## 5 Lake Trials
However, due to the property of the chirp signal, the multipath has little effect on the estimation. Figure 17a shows the correctly estimated symbol, which is on the right side in the FrFT bin and fixed on the number 240 of the FrFT bin point. We can determine from this symbol that the down-slope chirp signal has a center frequency of 1.2 kHz. On the other hand, Figure 17b shows the incorrectly estimated symbol. The receiver estimates to a different point. The transmitted signal has the transform order \(\alpha_{opt}=0.996\), but the received signal has the approximate value \"1\". This means that the receiver cannot distinguish between left and right. Therefore, we determined this symbol as the correct center frequency and the incorrect slope. In this case, the received symbol was affected by the serious Doppler effect beyond the Doppler tolerance of the chirp signal. Figure 18 shows the cumulative error rate per elapsed symbol to analyze the synchronizing effect of the proposed method. If the symbol synchronization is not performed, the cumulative errors increase because the symbol is compressed or expanded over time. Conversely, the performance of  
Figure 16: Spectrogram of (**a**) the transmitted signal, and (**b**) the received signal in the second trial. Figure 17: Color intensity of the FrFT results in (**a**) the correctly estimated symbol and (**b**) the incorrectly estimated symbol. the synchronized symbol is good. As a result, the FrFT receiver can be a system used to efficiently demodulate chirp signals that are robust and resistant to fading and Doppler effects. Tables 2 and 3 show the results of the uncoded bit error rate obtained by demodulating the received signal from the lake experiment, where the transmitted signal is not applied to the channel coding. Tables 2 and 3 show the cases where the moving speed of the transmitter is approximately 2 knots and 4 knots, respectively. Some experiments were performed in an environment where the weather condition was relatively poor, and the corresponding performance may be lower than those of other cases. Nevertheless, according to the proposed method, the bit error rate is reduced to 36.4% after iterative symbol synchronization. Therefore, we verified that the proposed method improves the performance. The chirp signal has properties that are robust to fading and Doppler effects, and the proposed method showed that it was possible to maintain these advantages through efficient demodulation. To compare the computations, we compared the computation times for each of the arbitrary symbols designed with the proposed method and the conventional method. The parameters of symbols were set to be similar to those shown in Figure 6. In this case, by using the MATLAB function \"\(\mathrm{Tic}-\mathrm{Toc}\)\", the running program time was measured and compared. The proposed method used a single receiver and scanned the value around the transform order to obtain the spectrum (see Figure 17). On the other hand, the conventional method used at least two receivers, which means that two transform orders must be calculated. As a result of a relative computation time comparison with a computer equipped with an Intel Core i7-6700/4 GHz processor, the proposed method took 0.2277 s while the conventional method took 0.4717 s. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Trial**} & \multicolumn{2}{c}{**First Trial**} & \multicolumn{2}{c}{**Second Trial**} \\ \cline{2-5}  & **Non-Sync Method** & **Sync Method** & **Non-Sync Method** & **Sync Method** \\ \hline
1 & 0.23 & 0.09 & 0.19 & 0.08 \\
2 & 0.24 & 0.09 & 0.35 & 0.11 \\
3 & 0.28 & 0.001 & 0.10 & 0.09 \\
4 & 0.12 & 0.09 & 0.08 & 0.07 \\
5 & 0.22 & 0.11 & 0.10 & 0.01 \\
6 & 0.09 & 0.01 & 0.07 & 0.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Uncoded bit error rate at 2 knots. Figure 18: Synchronization performance.