# Integrated Radar and Communications Waveform Design Based on Multi-Symbol OFDM
## 3 IRC Waveform Design Objective
### Joint Optimization of PMEPR and \(\mathrm{PSLR}\)  
High \(\mathrm{PMEPR}\) and poor \(\mathrm{PSLR}\) are two inherent problems for OFDM-based IRC waveforms, and both can be improved by waveform design. It can be seen from (9) and (13) that both \(\mathrm{PMEPR}\) and \(\mathrm{PSLR}\) are expressions of the waveform \(s(q)\), and according to (3), we also know that if the communication symbol vector \(\mathbf{X}\) is given, \(s(q)\) is uniquely determined by  
Figure 3: CCDFs of the \(\mathrm{PMEPR}\) for OFDM waveforms with different numbers of subcarriers for 4 QAM modulation and oversampling factor \(J=4\). the reserved symbol vector \(\mathbf{C}\). Therefore, PMEPR and PSLR are the images of the reserved symbol vector \(\mathbf{C}\) in two functions respectively, i.e.,  
\[f_{1}:\quad\mathbb{C}^{MN}\to\mathbb{R},\quad\mathbf{C}\mapsto\text{PMEPR}\qquad \qquad f_{2}:\quad\mathbb{C}^{MN}\to\mathbb{R},\quad\mathbf{C}\mapsto\text{PSLR} \tag{14}\]  
It is obvious from (14) that there is a new mapping \(O\) as  
\[O:\quad\mathbb{C}^{MN}\to\mathbb{R},\quad\mathbf{C}\mapsto\alpha\cdot 10\log_{10} \text{PMEPR}-10\log_{10}\text{PSLR}^{2} \tag{15}\]  
where \(\alpha(\alpha>0)\) is a constant weight to balance the relative magnitude between PMEPR and PSLR. Based on (15), a new joint optimization objective function is proposed, namely,  
\[ \begin{split} O(\mathbf{C})&=\alpha\cdot 10\log_{10} \text{PMEPR}-10\log_{10}\text{PSLR}^{2}\\ &=\alpha\cdot 10\log_{10}\frac{\max\limits_{0\leq q\leq MQ-1}|s(q)|^{ 2}}{E[|s(q)|^{2}]}+10\log_{10}\frac{\max\limits_{k\in\Omega}\{|R(k)|^{2}\}}{( MQ)^{2}(E[|s(q)|^{2}])^{2}}\end{split}  \tag{16}\]  
where PMEPR and PSLR are both calculated in dB. Therefore, the problem of interest in this paper is the following \(O(\mathbf{C})\) minimization problem:  
\[\min\limits_{\mathbf{C}\in\mathbb{C}^{MN}}O(\mathbf{C}) \tag{17}\]  
which includes both the PMEPR minimization problem and the PSLR maximization problem, and the performance of PMEPR and PSLR can be traded off by \(\alpha\). It can be found from (16) that the objective function is a complex function of \(s(q)\), and finding the exact solution of (17) is a tedious task. Thus, it is necessary to convert the objective function to an easily solvable form. Since the instantaneous power \(|s(q)|^{2}\) is a real-valued vector, the \(L_{p}\)-norm for it is defined as  
\[\||s(q)|^{2}\|_{p} = \left(\sum\limits_{q=0}^{MQ-1}\left(|s(q)|^{2}\right)^{p}\right) ^{1/p},\quad\forall p\geq 1,p\in\mathbb{R} \tag{18}\]  
Note that the above expression is monotonically decreasing versus \(p\), and has the commonly adopted peak power measure as its special case:  
\[\||s(q)|^{2}\|_{\infty}=\lim\limits_{p\to\infty}\||s(q)|^{2}\|_{p}=\max\limits _{0\leq q\leq MQ-1}|s(q)|^{2} \tag{19}\]  
Therefore, the objective function \(O(\mathbf{C})\) can be transformed by an approximation method based on the \(L_{p}\)-norm approximating the infinite-norm, and then the new optimization problem is given by  
\[ \begin{split}\min\limits_{\mathbf{C}\in\mathbb{C}^{MN}}\widetilde{O }(\mathbf{C})&=\ \alpha\cdot 10\log_{10}\frac{\left(\sum\limits_{q=0}^{MQ-1}|s(q)|^{ 2p}\right)^{1/p}}{E[|s(q)|^{2}]}+10\log_{10}\frac{\left(\sum\limits_{k\in \Omega}|R(k)|^{2p}\right)^{1/p}}{(MQ)^{2}(E[|s(q)|^{2}])^{2}}\\ &=\!\frac{10}{p}\!\left(\alpha\log_{10}\!\left(\sum\limits_{q=0} ^{MQ-1}|s(q)|^{2p}\right)+\log_{10}\!\left(\sum\limits_{k\in\Omega}|R(k)|^{2p }\right)\right)\!-\\ &\ 10(\alpha+2)\log_{10}\!\left(E[|s(q)|^{2}]\right)-20\log_{10} (MQ)\end{split}  \tag{20}\]  
where the peak power of \(R(k)\) in (16) is also approximated by \(L_{p}\)-norm. The constant terms 10 and \(20\log_{10}(MQ)\) in (20) can be ignored since they do not contribute to the optimization problem. Then, the optimization problem in (20) can be rewritten as  
\[