## 2 Signal Model in the 5G NR System  
In the 5G NR system, the communication information is transmitted into the wireless frame, as shown in Figure 1, which contains 10 subframes. Each subframe consists of \(2^{\mu}\) slots, where \(\mu\) is one of the parameters in the 5G NR system. It can take values of 0, 1, 2, 3, or 4. For each slot, there are 14 completed OFDM symbols with lengths of \(T_{s}\). Moreover, the completed OFDM symbol can be divided into two parts, the effective OFDM symbol with length \(T\) and the cyclic prefix (CP) with length \(T_{cp}\). Specifically, \(T_{s}=T_{cp}+T\), \(T_{cp}=7\%\times T\), and \(T=1/\Delta f\), where \(\Delta f=2^{\mu}\times 15\) kHz is the subcarrier interval. Hence, there are totally \(N=10\times 14\times 2^{\mu}\) OFDM symbols within a wireless frame. The baseband signal \(s_{n}(t)\) of the \(n\)th OFDM symbol within a wireless frame can be expressed as  
\[s_{n}(t)=\sum_{m=0}^{N_{c}-1}\alpha_{m,n}e^{2\pi m\Delta f(t-nT_{s})}\mathrm{ rect}\left(\frac{t-nT_{s}}{T_{s}}\right), \tag{1}\]  
where \(\alpha_{m,n}\) is the communication information modulated on the \(m\)th subcarrier of the \(n\)th OFDM symbol; \(n=0,\ 1,\ \ldots,\ N-1\), \(N_{c}\) is the number of subcarriers; and \(\mathrm{rect}(t)\) is the rectangular function, defined as  
\[\mathrm{rect}(t)= \begin{cases}1,&0\leq t\leq 1\\ 0,&\mathrm{otherwise}.\end{cases}  \tag{2}\]  
Then, the final transmitted OFDM waveform can be given by  
\[\overline{s}_{n}(t)=s_{n}(t)e^{2\pi f_{0}t}, \tag{3}\]  
Figure 1: 5G NR frame structure. where \(f_{0}\) is the carrier frequency. According to the 5G communication protocol [31], 5G communication supports the time division duplex (TDD). The TDD is divided into the uplink mode and downlink mode. When the 5G BS works in the uplink mode, it receives the OFDM communication signals sent by the users. When the BS works in the downlink mode, it transmits OFDM communication signals to the users. The BS switches continuously between uplink and downlink modes within one wireless frame. According to different communication demands, the uplink and downlink modes have different time ratios. We assume that the time ratio of the uplink and downlink modes is AS1, which is one of the time ratios specified in the 5G communication protocol, to provide clear constraints for the subsequent waveform design. Without loss of generality, the proposed method's optimization approach is similar to other cases. Furthermore, there are common channels in the transmission of OFDM signals. The common channels are used to transmit various communication-related important parameters, including downlink control information and achieving synchronization between communication parties. The common channels include the synchronization signal/PBCH block (SSB), physical downlink control channel (PDCCH), and physical downlink shared channel (PDSCH), which are only distributed in the first half of a wireless frame with different modes. For the same reason as the AS1, we assume that the distribution of the common channel OFDM symbols is AS2, which is also derived from the 5G communication protocol, to provide clear constraints for the subsequent waveform design. AS1 and AS2 are as follows:  
AS1: Downlink constraint. The first four of every five consecutive slots are fixed as the downlink, and the last slot is fixed as the uplink, as shown in Figure 2. Specifically, for the \(n\)th OFDM symbol, if \(\text{mod}(\lfloor n/14\rfloor,5)\in\epsilon_{1}\), it means that the \(n\)th OFDM symbol belongs to the downlink, otherwise it belongs to the uplink, where \(\epsilon_{1}=\{0,1,2,3\}\). AS2: Common channel constraint. The resource allocation for the common channel is considered to be the \(<120\) kHz, \(120\) kHz \(>\) mode, as shown in Figure 3. Specifically, for the \(n\)th OFDM symbol in the first half of a wireless frame, if \(\text{mod}(n,(N/2))=0\) and \(\text{mod}(n,28)\in\epsilon_{2}\), it means that it is a common channel symbol, otherwise, it is not, where \(\epsilon_{2}=\{4,5,6,7,8,9,10,11,16,17,18,19,20,21,22,23\}\).