# A Time-Varying Filter for Doppler Compensation Applied to Underwater Acoustic OFDM
## 3 Doppler Compensation for Multicarrier Modulation
### Compensation at the Receiver  
In this section, an algorithm to address time-variance and optimize link reliability at the receiver is proposed. First, a short discussion is offered on the optimum time-scaling factor in a mobile multipath environment, then an algorithm to rescale the received signal in real-time is described, and, finally, a full receiver architecture is provided to decode the data. Note that, while the Doppler scaling estimator is described in [14], the algorithm is modified to track the estimate between each OFDM symbol and interpolate the time-scaling on a sample-by-sample basis. In mobile underwater conditions, the signal may arrive through path arrivals that are subject to different time-scaling factors. While it is often assumed that all paths have equal time-scaling factor, measurements have shown that, in a real environment, each path may experience a different time-scaling factor [19]. In this case, the optimal resampling factor is not clearly defined. Specifically, in [14], the authors analyze the effect of resampling using a single time-scaling factor on the performance of the receiver in a multi-scale multipath channel. They show that, when the spread \(D_{s}\) of the channel time-scaling factor, called scaling spread, is large (\(D_{s}>1\times 10^{-3}\)), resampling with respect to a single scaling factor can cause degradation of reliability, because of the residual Doppler shift after resampling. In this case, the optimal time-scaling factor is equal to the time-scaling factor of the path which has the largest magnitude. In contrast, when the time-scaling factors are very close to each other, i.e., when the Doppler spread \(D_{s}<10^{-3}\), it is shown that the optimum resampling factor is a weighted average of the time-scaling factors of all paths. Earlier research on channel parameters estimation utilizes pilot sub-carriers to compensate for Doppler [12; 14; 20]. In most of these works, the Doppler shift is assumed to be constant during the packet and the estimation is performed using a block-type pilot arrangement. Using this methodology, it is not possible however to track a fast time-varying channel. Here, at the beginning of a given frame, a preamble is used to obtain a coarse estimation of the leading edge of the frame, and the following OFDM symbols have pilot tones as well as null sub-carriers strategically inserted between the payload carriers. In [14], time-scaling is also realized for each OFDM symbol. However, the time-scaling compensation relies on a constant interpolation factor. Consequently, the receiver needs to re-synchronize between each OFDM symbol. Instead, in this work, the time-scaling factor estimation and OFDM block synchronization components at the receiver are interpolated to mitigate distortion due to time scaling on a sample-by-sample basis throughout the duration of the received frame. To estimate the time-scaling factor, the received \(m\)th OFDM symbol can be expressed from Equation (16) using a single optimum resampling factor, \(\alpha\). The approximated received symbol is  
\[r_{m}[n]=\sum_{p=0}^{p}h_{p}\mathrm{e}^{-j2\pi f_{k}\tau_{p,0}}\sum_{k=N/2}^{N /2-1}s_{k}\mathrm{e}^{j2\pi k(1+\alpha)\frac{\pi}{N}}\mathrm{e}^{j2\pi f_{ \alpha}\pi\frac{T}{N}}+n[n]. \tag{19}\]  
By separating the effect of multipath and Doppler, the approximated received vector is [14]  
\[\hat{\mathbf{r}}=\mathbf{D}_{\alpha}\mathbf{\tilde{A}}_{\alpha}\mathbf{\Gamma x}, \tag{20}\]  
where \(\hat{\mathbf{r}}\) is an \(N\times 1\) vector that represents the \(N\) samples of the received OFDM symbol. Note that \(\mathbf{x}=\mathbf{Hs}\) models the effect of multipath on the transmitted symbol, where \(\mathbf{H}\) and \(\mathbf{s}\) are the multipath channel matrix and transmitted signal vector, respectively. The matrix \(\mathbf{D}_{\alpha}\) models the frequency offset, and, in contrast to \(\mathbf{D}_{p,m}\) defined in Equation (17), the frequency offset is common to all paths since a single optimum time-scaling factor \(\alpha\) is estimated.