## 2 Proposed Method
### Proposed Bio-Mimetic TFSK Modulation Method  
For the large DoM and the low BER, all types of whistles including the zero-slop whistles need to be utilized and the orthogonal requirements of the short-time unit are crucial. The proposed method utilizes the sequence with a large autocorrelation to solve the orthogonality problem for the zero-slop-like whistles and the short-time unit problem for the high data rate. The sequence with the large autocorrelation performance is widely used in digital communications to detect the exact time-frequency location when multiple signals exist at the same time [21; 22; 23; 24; 25]. If the different good and long sequences are multiplied to the multiple whistles, the time location of each whistle is precisely detected when the multiplied sequence at the receiver is the same as that used in the transmitted whistle. This is because the autocorrelation value of the sequence is large only at the time zero. Thus, if the sequence is utilized in the TFSK, the receiver can detect the exact time location even though \(\Delta t\) is smaller than that in the minimum in Equation (6) and the zero-slop whistles exist. If the sequence is the vector whose size is \(1\times L\), the sequence (\(C\)) is represented as \(C=\left[c_{1},\ldots,c_{l},\ldots,c_{L}\right]\), where \(c_{l}\) denotes the \(l\)-th element of the sequence with a value of \(1\) or \(-1\), and \(L\) is the length of the sequence, i.e., cardinality. The ideal \(C\) satisfies the following property [21; 22; 23; 24; 25]:  
\[\sum_{l}c_{l}\times c_{l-n}\ =\ \left\{ \begin{array}{c}\sum|c_{l}|^{2},\ n=0 \\ 0,\ n\
eq 0\end{array} \right., \tag{7}\]  
where \(\sum_{l}c_{l}\times c_{l-n}\) denotes the autocorrelation value of the sequence and \(n\) denotes a time-lag. Equation (7) is only satisfied when the sequence length (\(L\)) is infinite. However, if the sequence with a small \(L\) has a good autocorrelation characteristic, the autocorrelation at \(n\
eq 0\) has a very small value which can be considered as zero in Equation (7). When the sequence \(C\) is multiplied to the whistle, the whistle is divided by the cardinality of \(C\) and each vector element is sequentially multiplied to the divided whistles. Since the cardinality of \(C\) is \(L\), the length (\(L_{w}\)) of the whistle is divided by \(L\), and a piece (\(\tau\)) of the whistle is obtained by \(L_{w}/L\). The proposed transmission signal multiplied by the sequence is modeled as:  
\[S(t)=\left[\delta(t-m\Delta t)\otimes\{c_{t/\tau}\times w(t)\}\right]\times e ^{\beta\tau m\Delta ft}\, \tag{8}\]  
where \(A\) denotes the ceiling function of \(A\). In Equation (8), since \(\tau\) is inversely proportional to the time resolution, if \(L\) increases, the time detection resolution of the whistle also increases, whereas the frequency bandwidth of the whistle is spread out, which distorts the originality of the whistle. Thus, the best \(L\) needs to be determined to maximize the time resolution and to minimize the whistle distortion. In this paper, the maximum \(L\) is determined by satisfying the undistorted whistle requirement that human does not recognize the distortion of the whistle. The human assessment method is described in Sections 3 and 4. Assuming the best length of the sequence is \(L_{c}\), the spread frequency (\(B_{c}\)) of the whistle is calculated as:  
\[B_{c}=L_{c}/L_{w}. \tag{9}\]  
To satisfy Equation (7), \(\Delta t\) needs to be larger than half of \(\tau\), and to satisfy the orthogonality in the frequency domain, \(\Delta f\) needs to be greater than the twice of \(B_{c}\). Thus, the time- and orthogonality-requirements of the proposed method are derived as:  
\[\left\{ \begin{array}{l}\tau/2\leq\Delta t\\ 2B_{c}\leq\Delta f\end{array} \right.. \tag{10}\]  
Note that if the sequence length is large, \(\Delta t\) in Equation (10) is smaller than the minimum value in Equation (6).