# A Time-Varying Filter for Doppler Compensation Applied to Underwater Acoustic OFDM
## 2 A Low-Complexity Digital Filter to Model Mobility
### Acoustic Propagation in Mobile Conditions  
In this section, the effect of mobility on the propagation model is quantified as a function of the node speed relative to the propagation speed. The focus is primarily on platform mobility in the network, and can also loosely be extended to the mobility of scatterers including waves at the surface. In an acoustic propagation channel, the channel impulse response, \(h(t,\tau)\), is a function of time \(t\) and delay \(\tau\). It is expressed as a sum of \(N_{p}\) time delayed and amplitude scaled versions of the transmitted impulse function [6], such that  
\[h(t,\tau)=\sum_{p=1}^{N_{p}}h_{p}(t)\delta\left(\tau-\tau_{p}(t)\right), \tag{1}\]  
where, for path \(p\), the amplitude \(h_{p}(t)\) and the delay \(\tau_{p}(t)\) vary as a function of time. For a propagation distance \(r_{p}(t)\) traveled by the \(p\)th ray, the path delay \(\tau_{p}(t)\) is  
\[\tau_{p}(t)=\frac{r_{p}(t)}{c_{w}}, \tag{2}\]  
where \(c_{w}\) is the speed of sound in water. Next, a simple geometrical model of the environment is proposed to obtain the path delay \(\tau_{p}(t)\) as a function of time. Let us assume a pair of platforms moving in a direction parallel to the surface, such that the relative velocity \(\overrightarrow{v(t)}\) between a transmitter moving at \(\overrightarrow{v_{tx}(t)}\) and receiver moving at \(\overrightarrow{v_{rx}(t)}\) is \(\overrightarrow{v(t)}=\overrightarrow{v_{tx}(t)}-\overrightarrow{v_{rx}(t)}\). Then, the time-varying distance between the transmitter and receiver measured at the surface is \(d(t)=\int_{0}^{t}\overrightarrow{v(t)}dt+d_{0}\) where \(d_{0}\) is the initial distance between the two nodes. In shallow water environments such as shown in Figure 1, the propagation delay of each ray after several bounces from the bottom and surface can be calculated geometrically as demonstrated in [15]. Using this model, the total propagation distance \(r_{p}(t)\) is evaluated using simple trigonometry. The model can also be extended to a 3D one to consider reflections from shore lines, or other reflectors that may enclose the channel. However, the 3D model is out of the scope of this paper. Referring to Figure 1, let us define at a mobile receiver platform an angle of arrival \(\phi_{p}(t)\) with respect to the sea surface for the \(p\)th path arrival. The platform receives at time \(t=0\) a path arrival from a transmitter at distance \(d_{0}\), and that at this initial time the propagation delay is \(\tau_{p}(0)\). Then, at time \(t\), the propagation delay \(\tau_{p}(t)\) can be roughly estimated as  
\[\tau_{p}(t)=\tau_{p}(0)+\frac{1}{\cos(\phi_{p}(t))}\left(\frac{v(t)}{c_{w}}t \right), \tag{3}\]  
where \(v(t)=|\overrightarrow{v(t)}|\) is the velocity between the transmitter and receiver, in the direction of propagation. This expression indicates that the different path delays vary somewhat differently because of the different angles of arrival \(\phi_{p}(t)\) on each path. Then, the time-scaling factor for the \(p\)th path \(\alpha_{p}(t)\) describes the rate of the received signal compression or dilation. It is defined as  
\[\alpha_{p}(t)=(1+\mu_{p}(t))=\left(1+\frac{v_{p}(t)}{c_{w}}\right), \tag{4}\]  
where \(v_{p}(t)=v/\cos(\phi_{p}(t))\) and \(\mu_{p}=v_{p}/c_{w}\) is defined as the Mach factor. When \(v_{p}(t)<0\) and \(\alpha_{p}(t)<1\), the path length is decreasing, which causes compression in the time domain. Similarly, the signal would be dilated in time domain when \(v_{p}(t)>0\) and \(\alpha_{p}(t)>1\). Then, the delay as a function of the time-scaling factor \(\alpha_{p}(t)\) for path \(p\) is  
\[\tau_{p}(t)=\tau_{0,p}-(1-\alpha_{p}(t))t. \tag{5}\]  
where \(\tau_{0,p}\) is the \(p\)th path delay at the reference time \(t=0\). Note that, in this model, only the motion of the vessel tangential to the sea surface is modelled; however, it would be possible to extend this model to also account for the vertical motion of the node.