# Asynchronous Chirp Slope Keying for Underwater Acoustic Communication
## 2 Materials and Methods
### Modulation
#### 2.2.2 Shaping  
The amplitude shaping window \(w[n]\) restricts the sequence to be non-zero in the interval between 0 and \(N_{\text{ref}}\) only. While this can be achieved through different window  
Figure 3: Resampling example for a linear chirp with \(f_{\text{c}}=3\,\text{kHz}\), \(B_{\text{f}}=5\,\text{kHz}\), and \(T=10\,\text{ms}\). **leftmost:** Base band signal \(y_{\text{th}}\) at the transmitter, **center left:** Transmission band \(y_{\text{lb}}\), **center right:** Undersampled signal on reception, **rightmost:** Down-converted base band signal \(y_{\text{lb}}\) at the receiver where the originally transmitted signal is overlayed in gray. The transmission band’s center frequency is at \(67.5\,\text{kHz}\). In the experiments, the intermediate band on reception is around \(20.5\,\text{kHz}\) due to undersampling with only \(f_{\text{s}}=88\,\text{kHz}\). Note the changed frequency scale for the Bode plots in the two columns on the right.  
functions, the tapered cosine, i.e., Tukey window is used in this work, because it can be varied easily between the rectangular, i.e., Dirichlet window and a sine, i.e., Hann window, by changing the single tuning factor \(a_{\text{t}}\) to 0 or 1, respectively [47]. This sets the main lobe width between \(4\pi/N_{\text{ref}}\) and \(8\pi/N_{\text{ref}}\), as well as the peak sidelobe between -13 dB and -31 dB [48]. The Tukey window is mathematically defined as [48]  
\[w[n]=\begin{cases}\frac{1}{2}\Big{[}1-\cos\Big{(}\frac{\pi}{N_{\text{tk}}}n \Big{)}\Big{]},&\text{for }0<n\leq N_{\text{tk}}\\ 1,&\text{for }N_{\text{tk}}<n\leq(N_{\text{ref}}-N_{\text{tk}})\\ \frac{1}{2}\Big{[}1-\cos\Big{(}\frac{\pi}{N_{\text{tk}}}(n-(N_{\text{ref}}-N_{ \text{tk}}))\Big{)}\Big{]},&\text{for }(N_{\text{ref}}-N_{\text{tk}})<n\leq N_{\text{ref}}\\ 0,&\text{otherwise},\end{cases}, \tag{11}\]  
where the threshold of the taper is set by  
\[N_{\text{tk}}=\frac{a_{\text{t}}N_{\text{ref}}}{2}.\]  
The amplitude shape of the chirp can thus be adapted to the channel, depending on the application, i.e., if a narrow autocorrelation peak width is required for spatial distinction of two close reverberations or wide smooth peaks are desired for more robust communication. The implementation we used in this work is described in Algorithm A2. A simplified comparison of a selection of window functions for a close echo is shown in Figures 4 and 5.  
Figure 4: Autocorrelation magnitude comparison of a selection of shaping window functions. All magnitudes are normalized by the Dirichlet shaped chirp power for comparison. Gaussian noise was added to a signal-to-noise ratio \(SNR=0\,\text{dB}\), as well as two echoes at \(n=9\) and \(n=19\). To the left the spatial resolution and peak power is higher, to the right the inter-signal interference and spectral leakage are reduced.  
Figure 5: Simulated spectrograms of the autocorrelations of a selection of shaping window functions. All magnitudes are normalized by the Dirichlet shaped chirp power for comparison. Gaussian noise was added to a signal-to-noise ratio of 0 dB, as well as two echoes at \(n=9\) and \(n=19\).