# Underwater Biomimetic Covert Acoustic Communications Mimicking Multiple Dolphin Whistles
## 2 Modulation
When TFSK and CV-CFM are combined, the \(\Delta t\) and \(\Delta f\) requirements of TFSK need to be derived to reduce detection errors during CV-CFM decoding. The one-frequency shift (\(\Delta f\)) of TFSK should be larger than the frequency spread (\(B_{s}\)) caused by CV-CFM to avoid overlapping frequency ranges between whistles and to ensure orthogonality. Thus, the value of \(\Delta f\) needs to be more than twice that of \(B_{s}\), considering the overlapping intervals between whistles. This can be expressed as:  
\[B_{s}\times 2\leq\Delta f. \tag{2}\]  
To determine the requirement of \(\Delta t\) for the TFSK modulation, the phase modulation of CV-CFM is considered. The requirement of \(\Delta t\) is derived based on the property that when two signals with different phases are demodulated at the receiver, their cross-correlation converges to zero [25; 26]. In the case of CV-CFM applied to a whistle, the phase changes every \(t_{s}\), which is a unit symbol for time of CV-CFM. For the overlapping intervals between whistles, if a time shift of two times \(t_{s}\) is applied, the whistle at the original position without time shift will have a different phase from the shifted whistle, resulting in a cross-correlation of zero. Therefore, the value of \(\Delta t\) that satisfies orthogonality between symbols during time shift modulation is given by the following equation:  
\[2t_{s}\leq\Delta t. \tag{3}\]  
The TFSK-modulated \(l\)-th whistle, denoted as \(x^{{}^{\prime}}_{l}(t)\), is shifted by the product of the arbitrary \(m^{\prime}\) and \(n^{\prime}\) values with \(\Delta t\) and \(\Delta f\), respectively. Therefore, \(x^{{}^{\prime}}_{l}(t)\) is expressed as:  
\[x^{{}^{\prime}}_{l}(t)=\left[\delta\big{(}t-m^{\prime}\Delta t\big{)}\otimes w _{l}(t)\right]\times e^{l2\pi n^{\prime}\Delta f}, \tag{4}\]  
where \(\otimes\) denotes a convolution operation. Using the single whistle in Equation (4), \(L\) whistles are shifted to their original positions by \(\mathbf{T}^{vp}_{l}=\left[T^{vp}_{1}\quad\cdots\quad T^{vp}_{l}\quad\cdots \quad T^{vp}_{l}\right]^{\mathsf{T}}\), and we add them together. Then, the TFSK-modulated whistles \(z(t)\) with \(L\) whistles become the dolphin group sound and are expressed as:  
\[z(t)=\sum_{l=1}^{L}x^{{}^{\prime}}_{l}(t-T^{vp}_{l}), \tag{5}\]  
Figure 3: An example of the proposed method (an individual whistle). where \(z(t)\) includes interfered multiple whistles in the time-frequency domain. The CV-CFM modulation method for individually modulated TFSK whistles is presented. Due to the characteristics of underwater acoustic communications, the frequency changes between symbols in a single whistle of CV-CFM are larger than the coherent bandwidth, and the conventional coherent modulation at the transmitter is inapplicable [12; 21]. For simple demodulation, non-coherent modulation schemes such as differential binary phase shift keying (DBPSK) are used, which utilize the phase difference between two adjacent symbols. In Equation (5), when interference occurs, the detection performance at the receiver decreases. This paper proposes a modulation method to detect interfered whistles and mitigate the interference by using the orthogonal codes during overlapped CV-CFM modulation. The orthogonal codes are used only for CV-CFM modulation in the interfered whistles, while they are not used for the non-interfered whistles. For mitigating the interference when TFSK-generated whistles are overlapped, the proposed interference detection method utilizes energy detection in that the energy of the overlapping part of the contour of an individual whistle is larger than that of non-overlapped whistles. An example of energy when individual whistles are overlapped and interfered with by two dolphin whistles is shown in Figure 4. In Figure 4, the red line represents the \(l\)-th whistle, and the red rectangular background represents the energy of the \(l\)-th whistle. The blue line represents the \((l+1)\)-th whistle, and the blue rectangular background represents the energy of the \((l+1)\)-th whistle.