## 4 DASH7 Communication System
### Air Interface
#### 4.1.3 DASH7 Modulation Scheme  
The DASH7 protocol adopts the widely used Frequency-Shift Keying (FSK) modulation scheme with a Gaussian pulse shape filter for the transmitted samples (hence, this modulation is referred to as GFSK). Accordingly, the transmitted waveform can be presented as follows:  
\[s(t)=Ae^{i\Phi(t)}\, \tag{7}\]  
where \(A\) is the transmitted signal's amplitude and \(\Phi(t)\) is the angular phase which can be expressed as follows:  
\[\Phi(t)=2\pi h\int_{0}^{t}\alpha(\tau)d\tau. \tag{8}\]  
\(\alpha(t)\) for an FSK signal is as follows:  
\[\alpha(t)=\sum_{i=0}^{L}a(i)q(t-iT_{s})\, \tag{9}\]  
where \(a\) is \(1\) or \(-1\) and represents the transmitted bit value. L is the number of transmitted bits, and \(q(t-iT_{s})\) is the shape filter response of the transmitted signal (i.e., for GFSK, the \(q(t)\) is a Gaussian filter response). Finally, \(h\) is the modulation index, and can be expressed as follows:  
\[h=\frac{2\Delta f}{R_{b}}, \tag{10}\]  
where \(\Delta f\) is the peak frequency deviation and \(R_{b}\) is the data rate.  
In general, GFSK has a higher spectral efficiency compared to FSK. The Gaussian pulse shaping filter smooths the symbol transitions and, therefore, is more bandwidth-efficient because it reduces the amount of sidelobes. In this way, it also reduces Inter-Channel Interference (ICI). The effect of FSK- and GFSK-modulated signals can be seen in a two-dimensional and three-dimensional view in Figure 3 and Figure 4, respectively. The energy is much more spread in the frequency domain when no pulse shaping is applied, whereas the energy is more concentrated when a Gaussian pulse shaping step is implemented.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Class** & **Symbol Rate** & \begin{tabular}{c} **Modulation** \\ **Index** \\ \end{tabular} & \
\begin{tabular}{c} **Frequency** \\ **Deviation (\(\Delta f\))** \\ \end{tabular} &
\begin{tabular}{c} **Channel** \\ **Spacing (\(c\))** \\ \end{tabular} \\ \hline Lo-Rate & 9.6 kbps & \(1\) & \(\pm 4.8\) kHz & 0.025 MHz \\ \hline Normal & 55.555 kbps & \(1.8\) & \(\pm 50\) kHz & 0.2 MHz \\ \hline Hi-Rate & 166.667 kbps & \(0.5\) & \(\pm 41.667\) kHz & 0.2 MHz \\ \hline \hline \end{tabular}
\end{table}
Table 4: DASH7 channel classes.  
Figure 4: _Cont_.  
Figure 3: Two\(-\)dimensional spectrogram of ten modulated DASH7 packets seen over a time span of 1.4 s and a bandwidth of 200 kHz.  
The spread of a Gaussian-shaped symbol is defined by the bandwidth-to-symbol time product (BT), which, in essence, defines the curvature and amplitude of a symbol in the time domain. This is the product of the duration of a symbol and its spectral width. Its value lies between 0.1 and 1. If the BT has a small value, then the symbol is smeared in the time domain, and therefore, less bandwidth is required. However, it increases the chance of ISI occurring. When the BT product is high, the shape of the curve is closer to a rectangular shape in the time domain. This indicates that there is a lower chance of ISI, but it requires a much higher bandwidth. Therefore, a good trade-off needs to be found. For the executed measurements, a BT of 0.5 is used.