# A Channel Compensation Technique Based on Frequency-Hopping Binary Offset Carrier Modulated Signal
## 3 Performance Analysis
### Matched Filters
The vertical lines represent the original data bit required to be upsampled, and the magenta line standing for zero inserting output is achieved through the IFFT process. The blue line is the output of the square-root raised cosine filter in the transmitter, which deviates from the transmitted data a distance. Meanwhile, the cyan line represents the perfect recovery for SRRC filter's output in the receiver without interference, which is approaching the green line, and the output of a raised cosine filter designed in front of the receiver, while overlapping for the most of time. We can identify that the comparison in waveform formation between SRRC filter's output and zero inserted output is quite prominent in presenting less fluctuation in the amplitude, referring to the capability of better handling the effect Doppler shift imposed on the sampling decision. Figure 11: Impulse response and frequency response of raised cosine filter. (**a**) Impulse Response, (**b**) Frequency Response. Supposing the Doppler effect has inflicted relative frequency shift \(\xi\) on the received signal, a series of sampling points in the time domain gradually deviate from the original ones. We use verify the alleviation implemented through a raised cosine filter, using a root-mean-square error (RMSE) function on deviated samples to comparing with the zero inserting method that is normally employed to achieve over-sampling. The \(RMSE\) function can be expressed as  
\[RMSE=\sqrt{\frac{\parallel S_{ori}-S_{1}\parallel_{2}^{2}}{N}}, \tag{27}\]  
where \(N\) is the length of original sampling sequence \(S_{ori}\), \(S_{1}\) is a sampling-lagged sequence waiting to be tested, and 2-norm function is performed on absolute difference. Established on over-sampled OFDM baseband signals within a single period and without stochastic noise, they have been aligned to original discrete OFDM symbols at the very beginning. To choose a balanced factor, \(\beta=0.5\) and verified results, in comparison between the filtered OFDM signal by matched filters and zero-inserted OFDM signal, are demonstrated in following Table 1.  
Under different Doppler shift conditions, it can be concluded that either value of \(RMSE\) has increased with larger relative shift \(\xi\). Such a tendency gradually slows down when the Doppler shift comes to above 1000 Hz, and at each level it always has \(RMSE1<RMSE2\). The comparative ratio \(\eta\), raised cosine filtered \(RMSE1\) over zero inserted \(RMSE2\), reaches its minimum value of 25.63% at 250 Hz Doppler frequency shift and maximum value of 96.58% at 20,000 Hz. Additionally, explaining comparative ratio \(\eta\) as another factor \((1-\eta)\) for measuring how much Doppler effect has been alleviated and reduced, e.g., at 400 Hz raised cosine filtered \(RMSE1\) is less interfered with by \(1-\eta=51.83\%\) than zero inserted \(RMSE2\) \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Doppler Shift (Relative Shift \(\xi\))** & **Raised Cosine Filtered \(RMSE1\)** & **Zero Inserted \(RMSE2\)** &
\begin{tabular}{c} **Comparative Ratio \(\eta=\)** \\ \(RMSE1/RMSE2)\times 100\% \\ \end{tabular} \\ \hline
200 Hz (2.4438 \(\times\) 10\({}^{-5}\)) & 0.1376 & 0.5368 & 25.63\% \\
250 Hz (3.0547 \(\times\) 10\({}^{-5}\)) & 0.1759 & 0.5554 & 31.69\% \\
400 Hz (4.8876 \(\times\) 10\({}^{-5}\)) & 0.3041 & 0.6314 & 48.17\% \\
500 Hz (6.1095 \(\times\) 10\({}^{-5}\)) & 0.3994 & 0.6906 & 57.84\% \\
600 Hz (7.3314 \(\times\) 10\({}^{-5}\)) & 0.4978 & 0.7521 & 66.20\% \\
800 Hz (9.7752 \(\times\) 10\({}^{-5}\)) & 0.6929 & 0.8753 & 79.16\% \\
1000 Hz (1.2219 \(\times\) 10\({}^{-4}\)) & 0.8600 & 0.9875 & 87.10\% \\
1500 Hz (1.8328 \(\times\) 10\({}^{-4}\)) & 1.0894 & 1.1799 & 92.33\% \\
2000 Hz (2.4438 \(\times\) 10\({}^{-4}\)) & 1.1778 & 1.2576 & 93.66\% \\
4000 Hz (4.8876 \(\times\) 10\({}^{-4}\)) & 1.2737 & 1.3328 & 95.57\% \\
8000 Hz (9.7752 \(\times\) 10\({}^{-4}\)) & 1.3237 & 1.3758 & 96.22\% \\
20000 Hz (2.4 \(\times\) 10\({}^{-3}\)) & 1.3508 & 1.3986 & 96.58\% \\ \hline \hline \end{tabular}
\end{table}