# A Time-Varying Filter for Doppler Compensation Applied to Underwater Acoustic OFDM
## 2 A Low-Complexity Digital Filter to Model Mobility
### Time Varying Fractional Delay Line  
In this section, a filter that relies on a fractional delay line is proposed to model the channel time-variance. In [8], a similar filter is proposed, and is commonly used in delay locked loops. In this work, a low complexity implementation is applied, in a first instance, to model mobility in the channel, and, in a second instance, to realize Doppler compensation. Time-scaling is usually implemented by interpolating the data. Let us assume a signal \(x(t)\) sampled at a frequency \(F_{s}\). Then, if \(x(t)\) is scaled by a constant factor, \(\alpha\), it can be resampled at frequency  
\[F_{interp}=\frac{1}{\alpha}F_{s}. \tag{8}\]  
In other words, time-scaling is implemented by interpolating the signal at a frequency \(F_{interp}\). The interpolated discrete signal is discretized at the original sample rate \(F_{s}\) to model time-scaling. The time-scaling factor can be approximated as a rational number, \(M/N\), that is typically very close to unity (for a relative speed \(v\) much smaller than the speed of propagation). Then, resampling can be implemented by: (1) up-sampling and interpolating at the rate \(M\); and (2) down-sampling (decimating) the up-sampled signal at rate \(N\). When the time-scaling factor is close to 1, \(M\) and \(N\) will be large and the resampling process is computationally expensive, which makes it inappropriate in hardware implementation when the memory is limited. In this work, a _Time-Varying Fractional Delay Line_ (TVFDL) is introduced as an alternative technique to realize time-variant resampling [16]. Using this method, the time scaling is implemented by sampling the original signal with a time-varying delay \(\tau[n]\) at discrete time \(n\). Using this approach, once the time-varying delay is calculated using Equation (2), it can be implemented using a dynamic resampling process. To describe the TVFDL mathematically, assume \(x(\alpha t)\) is a continuous-time time-scaled signal. The signal \(x_{s}[n]=x(\alpha t)|_{t=nT_{s}}\) is the discrete-time representation with a sampling period of \(T_{s}\) and for which \(n\) is a positive integer between 0 and \(N_{max}-1\). Then, the signal \(x_{s}[n]\) is  
\[x_{s}[n]=x(\alpha t)|_{t=nT_{s}} =x((1-(1-\alpha))nT_{s}) \tag{9}\] \[=x(nT_{s}-(1-\alpha)nT_{s}).\]  
The time-scaled signal can also be written as \(x_{s}[n]=x(nT_{s}-\tau[n])\), where \(\tau[n]=(1-\alpha)nT_{s}\). The interpolation filter is a sinc function and is implemented by a tapped delay line, as shown in Figure 2. The sinc filter bandwidth \(B_{f}\) is defined to be equal to that of the signal bandwidth, i.e., \(B_{f}=1/T_{s}\). As such, the time delayed signal is  
\[x(nT_{s}-\tau_{s})=x[n-\tau_{s}]=\sum_{m=-\infty}^{\infty}\mathrm{sinc}[B_{f} \cdot(m-\tau_{s})]x[n-m], \tag{10}\]where \(\tau_{s}\) is the normalized delay with respect to the sampling period, \(T_{s}\). The normalized delay \(\tau_{s}\) can be broken into the integer part, \(D_{\tau_{k}}\), and the fractional part, \(d_{\tau_{k}}\), such that \(\tau_{s}\) is  
\[\tau_{s}=\frac{\tau}{T_{s}}=D_{\tau_{s}}+d_{\tau_{s}}. \tag{11}\]  
The TVFDL is realized by implementing an FIR filter that approximates the sinc function. To reduce its order to a finite length, the sinc filter is truncated. The order of the filter limits the filter accuracy and the length is a trade-off between the precision of the estimated signal value at the delay of interest and the computational complexity associated with this estimation. If the length of the FIR filter is chosen as \(2M+1\), then higher values of \(M\) will lead to a longer computational time, but more accurate results [17]. As shown in Figure 2, in discrete-time, the total delay \(\tau[n]\) at time \(n\) is implemented by an integer and a fractional part, such that \(\tau[n]=\tau_{D}[n]+\tau_{F}[n]\). The integer part \(\tau_{D}[n]\) indicates the number of taps by which the input signal will be delayed, and the filter coefficients take into account the fractional part \(\tau_{F}[n]\) at a given time instant \(n\).