## 2 First-Order Primal-Dual Algorithm-Based Sparse Time-Frequency Analysis
### Solution of the Model Based on the First-Order Primal-Dual Algorithm
\right.\) is the convex conjugate function of \(F(\mu\mathbf{X}_{m})\), \(\mathbf{Y}_{m}\in\mathbf{Y}\) is the dual variable, and \(\mathbf{Y}\) is the convex set defined as follows:  
\[\mathbf{Y}=\left\{\mathbf{Y}_{m}\in\mathbf{Y}\colon\|\mathbf{Y}_{m}\|_{\infty}\leq 1\right\}. \tag{11}\]  
The resolvent operator with respect to \(F^{*}\) can be obtained as follows:  
\[\mathbf{Y}_{m}{}^{(n+1)}=(\mathbf{I}+\sigma\,\partial F^{*})^{-1}\Big{(} \mathbf{Y}_{m}{}^{(n)}+\mu\sigma\overline{\mathbf{X}}_{m}{}^{(n)}\Big{)} \tag{12}\] \[=\underset{\mathbf{Y}_{m}{}^{(n+1)}}{\text{arg min}}\left\{\frac{1}{2} \|\mathbf{Y}_{m}{}^{(n+1)}-\Big{(}\mathbf{Y}_{m}{}^{(n)}+\mu\sigma\overline{\mathbf{X}}_{m }{}^{(n)}\Big{)}\|_{2}^{2}+F^{*}(\mathbf{Y}_{m}{}^{(n+1)})\right\}\.\]  
As \(F^{*}(\mathbf{Y}_{m})\) is the indicator function for a convex set, the resolvent operator with respect to \(F^{*}\) reduces to pointwise Euclidean projectors:  
\[\mathbf{Y}_{m}{}^{(n+1)}=\frac{\mathbf{Y}_{m}{}^{(n)}+\mu\sigma\overline{\mathbf{X}}_{m}{} ^{(n)}}{\text{max}(1,\big{|}\mathbf{Y}_{m}{}^{(n)}+\mu\sigma\overline{\mathbf{X}}_{m}{ }^{(n)}\big{|})}. \tag{13}\]  
The resolvent operator with respect to \(G\) can be expressed as follows:  
\[\mathbf{X}_{m}{}^{(n+1)}=(\mathbf{I}+\tau\,\partial G)^{-1}\Big{(}\mathbf{X} _{m}{}^{(n)}-\mu\tau\mathbf{Y}_{m}{}^{(n+1)}\Big{)} \tag{14}\] \[=\underset{\mathbf{X}_{m}{}^{(n+1)}}{\text{arg min}}\left\{\frac{1}{2} \|\mathbf{X}_{m}{}^{(n+1)}-\Big{(}\mathbf{X}_{m}{}^{(n)}-\mu\tau\mathbf{Y}_{m}{}^{(n+1)} \Big{)}\|_{2}^{2}+\frac{\tau}{2}\|\mathbf{Z}_{m}\mathbf{W}_{N}\mathbf{X}_{m}{}^{(n+1)}-\mathbf{ y}_{m}\|_{2}^{2}\right\}\.\]  
The resolvent operator with respect to \(G\) is a simple convex quadratic programming problem, the solution to which is given as follows:  
\[\mathbf{X}_{m}{}^{(n+1)}=\frac{\Big{(}\mathbf{X}_{m}{}^{(n)}-\mu\tau\mathbf{Y}_{m}{}^{(n+1)} \Big{)}+\tau(\mathbf{Z}_{m}\mathbf{W}_{N})^{H}\mathbf{y}_{m}}{\mathbf{I}+\tau(\mathbf{Z}_{m}\mathbf{W}_ {N})^{H}(\mathbf{Z}_{m}\mathbf{W}_{N})}. \tag{15}\]The sparse time-frequency spectrum \(\hat{\mathbf{X}}=\left[\hat{\mathbf{X}}_{0},\hat{\mathbf{X}}_{1},\cdots,\hat{\mathbf{X}}_{N-1}\right]\) of the signal can be finally obtained by solving a group of optimization problems, such as Equation (9), using the first-order primal-dual algorithm for \(m=0,1,\ldots,N-1\). The pseudocode of the STFA-PD is summarized in Algorithm 1. ```
Input:\(\mathbf{y}_{m},\ Z_{m}\mathbf{W}_{N},\ \mu,\ \tau,\ \sigma>0,\ \theta\in[0,1]\) Output:\(\mathbf{X}_{m}\) Initialize:\(\mathbf{X}_{m}{}^{(1)}=\mathbf{0},\ \mathbf{Y}_{m}{}^{(1)}=\mathbf{0},\ \mathbf{\overline{X}}_{m}{}^{(1)} \!=\mathbf{X}_{m}{}^{(1)},\ \text{{Maxiter}}=100\)
1:For\(m=0:N-1\)do
2:For\(n=1:\ \text{{Maxiter}}\)do
3:\(\mathbf{Y}_{m}{}^{(n+1)}=\frac{\mathbf{Y}_{m}{}^{(n)}+\mu\mathbf{\overline{X}}_{m}{}^{(n )}}{\max(1,\mathbf{Y}_{m}{}^{(n+1)}+\mu\mathbf{\overline{X}}_{m}{}^{(n)})}\)
4:\(\mathbf{X}_{m}{}^{(n+1)}=\frac{\left(\mathbf{X}_{m}{}^{(n)}-\mu\mathbf{r}\mathbf{Y}_{m}{}^{(n+ 1)}\right)+\tau(\mathbf{Z}_{m}\mathbf{W}_{N}){}^{H}\mathbf{y}_{m}}{I+\tau(\mathbf{Z}_{m}\mathbf{W} _{N}){}^{H}\mathbf{y}_{m}}\)
5:\(\mathbf{X}_{m}{}^{(n+1)}=\mathbf{X}_{m}{}^{(n+1)}+\theta(\mathbf{X}_{m}{}^{(n+1)}-\mathbf{X}_{m}{ }^{(n)})\)
6:End For
7:End for
8:Return\(\mathbf{X}_{m}{}^{(n+1)}\) as \(\mathbf{X}_{m}\)
```  
**Algorithm 1** Sparse time-frequency analysis method based on the first-order primal-dual algorithm (STFA-PD)