# Integrated Sensing and Communication Using Random Padded OTFS with Reduced Interferences
## 5 Interference Reduction
### Proposed Pilot-to-Data Interference Reduction  
Here we will discuss the interference between the pilot zone and the data zone and the ways to reduce it. Usually, the biggest part of reflections in a multi-path environment comes from the objects that have similar Doppler frequency shift, for example, from huge static objects like buildings in the cities. Eliminating the influence of these reflections could significantly reduce the level of ICI between pilots and data. The method introduced here is based on DFT properties for periodic signals [45]. If a signal consists of \(K\in\mathbb{N}\) repetitions of some basis \(P\) samples long sequence \(z(p)\), \(p=0\cdots P-1\), that is  
\[\widetilde{z}(n)=\{\underbrace{z(0),z(1),\cdots,z(P-1)}_{z(p)},\underbrace{z (0),z(1),\cdots,z(P-1)}_{z(p)},\cdots\}, \tag{17}\]  
and if \(\widetilde{z}(n)\) has zero mean, then the DFT of \(\widetilde{z}(n)\), i.e., \(\widetilde{Z}_{k}\), will be equal to the \(K-1\) times up-sampled the DFT of \(z(n)\), i.e., \(Z_{k}\):  
\[\widetilde{Z}_{k}=\left\{ \begin{array}{ll}Z_{k},&k\in\{\,0,\,K,\,2K,\cdots, PK\,\},\\ 0,&\text{otherwise.}\end{array} \right. \tag{18}\]  
In fact, this effect has already been presented in Figure 3: it was assumed there that the pilot samples repeat \(K=3\) times and, as a result, \(K-1=2\) zeros are present between non-zero values of the pilot DFTs in the resulting matrix. On the receiver's side, in case of a fractional delay, the pilot of the RP-OTFS will interfere with the data. In case of pilot repetition in the time (slow-time, long-time) direction in the TD domain (corresponds to Doppler direction in the DD domain), the interference signal will also repeat. Hence, using the above property, the ICI energy distribution on the DD grid could be controlled. To ensure that the number of repetitions remains an integer within a single RP-OTFS frame, the condition \(PK=N\) must be satisfied. As an example, the DD grid without data (that is, for pilots only) is shown for the new RP-OTFS in two figures: for \(K=\frac{N}{2}\) in Figure 9 and for \(K=\frac{N}{8}\) in Figure 10. In both cases, the total power of the pilots is the same, and fractional signal reflections are present. It is clearly visible that interference from pilot to data zone is distributed in a completely different manner: in the first case, it is smeared equally in the data zone, while in the second case, it is concentrated locally only in several stripes (see Figure 11). The columns in the DD grid, where the ICI will be concentrated, could be chosen by multiplying the rows of the TD matrix (only the part of the matrix that contains the pilot samples) by \(e^{2\pi ki/N}\), \(i=0,1,\ldots,N-1\), \(k\in 0\ldots K-1\). This will shift the ICI stripes in the Doppler direction by the \(i\) columns of the DD grid and gives additional flexibility in controlling the positions of the reflections. Only the following columns of the DD grid will be corrupted by ICI :  
\[n=i+Kj,\ j\in\{0,1,\ldots,N/K\}. \tag{19}\]  
It should be noted that the number of repetitions can be equal to any integer from \(0\ldots N\). The boundary case, where \(K=N\), means that all pilot symbols are the same. In this case, all ICIs from objects with the same Doppler will be concentrated in one single column. On the other hand, case \(K=0\) means that all OFDM pilots are different. In this situation, all ICIs from the objects are uniformly spread across all carriers of the DD grid. By choosing the parameters \(K\) and \(i\), the ICI caused by the pilots can be adapted to different scenarios. For example, \(K=0\) could be useful when there is no expectation of the optimal solution. Figure 11: The cut of the DD grid at the beginning of the data zone (filled with zeros) along the Doppler axis for the RP-OTFS with \(K=\frac{N}{2}\) (up) and \(K=\frac{N}{8}\) (down). Figure 9: The DD map for RP-OTFS with \(K=\frac{N}{2}\). strong reflections with fractional delays.