# The Long-Range Biomimetic Covert Communication Method Mimicking Large Whale
## 3 The Proposed Biomimicking Communication Method
### The Proposed Biomimicking Demodulation Method  
The transmitted signals generated using Equation (4) pass through the underwater channel and are received at the receiver. The received signal \(y(t)\) is modeled as  
\[y(t)=\sum_{i=1}^{l}H_{i}\times W(t-\tau_{i})+n(t). \tag{5}\]  
Figure 5: Block diagram of the proposed biomimicking modulation. Figure 6: The frame structure of the proposed biomimicking signal. In Equation (5), \(H_{i}\) is the channel gain of the \(i\)-th path and \(n(t)\) denotes an additive white Gaussian noise (AWGN). The number of the path is \(I\). The receiver generally finds the received signal on the path with the greatest channel gain using the preamble. Thus, we assume that \(\tau_{1}\) is zero and \(H_{i}\) has the maximum value when \(i\) is 1. For the precise detection of received bits, a maximum likelihood sequence detection (MLSD)-based receiver is proposed. In Equation (2), the \(k\)-th symbol is modulated depending on the \((k-1)\)-th symbol. Therefore, ML detecting for the sequence with \(K\) symbols shows a lower error rate than detecting each of the \(K\) symbols as a single one. When the sequence of the transmitted \(K\) symbols is \(\mathrm{S}_{\mathrm{Tx}}^{*}=\left\{s_{1}^{*},\ldots,s_{k}^{*},\ldots,s_{K}^{*}\right\}\), detecting the sequence of symbols with the maximum likelihood probability for \(\mathrm{S}_{\mathrm{Tx}}^{*}\) is expressed as  
\[ \begin{array}{rl}\hat{\mathrm{S}}_{\mathrm{Rx}}&=\max_{S_{1},\ldots,S_{K}} \mathrm{P}\big{(}\,\mathrm{G}_{1}\big{(}s_{1}^{*}\big{)}=\,\mathrm{G}_{1}(s_{ 1}),\ldots,\mathrm{G}_{K}\big{(}s_{K}^{*}\big{)}=\,\mathrm{G}_{k}(s_{K}), \ldots,\mathrm{G}_{K}\big{(}s_{K}^{*}\big{)}=\,\mathrm{G}_{K}(s_{K})\big{)}\\ &=\max_{S_{1},\ldots,S_{K}}\prod_{k=2}^{K}\mathrm{P}\big{(}\,\mathrm{G}_{k-1} \big{(}s_{k-1}^{*}\big{)}=\,\mathrm{G}_{k-1}(s_{k-1})\big{|}\,\mathrm{G}_{k} \big{(}s_{k}^{*}\big{)}=\,\mathrm{G}_{k}(s_{k})\big{)}\end{array} . \tag{6}\]  
To find a symbol sequence that satisfies Equation (6), the proposed demodulation method calculates a correlation between the transmitted whistle and all whistles that the model can generate. When the generated whistle is \(\hat{W}(t)\) with the given model and \(K\) symbols \(\{s_{1},\ldots,s_{K}\}\), the correlation between the generated whistle \(\hat{W}(t)\) and transmitted whistle \(W(t)\) is expressed as  
\[ \begin{array}{rl}R(\tau)&=\int\limits_{K=2}^{K}\int\limits_{K=2}^{K}\int \limits_{W_{k}}(t)\times\hat{W}_{k}(t)dt\\ &=\sum_{k=2}^{K}R_{k}(\tau)\end{array} . \tag{7}\]  
As derived from Equation (7), the correlation between \(\hat{W}(t)\) and \(W(t)\) is equal to the summation of \(K\) symbol correlation values \(R_{k}(\tau)\) that are calculated with the transmitted and generated whistle for each symbol. Therefore, if the symbol sequence satisfies Equation (6), the signal generated by the corresponding sequence has the maximum correlation with the received signal. The correlation between \(y(t)\) and \(\hat{W}(t)\) is expressed as Equation (8). \[y(t)*\hat{W}(t)=\sum_{i=1}^{I}H_{i}\times R(\tau-\tau_{i})+n^{\prime}(\tau), \tag{8}\]  
where \(*\) denotes a correlation operation. As mentioned above, when \(\hat{W}_{k}(t)\) is equal to \(W_{k}(t)\), \(R(\tau)\) is Kronecker delta function \(\delta(\tau)\) and the result of Equation (8) is the same as the delay profile. Note that \(H_{i}\) has the maximum value when \(i\) is 1. Thus, the maximum value of Equation (8) is the gain of the first path \(H_{1}\), whereas the proposed demodulation method obtains an additional signal gain from the received signal through the other paths. Let us assume that the preamble has gone through the same multipath as the received signal. The estimated channel using the preamble is \(\sum_{i=1}^{I}\;H_{i}\times\delta(\tau-\tau_{i})\). To obtain the additional signal gain, integrate and dump the estimated channel and Equation (8). The result of the integration and dump is expressed as follows:  
\[\left\{