# Underwater Biomimetic Covert Acoustic Communications Mimicking Multiple Dolphin Whistles
## 2 Modulation
The orange rectangular background shows the energy of the overlapped whistles. For energy detection, assume that the two-dimensional value of time-frequency by STFT in Equation (4) is \((\textbf{{S}}^{{}^{\prime}}(\tau,\omega))\), where \(\tau\) represents a time and \(\omega\) is a frequency, and the signal strength of a single signal \(x_{i}^{{}^{\prime}}(t)\) in Equation (4) is \(E\). Assume that the time position of the \(l\)-th original whale whistle is \(T_{l}^{in}\) and the time length of each whistle is \(L_{l}\). Then, by comparing the value of \(\textbf{{S}}^{\prime}(\tau,\omega)\) with \(E\), the location of the interfering signal can be easily found. If \(H_{0}\) is the interfered whistle and \(H_{1}\) is the non-interfered whistle, the interference detection criteria are expressed as follows:  
\[ \begin{cases}H_{0}:J_{\min f_{l}(t)}^{\max f_{l}(t)}\int_{T_{l}^{in}}^{T_{l}^{ \tau}+L_{l}}\textbf{{S}}^{\prime}(\tau,\omega)\;d\tau\;dt>E\\ H_{1}:J_{\min f_{l}(t)}^{\max f_{l}(t)}\int_{T_{l}^{\tau}}^{T_{l}^{\tau}+L_{l}} \textbf{{S}}^{\prime}(\tau,\omega)\;d\tau\;dt=E\end{cases} . \tag{6}\]  
If no interference \((H_{1})\) at the \(l\)-th whistle occurs, the whistle is modulated as the conventional CV-CFM. \(K-1\) information bits \((\textbf{{B}}_{l}=[b_{1},\ldots,b_{k},\ldots,b_{K}]^{T})\) are transmitted. Note that the first bit \((b_{1})\) as a dummy bit for differential modulation is allocated to the first  
Figure 4: An example of energy comparison to find overlapped whistles. DBPSK symbol (\(s_{1}\)). The \(k\)-th symbol (\(s_{k}\)) is represented as in Equation (7) and the symbol transmitted at the \(l\)-th whistle is \(\mathbf{S}_{I}=\left[s_{1},\ldots,s_{k},\ldots,s_{K}\right]^{T}\). \[s_{k}=\left(b_{k}+s_{k-1}\right)\,\,mod\,\,2,\,\,for\,\,k=2,\cdots,K. \tag{7}\]  
If the \(l\)-th whistle is overlapped, i.e., interfered \(\left(H_{0}\right)\), the proposed CV-CFM that mitigates the interference by interleaving with an orthogonal code is utilized. If the length of the orthogonal code is \(L_{c}\), the orthogonal code (\(\mathbf{C}_{l}=\left[c_{1},\ldots,c_{l_{c}},\ldots,c_{L_{c}}\right]^{T}\)) is used to modulate the \(l\)-th whistle where the interference occurs. The spread symbol \(\mathbf{S}_{I}\) with \(\mathbf{C}_{l}\) is defined as \(\mathbf{S}_{I}^{sp}=\left[s_{1},\ldots,s_{K/L_{c}}\right]^{T}\). When the interference occurs, the interference is concentrated on a part of a symbol. Since the overlapped length of the interfered whistles is short enough to mitigate the interference, the interference may not be completely erased. If the interleaving is applied to the symbols in an interfered whistle \(\mathbf{S}_{I}^{sp}\), the effect of the interference is spread over, which improves the interference mitigation performance. Assume that the interleaved symbols are denoted by \(\mathbf{S}_{I}^{int}\in\mathbb{R}^{K/L_{c}}\). The \(l\)-th proposed modulated whistle \(x_{l}(t)\) with the interference mitigated CV-CFM to the TFSK in Equation (4) is represented as:  
\[x_{l}(t)=\left[\left(\delta\left(t-m^{\prime}\Delta t\right)\otimes w_{l}(t) \right)+s_{k}\right]\times e^{i2\pi n^{\prime}\Delta ft} \begin{cases}if\,\,H_{ 0},&\quad s_{k}\in\mathbf{S}_{I}^{int}\\ if\,\,H_{1},&\quad s_{k}\in\mathbf{S}_{I}\end{cases} . \tag{8}\]  
If all \(L\) whistles are moved to the original whistle position \(\left(\mathbf{T}_{I}^{sp}\right)\) and added together, the multiple whistle signal that mimics the group dolphin whistles proposed in this paper is represented as:  
\[S_{prop}(t)=\sum_{l=1}^{L}x_{l}(t-T_{l}^{sp}). \tag{9}\]  
The following section describes how to demodulate the proposed modulation signal \(S_{prop}(t)\) consisting of TFSK and CV-CFM with interference mitigation methods.