# R&D of an Innovative OFDM Communication Payload for Small-Size AUV
## 2 Systems and Methods
### Time-Frequency Differential OFDM Scheme  
The common time-frequency differential OFDM scheme with strong anti-multipath ability and good channel adaptability is applied to the proposed UWA communication payload. The time-domain differential modulation is completed at adjacent OFDM symbols, and then the data information and its inverse data are loaded into the adjacent subcarriers to accomplish the frequency-domain modulation. The signal model of the above scheme is introduced briefly as follows.  
Assuming that the data on the \(n\)-th subcarrier of the \(i\)-th OFDM symbol before differential modulation can be written as  
\[D_{i,n}=e^{i\phi_{i,n}}, \tag{7}\]  
where \(\phi_{i,n}0,2\tau\) denotes the phase of the \(n\)-th subcarrier of the \(i\)-th OFDM symbol, \(n=\left(0,2,\ldots,2\left(\frac{l}{2}-1\right)\right)\) is the index of data subcarrier. The data after differential modulation can be expressed as  
\[S_{i,n}=e^{i\Delta\phi_{i,n}}, \tag{8}\]  
Hence, the phase information at the same index subcarriers on the adjacent OFDM symbols can be formulated as  
\[\Delta\phi_{i,n}=\phi_{i,n}+\Delta\phi_{i-1,n}, \tag{9}\]  
Furthermore, the phase information at the adjacent index subcarriers on the same OFDM symbol can be written as  
\[\Delta\phi_{i,n+1}=-(\phi_{i,n}+\Delta\phi_{i-1,n}), \tag{10}\]As a consequence, the phase in (9) is loaded into \((n+1)\)-th subcarrier as the pilot symbol  
\[S_{i,n+1}=e^{\mathrm{j}\Delta\phi_{i,n+1}}=e^{-\mathrm{j}\Delta\phi_{i,n}}. \tag{11}\]  
At the receiving end, the data of the \(n\)-th and \((n+1)\)-th of the \(i\)-th OFDM symbol can be formulated as  
\[R_{i,n}=A_{i,n}H_{i,n}e^{\mathrm{j}\phi_{i,n}(\epsilon)}+n_{i,n}=A_{i,n}H_{i,n} e^{\mathrm{j}(\Delta\phi_{i,n}+\theta_{i,n}(\epsilon))}+n_{i,n}, \tag{12}\]  
\[R_{i,n+1}=A_{i,n+1}H_{i,n+1}e^{\mathrm{j}\phi_{i,n+1}(\epsilon)}+n_{i,n+1}=A_{ i,n+1}H_{i,n+1}e^{\mathrm{j}(\Delta\phi_{i,n+1}+\theta_{i,n+1}(\epsilon))}+n_{i,n+1}, \tag{13}\]  
where \(A_{i,n}\) and \(A_{i,n+1}\) denote the amplitude factor, \(H_{i,n}\) and \(H_{i,n+1}\) represent the channel impulse response, \(\theta_{i,n}\) and \(\theta_{i,n+1}\) are the phase deviation, and \(n_{i,n}\) and \(n_{i,n+1}\) denote the white Gaussian noise.  
The phase information in (12) and (13) is extracted to perform the differential demodulation, the demodulation results are presented as follows:  
\[\hat{\phi}_{i,n}(\epsilon)=(\Delta\phi_{i,n}+\theta_{i,n}(\epsilon))-(\Delta \phi_{i-1,n}+\theta_{i-1,n}(\epsilon))=\phi_{i,n}+(\theta_{i,n}(\epsilon)- \theta_{i-1,n}(\epsilon)), \tag{14}\]  
\[\hat{\phi}_{i,n+1}(\epsilon)=(\Delta\phi_{i-1,n+1}+\theta_{i-1,n+1}(\epsilon) )-(\Delta\phi_{i,n+1}+\theta_{i,n+1}(\epsilon))=\phi_{i,n}+(\theta_{i-1,n+1}( \epsilon)-\theta_{i,n+1}(\epsilon)), \tag{15}\]  
Thus, the data symbols are corrected by the pilot symbols to recover the phase information on the \(n\)-th subcarrier of the \(i\)-th OFDM symbol  
\[\overline{\phi}_{i,n}(\epsilon)=\frac{\hat{\phi}_{i,n}(\epsilon)+\hat{\phi}_{ i,n+1}(\epsilon)}{2}=\phi_{i,n}+\Delta\theta(\epsilon), \tag{16}\]  
where \(\Delta\theta\) is the phase bias introduced by the complicated characteristic of an UWA channel. According to the above derivation, the initial phase information can be recovered utilizing the differential detection.  
In conclusion, there is no requirement for channel estimation and compensation by employing the above time-frequency differential OFDM scheme to eliminate the interference of Doppler effect on UWA OFDM communication, further facilitating the low complexity design and implementation of an UWA OFDM communication payload for small-size AUV platforms.