# LoRa 2.4 GHz Communication Link and Range
## 3 LoRa in the 2.4 GHz Band  
The physical layer of LoRa is a proprietary and closed source. Therefore, there are no official references or protocol specifications for the transmitted RF signal [21]. Accordingly, several research groups have been working to provide an understanding of the LoRa modulation scheme in the sub-GHz frequency band. [PERSON] [22], for instance, has provided a mathematical model, called Frequency Shift Chirp Modulation (FSCM), that describes the LoRa modulation process. The same model has been adopted by [PERSON] et al. [9]. On the other hand, [PERSON] [21], [PERSON] et al. [23] and [PERSON] et al. [24] have provided a model called Chirp Spread Spectrum (CSS) modulation based on the reverse engineering of LoRa's physical layer. Even though the formulation of the LoRa modulation scheme in the literature has been provided for the sub-GHz frequency band, the basic response of the modulation is expected to be the same for LoRa signals in the 2.4 GHz frequency band. Therefore, in this section, we modify the available physical layer models of sub-GHz LoRa to make them suitable for use in the 2.4 GHz frequency band. Assume \(x_{s}(k)\) is the transmitted LoRa sample; then, the received sampled signal \(x_{r}(k)\) with index \(k\) can be expressed as  
\[x_{r}(k)=a_{r}x_{s}(k-\tau)e^{i2\pi\Delta/k}+\omega(k), \tag{1}\]  
where \(a_{r}<1\) is the received signal amplitude, \(\tau\) is the time delay of the sample \(x_{s}(k)\) at the receiver, \(\Delta f\) is the frequency offset between the transmitter and the receiver, and \(\omega(k)\) is the identically independently distributed (i.i.d.) complex-valued Gaussian noise with zero-mean and variance \(\sigma^{2}\); i.e., \(CN(0,\sigma^{2})\). The time and frequency synchronization are beyond the scope of this paper. Therefore, in the following, we will consider a simplified version of (1), shown in (2). \[x_{r}(k)=a_{r}x_{s}(k)+\omega(k) \tag{2}\]The LoRa standard linear upchirp--also called a base chirp--can be expressed as [23; 24]  
\[x_{s}(k)=e^{l2\pi\left(\frac{\text{BW}}{2K^{2}}+f_{\circ}k\right)}, \tag{3}\]  
where BW is the operational bandwidth of the LoRa signal in the 2.4 GHz frequency band (as shown in Table 1) and \(K=2^{\text{SF}}/\text{BW}\) is the symbol duration, with SF representing the spreading factor (also shown in Table 1). Finally, \(f_{\circ}\) is the initial frequency, which can be expressed as  
\[f_{\circ}=s\frac{\text{BW}}{2^{\text{SF}}}, \tag{4}\]  
where \(s\in\left\{0,1\dots 2^{\text{SF}}\right\}\) is the transmitted data symbol. Setting \(s=0\) results in an upchirp, in which the frequency continuously increases during the symbol duration \(K\). We can also present (3) as  
\[x_{s}(k)=W_{K}^{\frac{\text{BW}}{2^{\text{SF}}}k^{2}+Kf_{\circ}k},\ \ W_{K}=e^{l2\pi/K}. \tag{5}\]  
The model in (5) is the linearly cyclically shifted version of a base Zadoff-Chu (ZC) sequence [25]. The ZC sequence possesses a unique autocorrelation property, in which the periodic autocorrelation is orthogonal (i.e., equal to zero) for all shifted replicas [26]. Therefore, the LoRa communication protocol uses this unique property to impose a random multiple access technique. Accordingly, an efficient utilization of the unlicensed spectrum can be obtained. The correlation between the received signal and the base chirp leads to  
\[z(k)= \frac{1}{K}\sum_{p=0}^{K-1}x(k+p)x_{s}^{*}(k)_{\text{mod}\,K} \tag{6}\] \[= \frac{1}{K}\sum_{p=0}^{K-1}\left(a_{r}x_{s}(k+p)+\omega(k+p) \right)x_{s}^{*}(k),\] \[= \begin{cases}a_{r}E_{s}+\
u_{\omega}&\text{for}\,p=0\\ \
u_{\omega}&\text{for}\,p\
eq 0\end{cases} \]  
where \(E_{s}\) is the energy of the symbol \(x_{s}\). Furthermore, \(\
u_{\omega}\) is the correlation between complex noise and the base chirp, which can be expressed as  
\[\
u_{\omega}=\frac{1}{K}\sum_{p=0}^{K-1}x_{s}^{*}(k)\omega(k+p), \tag{7}\]