# The Long-Range Biomimetic Covert Communication Method Mimicking Large Whale
## 2 Modeling Whistles of the Right Humpback Whale
It is assumed that there are \(M\) whistles with a frequency value of \(\Delta F\times l\) at an arbitrary time \(\Delta T\times k\). Among these \(M\) whistles, only \(M^{\prime}\) whistles have the increased frequency value \(\Delta F\times(l+1)\) at the next time \(\Delta T\times(k+1)\). When the frequency value of the whistle  
Figure 2: Example of the STFT results of whistles with various \(\Delta T\) and \(\Delta F\): (**a**) STFT with \(\Delta T\) and (**b**) STFT with \(\Delta T\). is \(\Delta F\times l\) at \(\Delta T\times k\), the probability that the frequency value is \(\Delta F\times(l+1)\) in the next time interval \(\Delta T\times(k+1)\) becomes \(M^{\prime}/M\). This conditional probability can be expressed as \(\mathrm{P}(\,f_{k}=\Delta F\times l\mid f_{k+1}=\Delta F\times(l+1))\). If this conditional probability is 0, the whale does not use the whistle pattern that changes from \(\Delta F\times l\) to \(\Delta F\times(l+1)\). By applying the Markov chain to the calculated conditional probabilities, we can model the time-dependent frequency-change rule of the whistle of the RHW. As a result of modeling, when the frequency value is \(\ f_{k-1}\) at \(\Delta T\times k\), the set of the frequency values generated by the whale at \(\Delta T\times(k+1)\) is expressed as Equation (1):  
\[\mathrm{F}_{k+1}=\{\ \Delta F\times l\mid\mathrm{Pr}(\,f_{k}\mid f_{k+1}= \Delta F\times l)>\mathrm{Thr},\ l=1\sim L,\ f_{k}\in\mathrm{F}_{k}\ \},\ k=1\sim K. \tag{1}\]  
In Equation (1), \(\mathrm{F}_{k+1}\) is the set of frequency values used by a whale at \(\Delta T\times(k+1)\). Thr is a threshold value for selecting only frequency values usually generated by whales, and we set it as 0.5. Note that frequency values in \(\mathrm{F}_{k+1}\) are quantized by \(\Delta F\), which guarantees symbol orthogonality. On the other hand, the actual whale whistle has time-frequency ambiguity, as shown in Figure 1. The time-frequency ambiguity of the whistle is also considered by modeling the time-dependent frequency change of the whistle with various lengths of the STFT window. Figure 2b shows the STFT results with another window length. In Figure 2, \(\Delta T\) is smaller than \(\Delta\hat{T}\). For example, let us assume that \(\Delta\hat{T}\) (e.g., \(\Delta\hat{T}\): 0.2) is \(2\Delta T\) (e.g., \(\Delta T\): 0.1). The time resolution of Figure 2a is higher than that of Figure 2b. However, the frequency resolution of Figure 2a is lower than that of Figure 2b. As shown in Figure 2, the time-frequency ambiguity of the whistle is considered by conducting STFT with various lengths of the window. In this paper, we modeled the whistle-generating habit using 6700 whistles of the RHW [38] and six lengths of the STFT window (i.e., 0.075 s, 0.1 s, 0.125 s, 0.15 s, 0.175 s, 0.2 s). Figure 3 shows some examples of the modeling results. In Figure 3, blue lines show the possible variation of whistle frequency. As shown in Figure 3, whistle lengths are quantized to the mean value of each time length of \(T_{w}\). Since \(\Delta T\) of Figure 3a is shorter than that of Figure 3b, whistles are divided into three parts in (a) and two parts in (b). In Figure 3, the RHWs are used to start within 130-165 Hz when generating a whistle. Thus, in Figure 3b, starting frequency values (\(\mathrm{F}_{1}\)) are quantized as (135 Hz, 145 Hz, 155 Hz) and frequency values at the half-time of the whistle (\(\mathrm{F}_{2}\)) are quantized as (145 Hz, 155 Hz, 165 Hz, 175 Hz, 185 Hz). As a result of the model in Figure 3b, the RHW does not generate exceeding 165 Hz at half of the whistle when the start frequency of the whistle is 135 Hz. The proposed modulation method utilizes this whistle-generating habit of the RHW to modulate bits. A simple example is represented in Figure 3b.