# A Time-Varying Filter for Doppler Compensation Applied to Underwater Acoustic OFDM
## 4 Validation Using Real Data
### Short-Range Ultra-Sonic Transmission
The decoded bits were compared to the transmitted bits to calculate the detection error. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Config. \#1** & **Config. \#2** \\ \hline Number of subcarriers (\(N\)) & 512 & 1024 \\ Data subcarriers (\(K_{d}\)) & 64 & 160 \\ Pilot subcarriers (\(K_{p}\)) & 64 & 96 \\ Null subcarriers (\(K_{n}\)) & 384 & 768 \\ Data subcarriers per frame (\(P_{d}\)) & 2 & 4 \\ Pilot subcarriers per frame (\(P_{p}\)) & 2 & 3 \\ Null subcarriers per frame (\(P_{n}\)) & 6 & 12 \\ Bandwidth (B) & 19 kHz & 20 kHz \\ Carrier frequency (\(f_{c}\)) & 121 kHz & 130 kHz \\ Fractional bandwidth (\(B/f_{c}\)) & 15.7\% & 15.4\% \\ Frequency separation (\(\Delta f\)) & 37.1 Hz & 19.5 Hz \\ Max. tolerable freq. shift (\(P_{n}\cdot\Delta f\)) & 222.6 Hz & 234.4 Hz \\ Block duration (\(T_{b}\)) & 33.6 ms & 60.2 ms \\ Cyclic prefix duration (\(T_{cp}\)) & 6.4 ms & 9 ms \\ Symbols/packet (\(M\)) & 6 & 6 \\ Length of preambles & 27.2 ms & 51.2 ms \\ \hline \hline \end{tabular}
\end{table} 
Table 2: OFDM parameters for the two short-range configurations. Figure 8: Measured and simulated channel impulse responses. As mentioned in Table 1, the maximum velocity was equal to \(v_{\max}=0.9\) m/s. The maximum value for the subcarrier frequencies for Config. #1 and Config. #2 were 130.5 kHz and 150 kHz, respectively. As such, using Equation (31) described in Section 3.2, the respective maximum Doppler shifts were equal to 78.3 Hz and 90 Hz. These frequency shifts are smaller than the maximum tolerable Doppler shift of either of the proposed systems, as calculated in Table 2. Therefore, the pilot placement should work efficiently in this scenario. To improve the performance of the OFDM communication link subject to frequency selectivity, a simple 1/3 repetition code was applied. At the receiver side, repetition decoding was done using majority logic detection. Although this technique provides limited gain, the purpose was to demonstrate the reliability of the Doppler compensation method. To predict the reliability of the proposed Doppler compensation method, a model of the communication link was developed. Initially, the model assumed that the time-varying delay of the strongest path is known and used for Doppler compensation. In a second model, the channel estimation algorithm was also included to accurately represent a fully automated receiver physical layer. The simulated BER for Config. #1 and Config. #2 are shown in Figures 8(a) and 9(a), respectively. As observed, the proposed pilot aided (PA) algorithm shows a BER performance better than 0.05 at high SNR (higher than 15 dB). As expected, the optimum equalization generally has a much better BER performance than the pilot aided compensation scenario. In addition, a comparison between the BER curves for Config. #1 with 64 pilot sub-carriers and Config. #2 with 96 pilot sub-carriers shows that the Doppler and channel estimation was improved by increasing the number of pilot sub-carriers. In the simulation, note that the channel model represents a multi-scale multipath scenario in which each path arrival has a different time-scaling factor. Therefore, the Doppler compensation with respect to the time-varying delay of the strongest path (optimum scenario) is not perfect in either of these systems. As a result, the BER curves reach an error floor because the inter-carrier interference cannot be entirely removed. The BER of both systems was evaluated through measurements and the results are presented in Figures 8(b) and 9(b). For Config. #1, the bit rate is 560 bps and 1680 bps with and without coding, respectively. According to Figure 8(a), the corresponding BERs for this configuration using PA compensation at SNR = 5.5 dB are simulated to be 0.102 and 0.159. These values are very close to the average measured BERs, which are 0.096 and 0.162, respectively, for a system with and without  
Figure 9: BER analysis of Config. #1. The red dotted lines in (**a**) show the value of BER at 5.5 dB for system with PA compensation (with and without coding).