# LoRa Resource Allocation Algorithm for Higher Data Rates
## 3 LRA Algorithm Framework  
### Physical Layer  
LoRa uses CSS modulation at various ISM bands such as 915 MHz for North America. LoRa transceivers, such as the SX1276, support programmable bandwidths (BW) from 7.8 to 500 kHz. Some of the most used BW settings are 125 kHz, 250 kHz, and 500 kHz. SF represents the number of bits carried by each symbol and is directly proportional to the symbol time or chirp duration. Longer SFs result in longer symbol times and vice versa. Signals modulated with different SFs are said to be quasi-orthogonal [4], thus they can co-exist and be decoded successfully at the same time. The quasi-orthogonality can be proved by following the principle of orthogonal functions. Assume that \(s_{n}(t)\) is the time domain representation of a chirp signal with \(SF=n\).  
\[s_{n}(t)=Ae^{j2\pi(f_{0}t+\frac{B}{2f_{0}s_{n}}t^{2})} \tag{1}\]  
\(B\) is the BW, \(T_{s,n}\) is the symbol time, and \(f_{0}\) is the frequency offset that refers to the initial frequency of the chirp for \(SF=n\) and are defined as below:  
\[T_{s,n}=\frac{2^{n}}{B} \tag{2}\]  
\[f_{0}=\frac{B}{2^{n}}k \tag{3}\]where \(k\in\{0,\ldots,2^{n}-1\}\) is the chip decimal value. Two signals with \(SF=n\) and \(SF=p\), \(n\
eq p\) are quasi-orthogonal if their inner product is near zero.  
\[<s_{n}(t).s_{p}^{*}(t)>=\int s_{n(t)}s_{p}^{*}(t)dt\cong 0 \tag{4}\]  
Integral in (4) can be expanded as (5).  
\[A^{2}\!\!\int e^{2\pi[(f_{0n}-f_{0n})t]}dt+A^{2}\!\!\int e^{2\pi[(\frac{1}{ts_{ 0}}-\frac{1}{ts_{p}})\frac{8}{2}t^{2}]}dt \tag{5}\]  
Integrating the first term over interval \([0,T]\), \(T=\max\bigl{(}T_{s,n},T_{s,p}\bigr{)}\) is zero. The phase of the second term is generally a large value (61.0352 \(\times\) 10\({}^{6}\) for \(B=125\) kHz, \(n=7\), \(p=8\)) causing rapid oscillations throughout the interval. Using the stationary phase approximation will yield the magnitude of \(\sqrt{(\pi/a)}\) where \(a=2\pi\bigl{(}1/T_{s,n}-1/T_{s,p}\bigr{)}B/2\), indicating a near-zero value for the inner product.  
The actual BR depends on the SF, Coding Rate (CR), and BW, and is calculated using (6) [24], which gives a maximum BR of 21.875 kbps for BW = 500 kHz, CR = 1, and SF = 7.  
\[BR=SF\frac{4}{4+CR}\frac{B}{2^{SF}} \tag{6}\]  
The transmission power supported by LoRa is adjustable from \(-4\) dBm to +20 dBm. The lowest receiver sensitivity is around \(-130\) dBm, and it performs better at higher SF and lower BW (\(-136\) dBm at SF = 12 and BW = 125 kHz) [24]. This sensitivity increases LoRa's range at higher SF and lower BW over other LPWAN technologies [25]. The range of LoRa transmission depends on the radio link conditions and the antenna height to reduce signal reflection and absorption by the ground.  
Keeping the current LoRa modulation unchanged, its capacity in terms of the BR can be increased with parallel transmissions and receptions of the payload. Unlike the GW LoRa modem (such as SX13 XX or similar), the existing LoRa EDs do not support multiple channels and SFs for simultaneous communication. However, LoRa link capacity and network goodput can be improved by utilizing its orthogonality features and by improving the MAC protocol. As shown in Figure 1a, the LoRa physical layer packet consists of two mandatory blocks called the Preamble and Payload, along with two optional blocks for the header and Payload-CRC (Cyclic Redundancy Check). Figure 1b shows our proposed parallel setup by grouping two LoRa physical channels. These channels have the same configuration (BW, RF and CR), except for the different SFs, and are highly synchronized to transfer the data stream.  
Due to the similarity of configuration and operation, in the rest of this paper this channel is called a logical channel (LC), to facilitate describing the proposed method. All  
Figure 1: LoRa structure. (**a**) Physical channel, and (**b**) logical channel used in LRA algorithm.  
the block sizes are variable according to the transceiver configuration and the payload size ranges from 1 to 255 bytes.