## 3 Time Frequency Transforms
From Figure 2, there exist four peaks, which respectively denote the corresponding auto-terms of the four Gaussian components in the TF plane; in addition, the WVD also presents six cross-terms, which occur between each pair of Gaussian signal components, and among them, two cross-terms overlap in the diagonal intersection point of the rectangle connected by the four vertices (_i.e._, four Gaussian components in the joint TF plane). These extra cross-terms have large oscillating amplitudes due to the interaction of the different signal components. The magnitudes of the oscillatory cross-terms can be twice as large as the auto-terms, but in the diagonal intersection position, the intensity of the corresponding cross-term can be four-times as large as the auto-terms, since two cross-terms overlap in this intersection point. In Figure 2b, the serious cross-terms apparently present in the regions of the TF plane where we expect no energy at all, which make proper interpretation impossible. This is the main drawback of the WVD approach. A feasible method to depress the effect of the cross-terms is to introduce windows in WVD; therefore, the concept of PWVD is educed. The WVD weighs equally all times of the future and past for a given time, but in the practical calculation of the distribution for a time instant \(t\), we may concentrate the properties of the signal near the time of interest rather than from minus to plus infinity. Therefore, if we want to emphasize the signal around time \(t\), a window function \(h(\uptau)\) can be multiplied with the instantaneous ambiguity function \(y_{a}(t+\frac{\uptau}{2})y_{a}^{*}(t-\frac{\uptau}{2})\) in order to define the PWVD, which is the WVD windowed in the time direction [20], written as follows:  
\[STFT(t,\upomega)=\frac{1}{2\uptau}e^{-j\upomega t}\int_{-\infty}^{+\infty}Y_{ a}(\upomega^{\prime})H(\upomega^{\prime}-\upomega)e^{j\upomega^{\prime}t} \text{d}\upomega^{\prime} \tag{13}\]2  
where \(PWVD(t,\upomega)\) is controlled by the window \(h(\uptau)\), which is an even function and peaked around \(\uptau=0\). \[STFT(t,\upomega)=\frac{1}{2\uptau}e^{-j\upomega t}\int_{-\infty}^{+\infty}Y_{ a}(\upomega^{\prime})H(\upomega^{\prime}-\upomega)e^{j\upomega^{\prime}t} \text{d}\upomega^{\prime} \tag{13}\]3 Time-domain trace of four Gaussian components’ signal. \[STFT(t,\upomega)=\frac{1}{2\uptau}e^{-j\upomega t}\int_{-\infty}^{+\infty}Y_{ a}(\upomega^{\prime})H(\upomega^{\prime}-\upomega)e^{j\upomega^{\prime}t} \text{d}\upomega^{\prime} \tag{13}\]4 Wigner–Ville distribution of four Gaussian components’ signal. Since the WVD is highly nonlocal, the window function is adopted in the PWVD to make it local. The overlapped window function contributes the cross-term suppression for multi-component signals. The operation of time-window overlay is equal to the frequency filtering in WVD; consequently, the interference between time-shifted signals is usually attenuated. An example of PWVD of the four Gaussian components signal is provided in Figure 3a, and correspondingly, the contour of the computed PWVD is presented in Figure 3b. From the results, it is easy to find that only two cross-terms remain in the TF plane, and the number of the extra cross-terms decreases from six to two, since the the cross-interfering term between each pair of time-shifted signal components is attenuated. In addition, from Figure 3b, it is clear to observe that the TF localization properties of the auto-terms of the analyzed signal are deteriorated in comparison with the WVD approach. In the PWVD, the cross-interfering terms have been partially mitigated by the adoption of the window function. The time window operation is equal to frequency filtering in WVD, which can reduce the number of cross-interfering terms by suppressing the interferences between signal components sufficiently separated in time. However, these advantages are achieved at the price of a blurring of the auto-terms of the signal and a loss of many desirable theoretical properties.