specimen, the standard deviation of the GST algorithm is less than that of the FFT algorithm. Conversely, the difference in the standard deviation of those two algorithms is similar for the aluminum specimen with stable structural performance.