## 2 The Proposed Pulse-OTFS Signal
### Signal Model
#### 2.1.1 Pulse-OTFS  
Pulse-OTFS utilized as a navigation signal modulates the PRN code sequence spread spectrum navigation data on a two-dimensional delay-Doppler grid:  
\[\mathbf{X}_{\text{dd}}[m,n]=\text{vec}_{M,N}^{-1}(d_{k}a_{k}) \tag{5}\]  
Figure 1: Block diagram of OTFS modulation.  
where \(k=1,2,\cdots,MN\), \(a_{k}\) is the PRN code sequence with a code length of \(MN\), \(d_{k}\) is the navigation data, and usually remains constant for several signal frame periods, and \(\text{vec}_{M,N}^{-1}(\cdot)\) is the inverse vectorization operator.  
The navigation data of the proposed Pulse-OTFS is spread by \(N\) identical PRN code sequences in a frame, that is, the code length of the PRN code sequence is \(M\), and the navigation data are the same as (5). Its matrix form can be expressed as \(\mathbf{X}_{\text{dd}}=\mathbf{a}_{M\times 1}\otimes\mathbf{d}_{1\times N}\), where \(\otimes\) is the Kronecker product. According to the matrix form of ISFFT [25], the equivalent time-frequency matrix is:  
\[\mathbf{X}_{\text{tf}} =\mathbf{F}_{M}\mathbf{X}_{\text{dd}}\mathbf{F}_{N}^{\text{t}} \tag{6}\] \[=\mathbf{F}_{M}(\mathbf{a}_{M\times 1}\otimes\mathbf{d}_{1\times N}) \mathbf{F}_{N}^{\text{t}}\] \[=(\mathbf{F}_{M}\mathbf{a}_{M\times 1})\otimes\left(\mathbf{d}_{1 \times N}\mathbf{F}_{N}^{\text{t}}\right)\] \[=\left[\sqrt{N}\mathbf{A}_{M\times 1},\mathbf{0}_{M\times(N-1)}\right]\]  
where \(\mathbf{F}_{M}\) and \(\mathbf{F}_{N}\) are the discrete Fourier transform matrices of dimensions \(M\times M\) and \(N\times N\), respectively, and \(\mathbf{A}_{M\times 1}\) is the Fourier transform of the PRN code sequence.  
Equation (6) illustrates that only the first column element in the time-frequency matrix \(\mathbf{X}_{\text{tf}}\) is non-zero, i.e., the time-domain baseband signal is only non-zero for the duration of the first symbol in a frame signal period, and the duty cycle is \(1/N\). This is why we call it Pulse-OTFS.  
Consider a Pulse-OTFS system with \(M\) delay bins and \(N\) Doppler bins. The signal has a total bandwidth of \(B\) and occupies a frame duration (signal period) of \(NT\), and the number of transmitted PRN code chips is \(M\times N\). Then, the following relationship can be established:  
\[\left\{\begin{array}{l}f_{c}=MN/(NT)=M\Delta f=B\\ T_{c}=1/f_{c}=1/B=T/M\end{array}\right. \tag{7}\]  
where \(f_{c}\) is the PRN code rate and \(T_{c}\) is the code chip duration.  
Note that, unlike traditional GNSS signals where the bandwidth of the main lobe is twice the PRN code rate, the main lobe bandwidth of OTFS/Pulse-OTFS is equal to the PRN code rate, which has a significant impact on its PSD and ACF properties; we will analyze this later in this section.  
In this paper, the OTFS and Pulse-OTFS signals are simply denoted by OTFS\((\alpha,\beta)\) and Pulse-OTFS\((\alpha,\beta)\), where  
\[\left\{\begin{array}{l}\alpha=f_{c}/f_{0}\\ \beta=f_{c}/\Delta f\end{array}\right. \tag{8}\]  
where \(f_{0}=1.023\) MHz is the reference frequency commonly used in the GNSS signal, \(\alpha,\beta\) are positive integers, and \(\beta\) is equal to the delay bins/subcarrier numbers \(M\).