## 2 Signal Model and Problem Formulation
### Motivation and Problem Statement  
As described in the previous section, an ISAC system can be designed for vehicle sensing under 5G communication base stations. However, the distribution of sensing symbols needs to be subject to the constraints of the 5G NR communication frame structure as shown in Figure 1, which takes subcarrier interval \(\Delta f\) = 120 kHz, for example. Generally, there are a total of 80 slots in a frame with settable downlink slot and uplink slot, which are allocated by the communication system parameters, and each slot contains 14 OFDM symbols. Obviously, the 5G NR frame structure will lead to a nonuniform distribution of sensing symbols. The reasons are as follows: (1) the communication frame structure contains uplink slots. However, the object sensing is realized based on the downlink signal. Therefore, it cannot satisfy the timing uniform structure of (9); and (2) the communication frame structure contains a common channel and it needs to occupy partial subcarriers in some OFDM symbols. Then, it cannot satisfy the subcarrier uniform structure of (8) with the common channel [31]. Due to the above two nonuniform structures, ISAC cannot work well with the existing time division duplex 5G NR communication base station. Otherwise, the sharing model of communication symbols and sensing symbols leads to the problems of interference and high peak side-lobe ratio (PSLR) [32]. Therefore, we propose the TD-ISAC based on a 5G NR communication base station. That is, we allocate some OFDM symbols in a communication frame designed especially for sensing. Furthermore, we use fewer OFDM symbols for sensing in order to keep a high communication efficiency. In this paper, the waveforms are designed under the following assumptions. As1: Sensing symbol cannot be allocated at uplink slot. There is one uplink slot in every five consecutive slots satisfying the max uplink demand in a 5G NR communication protocol. And the uplink slot can be set in any slot. If the uplink slot is set in any slot as flexible mode, we call it uplink-flexible mode. On the contrary, if the uplink slot is set in one fixed position, we call it uplink-fixed mode. As2: One sensing symbol can be allocated at most in a slot under uplink-flexible mode, and it can be slightly adjusted under uplink-fixed mode. As3: Sensing symbol cannot be allocated at common channel. As shown in Figure 1, the common channel occupies only partial OFDM symbols in one slot. Considering the assumption As2 and As3, we allocate the 13 th OFDM symbol in a downlink slot for sensing. Then, the nonuniform sensing symbol timing can be illustrated as Figure 2, where the uniform sensing timing represents the ideal situation with no assumption considering for comparison. Obviously, the sensing symbol timing in the TD-ISAC system is similar to the radar sensing pulse. And \(T_{r}\) is the pulse repetition interval (PRI) of the uniform sensing timing equal to one slot time. Then, \(\mathbf{b}_{fd}\) of the uniform sensing timing in the received matrix can be rewritten as  
\[\mathbf{bu}_{fd}=\Big{[}1,\ldots,e^{j2\pi f_{d}kT_{r}},\ldots,e^{j2\pi f_{d}(N_{u}-1 )T_{r}}\Big{]}^{\mathsf{T}}, \tag{10}\]  
where \(N_{u}\) is the total number of uniform sensing symbols and \(k=\{0,1,\ldots,(N_{u}-1)\}\). Let \(J=[j(0),\ldots,j(i),\ldots,j(N_{n}-1)]\) be the selected matrix, where \(j(i)=[0,\ldots,1,\ldots,0]^{\mathsf{T}}\) with almost 0 except for the \(k\)th element of the uniform sensing symbol corresponding to the \(i\)th  
Figure 1: 5G NR communication frame structure. Figure 2: Sensing symbol timing (where the orange color represents uplink slot). nonuniform sensing symbol. And \(N_{n}\) is the total number of nonuniform sensing symbols. Then, \(\mathbf{b}_{fd}\) of the nonuniform sensing timing in the received matrix can be rewritten as  
\[\mathbf{bn}_{fd}=\mathbf{bu}_{fd}\mathbf{J}. \tag{11}\]  
In this paper, we will design the nonuniform sensing timing of TD-ISAC for vehicle sensing based on the 5G base station under urban traffic scenarios in the following section.