# Adaptive Parameters for LoRa-Based Networks Physical-Layer
## 3 Proposal
### Network and System Model
LoRa modulation is resistant to Doppler and multi-path effects. It consists of 0- and 1-bit representations as linear variations of frequencies. Figure 2 illustrates an example of LoRa 0- and 1-bit representation, the corresponding symbol for each bit, and a representation of the frequency variation of the symbol. The frequency shift rate is controlled by the Spreading Factor (SF) parameter, where higher values mean better signal immunity against noise. Figure 1: LoRaTM architecture using the LoRaWAN protocol. Figure 2: LoRa-based network bit representation. The left side of Equation (1) illustrates the Shannon-Hartley theorem, representing the capacity \(C\) of a communication link (in bits per second--bps) as a function of _Bandwidth_ (\(B\)) (Hz) (i.e., the amount of data (_packet_) that can be sent and received in an instant \(t\)), and _Strength_\(S\) (mW) and _Noise_\(N\) (mW) ratio (i.e., \(S/N\)) of the signal. The right side of Equation (1) illustrates a simplification of the Shannon-Hartley theorem demonstrating the higher the Bandwidth (\(B\)), the lower the SNR (Signal to Noise Ratio) of the link (\(C\)). \[C=B\times\log_{2}\biggl{(}1+\frac{S}{N}\biggr{)}\ \rightarrow\ \frac{S}{N}\approx\frac{C}{B} \tag{1}\]  
The propagation factor in LoRa is built upon base 2 logarithms of the _number of chirps per symbol_, where its settings have a huge influence on the effective network resilience and characteristics. For example, its setting parameters are responsible for coordinating the bit rate of the modulation, the resistance to noise interference, and its decoding. It is possible to highlight the most relevant parameter in LoRa modulation, the bandwidth, or \(BW\). So, a LoRa symbol consists of \(2^{SF}\) chirps, responsible for covering the entire frequency band. Considering it, as there are \(2^{SF}\) chirps in a symbol, we can conclude that one symbol can effectively encode SF bits of information. The bandwidth \(BW\) in LoRa is associated directly with the chirp rate on a network. It is possible to demonstrate it as follows, i.e., \(\frac{chirp\ per\ second}{BW\ m\ Hz}\). The chirp has several consequences on modulation behavior, from the signal strength to the adequate bandwidth performed. The symbol rate and bit rate in a given SF are proportional to the bandwidth frequency. Equation (2) correlates the duration of a symbol (\(T_{S}\)) to \(BW\) and \(SF\). \[T_{S}=\frac{2^{SF}}{BW} \tag{2}\]  
LoRa includes a Forward Error Correction (FEC) to detect and recover from possible bit errors. Thus, \(CR\) is equal to \(4/(4+n)\), with \(n\in 1,2,3,4\), adding redundant bits to perform the error correction/detection. The useful bit rate (\(Rb\)) is calculated based on \(BW\), \(SF\), and \(CR\) link parameters as illustrated in Equation (3). For example, considering a configuration with \(BW\) = 125 kHz, \(SF=7\), and \(CR=4/5\) values, the bit rate is equal to \(Rb\) = 5.5 kbps. Following Equation (3), Table 2 shows the bit rate for all CR values supported, with the three biggest \(BWs\) and the available \(SFs\). \[Rb=SF\times\frac{BW}{2^{SF}}\times CR \tag{3}\]  
According to Semtech [30], the definition of a LoRa channel rate (\(R_{b}\)), which is measured in _bits/s_, is represented by the Equation (4). [PERSON] et al. [32] introduced an equation  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & & \multicolumn{8}{c}{**Bandwidth (kHz)**} \\ \cline{2-13}  & & \multicolumn{8}{c}{**125**} & \multicolumn{8}{c}{**250**} & \multicolumn{8}{c}{**500**} \\ \cline{2-13}  & & \multicolumn{8}{c}{**CR**} \\ \cline{2-13}  & **4/5** & **4/6** & **4/7** & **4/8** & **4/5** & **4/6** & **4/7** & **4/8** & **4/5** & **4/6** & **4/7** & **4/8** \\ \hline \multirow{5}{*}{**SF**} & **7** & 5.5 & 4.6 & 3.9 & 3.4 & 10.9 & 9.1 & 7.8 &  
that correlates the Bit Error Rate (BER), on an Additive White Gaussian Noise (AWGN) channel, with the energy per bit-to-noise power spectral density ratio.