# Integrated Sensing and Communication Using Random Padded OTFS with Reduced Interferences
## 3 The Random Padded OTFS
### The RP-OTFS Application Limitations  
The RP-OTFS pilots have the standard OFDM structure. The length of the OFDM pilot symbols, \(L\), defines the maximum delay that can be estimated with its help, including fractional delay side lobes. However, since RP-OTFS is intended to work in a strong Doppler environment, the OFDM-based pilot will suffer from the violation of orthogonality between subcarriers. From this point of view, it is necessary to keep the pilot symbol as short as possible. The amount of distortion of the OFDM-based pilot symbol caused by the Doppler effect is called the carrier-to-interference ratio (C/I ratio). The interference coming to the \(i\)-th subcarrier from all other subcarriers (\(j\
eq i\)) inside one OFDM pilot symbol can be estimated using the following formula derived in [38]:  
\[CI(i)=\frac{1}{\frac{(L/f_{e}-f_{d})^{2}}{2}\frac{\sum_{j=1}^{L}\frac{1}{(j-i)^{ 2}}}{\sum_{j\
eq i}^{L}\frac{1}{(j-i)^{2}}}}, \tag{5}\]  
where \(f_{d}\)--the Doppler frequency shift, \(f_{s}\)--the sampling frequency.  
In Figure 4 the dependence of the C/I ratio on \(f_{d}\) is shown for different lengths \(L\) of the pilot symbol. The carrier frequency in this example is 70 GHz. This frequency was chosen to be very close to the maximum carrier frequency (71 GHz) from the 5G specification [11]. It results from Figure 4 that RP-OTFS usage is possible in scenarios with a short impulse response. In Table 1 maximum velocities of moving objects are shown, for which the C/I ratio of the pilots is kept at least at the level of \(-40\) dB and \(-50\) dB.  
It was mentioned in the introduction and also will be shown in the system model (Section 4) that in case of fractional reflections, the OTFS can lose its orthogonality and the carriers begin to interfere. This effect is also observed between the data carriers and between the pilot and the data carriers. In most modern communication standards, the pilot power is usually higher than the power of the data symbols [39; 40]. This means that even stronger interference can be expected between pilots and data carriers. However, unlike the ZP-OTFS, where the whole power of the pilot zone is concentrated in only one active pilot carrier, the RP-OTFS is devoid of this drawback because it distributes the total energy  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{\(\mathbf{-40}\) dB Threshold} & \multicolumn{3}{c}{\(\mathbf{-50}\) dB Threshold} \\ \hline \(L\) & \(f_{d}\), Hz & v, km/h & \(f_{d}\), Hz & v, km/h \\ \hline
8 & 20,000 & 431 & 6900 & 148 \\ \hline
16 & 9930 & 214 & 3089 & 67 \\ \hline
32 & 5000 & 108 & 1567 & 34 \\ \hline
64 & 2465 & 53 & 0 & 0 \\ \hline
128 & 1336 & 29 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Maximum operating velocity of the 70 GHz RP-OTFS.  
Figure 4: The C/I ratio as a function of Doppler frequency shift \(f_{d}\) for different lengths of the pilot symbol \(L\).  
among a large number of active pilot carriers. With the same total power in the pilot zone as in ZP-OTFS, the power of each active pilot carrier of the RP-OTFS will be \((NL)^{2}\) times lower. This makes RP-OTFS attractive from the ICI reduction and Peak-to-Average Power Ratio points of view. In Figure 5 an exemplary RP-OTFS and ZP-OTFS in the DD, TD, and time domains are depicted. The power of the pilots for both waveforms is the same.  
Consequently, the OTFS pilot pattern is optimized for channels characterized by moderate Doppler shifts and short delay spreads. The dispersion of pilot energy across multiple subcarriers facilitates the detection of targets with lower radar cross sections (RCS), as will be demonstrated in the simulation section. One representative example of such a channel is an automotive ISAC scenario [41].