## 4 Data Communication Scheme Derivation
### The OTFDM Scheme  
First, we introduce tileband to refer to the frequency band occupied by a subtile during a fraction of the period of a \(\widetilde{N}\)-length symbol block.  
The OTFDM, proposed in [21], is a data communication scheme that uses the DOST [22] to divide the time-frequency space into orthogonal subtiles bounded by a specific arrangement. This study considers the \(\mathcal{B}\)-geometry to obtain the subtiles distributions in the time-frequency domain, where \(\mathcal{B}\) is the geometry order. To obtain the OTFDM scheme from the proposed formulation, we have to consider \(\mathbf{Q}=\overline{\mathbf{D}}\). The \((k,i)\)th element of the matrix \(\overline{\mathbf{D}}\) is given by  
\[\overline{D}[k,i]=\begin{cases}\frac{1}{\sqrt{\beta[k]}}\mathrm{e}^{\mathrm{j }(2\pi\frac{i[k]}{\beta[k]}k-\pi\tau[k])},&u\leq i\leq U\\ 0,&\text{otherwise}\end{cases}, \tag{33}\]  
where \(u=\
u[k]-\lfloor\beta[k]/2\rfloor\), \(U=\
u[k]+\lceil\beta[k]/2\rceil-1\), and \(\lfloor z\rfloor=\max\{n\in\mathbb{Z}|n\leq z\}\) is the floor function. The parameters \(\
u[k]\), \(\beta[k]\), and \(\tau[k]\) are elements of the vectors \(\mathbf{v}\), \(\mathbf{\beta}\), and \(\mathbf{\tau}\), respectively, and are responsible for the creation of each subtile, ensuring that it follows the specific arrangement. In this regard, the \(i\)th central frequency, subband group size, and position in time are given by \(\
u[k]\), \(\beta[k]\), and \(\tau[k]\), respectively. Subtiles that share the same value of \(\
u[k]\) are called \"voice\" and an OTFDM scheme using the \(\mathcal{B}\)-geometry has a voice equal to \(\
u=2\log_{2}(\mathcal{B})+2N/\mathcal{B}\) and the subband group related to the \(i\)th voice is given by \(\beta_{\
u}[i]\), \(i=0,\ldots,\
u-1\).  
A more detailed explanation of these parameters can be found in [21]. Furthermore, the mapping and demapping process used is the Hermitian symmetric can be found in [8].  
Figure 4e illustrates the block diagram for the OTFDM scheme. The transmitter matrix is given by  
\[\mathbf{T}_{\text{OTFDM}}=\mathbf{\
abla}_{\mathbf{t}}\mathbf{F}^{\dagger} \overline{\mathbf{D}}^{\dagger}, \tag{34}\]  
while the receiver matrix is expressed as  
\[\mathbf{R}_{\text{OTFDM}}=\overline{\mathbf{D}}\mathbf{E}\mathbf{F}\mathbf{y}_ {\mathbf{r}}. \tag{35}\]  
The use of (34) and (35) allow us to obtain matrices \(\mathbf{A}_{i},i=-1,\ldots,I\) and \(\mathbf{G}\), the estimated mapped vector using (9), and the SINR using (19) for the OTFDM scheme.  
For the data communication system that operates free of interference, the estimated symbol associated with the \(k\)th subtile of the \(2N\)-length symbol block, with the vector representation given by (16), is expressed as  
\[\widehat{X}_{\zeta,I}[k]=X_{\zeta,I}[k]+\frac{1}{\sqrt{\beta[k]}}\sum_{i=u}^{ U}\mathrm{e}^{\mathrm{j}2\pi\frac{i[k]}{\beta[k]}i}\mathrm{e}^{-\mathrm{j}\pi \tau[k]}\Lambda_{\mathbf{\mathcal{H}}}^{-1}[i,i]V_{i}[i]. \tag{36}\]  
As a result, the SNR in the \(k\)th tileband is given by  
\[\underline{\gamma}_{\text{SNR}}[k]=\frac{P_{\mathrm{x}}[k,k]}{\frac{1}{\beta[ k]}\sum_{i=u}^{U}|\Lambda_{\mathbf{\mathcal{H}}}^{-1}[i,i]|^{2}P_{\mathrm{n}}[i,i]}, \tag{37}\]  
and, consequently, the corresponding nSNR is equal to  
\[\underline{\gamma}_{\text{eSNR}}[k]=\left(\frac{1}{\beta[k]}\sum_{i=u}^{U} \gamma_{\text{nSNR}}^{-1}[i]\right)^{-1}. \tag{38}\]  
Following the comments in Sections 4.1 and 4.2, we can state that each tileband's nSNR is a harmonic mean of the subband group of size \(\beta[k]\) centralized at \(\
u[k]\)Furthermore, OTFDM can be interpreted as a midterm of the SCCP scheme and HS-OFDM because the choice of the subtile geometry can result in the subband or subslot geometries. It means that the performance of OTFDM is delimited by the performance of HS-OFDM.