# ICARUS--Very Low Power Satellite-Based IoT
## 3 Physical Layer and Doppler Compensation
### Joint Timing and Frequency Synchronization  
The joint time and frequency synchronization is difficult under the presence of large random Doppler shifts at low SNR. Conventional cross-correlation-based time and frequency synchronization typically uses pseudo-noise bipolar sequences, so called M-sequences [9; 11]. In ICARUS, we use a CAZAC (constant amplitude zero auto-correlation) sequence. Both sequence type options are compared in the following Table 2.  
When using bipolar M-sequences the correlation peak amplitude strongly depends on the Doppler shift of the received signal. The Doppler shift introduces a carrier frequency offset (CFO) of the received signal to its nominal center frequency which causes significant degradations. Figure 7 shows the cross-correlation peak degradation at high-carrier frequency offset values which makes M-sequences unsuitable for space IoT systems when operated with LEO (low earth orbit) satellite systems. When using CAZAC sequences, the performance changes [11; 12]. Figure 8 exhibits a preferable immunity of the cross-correlation peak amplitude against CFO. However, we observe a linear timing lag of the peak versus CFO. The linear timing lag appears quite impractical in the uplink when considering random Doppler shifts of the incoming tag signals. The ICARUS solution to jointly estimate timing and Doppler shift is to use a form of CAZAC sequences, widely known as Frank-Zhadoff-Chu (FZC) sequences. The UL and DL synchronization preamble sequences are composed of two consecutive complementary FZC subsequences. Subsequences 1 and 2 are defined as follows where the ICARUS parameters are listed in Table 3.  
\[\mathrm{z_{FZC}(n)}=\mathrm{\ exp}\bigg{(}\mathrm{jd}\pi\frac{\mathrm{un}(1+n+2 \mathrm{q})}{\mathrm{N_{Z}}}\bigg{)} \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline \hline  & **M-Sequences (Bipolar)** & **CAZAC (Complex Sequences)** \\ \hline Timing offset estimation & & \\ accuracy & High & Average \\ \hline Frequency offset tolerance & Low & High \\ \hline \multirow{2}{*}{
\begin{tabular}{c} Cross-correlator \\ complexity \\ \end{tabular} } & Low, binary sequences do not & High, complex sequences require \\  & require MULT operations & complex MULT operations during \\  & during correlation & correlation \\ \hline Signal generation & Easy (bipolar BPSK & Average (complex valued \\  & type sequence) & IQ sequence) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design Tradeoffs for Different Preamble Types. Figure 6: ICARUS downlink burst structure. where N\({}_{\text{Z}}\) denotes the sequence length. Variable n denotes the sample time domain index ranging from [0,..., N\({}_{\text{Z}}\)\(-\)1]. Value d is either set to 1 or \(-\)1, where u and q are carefully chosen signed integer values. The benefit of using two FZC sequences that are conjugate complex copies of each other is the symmetric timing shift property depicted in Figure 9. The correlation peak timing lag of both subsequences is inverted equal at a given CFO value. With this property in mind, an easy CFO and timing algorithm could be generated which allows the ICARUS system to be jointly synchronized in time and frequency. This synchronization algorithm is described in Figure 10. \begin{table}
\begin{tabular}{c c c} \hline \hline
**FZC Parameter** & **Subsequence 1** & **Subsequence 2** \\ \hline d & \(-\)1 & 1 \\ u & 1 & 1 \\ q & 0 & 0 \\ Nz Downlink & 255 = \(2^{8}-1\) & 255 = \(2^{8}-1\) \\ Nz Uplink & 16,383 = \(2^{14}-1\) & 16,383 = \(2^{14}-1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Downlink Beacon Sync Fields. Figure 8: Cross-correlation performance of a single CAZAC sequence used for time–frequency synchronization. Figure 7: Cross-correlation performance of bipolar M-sequences used for time–frequency synchronization. After the first correlation peak (subsequence 1) is detected, the receiver cross-correlator switches to subsequence 2. When detecting the cross-correlation peak of subsequence 2, the timing difference of both peaks is measured.