# Underwater Biomimetic Covert Acoustic Communications Mimicking Multiple Dolphin Whistles
## 3 Demodulation
For the detection of the interfered whistle at the receiver, the energy detection method in Equation (6) is used: If the TFSK modulated positions have been identified for \(L\) whistles, the \(L\) whistles are generated to reconfigure the received whistles. The time--frequency domain energy of these generated whistles is then calculated. If the energy of the whistles is greater than others, the whistles are considered overlapped whistles, i.e., interfered whistles. For the interfered whistle demodulation, the despreading is executed using the known orthogonal code. For whistle demodulation without interference, conventional CV-CFM phase demodulation is applied. This process is shown in Figure 6.  
To demodulate the phase-modulated bits of CV-CFM at the receiver, the \(l\)-th individual whistle is extracted from the received dolphin whistle \(y(t)\) using the TFSK demodulation results of Equation (10). This extraction process obtains the received individual whistle \(\hat{y}_{l}(t)\) as:  
\[\hat{y}_{l}(t)=|h(t)|\cos(2\pi\int f_{l}(t)dt+s_{k}+\theta_{h})\times e^{j2\pi n ^{\prime}\Delta ft}+n(t),k=1,\ldots K. \tag{12}\]The received extracted single whistle is multiplied by the complex conjugate of the frequency-shifted \(w_{l}(t)\), denoted as \(w_{l}^{\ast}(t)\times e^{-j2\pi n^{\prime}\Delta ft}\) in Equation (12), and the low-pass filtering is executed. Then, the whistle signal modulated by CV-CFM becomes a conventional baseband phase-modulated signal. The phase information of CV-CFM can be obtained as follows:  
\[r_{k}=|h|\cos(s_{k}+\theta_{h})+\hat{n},k=1,\ldots K \tag{13}\]  
In the case of whistles without interference (\(H_{1}\)), the symbols are demodulated by a conventional differential detection method. \(\angle r_{k}\) in Equation (13) represents the phase information of the \(k\)-th interval of the whistle. The \(K\) phase values on the \(l\)-th th received whistle are calculated as \(\hat{S}_{l}=\left[\angle r_{1},\ldots,\angle r_{k},\ldots,\angle r_{K}\right]\). Since the CV-CFM utilizes DBPSK, conventional differential detection can be used for detecting the \(K\)-1 transmitted bits. For the interference case (\(H_{0}\)), the transmitted signal is detected by deinterleaving and despreading the bits obtained by differential detection. The obtained phase value \(\angle r_{k}\) from the interfered whistle is the result of the multiplication between the code (\(c_{l_{c}}\)) and the transmitted symbol (\(s_{k}\)). Therefore, the transmitted symbol (\(\textbf{S}_{sp\_int}\)) is obtained by multiplying the deinterleaved symbol by \(\textbf{C}_{l}\) for despreading as:  
\[\textbf{S}_{sp\_int}=\left[c_{1},\ldots,c_{L_{c}}\right]\times \begin{bmatrix} c_{1}\times s_{1}&\cdots&c_{1}\times s_{\lfloor\frac{K}{L_{c}}\rfloor}\\ \vdots&\ddots&\vdots\\ c_{L_{c}}\times s_{1}&\cdots&c_{L_{c}}\times s_{\lfloor\frac{K}{L_{c}}\rfloor }\end{bmatrix} =\left[\sum\
olimits_{d=1}^{L_{c}}c_{1}^{2}\times s_{1},\cdots,\sum\
olimits_{l=1}^{L_{c}}c_{l}^{2}\times s_{\lfloor\frac{K}{L_{c}}\rfloor }\right] \tag{14}\]  
Since \(c_{l_{c}}\) takes a value of \(1\) or \(-1\) in Equation (14), the value of \(\sum\
olimits_{l_{c}=1}^{L_{c}}c_{l_{c}}^{2}\times s_{k}\) is equal to \(L_{c}\times s_{k}\). When spreading is used, an additional SNR gain of \(L_{c}\) is obtained compared to the case without spreading. This gain helps to mitigate inter-symbol interference. The data rate of the proposed technique without interference is calculated as follows: the transmission bits of TFSK by time-shifting and frequency-shifting are \(\left[log_{2}M\right]\) and \(\left[log_{2}\frac{Bw}{\Delta f}\right]\), respectively. The total TFSK transmission bits per whistle are given as \(\left(\left[log_{2}\frac{Bw}{\Delta f}\right]+\left[log_{2}M\right]\right)\). Let \(f_{max}\) be the maximum modulation bandwidth that preserves the DoM, and let the average length of a whistle be \(L_{w}\). The maximum number of symbols  
Figure 6: Despreading phase demodulation method. in CV-CFM is calculated as \(K=f_{max}L_{w}\).