## 2 First-Order Primal-Dual Algorithm-Based Sparse Time-Frequency Analysis
### Simulation Analysis  
To validate the effectiveness and superiority of the STFA-PD, a multi-component simulated signal, defined by Equation (16), was employed:  
\[\left\{ \begin{array}{l}s_{1}(t)=\sin(30\pi t,)\\ s_{2}(t)=\sin(30+75\pi t\cdot 3^{t})\\ s_{3}(t)=\sin[200\pi t+200\pi t^{2}+20\pi t\cos(2\pi t)]\\ s(t)=s_{1}(t)+s_{2}(t)+s_{3}(t)\end{array} \right. \tag{16}\]  
where \(s_{1}(t)\) was a sinusoidal signal and both \(s_{2}(t)\) and \(s_{3}(t)\) were nonlinear frequency-modulated signals. The sampling frequency was set to 1000 Hz, and the sample length was 1 s.  
Figure 1(a) shows the time-domain waveform of the simulated signal \(s(t)\), which was a non-stationary signal with a frequency that varied with time. Figure 1(b),e show the TFIs obtained using the STFT, pseudo WVD (PWVD), WT, and the STFA-PD. In this study, the bump mother wavelet was used in WT of the simulated signal, as well as bearing vibration signals. Figure 1(b) shows that the resolution of the traditional STFT was low either in the lower frequency band or in the higher frequency band, and the time-frequency distribution of the signal was not accurately reflected. By contrast, the TFI obtained using the PWVD was good in terms of the time-frequency resolution, but suffered from cross-term interferences, and thus, did not precisely reflect the time-frequency characteristics of the signal, as shown in Figure 1(c). As shown in Figure 1(d), the TFI obtained using the WT had a good time-frequency resolution in the lower frequency band, but a poor resolution in the higher frequency band. As the WT was essentially a type of time-scale transform and as there was no good correspondence between the scale and the frequency of the signal, the WT did not accurately express the time-based variations in the frequency of each component of the signal. Figure 1(e) shows the TFI obtained using the STFA-PD method. Comparing Figure 1(e) with Figure 1(b)-d, we see that the STFA-PD method had higher time-frequency concentration and resolution than the other methods. To validate the denoising performance of the STFA-PD method, the Gaussian white noise was artificially added to the simulated signal \(s(t)\). The noisy signal was denoted as \(z(t)\). Figure 3a shows the time-domain waveform of the noisy signal \(z(t)\) with the signal-to-noise ratio (SNR) of \(-2\) dB. Figure 3b-e show the TFIs obtained using STFT, PWVD, WT, and STFA-PD. It can be seen there were different degrees of noise interferences in the TFIs. As displayed in Figure 3c, The TFI obtained by PWVD seemed more sensitive to the noise than others. The time-frequency characteristic of the signal \(s(t)\) was almost masked by the noise and cross-term interferences. From Figure 3e, we can find that The TFI obtained by STFA-PD still had a high time-frequency resolution even if there was a high level of noise. Figure 2: (**a**) Time-domain waveform of the simulated signal \(s(t)\) and its time–frequency images (TFIs) obtained using (**b**) short-time Fourier transform (STFT), (**c**) pseudo Wigner–Ville distribution (PWVD), (**d**) wavelet transform (WT), and (**e**) the sparse time–frequency analysis method based on the first-order primal-dual algorithm (STFA-PD). To further quantitatively analyze and evaluate the denoising performance of the STFA-PD method, the peak signal-to-noise ratio (PSNR) was adopted as a denoising performance evaluation measurement. The PSNR is defined as follows:  
\[\text{PSNR}(\mathbf{J},\mathbf{K})=10\log_{10}\frac{MAX_{J}{}^{2}}{\frac{1}{\text{M} \times N}\sum\limits_{i=1}^{M}\sum\limits_{j=1}^{N}\left(\mathbf{J}_{ij}-\mathbf{K}_{ ij}\right)^{2}}, \tag{17}\]  
where \(\mathbf{J}\) is the TFI of the simulated signal \(s(t)\), and \(\mathbf{K}\) is the TFI of the noisy signal \(z(t)\); the bigger the PSNR is, the closer \(\mathbf{J}\) and \(\mathbf{K}\) are; \(MAX_{J}\) is the maximal value of \(\mathbf{J}\).