# A Time-Varying Filter for Doppler Compensation Applied to Underwater Acoustic OFDM
## 3 Doppler Compensation for Multicarrier Modulation
### Impact of Mobility on Signal Integrity  
In a wideband communication channel, time-scaling due to time-variance introduces a frequency-dependent frequency shift. Because of this, during the transmission of a multicarrier modulated signal, the energy of a sub-carrier spills over to adjacent sub-carriers. This phenomenon is called _inter-carrier interference_ (_ICI_). In this section, the analytical model used to represent the signal at the receiver is provided. The model is similar to that presented in [14], but uses as its input the relative speed and acceleration of the platforms. In addition, in this work, pilot tones inserted in each OFDM symbol serve to refine the Doppler estimation even at high mobility. For a time-scaling factor that is dependent on platform speed, the time-varying channel impulse response defined by Equation (1) can be expressed as  
\[h(t,\tau)=\sum_{p=0}^{P-1}h_{p}\delta\left(\tau-\tau_{p,0}+\frac{1}{c_{w}} \left(v_{p,0}+\frac{a_{p}}{2}t\right)t\right), \tag{12}\]  
where \(v_{p,0}\) is the initial velocity, \(a_{p}\) is the average acceleration of the \(p\)th path and \(h_{p}\) is assumed to be constant as a function of time. To analyze the effect of a constant time-scaling on an OFDM symbol, the transmitted OFDM symbol is rewritten as a sum of a number of orthogonal complex sinusoidal functions, such that  
\[s_{m}(t)=\sum_{k=-N/2}^{N/2-1}s_{k}\mathrm{e}^{j2\pi f_{k}t}u(t), \tag{13}\]  
where \(s_{k}\) is the \(k\)th data symbol and \(u(t)\) is a rectangular pulse shaping function, expressed as  
\[u(t)= \begin{cases}\sqrt{\frac{1}{T}}&\text{if }t\in[0,T].\\ 0,&\text{otherwise}.\end{cases}  \tag{14}\]  
The \(k\)th sub-carrier frequency is \(f_{k}=f_{c}+k\Delta f\), where \(\Delta f\) is the frequency separation between the sub-carriers and \(k=-\frac{N}{2},\cdots,\frac{N}{2}-1\). The \(m\)th received OFDM symbol can be expressed as [14]  
\[r_{m}(t) =\sum_{p=0}^{P-1}h_{p}s_{m}((\mu_{p,m}+1)t-\tau_{p,0})+n(t) \tag{15}\] \[=\sum_{p=0}^{P-1}h_{p}\sum_{k=-N/2}^{N/2-1}s_{k}\mathrm{e}^{j2 \pi f_{k}((\mu_{p,m}+1)t-\tau_{p,0})}u(t)+n(t)\] \[=\sum_{p=0}^{P-1}h_{p}\sum_{k=-N/2}^{N/2-1}s_{k}\mathrm{e}^{j2 \pi f_{k}t}\mathrm{e}^{-j2\pi f_{k}\tau_{p,0}}\mathrm{e}^{j2\pi(f_{k}\mu_{p,m })t}u(t)+n(t).\]  
where \(\mu_{p,m}\) is the Mach factor at path delay \(p\) at OFDM symbol \(m\). At the receiver, the baseband signal is obtained by down-converting and then sampling \(r_{m}(t)\) at \(T/N\) seconds [14]. It is described in discrete-time by \[\mathrm{r}_{m}[n] =r_{m}(t)e^{|2\pi f_{\xi}t}|_{t=nT/N} \tag{16}\] \[=\sum_{p=0}^{p-1}h_{p}\sum_{k=-N/2}^{N/2-1}\left(s_{k}\mathrm{e}^{- j2\pi f_{\xi}\tau_{p,0}}\cdot\mathrm{e}^{j2\pi k(1+p_{p,m})\frac{n}{N}}\cdot \mathrm{e}^{-j2\pi f_{\xi}t_{p,m}m\frac{n}{N}}\right)+\mathrm{n}[n]\]  
where \(n=0,1,\cdots,N-1\). The discrete-time \(m\)th received OFDM symbol sampled at the data rate is represented in a vector form as [14]  
\[\mathrm{r}_{m} =[\mathbf{r}(0),\mathbf{r}(1),\cdots,\mathbf{r}(N-1)],\] \[\mathrm{r}_{m} =\sum_{p}h_{p}\mathbf{D}_{p,m}\mathbf{A}_{p,m}\mathbf{T}\mathbf{ B}_{p,m}\mathbf{s}+\mathbf{n}, \tag{17}\]  
where \(\mathbf{D}_{p,m}\) and \(\mathbf{B}_{p,m}\) are \(N\times N\) diagonal matrices, which model, respectively, the frequency offset and phase shift in the \(p\)th path at the \(m\)th OFDM symbol. \(\mathbf{A}_{p,m}\) is an \(N\times N\) Vandermonde matrix that models frequency distortion due to time-scaling and \(\mathbf{\Gamma}\) is an \(N\times N\) selection matrix that selects the active sub-carriers (\(N_{a}\)). The matrices are populated using  
\[\mathbf{D}_{p,m}(n,n) =\mathrm{e}^{-j2\pi f_{\xi}t_{p,m}(nT/N)}, n=0,1,\cdots,N-1,\] \[\mathbf{B}_{p,m}(n,n) =\mathrm{e}^{-j2\pi f_{\xi}\tau_{p,0}}, m\in N_{a},\] \[\left[\mathbf{A}_{p,m}\right](n,k) =\mathrm{e}^{j2\pi nk(1+p_{p,m})/N}, n,k=0,1,\cdots,N-1,\]  
and  
\[\Gamma_{n,k} = \begin{cases}1,&\text{if $n=k$ and $k\in N_{a}$},\\ 0,&\text{otherwise}\end{cases}  n,k=0,1,\cdots,N-1.