## 3 Proposed Sensing Waveform
### Proposed Waveform Design Strategy  
Due to the shared hardware of the existing 5G BSs for ISAC, the 5G BSs are required to work in downlink mode. To minimize the impact on communication, waveform design  
Figure 3: One of the common channel constraints. Figure 2: Downlink constraint. methods need to avoid modifications to common channel symbols. In this paper, the above requirements are assumed to be AS1 and AS2, respectively. This will result in the sensing waveform being nonuniform. Additionally, to avoid the interference of communication signals on sensing and ensure minimal impacts on the communication, we focus on the inner-frame time division multiplexing of ISAC to exploit the millimeter-wave bands in 5G NR systems. Specifically, we extracted a few OFDM symbols for sensing within the wireless frame, and would like the sensing waveform to be as sparse as possible while ensuring acceptable sensing performance. Regarding sensing performance, range resolution and Doppler resolution are two basic metrics. Since we use the inner-frame time division multiplexing, it is easy to obtain a high-range resolution by selecting a large bandwidth. But for the Doppler resolution, it relies on the slow-time sampling and observation time. The uniformly spaced intervals cannot be realized, which is a major challenge when designing sensing waveforms within the 5G NR frame structure. Therefore, this paper focuses on designing a sparse and nonuniform sensing waveform with acceptable detection and estimation performance. Generally, the usual waveform design methods include detection probability-based, signal-to-noise ratio (SNR)-based, signal-to-interference-plus-noise ratio (SINR)-based, ambiguity function-based, and Kullback-Leibler scatter-based methods [32]. The ambiguity function is an important tool for evaluating the sensing parameters in the radar system. In order to remove the interference of modulation information on sensing, we set \(\alpha_{m,n}=1\) in (1) and propose the sensing symbolic baseband model, given by  
\[s_{n}^{\prime}(t)=\sum_{m=0}^{N_{c}-1}e^{|2\pi m\Delta f(t-nT_{s})}\text{rect }\bigg{(}\frac{t-nT_{s}}{T_{s}}\bigg{)}. \tag{4}\]  
Thus, the sparse and nonuniform sensing waveform can be expressed as  
\[S_{t}=U\mathbf{s}_{t}, \tag{5}\]  
where \(U\) is an \(L\times N\) sparse matrix whose \(l\)th row and \(n\)th column element \(u_{l,n}\in\{0,1\}\), and \(\mathbf{s}_{t}=[s_{0}^{\prime}(t),s_{1}^{\prime}(t),\ldots,s_{N-1}^{\prime}(t)]^{T}\) is an \(N\times 1\) vector. Specifically, each row of the \(U\) has one and only one \(u_{l,n}=1\), and each column of the \(U\) has, at most, one \(u_{l,n}=1\). For convenience, we define the column numbers of the non-zero elements in the \(l\)th row of \(U\) as \(n_{l}\), for \(l=0,1,\ldots,L-1\). Hence, we compute the ambiguity function, expressed by  
\[ \begin{split}\chi u(\tau,f_{d})=&\int_{-\infty}^{ \infty}S_{t-\tau}^{H}S_{t}e^{i2\pi f_{d}t}dt\\ =&\sum_{l=0}^{L-1}\sum_{m=0}^{N_{c}-1}\sum_{m^{ \prime}=0}^{N_{c}-1}(T_{s}-|\tau|)e^{i\pi[(m-m^{\prime})\Delta f+f_{d}](T_{s}+ \tau)}\\ & e^{i2\pi m^{\prime}\Delta f\tau}\operatorname{sinc}\{\pi[(m-m^ {\prime})\Delta f+f_{d}](T_{s}-|\tau|)\}e^{j2\pi f_{d}n_{l}T_{s}}.\end{split}  \tag{6}\]  
where \(\tau\) is the round-trip propagation time, \(|\tau|\leq T_{s}\), and \(f_{d}\) is the Doppler frequency shift. Moreover, \(\chi u(\tau)\) represents the range ambiguity function when \(f_{d}\) is fixed to 0 in (6), and \(\Delta f\) and \(N_{c}\) are important parameters affecting the range estimation performance. Similarly, \(\chi u(f_{d})\) represents the velocity ambiguity function when \(\tau\) is fixed to 0 in (6), and \(U\) is an important parameter affecting the velocity estimation performance. Since we use inner-frame time division multiplexing, we can easily obtain an acceptable range estimation performance by picking appropriate parameters, but it is not easy to obtain an acceptable velocity estimation performance because of the nonuniform and sparse waveform.