# Asynchronous Chirp Slope Keying for Underwater Acoustic Communication
## 2 Materials and Methods
### Demodulation
#### 2.4.1 Digital Down-Converter  
Before the signal is fed into the resource intensive compression algorithm, we exploit the bandlimited nature of the signal and bring it down into the baseband, by calculating  
\[y_{\mathrm{fb}}[n,c] =\mathrm{BPF}\{y_{\mathrm{rc}}[n]\}, \tag{18}\] \[y_{\mathrm{fb}}[n,c] =y_{\mathrm{fb}}[n]\,y_{\mathrm{lo}},\] \[y_{\mathrm{bb}}[n,c] =\mathrm{LPF}\{y_{\mathrm{fb}}[n,c]\},\]  
where the functions LPF denotes an arbitrary lowpass filter, and BPF any suitable bandpass filter. The implementation is attached in Algorithm A4. The signal content outside of the band is suppressed by the analog bandpass filter of the receivers signal conditioning before the sampling. This is especially important for undersampling a signal to limit the aliasing effect of noise. To achieve the downconversion we first multiply the bandpass filtered raw signal \(y_{\mathrm{fb}}\) of each transmission band with sine waves \(y_{\mathrm{lo}}\) of frequency \(f_{\mathrm{lo}}\) to create the intermediate signal \(y_{\mathrm{lb}}\). This operation shifts the content of each of the \(N_{\mathrm{lo}}\) channels into the baseband, where a lowpass filter removes the higher harmonics and produces the baseband sequence \(y_{\mathrm{bb}}\). In doing so, the memory consumption increases by the number of channels \(N_{\mathrm{lo}}\), a one-dimensional real-valued input sequence of \([N_{\mathrm{rc}},1]\) gets mapped onto an \([N_{\mathrm{rc}},N_{\mathrm{lo}}]\) output array. The sequence can be truncated in frequency domain to an interval around the center, since most of the frequency bands ideally contain no information about the signal and one loses only information about noise and interference. We effectively resample the sequence to  
\[y_{\mathrm{ddc}} =\mathrm{resample}\{y_{\mathrm{bb}},\,f_{\mathrm{s}},\,f_{\mathrm{ s1}}\}, \tag{19}\] \[f_{\mathrm{s1}} =\frac{f_{\mathrm{s}}}{N_{\mathrm{res1}}}.\]  
As we use the single sideband approach, a minimal interval is limited by the center frequency  
\[f_{\mathrm{c}}=\tfrac{1}{2}(f_{1}+f_{0}) \tag{20}\]  
and half the bandwidth  
\[B_{\mathrm{f}}=(f_{1}-f_{0}). \tag{21}\]  
Assuming a sampling rate of, e.g., \(f_{\mathrm{s}}=88\,\mathrm{kHz}\), a bandwidth of \(B_{\mathrm{f}}=2.5\,\mathrm{kHz}\) and a sub-band center frequency of \(f_{\mathrm{c}}=3.0\,\mathrm{kHz}\) as used in the dual-band case, the minimal one-sided base band is  
\[B_{\mathrm{fbbm}}=f_{\mathrm{c}}+\tfrac{1}{2}B_{\mathrm{f}}, \tag{22}\]  
which is for the given example \(B_{\mathrm{fbbm}}=4.25\,\mathrm{kHz}\). Considering the original sample bandwidth and unchanged frequency bin width, the computation is reduced to \(B_{\mathrm{fbbm}}/\tfrac{1}{2}f_{\mathrm{s}}\), here by about 90% at most. The minimal interval truncation also removes information about the noise, so a trade-off is feasible that implements a larger interval of several bandwidths. Moreover, the whole band-shifting and resampling can effectively be done in the frequency domain with a shift and truncate operation, as described in detail in Algorithm A4. An example for a result of this operation is shown in Figure 3.