# Modulus Waveform Design Based on Manifold ADMM Idea in Dual-Function Radar-Communication System
## 3 DFRC Waveform Design by Alternating Sequential Optimization Manifold-ADMM Idea
### Auxiliary Variable Calculation  
To solve the problem of (13), we decouple the fourth-order polynomial by introducing an auxiliary primal variable \(\mathbf{h}\in\mathbb{C}^{NL+1}\), which is defined in the manner of the Riemannian manifold [21]; thus, the problem (9) is equivalent to  
\[ \begin{array}{ll}\underset{\mathbf{z,h}}{\text{min}}&F(\mathbf{z,h})=\frac{1} {\mathcal{K}}\sum\limits_{k=1}^{K}w_{k}\Big{|}\mathbf{h}^{\text{H}}\mathbf{A}( \theta_{k})\mathbf{z}\Big{|}^{2}\\ s.t.&\mathbf{h}\!-\!\mathbf{z}\!=\!\!\mathbf{0},\\ &\mathbf{h}\in S_{h},\mathbf{z}\in S_{z},\end{array}  \tag{15}\]  
where \(S_{h},S_{z}\) denotes the Riemannian manifold sets with CM. Notice that the objective function of (15) is bi-convex, i.e., it is convex in \(\mathbf{h}\) for each \(\mathbf{z}\) and convex in \(\mathbf{z}\) for each \(\mathbf{h}\). So that we can use the ADMM idea in Riemannian manifold framework, to solve (15), we introduce one dual variable \(\boldsymbol{\lambda}_{1}\in\mathbb{C}^{NL+1}\), and then the augmented Lagrange function can be expressed as  
\[L(\mathbf{z,h,\lambda}_{1})=F(\mathbf{z,h})+\frac{\rho}{2}\|\mathbf{h}\!-\! \mathbf{z}\!+\!\boldsymbol{\lambda}_{1}\|_{2}^{2}. \tag{16}\]  
At the \((m+1)\)-th iteration, the manifold-ADMM consists of the following iterations:  
\[\mathbf{h}^{(m+1)}:=\underset{\mathbf{h}\in S_{h}}{\text{argmin}}L\left( \mathbf{z}^{(m)},\mathbf{h}\boldsymbol{\lambda}_{1}{}^{(m)}\right), \tag{17}\]  
\[\mathbf{z}^{(m+1)}:=\underset{\mathbf{z}\in S_{z}}{\text{argmin}}L\left( \mathbf{z,h}^{(m+1)},\boldsymbol{\lambda}_{1}{}^{(m)}\right), \tag{18}\]  
\[\boldsymbol{\lambda}_{1}{}^{(m+1)}:=\boldsymbol{\lambda}_{1}{}^{(m)}\!+\! \mathbf{h}^{(m+1)}\!-\!\mathbf{z}^{(m+1)}. \tag{19}\]  
Taking (16) into (17), we could obtain  
\[\mathbf{h}^{(m+1)}:=\underset{\mathbf{h}\in S_{h}}{\text{argmin}}\!F\Big{(} \mathbf{z}^{(m)},\mathbf{h}\Big{)}+\frac{\rho}{2}\Big{\|}\mathbf{h}\!-\! \mathbf{z}^{(m)}+\boldsymbol{\lambda}_{1}{}^{(m)}\Big{\|}_{2}^{2}, \tag{20}\]  
Since the problem of (20) is convex about \(\mathbf{h}\), its closed-form solution can be obtained by the first derivative condition  
\[\mathbf{0}=\text{grad}_{\mathbf{h}^{*}}L\left(\mathbf{z}^{(m)},\!\mathbf{h} \boldsymbol{\lambda}_{1}{}^{(m)}\right), \tag{21}\]  
where \(\text{grad}_{\mathbf{h}^{*}}L\left(\mathbf{z}^{(m)},\!\mathbf{h}\boldsymbol{ \lambda}_{1}{}^{(m)}\right)\) is the Riemannian gradient of augmented Lagrange function with respect to \(\mathbf{h}^{*}\) in Riemannian space. \(S_{h}\) is a Riemannian submanifold of Euclidean space \(\mathbb{C}^{NL+1\times 1}\). The relationship between Riemannian gradient and Euclidean gradient can be obtained by orthogonal projection of Euclidean gradient \(\
abla_{\mathbf{h}^{*}}L\left(\mathbf{z}^{(m)},\!\mathbf{h}\boldsymbol{ \lambda}_{1}{}^{(m)}\right)\) to tangent space \(T_{h}S_{h}\), i.e.,  
\[ \begin{array}{ll}\text{grad}_{\mathbf{h}^{*}}L\left(\mathbf{z}^{(m)},\! \mathbf{h}\boldsymbol{\lambda}_{1}{}^{(m)}\right)&=\text{Proj}_{\mathbf{h}^{*} }\Big{(}\
abla_{\mathbf{h}^{*}}L\left(\mathbf{z}^{(m)},\!\mathbf{h} \boldsymbol{\lambda}_{1}{}^{(m)}\right)\Big{)}\\ &=\
abla_{\mathbf{h}^{*}}L\left(\mathbf{z}^{(m)},\!\mathbf{h}\boldsymbol{ \lambda}_{1}{}^{(m)}\right)-\Re\Big{\{}\
abla_{\mathbf{h}^{*}}L\left(\mathbf{ z}^{(m)},\!\mathbf{h}\boldsymbol{\lambda}_{1}{}^{(m)}\right)\odot\mathbf{h}\Big{\}} \odot\mathbf{h}^{*}.\end{array}  \tag{22}\]  
Solving the European gradient, we could have  
\[ \begin{array}{l}\
abla_{\mathbf{h}^{*}}F\Big{(}\mathbf{z}^{(m)},\!\mathbf{h} \Big{)}=\left(\frac{1}{\mathcal{K}}\sum\limits_{k=1}^{K}w_{k}\mathbf{A}(\theta _{k})\mathbf{z}^{(m)}\mathbf{z}^{(m)}\mathbf{h}^{\text{H}}\mathbf{A}^{\text{H} }(\theta_{k})\right)\!\mathbf{h},\\ \
abla_{\mathbf{h}^{*}}\Big{\|}\mathbf{h}\!-\!\mathbf{z}^{(m)}+\boldsymbol{ \lambda}_{1}{}^{(m)}\Big{\|}_{2}^{2}=\mathbf{h}-\left(\mathbf{z}^{(m)}- \boldsymbol{\lambda}_{1}{}^{(m)}\right),\end{array}