## 4 Data Communication Scheme Derivation
### The OCDM Scheme  
First, we introduce the chirp band to refer to the dynamic frequency band occupied by a subchirp in the time interval of a \(\widetilde{N}\)-length symbol block.  
Let the block diagrams for the OCDM scheme transmitter and receiver be shown in Figure 4c. the OCDM scheme is obtained through the generic formulation by considering \(\mathbf{Q}=\mathbf{\Gamma}_{\text{HS}}^{\mathbf{t}}\mathbf{F}\), see (3), with the \((k,k)\)th element of \(\mathbf{\Gamma}_{\text{HS}}\) given by  
\[\Gamma_{\text{HS}}[k,k]=\begin{cases}\text{e}^{-\frac{1}{2N}k^{2}},&\text{for } 0\leq k\leq N-1\\ \text{e}^{\frac{1}{2N}k^{2}},&\text{for }N-1<k\leq 2N-1\end{cases}, \tag{21}\]  
and using the so-called type-IV mapping presented in [16], since it ensures that the OCDM scheme is fit for performing in baseband data communication systems. Therefore, the OCDM scheme transmitter and receiver matrix are equal to  
\[\mathbf{\Gamma}_{\text{OCDM}}=\mathbf{\overline{\mathbf{v}}}_{\mathbf{i}}\mathbf{F}^{ \mathbf{t}}\mathbf{\Gamma}_{\text{HS}}^{\mathbf{t}}\mathbf{F} \tag{22}\]  
and  
\[\mathbf{\mathbf{R}}_{\text{OCDM}}=\mathbf{F}^{\mathbf{t}}\mathbf{\Gamma}_{\text{HS}} \mathbf{E}\mathbf{\overline{\mathbf{v}}}_{\mathbf{r}}, \tag{23}\]  
respectively. With the possession of (22) and (23), we obtain the matrices \(\mathbf{\Lambda}_{i}\), \(i=-1,\ldots,I\) and \(\mathbf{G}\) for the OCDM scheme. Applying (22) and (23) in (19) results in the SINR matrix for the OCDM scheme.  
To find the signal-to-noise ratio (SNR) for the OCDM scheme, we must assume that \(L_{\text{CP}}\geq L_{h}-1\) and \(\Delta=0\). Consequently, the estimated symbol associated with the \(k\)th subchirp of the \(2N\)-length symbol block, with the vector representation given by (16), is equal to  
\[\widehat{X}_{\mathcal{L}}[k]=X_{\mathcal{L}}[k]+\frac{1}{\sqrt{2N}}\sum_{i=0}^ {2N-1}\text{e}^{\frac{2\pi}{2N}(ki-\hat{t}^{2})}\Lambda_{\mathbf{\mathcal{H}}}^{-1 }[i,i]V_{i}[i], \tag{24}\]  
with \(\mathbb{E}\{V_{i}[i]\}=0\), \(\mathbb{E}\{V_{i}[i]V_{i}^{*}[i^{\prime}]\}=\mathbb{E}\{V_{i}[i]\}\mathbb{E} \{V_{i}^{*}[i^{\prime}]\}\), \(\forall i\
eq i^{\prime}\), and \(\mathbf{\Lambda}_{\mathbf{\mathcal{H}}}\) standing for the diagonal matrix containing the elements of \(\mathbf{\mathcal{H}}\). Therefore, the SNR in the \(k\)th chirp band is expressed as  
\[\underline{\gamma}_{\text{SNR}}[k]\triangleq\frac{P_{\text{x}}[k,k]}{\frac{1} {2N}\sum_{i=0}^{2N-1}|\Lambda_{\mathbf{\mathcal{H}}}^{-1}[i,i]|^{2}P_{\text{in}}[i,i]}. \tag{25}\]Hence, the nSNR in the \(k\)th chirp band is given by  
\[\underline{\gamma}_{\text{nSNR}}[k]\triangleq\left(\frac{1}{2N}\sum_{i=0}^{2N-1} \gamma_{\text{nSNR}}^{-1}[i]\right)^{-1}, \tag{26}\]  
where  
\[\gamma_{\text{nSNR}}[i]=|\Lambda_{\mathbf{\mathcal{H}}}[i,i]|^{2}P_{\text{n}}^{-1} [i,i] \tag{27}\]  
is the nSNR associated with the \(i\)th subcarrier of the well-known HS-OFDM scheme and \(P_{\text{n}}[i,i]\) is the element in the position \((i,i)\) of the matrix \(\mathbf{P}_{\text{n}}\). In other words, the nSNR for using the OCDM scheme can be interpreted as a harmonic mean of the nSNR for the HS-OFDM scheme, which raises a few interesting comments. First, (26), being a harmonic mean of nSNR in the HS-OFDM scheme, confirms that a single chirp occupies the whole spectrum during the period of a \(\tilde{N}\)-length symbol block, while the HS-OFDM scheme states that one subcarrier occupies the same frequency subband during the period of a \(\tilde{N}\)-length symbol block. Moreover, since all chirp bands have the same nSNR, as shown in (26), the same number of bits are transmitted by all subchirps. In other words, the use of OA and UA results in the same power and bit allocations in all subchirps of an OCDM scheme.