# 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
### DFRC Waveform Design  
The objective function of (14) can be represented as  
\[ \begin{split}& w\cdot\|\mathbf{H}\mathbf{X}-\mathbf{S}\|_{F}^{2}+(1-w) \cdot\|\mathbf{X}-\mathbf{X}_{0}\|_{F}^{2}\\ &=\bigg{\|}\big{[}\sqrt{w}\mathbf{H}^{\mathsf{T}},\sqrt{1-w} \mathbf{I}_{N}\big{]}^{\mathsf{T}}\mathbf{X}-\Big{[}\sqrt{w}\mathbf{S}^{ \mathsf{T}},\sqrt{1-w}\mathbf{X}_{0}^{\mathsf{T}}\Big{]}^{\mathsf{T}}\bigg{\|}_ {F}^{2}.\end{split}  \tag{32}\]  
We define \(\mathbf{A}=\big{[}\sqrt{w}\mathbf{H}^{\mathsf{T}},\sqrt{1-w}\mathbf{I}_{N} \big{]}^{\mathsf{T}}\),\(\mathbf{B}=\big{[}\sqrt{w}\mathbf{S}^{\mathsf{T}},\sqrt{1-w}\mathbf{X}_{0}^{ \mathsf{T}}\big{]}^{\mathsf{T}}\), then (32) can be expressed as  
\[ \begin{split}&\underset{\mathbf{X}}{\text{min}}\|\mathbf{A} \mathbf{X}-\mathbf{B}\|_{F}^{2},\\ & s.t.\big{|}x_{n,l}\big{|}=\sqrt{\frac{P_{0}}{NL}},n=1,2,\ldots, N;l=1,2,\ldots,L,\end{split}  \tag{33}\]  
and we could further have  
\[ \begin{split}\|\mathbf{A}\mathbf{X}-\mathbf{B}\|_{F}^{2}& =\text{tr}\Big{(}(\mathbf{A}\mathbf{X}-\mathbf{B})^{\mathsf{H}}( \mathbf{A}\mathbf{X}-\mathbf{B})\Big{)}\\ &=\text{tr}\Big{(}\mathbf{X}^{\mathsf{H}}\mathbf{A}^{\mathsf{H}} \mathbf{A}\mathbf{X}\Big{)}-\text{tr}\Big{(}\mathbf{X}^{\mathsf{H}}\mathbf{A}^ {\mathsf{H}}\mathbf{B}\Big{)}-\text{tr}\big{(}\mathbf{B}^{\mathsf{H}}\mathbf{ A}\mathbf{X}\big{)}+\text{tr}\big{(}\mathbf{B}^{\mathsf{H}}\mathbf{B}\big{)}. \end{split}  \tag{34}\]  
Letting \(\mathbf{D}\)=\(\mathbf{A}^{\mathsf{H}}\),\(\mathbf{A}\),\(\mathbf{G}\)=\(\mathbf{A}^{\mathsf{H}}\),\(\mathbf{B}\), (34) can be expressed as  
\[ \begin{split}&\underset{\mathbf{X}}{\text{min}}\text{tr}\Big{(} \mathbf{X}^{\mathsf{H}}\mathbf{D}\mathbf{X}\Big{)}-2\Re\Big{(}\text{tr}\Big{(} \mathbf{X}^{\mathsf{H}}\mathbf{G}\Big{)}\Big{)}.\\ & s.t.\big{|}x_{n,l}\big{|}=\sqrt{\frac{P_{0}}{NL}},n=1,2,\ldots, N;l=1,2,\ldots,L,\end{split}  \tag{35}\]  
Similarly, we introduce an auxiliary primal variable \(\mathbf{Y}\in\mathbb{C}^{N\times L}\), and (35) can be expressed as  
\[ \begin{split}&\underset{\mathbf{X}\in\mathbb{S}_{X}\mathbf{X} \mathbf{Y}\in\mathbb{S}_{X}}{\text{min}}F(\mathbf{X}_{\mathsf{Y}})=\text{tr} \Big{(}\mathbf{Y}^{\mathsf{H}}\mathbf{D}\mathbf{X}\Big{)}-2\Re\Big{(}\text{tr} \Big{(}\mathbf{Y}^{\mathsf{H}}\mathbf{G}\Big{)}\Big{)}.\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\big{|}x_{n,l} \big{|}=\sqrt{\frac{P_{0}}{NL}},n=1,2,\ldots,N;l=1,2,\ldots,L,\end{split}  \tag{36}\]  
where  
\[ \begin{split}& S_{X}=\bigg{\{}\mathbf{X}\in\mathbb{C}^{N\times L }\bigg{|}\left|x_{n,l}\right|=\sqrt{\frac{P_{0}}{NL}},n=1,2,\ldots,N;l=1,2, \ldots,L,\bigg{\}},\\ & S_{Y}=\bigg{\{}\mathbf{Y}\in\mathbb{C}^{N\times L}\bigg{|} \left|y_{n,l}\right|=\sqrt{\frac{P_{0}}{NL}},n=1,2,\ldots,N;l=1,2,\ldots,L \bigg{\}}.\end{split}  \tag{37}\]  
Here, we also use the manifold-ADMM algorithm to solve the problem of (36). Obviously, the augmented Lagrange function can be expressed as  
\[L(\mathbf{X}\text{,}\mathbf{Y}\text{,}\boldsymbol{\lambda})=F(\mathbf{X}_{ \mathsf{Y}}\text{)}+\frac{\rho}{2}\|\mathbf{Y}\text{-}\mathbf{X}+\boldsymbol{ \lambda}\|_{2}^{2}. \tag{38}\]  
Taking (38) into (17),  
\[\mathbf{Y}^{(m+1)}:=\underset{\mathbf{Y}\in\mathbb{S}_{Y}}{\text{argmin}}F \Big{(}\mathbf{X}^{(m)}\text{,}\mathbf{Y}\Big{)}+\frac{\rho}{2}\Big{\|} \mathbf{Y}\text{-}\mathbf{X}^{(m)}+\boldsymbol{\lambda}^{(m)}\Big{\|}_{2}^{2}. \tag{39}\]  
Since (39) is a convex function problem about \(\mathbf{Y}\), and the closed-form solution can be obtained by the first derivation on \(\mathbf{y}\), i.e.,  
\[ \begin{split}\
abla_{\mathbf{y}^{*}}L\Big{(}\mathbf{x}^{(m)} \text{,}\mathbf{y}\text{,}\boldsymbol{\lambda}^{(m)}\Big{)}&=\
abla _{Y^{*}}F\Big{(}\mathbf{X}^{(m)}\text{,}\mathbf{Y}\Big{)}+\frac{\rho}{2} \
abla_{Y^{*}}\Big{\|}\mathbf{Y}\text{-}\mathbf{X}^{(m)}+\boldsymbol{ \lambda}^{(m)}\Big{\|}_{2}^{2}\\ &=\frac{\rho}{2}\mathbf{Y}\text{-}\Big{(}\frac{\rho}{2}\Big{(} \mathbf{X}^{(m)}-\boldsymbol{\lambda}^{(m)}\Big{)}-\mathbf{D}\mathbf{X}\text{+} \mathbf{2}\mathbf{G}\Big{)}=\mathbf{0},\end{split}