# Integrated Radar and Communications Waveform Design Based on Multi-Symbol OFDM
## 4 Proposed Algorithm and Its Complexity
### Update Rules and Step Size  
It can be seen from (34) that it is an analytical expression for \(\mathbf{C}\), which provides strong support for CGA to be used to solve the joint optimization problem we established. In the iterative process, the \(l\)-th iteration point is assumed to be \(\mathbf{C}^{l}\), then it is updated by the new iteration as  
\[\mathbf{C}^{l+1}=\mathbf{C}^{l}+\mu^{l}\mathbf{d}^{l} \tag{35}\]  
where \(\mu^{l}\) represents the step size and \(\mathbf{d}^{l}\) represents the descent direction. The step size is recalculated at each iteration and plays a crucial role in the convergence rate of our algorithm. The classical line search method needs to compute the objective function many times to obtain the step size, which makes it have a high time cost [38]. In this paper, we will directly solve the step size by establishing a minimization problem about the step size, i.e., minimize the objective function \(\widetilde{O}(\mathbf{C})\) at the \((l+1)\)-th iteration point to obtain the optimal step size, as follows:  
\[ \begin{split}\min_{\mu^{l}}W(\mu^{l})&=\widetilde{ O}(\mathbf{C}^{l+1})\\ &=\frac{1}{p}\Big{[}\alpha\log_{10}f(\mathbf{C}^{l+1})+\log_{10}g( \mathbf{C}^{l+1})\Big{]}-(\alpha+2)\log_{10}L(\mathbf{C}^{l+1})\end{split}  \tag{36}\]  
Similar to (23), the first derivative \(W^{\prime}(\mu^{l})\) is given by  
\[W^{\prime}(\mu^{l})=\frac{1}{\ln 10}\left[\frac{1}{p}\left(\alpha\frac{f^{ \prime}(\mathbf{C}^{l+1})}{f(\mathbf{C}^{l+1})}+\frac{g^{\prime}(\mathbf{C}^{ l+1})}{g(\mathbf{C}^{l+1})}\right)-(\alpha+2)\frac{L^{\prime}(\mathbf{C}^{l+1})}{L( \mathbf{C}^{l+1})}\right] \tag{37}\]  
According to the expressions of \(f(\mathbf{C})\), \(g(\mathbf{C})\), and \(L(\mathbf{C})\), the derivatives of \(f(\mathbf{C}^{l+1})\), \(g(\mathbf{C}^{l+1})\) and \(L(\mathbf{C}^{l+1})\) with respect to \(\mu^{l}\) are obtained as follows, respectively,  
\[f^{\prime}(\mathbf{C}^{l+1})=p\sum_{q=0}^{MQ-1}|s^{l+1}(q)|^{2(p-1)}\frac{d|s^ {l+1}(q)|^{2}}{d\mu^{l}}=2p\sum_{q=0}^{MQ-1}s^{l+1}(q)s_{re}(q) \tag{38}\]  
where \(s_{re}(q)=\text{Re}\big{\{}s^{l+1}(q)(D^{l}(q))^{*}\big{\}}\) and \(D^{l}(q)=\boldsymbol{\Phi}_{q}\mathbf{d}^{l}\), \(\boldsymbol{\Phi}_{q}\) represents the \(q\)-th row of \(\boldsymbol{\Phi}\). \[g^{\prime}(\mathbf{C}^{l+1})=p\sum_{k\in\Omega}|R^{l+1}(k)|^{2(p-1)}\frac{d|R^ {l+1}(k)|^{2}}{d\mu^{l}}=2p\sum_{k\in\Omega}\tilde{R}^{l+1}(k)R_{re}(k) \tag{39}\]  
where \(R_{re}(k)=\text{Re}\Big{\{}(R^{l+1}(k))^{*}\Big{(}R_{s,D^{l}}(k)+R_{D^{l},s}( k)\Big{)}\Big{\}}\), \(R_{s,D^{l}}(k)=\sum\limits_{q=0}^{MQ-1-k}s^{l+1}(q)(D^{l}(q+k))^{*}\) is the cross-correlation function of \(s^{l+1}(q)\) and \(D^{l}(q)\), and \(R_{D^{l},s}(k)=\sum\limits_{q=0}^{MQ-1-k}(s^{l+1}(q+k))^{*}D^{l}(q)\) is the cross-correlation function of \(D^{l}(q)\) and \(s^{l+1}(q)\). \[ \begin{split} L^{\prime}(\mathbf{C}^{l+1})&=\frac{1}{M} \sum\limits_{m=0}^{M-1}\sum\limits_{n\in\mathcal{R}}\left(|\mathbf{C}^{l+1}_{n, m}|^{2}\right)^{\prime}=\frac{1}{M}\sum\limits_{m=0}^{M-1}\sum\limits_{n\in\mathcal{R}} \left(\mathbf{C}^{l+1}_{n,m}{{}^{\prime}}^{\prime}\mathbf{C}^{l+1}_{n,m}{{}^{ \prime}}+\mathbf{C}^{l+1}_{n,m}{{}^{\prime}}\left(\mathbf{C}^{l+1}_{n,m}{{}^{ \prime}}\right)^{\prime}\right)\\ &=\frac{2}{M}A_{d}\end{split}  \tag{40}\]  
where \(A_{d}=\sum\limits_{m=0}^{M-1}\sum\limits_{n\in\mathcal{R}}\text{Re}\Big{\{} \mathbf{C}^{l+1}_{n,m}{{}^{\prime}}^{*}\Big{\}}\). By substituting (38)-(40) into (37), we have  
\[W^{\prime}(\mu^{l})=\frac{2}{\ln 10}\left(\alpha\frac{\sum\limits_{q=0}^{MQ-1} {{}^{\prime}}^{l+1}(q)s_{re}(q)}{f(\mathbf{C}^{l+1})}+\frac{\sum\limits_{k\in \Omega}\hat{R}^{l+1}(k)R_{re}(k)}{g(\mathbf{C}^{l+1})}-\frac{\alpha+2}{M}\frac {A_{d}}{L(\mathbf{C}^{l+1})}\right) \tag{41}\]  
In (41), \(W^{\prime}(\mu^{l})\) is a cumbersome high-order function with respect to \(\mu^{l}\), which makes it a difficult task to directly compute the step size. Thus, the [PERSON]'s downhill method [23] is used to solve the approximate solution of the step size.