# 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
 \tag{23}\]  
\[\
abla_{\mathbf{h}^{*}}L\Big{(}\mathbf{z}^{(m)},\!\mathbf{h}\boldsymbol{ \lambda}_{1}{}^{(m)}\Big{)}=\mathbf{0}\Rightarrow\text{grad}_{\mathbf{h}^{*}}L \Big{(}\mathbf{z}^{(m)},\!\mathbf{h}\boldsymbol{\lambda}_{1}{}^{(m)}\Big{)}= \mathbf{0}, \tag{24}\]\[\
abla_{\mathbf{h}^{*}}L\left(\mathbf{z}^{(m)},\mathbf{h}\boldsymbol{ \lambda}_{1}{}^{(m)}\right) \tag{25}\] \[= \
abla_{\mathbf{h}^{*}}F\left(\mathbf{z}^{(m)},\mathbf{h}\right)+ \frac{\rho}{2}\
abla_{\mathbf{h}^{*}}\left\|\mathbf{h}-\mathbf{z}^{(m)}+ \boldsymbol{\lambda}_{1}{}^{(m)}\right\|_{2}^{2}\] \[= \left(\left(\frac{1}{K}\sum\limits_{k=1}^{K}w_{k}\mathbf{A}( \theta_{k})\mathbf{z}^{(m)}\mathbf{z}^{(m)}\mathbf{z}^{(m)}\mathbf{A}^{\text{ H}}(\theta_{k})\right)+\frac{\rho}{2}\mathbf{I}\right)\mathbf{h}-\left(\frac{ \rho}{2}\left(\mathbf{z}^{(m)}-\boldsymbol{\lambda}^{(m)}\right)\right)\] \[= \mathbf{0}.\]  
Finally, the closed-form solution can be represented as  
\[\widetilde{\mathbf{h}}^{(m+1)}=\left(\left(\frac{1}{K}\sum\limits_{k=1}^{K}w _{k}\mathbf{A}(\theta_{k})\mathbf{z}^{(m)}\mathbf{z}^{(m)}\mathbf{A}^{\text{ H}}(\theta_{k})\right)+\frac{\rho}{2}\mathbf{I}\right)^{-1}\left(\frac{\rho}{2} \left(\mathbf{z}^{(m)}-\boldsymbol{\lambda}^{(m)}\right)\right). \tag{26}\]  
That is, the updating of each point of (26) can be expressed as  
\[\mathbf{h}^{(m+1)}=\sqrt{\frac{P_{0}}{NL}}\left[\sqrt{\frac{NL}{P_{0}}} \widetilde{\mathbf{h}}_{1}^{(m+1)},\frac{\widetilde{\mathbf{h}}_{2}^{(m+1)}} {\left\|\widetilde{\mathbf{h}}_{2}^{(m+1)}\right\|_{2}^{2}},\dots,\frac{ \widetilde{\mathbf{h}}_{NL+1}^{(m+1)}}{\left\|\widetilde{\mathbf{h}}_{NL+1} \right\|_{2}^{2}}\right]. \tag{27}\]  
We could obtain \(\mathbf{z}^{(m+1)}:=\underset{\mathbf{z}\in S_{z}}{\text{argmin}}F\left( \mathbf{z},\mathbf{h}^{(m+1)}\right)+\frac{\rho}{2}\left\|\mathbf{h}^{(m+1)}- \mathbf{z}+\boldsymbol{\lambda}_{1}{}^{(m)}\right\|_{2}^{2}\) which is also a convex function problem about \(\mathbf{h}\), and its closed-form solution can be obtained by the first derivation. Similarly, the solution process of \(\mathbf{z}\) has  
\[\
abla_{z}L\left(\mathbf{z},\mathbf{h}^{(m+1)},\boldsymbol{\lambda }_{1}{}^{(m)}\right)=\
abla_{z}F\left(\mathbf{z},\mathbf{h}^{(m+1)}\right)+ \frac{\rho}{2}\
abla_{z}\left\|\mathbf{h}^{(m+1)}-\mathbf{z}+\boldsymbol{ \lambda}_{1}{}^{(m)}\right\|_{2}^{2} \tag{28}\] \[= \left(\left(\frac{1}{K}\sum\limits_{k=1}^{K}w_{k}\mathbf{A}^{ \text{H}}(\theta_{k})\mathbf{h}^{(m+1)}\mathbf{h}^{(m+1)}\mathbf{A}(\theta_{k })\right)+\frac{\rho}{2}\mathbf{I}\right)\mathbf{z}-\left(\frac{\rho}{2}\left( \mathbf{h}^{(m+1)}+\boldsymbol{\lambda}_{1}{}^{(m)}\right)\right)\] \[= \mathbf{0},\]  
Its closed-form solution has  
\[\
abla_{\mathbf{h}^{*}}L\left(\mathbf{z}^{(m)},\mathbf{h}\boldsymbol{ \lambda}_{1}{}^{(m)}\right) \tag{25}\]0  
Thus, the updating of each point of (29) can be expressed as  
\[\
abla_{\mathbf{h}^{*}}L\left(\mathbf{z}^{(m)},\mathbf{h}\boldsymbol{ \lambda}_{1}{}^{(m)}\right) \tag{25}\]1  
Once we have obtained each point \(\left(\mathbf{z}^{(m+1)},\mathbf{h}^{(m+1)}\right)\) on a given product manifold, we can update \(\boldsymbol{\lambda}\) by  
\[\
abla_{\mathbf{h}^{*}}L\left(\mathbf{z}^{(m)},\mathbf{h}\boldsymbol{ \lambda}_{1}{}^{(m)}\right) \tag{25}\]2  
We summarize the waveform design Algorithm 1 for pattern-matching (WAPM) as follows. ```
Initialize:\(\mathbf{z}^{(0)},\mathbf{h}^{(0)},\boldsymbol{\lambda}_{1}{}^{(0)},\rho\), and the tolerance \(\varepsilon\). While the termination criteria are not satisfied do. Update \(\mathbf{h}^{(m+1)}\) using (27). Update \(\mathbf{z}^{(m+1)}\) using (30). Update \(\boldsymbol{\lambda}_{1}{}^{(m+1)}\) using (31). \(m=m+1\). Endwhile
```  
**Algorithm 1**Applying manifold-ADMM to solve the problem (15).