# 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
\begin{split}& L(\mathbf{v},t,\mathbf{u},\boldsymbol{\lambda}, \boldsymbol{\gamma},\mu_{1},\mu_{2},\
u,\rho,\mu)=\lVert\mathbf{v}\rVert_{2}^{2 }+\mu t^{2}\\ &\quad+\frac{\rho}{2}\lVert\mathbf{C}^{*}\mathbf{B}^{-1/2} \mathbf{v}-\mathbf{g}+\boldsymbol{\lambda}\rVert_{2}^{2}-\frac{\rho}{2} \lVert\boldsymbol{\lambda}\rVert_{2}^{2}\\ &\quad+\frac{\rho}{2}\left(|\mathbf{x}^{H}\mathbf{B}^{-1/2} \mathbf{v}-\mathbf{x}^{H}\mathbf{x}+\gamma|^{2}-|\gamma|^{2}\right)\\ &\quad+\frac{\rho}{2}\left((t-\lVert\mathbf{u}\rVert_{2}+\mu_{1} )^{2}-\mu_{1}^{2}\right)\\ &\quad+\frac{\rho}{2}\left(\left|\mathbf{u}^{H}\mathbf{u}-|\mathbf{ v}^{H}\mathbf{B}^{-1/2}\mathbf{x}-b_{\text{max}}|^{2}+\mu_{2}\right|^{2}-|\mu_{2}|^{2} \right)\\ &\quad+\sum_{n=1}^{N}\frac{\rho}{2}\left((\mathbf{x}^{H}\mathbf{E }_{n}\mathbf{x}-\eta+\
u_{n})^{2}-\
u_{n}^{2}\right)\end{split}  \tag{32}\]  
where \(\rho\) is called the penalty parameter. To write the update iterations for this problem in the style of ADMM, we need to update \(\mathbf{v}\), \(t\), \(\mathbf{u}\), \(\boldsymbol{\lambda}\), \(\boldsymbol{\gamma}\), \(\boldsymbol{\mu}\), and \(\boldsymbol{\
u}\) iteratively. Here is a general outline of the update steps:  
\[ \begin{cases}\mathbf{v}^{k+1}=\arg\min_{\mathbf{v}}L(\mathbf{v},t^{k}, \mathbf{u}^{k},\boldsymbol{\lambda}^{k},\boldsymbol{\gamma}^{k},\mu_{1}^{k}, \mu_{2}^{k},\
u^{k},\rho,\mu)\\ \mathbf{u}^{k+1}=\mathbf{v}^{+1}+\mu_{2}^{k}\\ \mathbf{\lambda}^{k+1}=\mathbf{v}^{k+1}-\mathbf{g}+\boldsymbol{\lambda}^{k}\\ \boldsymbol{\gamma}^{k+1}=\mathbf{x}^{H}\mathbf{B}^{-1/2}\mathbf{v}^{k+1}- \mathbf{x}^{H}\mathbf{x}+\boldsymbol{\gamma}^{k}\\ \mu_{1}^{k+1}=t^{k+1}-\lVert\mathbf{u}\rVert_{2}+\mu_{1}^{k}\\ \mu_{2}^{k+1}=\mathbf{u}_{k+1}^{H}\mathbf{u}_{k+1}-\left|\mathbf{v}^{k+1} \mathbf{B}^{-1/2}\mathbf{x}-b_{\text{max}}\right|^{2}+\mu_{2}^{k}\\ \
u_{n}^{k+1}=\mathbf{x}^{H}\mathbf{E}_{n}\mathbf{x}-\eta+\
u_{n}^{k},\quad n= 1,2,\dots,N\end{cases}  \tag{33}\]  
where \(k\) represents the iteration index. First, the update for \(\mathbf{v}\) is addressed. By optimizing \(\mathbf{v}\) while keeping the auxiliary and dual variables constant, its minimization is derived through partial derivatives. Next, with \(\mathbf{v}\), \(t\), and other variables held constant, the auxiliary variable \(\mathbf{u}\) is updated using the latest values. The analytical solution for \(\mathbf{x}\) is given by  
\[\mathbf{x}^{k+1}=\mathbf{B}^{1/2}\left(\frac{b_{\max}\pm\sqrt{\mathbf{u}_{k+1}^ {H}\mathbf{u}_{k+1}-\Delta\mu_{2}}}{\mathbf{v}^{k+1}}\right). \tag{34}\]  
Thirdly, \(\mathbf{h}\) is updated for the fixed \(\mathbf{x}\), \(\mathbf{v}\), and other variables fixed. The analytical solution of \(\mathbf{h}\) is expressed as:  
\[\mathbf{h}^{k+1}=\frac{\mathbf{v}^{k+1}+\rho(\mathbf{B}^{-1/2}\mathbf{C} \mathbf{g}+\mathbf{B}^{-1/2}\mathbf{x}_{k+1}\mathbf{x}_{k+1}^{H}\mathbf{x}_{ k+1})}{\mathbf{B}\Big{(}\mathbf{I}+\rho(\mathbf{B}^{-1/2}\mathbf{C}\mathbf{C}^{*} \mathbf{B}^{-1/2}+\mathbf{B}^{-1/2}\mathbf{x}_{k+1}\mathbf{x}_{k+1}^{H} \mathbf{B}^{-1/2})\Big{)}} \tag{35}\]  
Based on prior knowledge of the environmental range, the desired reference template sidelobe shape is predefined. The desired reference template sidelobe shape is denoted by \(D(i)\), where \(i\) represents the \(i\)-th range cell. We further define the filter weight based on the environmental reference template as \(w(i)\). When \(i\in\mathfrak{G}\), \(w(i)=0\). When \(i\
otin\mathfrak{G}\), \(w(i)\) can be expressed as follows:  
\[w^{k+1}(i)=\ w^{k}(i)+\frac{\lambda w^{k}(i)\Big{[}|y_{k}(i)|-\mathrm{Pr}^{k} D(i)\Big{]}}{\mathrm{Pr}^{k}D(i)}+\beta\cdot(w^{k}(i)-w^{k-1}(i)) \tag{36}\]  
where \(\lambda\) is an iteration coefficient, \(\mathrm{Pr}^{k}\mathrm{D}(i)\) is the extent to which the reference template is linearly scaled to the minimum peak level, known as the absolute level [32], and \(\beta\) is a momentum coefficient used to smooth the iteration process and prevent getting stuck in local optima. Compare the level of a certain range cell with the levels of its adjacent cells on both sides.