# Long-Range Acoustic Communication Using Differential Chirp Spread Spectrum
## 3 Differential Chirp Spread Spectrum  
The differential chirp spread spectrum (DCSS) method, combining differential coding and CSS methods, is proposed here. Differential coding is encoded using the previous bit and current bit's exclusive or (XOR) operation [28]. When the original bit is \(b_{i}\), the differential coding bit is expressed as  
\[xb_{i}=b_{i-1}\oplus b_{i}\,\ 1\leq i\leq N, \tag{6}\]  
where \(xb_{i}\) is differential encoding bit and N is the number of bits. \(b_{0}\) is called the initial bit, and it is set to either 0 or 1. The differential encoding bits are decoded through XOR operation, so they can be decoded into original bits. The signal only uses differential encoding bits and generates the same equation as in Equation (4). This is expressed as  
\[s_{dcss}(t)=\left\{\begin{array}{l}s_{cu}(t),\ if\ xb_{n}=0\\ s_{cd}(t),\ if\ xb_{n}=1\end{array}\right.,\ nT\leq t\leq(n+1)T \tag{7}\]  
First, the received signal, cut into two symbol lengths, is expressed as \(s_{dt}\). There are four types of matched filters in DCSS. This is expressed as  
\[\left\{\begin{array}{l}s_{uu}(t)=s_{cu}(t_{1})+s_{cu}(t_{2})\\ s_{dd}(t)=s_{cd}(t_{1})+s_{cd}(t_{2})\\ s_{ud}(t)=s_{cd}(t_{1})+s_{cu}(t_{2})\\ s_{du}(t)=s_{cu}(t_{1})+s_{cd}(t_{2})\end{array}\right.,\ 0\leq t_{1}<T,\ T\leq t_{2}<2T, \tag{8}\]  
where \(s_{uu}(t)\) is an up chirp that comes after an up chirp; \(s_{dd}(t)\) is a down chirp that comes after a down chirp; \(s_{ud}(t)\) is an up chirp comes after a down chirp; and \(s_{du}(t)\) is a down chirp come after an up chirp. If the received signal is expressed as \(Rx(t)\) and is divided into two symbol lengths, it can be described as \(s_{nn}=Rx(t),\ (n-1)T\leq t<(n+1)T\). The cross-correlation between \(s_{nn}\) and the four symbols are checked, and the bit to the largest symbol is determined. We used these filters for demodulation. That is, in the conventional method, a matched filter is constructed using only one symbol of up or down chirp, but the size of this time-bandwidth product changes because the proposed method uses two symbols. This is expressed as  
\[d(n)=\max\!\left\{\begin{array}{l}d_{uu}=\max(F(s_{nn})*F(s_{uu}))\\ d_{dd}=\max(F(s_{nn})*F(s_{dd}))\\ d_{du}=\max(F(s_{nn})*F(s_{du}))\\ d_{ud}=\max(F(s_{nn})*F(s_{ud}))\end{array}\right., \tag{9}\]  
where \(d_{uu}\) is the maximum value of cross-correlation between \(s_{uu}\) and \(s_{nn}\); \(d_{dd}\) is the maximum value of cross-correlation between \(s_{dd}\) and \(s_{nn}\); \(d_{du}\) is the maximum value of cross-correlation between \(s_{ud}\) and \(s_{nn}\). When \(d_{uu}\) or \(d_{dd}\) have a maximum value, the bit is demodulated by 0, and when \(d_{du}\) or \(d_{ud}\) have the maximum value, the bit is demodulated by 1.  
The DCSS method is shown in Figure 3. Although it is similar to the overall common CSS method, it uses differential encoding in the modulation process and two symbols at the receiver. The time-bandwidth product doubles because the chirp of the two symbols is determined whether there is a change in the chirp or not, and the matched filter is applied using two symbols. As the time-bandwidth product increases, the filter output increases, and it is possible to have better performance even if the SNR becomes worse.