# Underwater Biomimetic Covert Acoustic Communications Mimicking Multiple Dolphin Whistles
## 3 Demodulation  
In this section, the demodulation process of multiple whistles modulated by TFSK and interference-mitigated CV-CFM is described. To simultaneously demodulate the TFSK and CV-CFM, the complexity of the demodulation increases due to the numerous decoding possibilities. Thus, we propose a simple sequential demodulation approach. To determine the decoding sequence of TFSK and CV-CFM, it is preferable to demodulate one method first in a way that is not influenced by the other decoding method. In the proposed dolphin whistle mimicking method, it is necessary to demodulate the TFSK signal first because the CV-CFM demodulation cannot be executed without knowledge of the time-frequency shifted positions of the whistles. When two modulation techniques are simultaneously demodulated, the searching space of the demodulation is given as \(M\times N\times K\), while the sequential detection provides only (\(M\times N\)) + \(K\). If the whistle information modulated by TFSK is obtained, the phase-modulated bits of CV-CFM can be detected. The structure of the proposed receiver demodulation block is shown in Figure 5.  
Figure 5: The block diagram of the proposed demodulation method. First, the method for TFSK-modulated bits is described. The conventional TFSK demodulation method used a maximum likelihood (ML) detection approach in that the conjugates of all possible TFSK-modulated whistles are multiplied by the received whistles. The detection rule is to select the largest energy at a point in the time-frequency shift. However, since the received whistles are modulated by the PSK in CV-CFM, the conventional demodulation method cannot be utilized. Therefore, this paper proposes a time-frequency energy detection method that is not affected by the phase modulation of CV-CFM. In the proposed approach, the received signal is transformed into the 2D time-frequency domain using the short-time Fourier transform (STFT), and the value of each bin in the time-frequency domain is squared to obtain the energy. As a result of the energy calculation, the values of the bins are not affected by the phase modulation. By comparing the whistle contour energy of the generated whistle at the receiver with that of the received signal, the TFSK-modulated whistle that has the closest energy contour to that of the generated whistle is determined. The received dolphin signal (\(y(t)\)) is obtained by assuming that the transmitted signal \((S_{prop}(t))\) passes through the underwater channel (\(h(t)\)). This can be represented by :  
\[y(t)=h(t)\otimes S_{prop}(t)+n(t), \tag{10}\]  
where \(n(t)\) denotes the underwater background noise. Let the energy values of the received whistle (\(x_{l}(t)\)) be \(\mathbf{X}_{l}\in\mathrm{I}^{M\times N}\) and let the energy values of the whistle (\(w_{l}(t)\)) generated at the receiver based on \(\mathbf{T}_{l}^{w}\) be \(\mathbf{W}_{l}\in\mathrm{I}^{M\times N}\). The time interval resolution of the STFT is assumed to be \(t_{s}\) and the frequency resolution is assumed to be \(B_{s}\). The window length and the discrete Fourier transform shift interval are set accordingly to ensure that both \(\mathbf{X}_{l}\) and \(\mathbf{W}_{l}\) have the same intervals in the time-frequency bins [27]. Then, \(\mathbf{W}_{l}\) is shifted by the time-frequency modulation, \(m^{\prime}\Delta t\) and \(n^{\prime}\Delta f\), with \(M\) and \(N\), respectively. The TFSK modulation indices \(m\) and \(n\) can be found when the time-frequency contour energy distribution of the received \(\mathbf{Y}_{l}\) is the closest distribution of the \(l\)-th whistle \(\mathbf{X}_{l}\), which is given as:  
\[TFSK(m,n)=\arg\min_{m,n}(W_{l}(m,n)-\mathbf{Y}_{l}). \tag{11}\]  
After the TFSK demodulation, the receiver has the time-frequency shift information of each whistle and proceeds to decode the phase-modulated values of the CV-CFM. Since the spread and the non-spread CV-CFM to the whistle are used in the presence of interference, different demodulation schemes need to be used whether the interference exists or not.