# A Time-Varying Filter for Doppler Compensation Applied to Underwater Acoustic OFDM
## 1 Introduction  
The development of underwater communication systems is being spearheaded by the investment of infrastructure including autonomous underwater vehicles (AUVs) along the coasts for commercial, environmental and military applications. Shallow water deployments are of particular interest because of the increased industrial activity in these regions. However, underwater wireless data transmission suffers from significant channel variations. To enable a spectrally efficient communication link underwater, orthogonal frequency division multiplexing (OFDM) is a multi-carrier transmission scheme that has received increased interest in wireless communications due to its low complexity channel estimation and equalization. The key feature that defines OFDM is that the sub-channels are separated by the minimum frequency separation while maintaining orthogonality. The orthogonality allows simultaneous transmissions on adjacent sub-channels to not interfere with each other. For underwater acoustic communications, OFDM has been proposed in different works [1; 2; 3], since it promises much greater data rates than single carrier modulation based schemes [4], and especially in comparison to non-coherent frequency shift keying techniques [5]. However, it is well recognized that OFDM is subject to time-variance. As such, the objective of this paper is to model and assess the performance of underwater acoustic communication systems relying on multi-carrier transmission subject to high mobility. Most wireless communication systems rely on a narrowband representation of the signal where the bandwidth is much smaller than the carrier frequency. Then, temporal variations due to mobility in the channel can be simply represented using a linear fading process on each tap arrival. However, as explained in [6], in wideband communication systems, mobility must be characterized by time/frequency scaling. In addition, in a multipath environment, the angle of arrival and departure are typically different for each path arrival, and as such induce different Doppler shifts in each path. This generates a frequency dispersion of the transmit signal [7]. In this paper, a low-complexity filter originally introduced in [8] is utilized to model instantaneous channel variations due to vessel velocity and acceleration. The proposed technique can be used to enhance currently existing channel models, including geometry based simulators relying on ray tracing for example [9], or statistical models described in [7]. The methodology proposed allows compression and dilation of the signal as a function of time since it implements a time-variant convolution, and the filter utilized re-samples the signal, in a similar fashion as those used in delay locked loops. It is generally accepted that temporal variations of the channel are adverse to the coherent reception of OFDM, because the energy of the subcarriers spills over into adjacent bands. The effect is even more difficult to address in mobile conditions because the coherence time is short and the channel signature may even change within the transmission of long OFDM symbols. A few techniques have been described in the literature to mitigate Doppler [1; 2; 10; 11; 12; 13]. In [12], a methodology to estimate the carrier frequency offset (CFO) on the individual subcarriers is proposed. However, this technique relies on accurate channel state information. In [1], a two-step approach is taken, in which a coarse estimation of the time-scaling factor over the packet is accomplished using the preamble and postamble. Then, the residual frequency shift is estimated by minimizing the spillover energy in null subcarriers. Since a common time-rescaling factor is used for the entire duration of the frame, this technique is not optimal when there are sudden variations in equivalent speed between the two platforms.