# Long-Range Acoustic Communication Using Differential Chirp Spread Spectrum
## 2 Chirp Spread Spectrum  
The method of transmitting data using chirp signals in underwater communication is studied [22; 23; 24]. The conventional CSS method transmits bit information in up or down chirps. The chirp signal is expressed as  
\[s_{\text{Chirp}}(t)=\exp\Bigl{[}-2j\pi f_{0}t+\pi\mu^{2}\Bigr{]}, \tag{1}\]  
where \(f_{0}\) is the initial frequency; \(\mu\) is the chirp rate; \(t\) is a time; and \(j\) is an imaginary number. Then, \(\mu>0\) becomes the up chirp, and \(\mu<0\) becomes the down chirp. We can express Equation (1) using maximum frequency and minimum frequency, which are expressed as  
\[s_{cu}(t) =\exp\Biggl{[}-2j\pi\Biggl{(}\frac{f_{\text{max}}-f_{\text{min}}} {T}t+f_{\text{min}}\Biggr{)}\Biggr{]}, \tag{2}\] \[s_{cd}(t) =\exp\Biggl{[}-2j\pi\Biggl{(}\frac{f_{\text{min}}-f_{\text{max}} }{T}t+f_{\text{max}}\Biggr{)}\Biggr{]}, \tag{3}\]  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Name** & **Distance** \\ \hline Very short & \(\sim\)0.1 km \\ Short & 0.1 km\(\sim\)1 km \\ Medium & 1 km\(\sim\)10 km \\ Long & 10 km\(\sim\)100 km \\ Very long & 100 km\(\sim\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Underwater acoustics communication classification.  
where \(s_{cu}\) is the up chirp signal; \(s_{cd}\) is the down chirp signal; \(f_{\max}\) is the maximum frequency; and \(f_{\min}\) is the minimum frequency. Then, we modulated it according to bit information and the data packet is expressed as  
\[s_{css}(t)=\left\{\begin{array}{ll}s_{cu}(t),\;if\;b_{n}=0\\ s_{cd}(t),\;if\;b_{n}=1\end{array}\right.,\;nT\leq t\leq(n+1)T \tag{4}\]  
where \(b_{n}\) is a bit sequence; \(n\) is the n-th bit; and \(T\) is a symbol length.  
There are three main methods to demodulate the CSS signal [14]: dump detector, matched filter and Hough transform. We used the matched filter method because the dump detector is very vulnerable to Doppler and because Hough transform is complex. The matched filter method seeks the correlation between the received signal and transmitted signal.  
The packet structure is represented in Figure 1, where linear frequency modulation (LFM) is used to determine whether a signal was present or not, even if it was in the worst channel, and preamble is used to accurately measure the synchronization of the signal. The signal used either \(s_{cu}(t)\) or \(s_{cd}(t)\) to construct symbols based on the bit information.  
The matched filter used for CSS demodulation compares the two chirp signals. If the received signal is expressed as \(Rx(t)\) and is divided into one symbol length, it can be described as \(s_{n}=Rx(t),\;(n-1)T\leq t<nT\). The cross-correlation between \(s_{n}\) and the two chirps is checked, and the bit to the larger chirp is determined. Equation (5) expresses the matched filter algorithm:  
\[d(n)=\max\!\left\{\begin{array}{ll}d_{u}=\max(F(s_{n})*F(s_{cu}))\\ d_{d}=\max(F(s_{n})*F(s_{cd}))\end{array}\right., \tag{5}\]  
where \(F(\cdot)\) is the value converted to the fast Fourier transform (FFT) of the input signal; \(d_{1}\) is the maximum value of cross-correlation between \(s_{cu}\) and \(s_{n}\); \(d_{2}\) is the maximum value of cross-correlation between \(s_{cd}\) and \(s_{n}\); and \(d(n)\) is the demodulated bits. If \(d_{u}>d_{d}\), the bits are demodulated by 0, else the bits are demodulated by 1.  
A CSS modulation block diagram is shown in Figure 2. The signal that passes through a channel leaves only the band energy using a band pass filter (BPF) and uses preamble for fine synchronization of the data packet. Finally, two matched filters are used to demodulate the bit information.  
Figure 1: The packet structure.  
Figure 2: Block diagram of a conventional chirp spread spectrum (CSS) method.