# The Long-Range Biomimetic Covert Communication Method Mimicking Large Whale
## 3 The Proposed Biomimicking Communication Method
### The Proposed Biomimicking Modulation Method  
The time-frequency units of the model (\(\Delta T\) and \(\Delta F\)) satisfy symbol orthogonality and become the symbol time frequency. The whistle is divided by the symbol length \(\Delta T\). Let us assume that whistle is divided into \(K\) symbols. Information bits are mapped to the available frequency values at each symbol period. Figure 4 shows an example. In the \((k-1)\)-th symbol period, there are four available frequency values and two bits are modulated. If 10 is modulated to the \((k-1)\)-th symbol, there are two available frequencies in the \(k\)-th symbol period. Thus, the \(k\)-th symbol modulates one bit, whereas the \((k-1)\)-th symbol modulates two bits. That is, the number of modulation bits in the \(k\)-th symbol depends on the frequency value selected in the \((k-1)\)-th symbol. \(M_{k}\) is the modulation order in the \(k\)-th symbol. In Figure 4, the proposed method selects the available whistle frequency according to the bits using gray coding and generates the entire biomimicking whistle using the selected frequency. The modulation process is described in detail. When \(k\) is 1, available frequency values are elements of the set \(\mathrm{F}_{1}\) in Equation (1) and the modulation order \(M_{1}\) is calculated as \(log_{2}(|\mathrm{F}_{1}|)\). Thus, \(M_{1}\) bits are transmitted with the first symbol. Assume that \(\mathrm{B}_{M_{1}}\) is the set of all \(M_{1}\) bit sequences. Each element of \(\mathrm{B}_{M_{1}}\) is mapped one by one to an element in a set \(\mathrm{F}_{1}\) using gray coding. When all the bit sequences in \(\mathrm{B}_{M_{1}}\) are converted to decimals, the converted result is a set of integers from 0 to \(2^{M_{1}}-1\). This set of integers becomes the symbol set (\(\mathrm{S}_{1}\)) for the first symbol and each element of \(\mathrm{S}_{1}\) is an index of \(\mathrm{F}_{1}\). Let us assume that \(b_{1}\) is the \(\mathrm{Tx}\) bits sequence and it is converted to the symbol \(s_{1}(s_{1}\in\mathrm{S}_{1})\). The whistle frequency of the first symbol is selected with \(\mathrm{F}_{1}(s_{1})\). When \(k\) is 2 or more, the available frequency in the \(k\)-th symbol depends on the selected frequency in the \((k-1)\)-th symbol. Therefore, all frequencies in the set \(\mathrm{F}_{k}\) cannot be used, as shown in Figure 4. When \(\mathrm{G}_{k}\) is the set of frequency values available in the \(k\)-th symbol and \(\mathrm{G}_{k}\) is expressed by Equation (2), then  
Figure 4: Example of a whistle-mimicking modulation signal. \[\left\{ \begin{array}{ll}k\geq 2,&\mathrm{G}_{k}=\left\{\Delta F\times l|\mathrm{Pr}(G _{k-1}(s_{k-1})=f_{k-1}|f_{k}=\Delta F\times l)>\mathrm{Thr},l=1\,\sim\,L\right\} \.\\ k=1,&\mathrm{G}_{1}=F_{1}\end{array} \right. \tag{2}\]  
In Equation (2), when \(k\) is 1, any frequency in the set F\({}_{1}\) is available. Thus, G\({}_{1}\) is the same as F\({}_{1}\). If \(k\) is greater than or equal to 2, a frequency value is selected from the set G\({}_{k}\) according to the Tx bits \(b_{k}\) and the modulation order \(M_{k}\) is calculated as \(log_{2}(|\mathrm{G}_{k}|)\). Therefore, likewise, when \(k\) is 1, the whistle frequency for the \(k\)-th symbol is selected as G\({}_{k}(s_{k})\) and the biomimicking communication signal in the \(k\)-th symbol period is expressed by Equation (3):  
\[W_{k}(t)=\cos\biggl{(}2\pi\biggl{(}\frac{(\mathrm{G}_{k-1}(s_{k-1}))-\mathrm{ G}_{k}(s_{k})}{2\Delta T}t^{2}+\mathrm{G}_{k-1}(s_{k-1})t\biggr{)}\biggr{)}. \tag{3}\]  
In Equation (3), the biomimicking communication signal in the \(k\)-th symbol period is LFM, as shown in Figure 4. The entire biomimicking communication signal is the sum of all the signals from 1 to the \(K\)-th symbol period, and it is obtained by Equation (4):  
\[W(t)=\sum_{k=2}^{K}W_{k}(t-\Delta Tk). \tag{4}\]  
The block diagram of the proposed biomimicking modulation is shown in Figure 5.  
A preamble is inserted in front of the biomimicking communication signal to find the received signal and estimate the channel delay at the receiver.