## 2 Proposed Method
### TFSK Performance Analysis According to Whistle Pattern
Since the whistle patterns are varied and non-linear function, the derivation of the orthogonality requirement of \(\Delta t\) for the whistles is difficult. If the non-linear whistle pattern is divided into short-intervals, a piece of the whistle can simply be modeled as a linear function (\(f_{w}(t)\)), e.g., \(1f_{w}(t)=at+b\) with a chirp rate (\(a\)), which is defined as \(a=\Delta f/\Delta t\) where \(\Delta t\) and \(\Delta f\) denote the time- and frequency-differences, respectively. Thus, if \(\Delta t\) is derived from the smallest chirp-rate of the whistle, the derived \(\Delta t\) satisfies the orthogonality of the whistle with the length (\(L_{w}\)). Assume that \(\theta\) is the decline of the whistle in the time-frequency domain and defined as \(\theta=tan^{-1}(\Delta f/\Delta t)\). To calculate the orthogonality requirement of the whistle with \(a\), we rotate two whistles to \(\theta\) clockwise in the time-frequency domain in Figure 3. The two modulated whistles are separated in the time-frequency domain keeping the orthogonality. In Figure 3a, if one whistle is shifted by \(\Delta t\) in the time domain, the overlapped time is given as \((L_{w}-\Delta t)\) and the frequency difference is given as \(df\) Hz. In Figure 3b, after the rotation by \(\theta\), the time duration of \((L_{w}-\Delta t)/cos\ \theta\) is overlapped in the time domain and the frequency gap of \(cos\theta\times df\) is obtained in the frequency domain. Figure 3: TFSK modulation whistle; (**a**) Normal, (**b**) Rotation by \(\theta\). If the frequency gap in Figure 3b is larger than the inverse of the overlapped time duration, two whistles satisfy the orthogonality in the time-frequency domain. Therefore, the orthogonality requirement for \(\Delta t\) is calculated as:  
\[cos\theta\times df\geq cos\theta/(L_{w}-\Delta t),0<\Delta t<L_{w}. \tag{4}\]  
Since \(df\) is equal to \(a\Delta t\) in Figure 3a, Equation (4) is rewritten as:  
\[a\geq 1/((L_{w}-\Delta t)\Delta t)\ and\ 0<\Delta t<L_{w}. \tag{5}\]  
For the whistle with the length of \(L_{w}\), the minimum \(a\) that distinguishes the whistle slops is calculated by \(\left(\frac{2}{L_{w}}\right)^{2}\)[20]. Thus, if \(a\) is smaller than \(\left(\frac{2}{L_{w}}\right)^{2}\), the whistle is considered as the zero-slop whistle and \(\Delta t\) is set as \(L_{w}\). If \(a\) is larger than \(\left(\frac{2}{L_{w}}\right)^{2}\), \(\Delta t\) is derived from Equation (5). In Equation (5), \(\Delta t\) is the variable of the 2 nd order convex function, and the solution of \(\Delta t\) is obtained as \(\frac{L_{w}}{2}-\sqrt{\frac{L_{w}^{2}}{4}-\frac{1}{a}}\leq\Delta t\leq\frac{L _{w}}{2}+\sqrt{\frac{L_{w}^{2}}{4}-\frac{1}{a}}\). Since the minimum \(\Delta t\) is of interest, the orthogonality requirement for the minimum \(\Delta t\) to \(a\) and \(L_{w}\) is obtained as:  
\[\left\{ \begin{array}{c}\Delta t=L_{w},\ a<\left(\frac{2}{L_{w}}\right)^{2} \\ \frac{L_{w}}{2}-\sqrt{\frac{L_{w}^{2}}{4}-\frac{1}{a}}\leq\Delta t,\ \left(\frac{2}{L_{w}}\right)^{2}\leq a \end{array} \right.. \tag{6}\]  
If \(\Delta t\) dissatisfies Equation (6), the different two whistles are not orthogonal in the time-frequency region. For the orthogonal requirement of the zero-chirp rate whistle, \(\Delta t\) needs to be equal to \(L_{w}\), which decreases the data rate of the biomimetic TFSK. Note that the whistle duration is several hundred msec to a few seconds. If the conventional biomimetic TPSK uses few available whistles for the small \(\Delta t\), the same whistles are frequently re-transmitted, which results in the low DoM. Therefore, this paper proposes the TFSK-based biomimetic modulation using all whistle patterns to increase the DoM.