# Joint Transmit Waveform and Receive Mismatched Filter Design to Suppress Range Sidelobe
## 2 Design of NLFM-CPM Integrated Waveform and Signal Processing Methods
### EWMF Design Based on the ADMM Algorithm  
Based on the minimum ISL criterion and subject to constraints on the mainlobe gain, it is framed as  
\[ \begin{split}\underset{\mathbf{h}}{\text{min}}& \mathbf{h}^{*}\mathbf{B}\mathbf{h}\\ \text{s.t}&\mathbf{x}_{0}^{*}\mathbf{h}=1\end{split}  \tag{26}\]  
In (26), the constrained mainlobe includes only one point. If the constrained mainlobe range is larger, the linear constraints can be extended to multiple range cells. Assuming the constraint on the mainlobe spans \(L\) range cells near the main peak, covering \(k_{l}\) range cells to the left and \(k_{r}\) range cells to the right, the constrained mainlobe range can be represented as  
\[ \begin{split}\mathbf{C}&=\text{span}(\mathbf{x}_{i }),i\in\mathfrak{G}\\ \mathfrak{G}&=[-k_{l},-(k_{l}-1),\cdots-1,0,1,\cdots,k_{r}]\\ k_{r}-k_{l}+1&=L.\end{split}  \tag{27}\]  
where \(i\in\mathfrak{G}\) is the mainlobe constraint matrix. (26) can be extended to the following optimization problem  
\[ \begin{split}\underset{\mathbf{h}}{\text{min}}& \mathbf{h}^{*}\mathbf{B}\mathbf{h}\\ \text{s.t}&\mathbf{C}^{*}\mathbf{h}=\mathbf{g}\end{split}  \tag{28}\]  
where \(\mathbf{g}\) is a constant vector of length \(L\times 1\) corresponding to the gain constraint. To regulate the LPG, the design problem incorporates constraints on waveform energy, filter energy, and pulse compression peak level. The filter's energy is controlled through a direct constraint [29], while the waveform is bound by a constant modulus ensuring total energy equals \(N\)[30; 31]. The peak constraint, \(\mathbf{h}^{\text{H}}\mathbf{x}\), depends on both the waveform and the filter. The objective function minimizes the squared deviation between the peak and a predefined threshold, \(|\mathbf{h}^{\text{H}}\mathbf{x}-b_{\text{max}}|^{2}\), where \(b_{\text{max}}\) is the target peak level. (28) is rewritten as  
\[ \begin{split}\underset{\mathbf{h}}{\text{min}}& \Big{\|}\mathbf{B}^{1/2}\mathbf{h}\Big{\|}_{2}^{2}+\mu|\mathbf{h}^{H} \mathbf{x}-b_{\text{max}}|^{2}\\ \text{s.t}&\mathbf{C}^{*}\mathbf{h}=\mathbf{g}\\ \mathbf{x}^{H}\mathbf{h}&=\mathbf{x}^{H}\mathbf{x}\\ \mathbf{x}^{H}\mathbf{E}_{n}\mathbf{x}&\leq\eta,n=1, 2,\ldots,N\end{split}  \tag{29}\]  
where the constraint \(\mathbf{x}^{H}\mathbf{h}=\mathbf{x}^{H}\mathbf{x}\) is mainly included to discard the null solution, \(\mathbf{g}\) is a constant vector of length \(L\) corresponding to the gain constraint, \(b_{\text{max}}\) is the predefined peak level, \(\mu\) denotes the weight between the two objective functions, and the matrixes \(\mathbf{E}_{n}\) is defined by \[\mathbf{E}_{n}(i,j)= \begin{cases}1&i=j=n\\ 0&\text{else}.\end{cases}  \tag{30}\]  
To transform the given optimization problem into a second order cone problem (SOCP), we need to appropriately reformulate the objective function and constraints. First, expand the absolute value term by introducing two new variables, \(\mathbf{v}=\mathbf{B}^{1/2}\mathbf{h}\) and \(t=|\mathbf{v}^{\text{H}}\mathbf{B}^{-1/2}\mathbf{x}-b_{\text{max}}|\). The objective function is rewritten as  
\[ \begin{split}&\min_{\mathbf{v},\mathbf{u}}\lVert\mathbf{v}\rVert _{2}^{2}+\mu t^{2}\\ &\text{s.t }\mathbf{C}^{*}\mathbf{B}^{-1/2}\mathbf{v}=\mathbf{g}\\ &\mathbf{x}^{H}\mathbf{B}^{-1/2}\mathbf{v}=\mathbf{x}^{H}\mathbf{x }\\ & t\geq\lVert\mathbf{u}\rVert_{2}\\ &\mathbf{u}^{H}\mathbf{u}\leq z\\ & z\leq|\mathbf{v}^{H}\mathbf{B}^{-1/2}\mathbf{x}-b_{\text{max}}|^{2}\\ &\mathbf{x}^{H}\mathbf{E}_{n}\mathbf{x}\leq\eta,n=1,2,\dots,N\end{split}  \tag{31}\]  
The ADMM method is ideal for this problem as it simplifies complex optimization tasks by breaking them into manageable subproblems and managing various constraints, including second-order cone and nonlinear equality constraints, by incorporating them into the augmented Lagrangian function. It is expressed as  
\[