## 3 The Proposed Delay-Doppler Block Division Multiplexing
### Signal Generation  
The block diagram of the DDBDM signal generation is shown in Figure 3. The generation of the transmitted signal includes the following steps. (1) Navigation message mapping. The navigation message is spread by the PRN code and placed in the DD matrix. The \(\mathbf{X}_{\text{nav}}\) in (5) is rewritten as follows:  
\[\mathbf{X}_{\text{nav}}[m,n]=\text{vec}_{M_{n},N}^{-1}(D_{k}C_{k}) \tag{9}\]  
where \(k=0,1,\ldots,M_{n}N-1\), \(\text{vec}_{M_{n},N}^{-1}(\cdot)\) converts a one-dimensional sequence into an \(M_{n}\times N\) size matrix \(\mathbf{X}_{\text{nav}}\). \(M_{n}\) is the number of subcarriers (delay bins) occupied by the navigation signal. \(C_{k}\) is a PRN code sequence, and the code division multiple access (CDMA) could be utilized for limiting MI between different satellite signals [43]. \(D_{k}\) is the transmitted navigation message that remains constant within a PRN code chips matrix duration. To provide the integrity of navigation services, navigation messages are still transmitted on the navigation subcarriers after being spread by PRN codes, because the amount of data required for positioning is not large. (2) Communication symbol mapping. The communication information symbols are taken from the quadrature amplitude modulation (QAM) alphabet \(\mathbb{A}=\left\{a_{1},\ldots,a_{\mathbb{Q}}\right\}\) of size \(Q\) and placed in the DD matrix \(\mathbf{X}_{\text{com}}\in\mathbb{C}^{M_{c}\times N}\), where \(M_{c}\) is the number of subcarriers/delay bins occupied by the communication signal. (3) Block-wise ISFFT. The modulated navigation and communication data \(\mathbf{X}_{\text{com}}\) and \(\mathbf{X}_{\text{nav}}\) are transformed to matrices \(\mathbf{X}_{\text{tf,c}}\) and \(\mathbf{X}_{\text{tf,n}}\) in the TF domain, respectively, by the ISFFT:  
\[\left\{ \begin{array}{l}\mathbf{X}_{\text{tf,c}}=\mathbf{F}_{M_{c}}\mathbf{ X}_{\text{com}}\mathbf{F}_{N}^{H}\\ \mathbf{X}_{\text{tf,n}}=\mathbf{F}_{M_{n}}\mathbf{X}_{\text{nav}}\mathbf{F}_ {N}^{H}\end{array} \right. \tag{10}\]  
Figure 3: The block diagram of the DDBDM signal generation. where \(\mathbf{F}_{M_{c}}\in\mathbb{C}^{M_{c}\times M_{c}}\), \(\mathbf{F}_{M_{m}}\in\mathbb{C}^{M_{n}\times M_{m}}\), and \(\mathbf{F}_{N}\in\mathbb{C}^{N\times N}\) are the normalized DFT matrices with different sizes, and \(\left(\cdot\right)^{H}\) is the Hermitian of the matrix. The above equation is the matrix form of the ISFFT. (4) Zero subcarrier padding. Having zero-padded subcarriers between the navigation and communication subcarriers creates a guard band for the front-end filter. This reserves a transition bandwidth for the receiving part of the signal, while mitigating possible MI between the navigation and communication blocks in high-mobility channels. Then, the TF matrix \(\mathbf{X}_{\text{tf}}\) becomes the following:  
\[\mathbf{X}_{\text{tf}}[k,l]= \begin{cases}\mathbf{X}_{\text{tf},\text{fn}}[k,l ],&0\leq k<M_{n}\\ 0,&M_{n}\leq k<M_{n}+M_{p}\\ \mathbf{X}_{\text{tf},\text{fc}}[k,l],&M_{n}+M_{p}\leq k<M\end{cases}  \tag{11}\]  
where \(M_{p}\) is the number of zero-padding subcarriers, and satisfies \(M_{n}+M_{c}+M_{p}=M\). Here, we assume that the power of navigation and communication signals is normalized. (5) Subcarrier power allocation. Let \(P\) denote the total signal power; \(P_{k}=P_{n,k}+P_{c,k}\) denotes the \(k\)-th subcarrier power, where at least one out of the navigation subcarrier power \(P_{n,k}\) and the communication subcarrier power \(P_{c,k}\) is zero for a specific \(k\). The subcarrier power and total power satisfy \(\sum_{k=0}^{M-1}P_{k}=P\). The power allocation of the communication and navigation signals determines the overall performance of the signal; the relationship between power allocation and performance will be discussed in detail in Section 5. (6) IFFT modulation.