## 2 Proposed Method
### TFSK Performance Analysis According to Whistle Pattern  
For the derivation of the time- and frequency-shifting unit requirements, the whistles are mathematically modeled, and the orthogonality requirements are derived. Assume that the whistle pattern is modeled as a function (\(f_{w}(t)\)), which is presented as [12]:  
\[w(t)=cos\left[\int f_{w}(t)dt\right]. \tag{1}\]  
Since the conventional TFSK modulation scheme shifts the whistle patterns by the time and the frequency according to transmit the binary bits, let the time- and frequency-shifting units be \(\Delta t\) and \(\Delta f\), respectively, and the total number of time and frequency grids be \(M\) and \(N\), respectively. If the whistle is modulated with input bits, a reference whistle is shifted from \(-\Delta t\frac{M-1}{2}\) to \(\Delta t\frac{M-1}{2}\) in the time domain and from \(-\Delta f\frac{N-1}{2}\) to \(\Delta f\frac{N-1}{2}\) in the frequency domain, respectively. Gray coding to the time-frequency mapping may be utilized to increase the BER. For \(M\) and \(N\) grids, the TFSK modulation conveys \((\ log_{2}(N)+log_{2}(M))\) bits per one whistle. If the whistle is modulated with \(m\) and \(n\) grid, which is an integer less than \(M\) and \(M\), respectively, the TFSK modulated whistle (\(S_{TF}(t)\)) is expressed as:  
\[S_{TF}(t)=\left[\delta(t-m\Delta t)\otimes w(t)\right]\times e^{\beta 2\pi m \Delta ft}, \tag{2}\]  
where \(\otimes\) denotes a convolutional operation. Figure 2 shows the two different TFSK modulated whistle examples. In Figure 2, \(M\) and \(N\) are four each and a total of four bits are allocated. The gray-colored whistle denotes a reference whistle, and blue and red whistles denote the TFSK modulated ones. In Figure 2a, a zero-chirp rate whistle is modulated. The red and blue  
Figure 1: Spectrograms of the various real Delphinus delphis whistles. Figure 2: Examples of Time Frequency Shift Keying (TFSK) modulated signals: (**a**) zero-chirp rate whistle pattern (chirp rate: 0 Hz/s), (**b**) Up-chirp whistle pattern (chirp rate is larger than 0 Hz/s). whistles are modulated by 01 and 11 in time, respectively, and by 10 in frequency. In Figure 2b, a whistle with a large-chirp rate is modulated. The red and blue ones are modulated by 01 and 11 in time, respectively, and by 10 in frequency. In Figure 2a, the blue and the red whistles are overlapped in the time-frequency domain because of the zero-chirp rate whistle and the small \(\Delta t\), and the receiver is unable to correctly determine the transmitted bits by the overlapped region. If \(\Delta t\) is larger than that of the whistle length, two whistles are separated in the time domain and the detection of the whistle has no ambiguity. Then, the receiver correctly decodes the transmitted bits. In Figure 2b, however, even though the same '\(\Delta t\) is utilized, two modulated whistles are orthogonal in the time-frequency domain, and the receiver has a low BER. Thus, the BER performance of the TFSK method depends on the whistle pattern, and to increase the BER, finding the good \(\Delta t\) and \(\Delta f\) satisfying the orthogonal requirements of the TFSK needs for the given whistle pattern. For \(\Delta f\), the calculation of the orthogonality requirement for an arbitrary whistle pattern is difficult. However, if the receiver demodulates the received signal using the multiplication of the complex conjugate to the transmitted whistle, the demodulated signal has a zero-chirp rate pattern. Then, the orthogonality requirement of \(\Delta f\) becomes similar to that of the conventional FSK modulation. When the time length of the whistle is \(L_{w}\), the \(\Delta f\) satisfying orthogonal requirements of the TFSK is given as:  
\[\Delta f>1/L_{w}. \tag{3}\]  
For \(\Delta t\), more manipulation is needed. Firstly, assume that \(\Delta t\) is less than \(L_{w}\). If \(\Delta t\) is larger than \(L_{w}\), no overlap and no misdetection occur, but the data rate decreases. Thus, this case is not considered in this paper.