## 2 System Model and Problem Formulation
### Problem Formulation  
Inspired by the orthogonal TF resource blocks allocated in OFDM, a similar approach is to deploy navigation PRN codes and communication symbols in the DD grid. In this case, the data modulated in the DD grid is written as follows:  
\[\mathbf{X}_{\text{dd}}[m,n]=\left\{ \begin{array}{l}\mathbf{X}_{\text{nav}}[m,n ],\ 0\leq m<M_{n}\\ \mathbf{X}_{\text{con}}[m,n],\ M_{n}\leq m<M\end{array} \right. \tag{4}\]  
where \(M_{n}\) is the delay bins occupied by the navigation signal. The navigation PRN code matrix is as follows:  
\[\mathbf{X}_{\text{nav}}=\text{vec}_{M_{n},N}^{-1}(C_{k}) \tag{5}\]  
where \(k=0,1,\dots,M_{n}N-1\), \(\text{vec}_{M_{n},N}^{-1}(\cdot)\) converts a sequence into an \(M_{n}\times N\) size matrix. \(C_{k}\) is a PRN code sequence. \(\mathbf{X}_{\text{com}}\) is the communication symbol matrix; the symbols are mapped from the transmitted serial data stream. In this case, the communication and navigation signals are orthogonal, and PRN codes with good correlation can be utilized as navigation downlink broadcast signals. Substituting (4) into the original ISFFT defined in (1) yields the INAC signal expression in the TF domain. Figure 2 provides a graphical representation of this process, from which it can be observed that each symbol in the TF domain of the INAC signal is derived from all the navigation PRN codes and communication symbols in the DD domain. This indicates that the navigation and communication signals are spread across the entire bandwidth. The positioning of the navigation receiver with TOA estimations, by correlating the received signal with a local replica, requires four or more satellites. A common solution to reduce the signal processing burden is to use a front-end filter with a bandwidth smaller than the main frequency band of the signal to process a portion of the bandwidth; this is a method that is widely employed in navigation signal processing [41]. However, this may lead to degradation in navigation performance. The autocorrelation function (ACF) corresponding to the time-domain signal in (2) is expressed as follows [42]:  
\[R(\tau)=\frac{\sigma_{X}^{2}}{M}\frac{\sin(\pi\Delta fM\tau)}{\sin(\pi\Delta f \tau)}\bigg{(}1-\frac{|\tau|}{T}\bigg{)}e^{j\pi\Delta f(M-1)\tau} \tag{6}\]  
where \(\sigma_{X}^{2}\) is the power of the navigation PRN code. Since the ISFFT does not alter the power and orthogonality of the signal [19], we have \(E\left[\overline{\mathbf{X}_{\text{df}}}\mathbf{X}_{\text{df}}\right]=E\left[ \overline{\mathbf{X}_{\text{dd}}}\mathbf{X}_{\text{dd}}\right]=\sigma_{X}^{2}\). Without loss of generality, we assume that the front-end filter is ideal, and that the bandwidth of the received signal is \(M_{n}\Delta f\) (corresponding to the delay bins occupied in the DD domain). In this case, the ACF is given by the following:  
\[R^{\prime}_{\text{OTFS}}(\tau)=\frac{\sigma_{X}^{2}}{M}\frac{\sin(\pi\Delta fM _{n}\tau)}{\sin(\pi\Delta f\tau)}\bigg{(}1-\frac{|\tau|}{T}\bigg{)}e^{j\pi \Delta f(M_{n}-1)\tau} \tag{7}\]  
The value of the ACF is maximized if \(\tau=0\). Therefore, the ratio of the peak values of the ACFs in (6) and (7) is as follows:  
\[\varsigma_{\text{OTFS}}=\frac{R^{\prime}_{\text{OTFS}}(0)}{R(0)}=\frac{M_{n}} {M} \tag{8}\]  
Typically, \(M_{n}<M\), which implies that \(\varsigma_{\text{OTFS}}<1\). This indicates that when receiving a portion of the bandwidth, the peak value of the ACF decreases, resulting in a loss of navigation signal power. This occurs because the PRN codes are spread across the entire bandwidth, and the receiver is unable to separate the navigation signal in the frequency domain, as illustrated in Figure 2.  
From the analyses in this section, it can be seen that directly splitting the DD grid into two halves for communication and navigation, respectively, suffers from the problem of overburdening the navigation signal processing, while reducing the navigation signal reception bandwidth will result in an obvious loss of navigation performance.