## 2 First-Order Primal-Dual Algorithm-Based Sparse Time-Frequency Analysis  
### Model of Sparse Time-Frequency Analysis  
The short-time Fourier transform (STFT) is one of the earliest and most widely used time-frequency analysis methods. The basic idea of the STFT can be summarized as described below. A short time window is added to the desired signal, and the local Fourier spectrum of the windowed signal is calculated. By sliding the window with respect to time, the time-frequency analysis of the signal can be carried out [20; 48]. The STFT of the continuous-time signal \(x(t)\) can be defined as follows:  
\[X(t,\omega)=\int_{-\infty}^{+\infty}x(\tau)w(\tau-t)\exp(-j\omega\tau)d\tau, \tag{1}\]  
where \(w(t)\) is the sliding window function. Correspondingly, for a discrete signal \(x(n)\) of period \(N\), where \(n=0,1,\ldots,N-1\), the discrete STFT can be defined as follows:  
Figure 1: Flowchart of the proposed approach. \[X(m,l)=\sum_{n=0}^{N-1}x(n)w(n-m)W_{N}^{nl}, \tag{2}\]  
where \(l=0,1,\ldots,N-1\), and \(W_{N}=\exp(-j2\pi/N)\). According to Equation (1), the STFT \(X(t,\omega)\) of the continuous-time signal \(x(t)\) can be regarded as a Fourier transform (FT) of the windowed signal \(x(\tau)w(\tau-t)\). On the contrary, when \(X(t,\omega)\) and window functions \(w(t)\) are given, the original signal can be directly recovered by inverse Fourier transform (IFT), which can be represented as follows:  
\[x(\tau)w(\tau-t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}X(t,\omega)\exp(j\omega \tau)d\omega. \tag{3}\]  
Correspondingly, for the windowed signal \(x_{m}(n)=x(n)w(n-m)\) of \(x(n)\) at time \(m\), the discrete STFT can be defined as follows:  
\[x_{m}(n)=\frac{1}{N}\sum_{l=0}^{N-1}X(m,l)W_{N}^{-nl}. \tag{4}\]  
The vector form of \(x_{m}(n)\) can be expressed as follows:  
\[\mathbf{x}_{m}=\mathbf{W}_{N}\mathbf{X}_{m}, \tag{5}\]  
where \(\mathbf{x}_{m}=\left[x_{m}(0),x_{m}(1),\ldots,x_{m}(N-1)\right]^{T}\) and \(\mathbf{X}_{m}=\left[X_{m}(m,0),X_{m}(m,1),\ldots,X_{m}(m,N-1)\right]^{T}\) are the windowed signal and Fourier spectrum at time \(m\), respectively; \(\mathbf{W}_{N}\) is the inverse Fourier transform matrix; and \(\left[\mathbf{W}_{N}\right]_{k,l}=1/N\cdot W_{N}^{-(k-1)(l-1)}\), where \(k,l=1,2,\ldots,N\). The short-time truncated sub-signal can be given as follows:  
\[\mathbf{y}_{m} =\mathbf{Z}_{m}\mathbf{x}_{m}=\mathbf{Z}_{m}\mathbf{W}_{N}\mathbf{X}_{m}, \tag{6}\]  
where \(\mathbf{y}_{m}\in\mathbf{R}^{U\times 1}\) is the short-time truncated sub-signal, \(\mathbf{Z}_{m}\in\mathbf{R}^{U\times N}\) is the sliding window at time \(m\), and the sliding window \(\mathbf{Z}_{m+1}\) at time \(m+1\) is the circular shift of \(\mathbf{Z}_{m}\) with the column values shifted to right by one. In Equation (6), \(\mathbf{Z}_{m}\mathbf{W}_{N}\) is a partial Fourier matrix, and its columns may be highly correlated, where this cannot guarantee the uniqueness of the solution of \(\mathbf{X}_{m}\). The windowed signal \(\mathbf{x}_{m}\) at time \(m\) is sparse in the frequency domain. Inspired by SR theory, the special spectrum \(\mathbf{X}_{m}\) can be easily obtained by adding the sparse constraint to fit the sparse prior knowledge of the spectrum. The sparse constraint problem can be modeled as Equation (7). \[\mathbf{\hat{X}}_{m}=\operatorname*{argmin}_{\mathbf{X}_{m}}\lVert\mathbf{X}_{m}\rVert_{ 0}s.t.\mathbf{y}_{m}=\mathbf{Z}_{m}\mathbf{W}_{N}\mathbf{X}_{m}, \tag{7}\]  
where \(\lVert\cdot\rVert_{0}\) denotes the L0-norm. By comparing Equations (6) and (7), we find that \(\mathbf{Z}_{m}\mathbf{W}_{N}\) plays the role of a dictionary of SR. Considering that the L0-norm minimization problem is NP-hard, the L1-norm is generally approximated to the L0-norm, i.e., the convex relaxation is considered an L1-norm minimization problem, or an interference in practical situations. Therefore, we can estimate the spectrum by solving the sparse constraint problem modeled as Equation (8).