# Integrated Sensing and Communication Using Random Padded OTFS with Reduced Interferences
## 6 Choosing DD Grid Parameters for Different Use Cases  
In this section, we will present three examples of choosing values of the global parameters (\(M\), \(N\) and \(L\)) of the DD grid in the radar context. We start a discussion of each example with an introduction of its use case scenario, and then present an adaptation of the grid parameter values. We define three use-cases for ISAC:  
1. very high speed: civil airplanes, high speed trains, etc; in this group, we consider flying objects with speed \(v\) in the range \(120\dots 340\) m/s;
2. medium speed: vehicular, drones; in this group, we have moving objects with speed \(v\) in the range \(10\dots 60\) m/s;
3. very low speed: pedestrians, low speed vernaculars; in this group, we have objects moving with speed \(v\) up to \(2\) m/s. As the Doppler frequency shift \(f_{d}\) depends not only on the movement velocity \(v\) of the objects but also on the carrier frequency \(f_{c}\), as an example, we choose the carrier frequency to be the same as in the previous section, i.e., equal to \(70\) GHz:  
\[f_{d}=\frac{v}{\lambda}=\frac{vf_{c}}{c}. \tag{21}\]  
where \(\lambda=\frac{c}{f_{c}}\) denotes a wavelength and \(c\) is the speed of light. For unambiguous Doppler measurements, the sampling rate in the Doppler direction should satisfy the Nyquisttheorem, and therefore, should be defined as \(f_{s}^{(Doppler)}=\frac{f_{s}}{M}\). Using Equation (21), the maximum value of the parameter \(M\) that meets this requirement is given by  
\[M_{max}=\frac{f_{s}}{2\cdot f_{d}}. \tag{22}\]  
Next, we take into account the range resolution of the DD grid. Communication systems operating in the range of tens of gigahertz typically make use of the bandwidth (BW) of tens or even hundreds of megahertz. As a sampling rate \(f_{s}=BW\) of the complex signal, this gives a very high range resolution. For example, \(BW=f_{s}=100\) MHz gives a resolution of up to 3 m. Such a high resolution will be useless for use-case A, but could be useful for use-case B. The same time resolution of 3 m may not be enough for some applications with targets from use-case C. In such cases, an increase in the BW should be considered. Another aspect that should be taken into consideration for ISAC systems in the context of their radar sections is the value of integration time, that is, how long targets from different use cases (A, B, or C) can stay in one resolution element of the DD grid. For resolution in range direction, the calculations are trivial: the time required for a moving object to cross the resolution element is \(\Delta t_{element}=\frac{c}{f_{s}\cdot 0}\). In order to be able to combine the reflection coherently during the Zak transform, the frame time duration should be much shorter than the time spent by the target in one resolution element [19]:  
\[\frac{NM}{f_{s}}\ll\Delta t_{element}. \tag{23}\]  
By changing the parameters of the DD grid \(M\) and \(N\) (keeping their product \(MN\) constant), different resolutions in Doppler and maximum unambiguous Doppler can be achieved. In terms of computational complexity, decreasing \(N\) (and increasing \(M\)) is more efficient than decreasing \(M\) (and increasing \(N\)). The computational complexity of the discrete Zak transform is \(\mathcal{O}(MN\log N)\). In terms of RP-OTFS, keeping \(N\) as small as possible is also more profitable because the pilot overhead of RP-OTFS is directly proportional to \(N\). \(N\) defines the number of OFDM pilots. On the other hand, the decrease in the value of \(N\) will result in worse Doppler resolution. Below is an example of how to choose the RP-OTFS parameter values in use case B. At the end of this part, for illustration purposes, we provide a general table with results for the three use cases. Before we start, it is important to note that applying the RP-OTFS to targets from use case A is possible in an environment where all propagation paths have practically the same delay because of the limited length of the OFDM-like pilot symbol.