## 3 Dual-PRF Waveform Design and Signal Processing
### Waveform Design under Uplink-Fixed Mode  
As shown in Figure 3, the uplink slot in every five consecutive slots is varying under uplink-flexible mode. In order to simplify the structure of TD-ISAC, we then modify the sensing timing based on the uplink-flexible mode with the constraints of uplink-fixed. Without loss of generality, we assume that the second slot in every five consecutive slots is fixed as uplink slot. Then, there are some slot conflicts that sensing symbols occupy the fixed uplink slot, such as 6, 16, 21, 26, 41, 46, 51, 56, 66, and 76. We move the sensing symbol of the conflict slot to the first OFDM symbol of the next slot in order to change less the structure as shown in Figure 5, where blue represents the OFDM symbol allocated for sensing, yellow represents part of the slot allocated for sensing, and orange represents the uplink slot (which is the same meaning as in the following text). Taking the sixth slot, for example, it needs to be set as uplink slot in uplink-fixed mode, and then its sensing symbol is moved to the first symbol of the seventh slot as shown in Figure 5a. Similarly, the twenty-first slot needs to be set as uplink slot in uplink-fixed mode, and then its sensing symbol is moved to the first symbol of the twenty-second slot as shown in Figure 5b. The difference between the two cases is that the seventh slot does not contain any sensing symbol in uplink-flexible mode. Hence, it has one sensing symbol in the uplink-fixed mode, while the twenty-first slot contains one sensing symbol in uplink-flexible mode, resulting in two sensing symbols in the uplink-fixed mode. Applying the same operation to all conflict slots, we can then obtain the sensing timing under uplink-fixed mode, as illustrated in Figure 6. Compared with sensing timing under uplink-flexible mode, the \(2T_{r}\) and \(3T_{r}\) pulses are no longer strict uniform pulses with a small time error that can be tolerated. Its theoretical performance will be analyzed in the next part. Similarly to (16) and (17), the Doppler dimension vector of the \(2T_{r}\) and \(3T_{r}\) pulses in the received matrix can be expressed as  
\[\mathbf{b}_{2 fix}=\mathbf{b}_{2 fix}\odot\left[1,1,\dots,e^{j2\pi f_{d}km2T_{r}T_{s}}, \dots\right], \tag{21}\]  
and  
\[\mathbf{b}_{3 fix}=\mathbf{b}_{3 fix}\odot\left[1,1,\dots,e^{j2\pi f_{d}lm3T_{r}T_{s}}, \dots\right], \tag{22}\]  
where \(\odot\) denotes Hadamard product, \(k_{m}\) denotes the number of conflict slot modifications of the \(2T_{r}\) sensing pulse, and \(l_{m}\) denotes the number of the conflict slot modifications of the \(3T_{r}\) sensing pulse.  
During the waveform-designed process, we avoid the common channel by allotting a sensing symbol at the last position of one slot under uplink-flexible mode. There are some symbol modifications under uplink-fixed mode, and we shift the symbol to the first symbol of the next slot. It is also allocated at the idle OFDM symbols, so the proposed waveform can satisfy the constraint of common channel which is assumed in As3. And from the designed waveform in Figure 6 where the red number represents the modification position compare with the sensing timing under uplink-flexible mode. We can see that the designed sensing waveform satisfies the maximum uplink needs that is one slot in five consecutive slots.