# A New LoRa-like Transceiver Suited for LEO Satellite Communications +
## 2 Synchronization Issues in LoRa
### Analysis of Imperfect Synchronization on Symbol Estimation  
In this section, our objective is to derive and analyze a closed-form expression of the signal used to estimate the symbol in the presence of a:  
* Time varying Doppler frequency shift, \(f_{d}(t)=c_{d}t+v_{d}\), with \(c_{d}\) the Doppler rate (DR) and \(v_{d}\) is the Doppler shift;
* Uniformly distributed STO \(\Delta\tau\in\left[-\frac{T}{2},\frac{T}{2}\right)\). It should be noted that the CFO \(\Delta f\) is equal to \(v_{d}+v_{o}\), with \(v_{o}\) being the frequency mismatch of LOs between the transmitter and the receiver. Based on the latter notations, the continuous-time baseband received signal is expressed as:  
\[y(t)=\sqrt{P_{\text{s}}}(t-\Delta\tau)e^{j(2\pi\Delta f+\theta_{0})}e^{j2\pi \int_{0}^{t}f_{d}(\tau)d\tau}+w(t) \tag{11}\]  
where \(s(t)\), \(t\in\mathbb{R}\), is the continuous-time version of \(s(n)\), \(P\) is the received signal power, \(\theta_{0}\) is the initial phase, and \(w(t)\) is the complex additive white Gaussian noise (AWGN) signal with \(\sigma_{w}^{2}\) its variance. It should be noted that we considered here a non-frequency selective channel, which makes sense when LPWANs are considered, and even more for LEO communications. To correctly obtain the radio frequency signal, and due to the CFO and the Doppler shift, the analog to digital converter (ADC) output signal should be sampled with \(f_{s}^{\prime}\) greater than the Nyquist rate \(f_{s_{min}}=B\), with an oversampling \(\alpha=f_{s}^{\prime}/f_{s_{min}}\). However, to be compliant with the low complexity of the CSS demodulation principle [13], we consider in the following the sub-sampled signal at \(T_{s}=\frac{1}{f_{s_{min}}}\). Indeed, as it will be explained in Section 3, our proposed synchronization algorithm works at \(T_{s}\) contrary to the solution proposed by [22]. The signal sampled at \(f_{s}^{\prime}\) is just used to perform a precise time alignment of the signal when the fractional part of the STO is estimated. The discrete-time version of the received received signal can be expressed as:  
\[y(n)=\sqrt{P_{S}}(nT_{s}-\Delta\tau)e^{2\pi(\Delta f\pi T_{s}+\frac{\zeta_{s}( nT_{s})^{2}}{2})}e^{i\beta_{0}}+w(n) \tag{12}\]  
To perform our analysis, we propose to focus our attention on the decoding process of the \(p\)th transmitted chirp. We notice that, in the presence of a timing offset, the signal processed by the FFT at the receiver is composed of two consecutive chirps as illustrated in Figure 1.  
Thus, the signal in the \(p\)th \(T\)-long sequence can be expressed, after the dechirping process, as follows:  
\[ \begin{split} z(n,p)&=y(n,p)x_{ref}^{*}(n)\\ &=\sqrt{P}(v_{p-1}(n)+v_{p}(n))+w(n)\end{split}  \tag{13}\]  
where \(y(n,p)=y(n+(p-1)M)\)\(\forall n\in\llbracket 0,M-1\rrbracket\). If we define \(L=\lfloor\frac{\Delta\tau}{T_{s}}\rfloor\) as the floor value of the discrete time offset, the two signal components of \(z(n,p)\) can be written as:  
* A contribution of the \((p-1)^{th}\) transmitted chirp during the time interval \(\llbracket 0,L-1\rrbracket\); \[ \begin{split} v_{p-1}(n)&=s_{p-1}(n+M-\frac{ \Delta\tau}{T_{s}})e^{j2\pi(\Delta f+\frac{\zeta_{s}(nT_{s})}{2})nT_{s}}x_{ref }^{*}(n)\\ &=e^{i\theta_{1}}e^{j2\pi n\left(\frac{\frac{S_{p-1}-\frac{ \Delta\tau}{T_{s}}+(\Delta f+\frac{\zeta_{s}(nT_{s})}{2})T}}{M}\right)}\end{split} \] (14)
* A contribution of the \(p^{\text{th}}\) transmitted chirp during the time interval \(\llbracket L,M-1\rrbracket\); \[ \begin{split} v_{p}(n)&=s_{p}(n-\frac{\Delta \tau}{T_{s}})e^{j2\pi(\Delta f+\frac{\zeta_{s}(nT_{s})}{2})nT_{s}}x_{ref}^{*} (n)\\ &=e^{i\theta_{2}}e^{j2\pi n\left(\frac{S_{p-1}-\frac{\Delta\tau} {T_{s}}+(\Delta f+\frac{\zeta_{s}(nT_{s})}{2})T}{M}\right)}\end{split} \] (15) where \(\theta_{1}\) and \(\theta_{2}\) represent two constant arguments, which have an impact on the symbol estimation in a presence of time synchronization errors due to the phase discontinuity created in the signal processed by FFT.