# Hybrid Chirp Signal Design for Improved Long-Range (LoRa) Communications
## 4 Proposed Modulation Scheme
### FFT-Based Demodulation  
We propose an FFT-based demodulation technique for the up- and down-chirp-based LoRa modulation. At the receiver, the received signal is first multiplied by a synchronized raw down-chirp and then by a synchronized raw up-chirp signal. Let us assume that the received signal corresponds to an up-chirp-based transmitted symbol (i.e., \(k<2^{S-1}\)). In the absence of noise and interference, the received signal becomes:  
\[y_{k}(t)=\begin{cases}Ae^{2\pi(f_{l}+\frac{kB}{2^{S-1}}+\frac{B}{T})t}&\text{ if }t\leq\tau_{k_{u}}\\ Ae^{2\pi(f_{l}+\frac{kB}{2^{S-1}}-B+\frac{B}{T})t}&\text{if }t>\tau_{k_{u}} \end{cases} \tag{8}\]  
When the received signal is multiplied by a synchronized raw up-chirp \(Ae^{j2\pi\left(f_{l}+\frac{kB^{2}}{2T}\right)}\), the resultant signal becomes:  
\[z_{k_{u}}(t)=\begin{cases}A^{2}e^{2\pi(2f_{l}+\frac{kB}{2^{S-1}}+2\frac{B}{T}) t}&\text{if }t\leq\tau_{k_{u}}\\ A^{2}e^{2\pi(2f_{l}+\frac{kB}{2^{S-1}}-B+2\frac{B}{T})t}&\text{if }t>\tau_{k_{u}} \end{cases} \tag{9}\]  
On the other hand, when the received signal is multiplied by a synchronized raw down-chirp, the frequency of the resultant signal becomes:  
\[z_{k_{d}}(t)=\begin{cases}A^{2}e^{2\pi(f_{l}+f_{h}+\frac{kB}{2^{S-1}})t}&\text {if }t\leq\tau_{k_{u}}\\ A^{2}e^{2\pi(f_{l}+f_{h}+\frac{kB}{2^{S-1}}-B)t}&\text{if }t>\tau_{k_{u}} \end{cases} \tag{10}\]  
By examining the peak of the frequency spectrum obtained from (9) and (10), one can determine that the received signal corresponds to an up-chirp-based transmitted signal. We observe that (10) is similar to the traditional demodulation scheme, where only two frequency components are observed (as described in Section 3.2), while (9) is composed of multiple frequency components, where the signal power is distributed among the frequency components. As a result, after FFT, the peak of the frequency spectrum obtained from (10) will be much higher than the peak of the frequency spectrum obtained from (9). Note that a similar phenomenon can be observed when the down-chirp-based received signal is multiplied by a synchronized raw down- and up-chirp, respectively. Mathematically, the process can be described in the following manner. Firstly, we find the estimates of the index (\(\hat{k}_{u}\)) of the up-chirp-based transmitted symbol and the corresponding amplitude (\(\hat{F}_{k_{u}}\)) by the following manner:  
\[\hat{k}_{u}=\operatorname{argmax}\mathcal{F}(z_{k_{u}}(t))-f_{l}-f_{h} \tag{11}\] \[\hat{F}_{k_{u}}=\max(\mathcal{F}(z_{k_{u}}(t)))\]  
Figure 2: Time-frequency representation of the proposed LoRa modulation for \(\mathcal{S}=2\). (a) \(k=0\), (b) \(k=1\), (c) \(k=2\), (d) \(k=3\).  
Similarly, for the down-chirp-based transmitted symbol, we find the index (\(\hat{k}_{d}\)) and the corresponding amplitude (\(F_{\hat{k}_{d}}\)):  
\[\hat{k}_{d} =\operatorname*{argmax}\mathcal{F}(z_{k_{d}}(t))-f_{l}-f_{h} \tag{12}\] \[F_{\hat{k}_{d}} =\max\bigl{(}\mathcal{F}(z_{k_{d}}(t))\bigr{)}\]  
The estimated index of the transmitted symbol is given by  
\[\hat{k}=\operatorname*{argmax}_{i\in\{\hat{k}_{u},\hat{k}_{d}\}}(F_{i}).\]