# On the Use of LoRa Technology for Logic Selectivity in MV Distribution Networks
## 3 LoRa Technology  
### LoRa Bit Rate and Sensitivity  
The capability of a communication system modeled as a channel corrupted by additive white Gaussian noise (AWGN) can be expressed by the parameter channel capacity C, given by:  
\[\mathrm{C}=\mathrm{B}\log_{2}\left(1+\frac{\mathrm{S}}{\mathrm{N}}\right) \qquad\mathrm{[bit/s]} \tag{1}\]  
where B is the bandwidth and S/N is the signal-to-noise ratio at the receiver. Equation (1) shows that communication reliability can be achieved, even with extremely low values of the signal-to-noise ratio, by increasing the bandwidth. This characteristic is the basis of spreading techniques, which enable the spectral density of the transmitted signal power to be brought down to below the noise floor through an enlargement of its bandwidth without power variation. The spread spectrum techniques yield different advantages, such as signal protection with respect to narrowband interference, multiple channel access, and coexistence with existing systems. LoRa technology is based on a special spreading technique called chirp spread spectrum (CSS), which encodes the digital information of the narrowband signal to be transmitted in the phase variation of an up-chirp signal, thus generating a broadband signal [14; 15]. This spreading technique is particularly efficient and simplifies the realization of the receiver; in fact, upon reception, the narrowband signal is recovered by multiplying the received broadband signal by a down-chirp, allowing demodulation with a low bit error rate (BER) and with very low signal-to-noise ratios [16; 17]. This allows LoRa technology to span very high communication distances that range from 2 km in a densely populated urban environment to over 15 km in a rural environment. In a chirp, the information of a symbol is encoded, and the number of bits that constitutes the encoded symbol is the spreading factor (SF). According to the specifications of the technology, the SF can assume integer values from 7 to 12 and, therefore, as a symbol, it can be encoded as information using a number between 7 and 12 bits. The duration of the chirp coincides with the time symbol [18]. The levels (chips) that can be distinguished in a single chirp are:  
\[Chips=2^{\mathrm{SF}} \tag{2}\]  
and the chips resolution is given by:  
\[T_{chip}=\frac{1}{\mathrm{BW}} \tag{3}\]  
where bandwidth (BW) can be assumed to have a value equal to 150, 250, or 500 kHz. Therefore, the time symbol is:  
\[T_{symbol}=Chips\cdot T_{chip} \tag{4}\]  
By substituting Equations (2) and (3) into Equation (4):  
\[T_{symbol}=\frac{2^{\mathrm{SF}}}{\mathrm{BW}} \tag{5}\]  
Considering that SF bits are encoded by one symbol, then the bit rate is equal to:  
\[R_{b}=\frac{\mathrm{SF}}{T_{symbol}}=\mathrm{SF}\cdot\frac{\mathrm{BW}}{2^{ \mathrm{SF}}} \tag{6}\]Equations (5) and (6) highlight that the increase in BW causes the doubling of the bit rate \(R_{b}\), while the increment of a bit of the symbol and therefore of one unit of the SF results in its halving. The technology allows for different combinations of SF and BW, some of which determine the pseudo-orthogonality conditions between signals [19], i.e., chirps that have different slopes in the time-frequency representation and can be transmitted simultaneously in the channel without interference. The receiver must be set to the same SF and BW values as the transmitter in order to generate the down-chirp signal that allows the extraction of the narrowband signal from the received noisy wideband signal. LoRa modulation introduces redundancy bits that improve the robustness of the transmitted signal at the expense of the bit rate, which decreases further. So, the bit rate becomes:  
\[R_{b}=\frac{\text{SF}}{T_{symbol}}=\text{SF}\cdot\frac{\text{BW}}{2^{\text{ SF}}}\cdot\frac{4}{4+\text{CR}} \tag{7}\]  
where CR is the code rate and can assume an integer value ranging from 1 to 4. In Table 2, the values of the bit rate depend on BW and SF, with CR = 1, are reported.