## 3 An Efficient Time Reversal Method  
In this section, an ETRM is proposed to improve the practical applicability and efficiency of the time reversal concept. In the ETRM, multiple reconstructed signals related to different excitations can be acquired through only one actuating-sensing step, and further, the determining of the two time intervals could be avoided. That significantly reduces the hardware manipulation. (1) A broadband excitation signal is actuated from PZT A, and the response is captured by PZT B;  
(2) Time-frequency analysis is performed on the broadband response signal. Then the ridge and the neighbor of the desired components are determined;  
(3) Reconstructing the desired components in the time-frequency domain into waveforms in the time domain;(4) The reconstructed signals related to narrowband excitations are obtained through deconvolution in frequency domain. The analysis of the ETRM is also given in the frequency domain. As shown in Figure 2, a broadband excitation signal \(V_{A}(\omega)\) is injected in actuator PZT A. The broadband response signal, \(V_{B}(\omega,r)\), is captured by sensor PZT B. Figure 2 gives its schematic diagram. The steps of the ETRM are as follows:  
Because of dispersion, attenuation and multi-modal propagation, broadband Lamb waves in composite laminates are difficult to interpret. Time-frequency representation (TFR) is an efficient method for the analysis of Lamb wave signal, because it provides a clear illustration for the temporal variation modal energy stream in the time-frequency domain. There are a variety of TFR algorithms that could map the Lamb wave signal from the time domain into time-frequency domain, e.g., short time Fourier transform [26], wavelet transform [27; 28], warped frequency transform [29; 30], and fractional Fourier transform [31]. Even though some of them cannot to extract and reconstruct the wave components, the time-varying filtering could be employed for breaking this limitation [32; 33]. In this paper, for simplicity, the short-time Fourier transform (STFT) is used for illustration. Mathematically, the STFT of a signal \(s(t)\) is formulated as  
\[S_{STFT}(t,\omega)=\int_{-\infty}^{+\infty}s(\tau)\cdot h(\tau-t)e^{-i\omega \tau}d\tau, \tag{8}\]  
where, \(t\) and \(\omega\) are the time and angular frequency, respectively, \(h(\tau-t)\) is a window function (commonly a Hanning or Gaussian window centered at \(t\)). STFT is actually frequency transform of signal pieces in the window. Therefore, the energy of an instantaneous component will spread on the time-frequency plane due to the influence of window. To extract the desired components, not only the points on the ridge but also the adjacent area of the ridge need to be reconstructed. The adjacent area influences the accuracy of mode extraction obviously. A small neighbor area will result in loss of energy, while a large neighbor area may arise noises in the extracted signal. In this paper, the left and right boundaries are determined by the decibels of the instantaneous frequency energy drop on the ridge. The percentage of energy drop is represented by _weight_. Figure 2 shows the schematic diagram of mode extraction, where the red solid line is the ridge of a Lamb wave mode. \(A_{R}\) is the amplitude of the ridge element corresponding to time \(t\) and instantaneous frequency \(f\) in the time-frequency matrix. \(t_{l}\) and \(t_{r}\) are the time range where the magnitude of elements at instantaneous frequency \(f\) are larger than _weight_ times \(A_{R}\), i.e.,  
Figure 2: Schematic diagram of the proposed efficient time reversal method (ETRM). \[A_{l}=A_{r}=weight\times A_{R}, \tag{9}\]  
Once the left and right boundaries of the neighbor area (the blue area in Figure 3) are determined, the desired components, \(v_{d}\)(t,r), could be extracted from the response signal by inverse STFT transform.