## 2 Background Theory
### Timing Acquisition by Time Reversal  
Next, we estimate the starting time of the preamble training sequences at the receiver. After the transmitted signals, \(s_{p}(t)\), propagate through the channel, the preambles arriving at the receiver become:  
\[r_{p}(t)=\sum_{n=0}^{N-1}y(-t-nT_{f})*h(t)=\sum_{n=0}^{N-1}p(-t+nT_{f})*h(-t)* h(t) \tag{14}\]  
Note that we omit the scaling factor, \(k\), given in Equation (12) in the derivation, because it approximates a constant for static channels. It is well known that by time reversal, \(h(-t)*h(t)\) is the auto-correlation function of \(h(t)\), which appears to be a highly focused waveform when \(h(t)\) is dispersive. Here, we employ only one sliding correlation detector combined with a single tap finger to capture and detect the received signals. To recover the starting time from the received preambles, a locally generated template, \(u(t)\), is applied to matching them in a sliding mode as follows:  
\[R_{p}(\tau) = \int_{0}^{NT_{f}}r_{p}(t)\cdot u(t-\tau)dt \tag{15}\] \[= \int_{0}^{NT_{f}}\left(\sum_{n=0}^{N-1}p(-t+nT_{f})*h(-t)*h(t) \right)\cdot u(t-\tau)dt \tag{16}\]  
where \(u(t)=\sum_{n=0}^{N-1}p(t-nT_{f})\). Inserting the elastic wave propagation channel model Equation (4) into Equation (16) yields:  
\[R_{p}(\tau)=\int_{0}^{NT_{f}}\left(\sum_{n=0}^{N-1}p(-t+nT_{f})*\sum_{i=0}^{L- 1}\alpha_{i}\delta(-t-\tau_{i})*\sum_{j=0}^{L-1}\alpha_{j}\delta(-t-\tau_{j}) \cdot\sum_{m=0}^{N-1}p(-t-mT_{f}-\tau)\right)dt\]  
Next, we let the autocorrelation function be \(R_{pp}(\tau)=\int_{-\infty}^{+\infty}p(-t)p(t-\tau)dt\) and \(\triangle\tau_{ij}=\tau_{j}-\tau_{i}\), \(\triangle T_{mn}=(n-m)T_{f}\); we can re-write \(R_{p}(\tau)\) as:  
\[R_{p}(\tau) = \sum_{i=0}^{L-1}\sum_{j=0}^{L-1}\sum_{n=0}^{N-1}\sum_{m=0}^{N-1} \alpha_{i}\alpha_{j}R_{pp}(\tau-\triangle\tau_{ij}-\triangle T_{mn}) \tag{17}\] \[= Q_{0}(\tau)+Q_{1}(\tau)+Q_{2}(\tau)+Q_{3}(\tau)\]  
where:  
\[Q_{0}(\tau)=\sum_{i=j=0}^{L-1}\sum_{n=m=0}^{N-1}\alpha_{i}^{2}R_{pp}(\tau)=NR _{pp}(\tau)\sum_{i=0}^{L-1}\alpha_{i}^{2} \tag{18}\]  
is the term that represents the autocorrelation between the multipath components. The rest of the three terms in Equation (17) are the cross-correlation terms:  
\[Q_{1}(\tau) = \sum_{i=j=0}^{L-1}\sum_{n\
eq m}^{N-1}\sum_{m\
eq n}^{N-1}\alpha _{i}^{2}R_{pp}(\tau-\triangle T_{mn}) \tag{19}\] \[= \sum_{i=0}^{L-1}\alpha_{i}^{2}\sum_{n\
eq m}^{N-1}\sum_{m\
eq n }^{N-1}R_{pp}(\tau-\triangle T_{mn})\] (20) \[Q_{2}(\tau) = \sum_{i\
eq j}^{L-1}\sum_{j\
eq i}^{L-1}\sum_{n=m=0}^{N-1} \alpha_{i}\alpha_{j}R_{pp}(\tau-\triangle\tau_{ij})\] (21) \[R_{p}(\tau) = \int_{0}^{NT_{f}}r_{p}(t)\cdot u(t-\tau)dt \tag{15}\]0  
Since the autocorrelation collects all the signal energies of the multipath components of the received preambles, \(Q_{0}(\tau)\) has the strongest intensity and possesses the most energy of the detector output. As a result, the last three terms can be neglected, because the cross-correlation results in destructive addition between the template and the received preambles. Thus, we have:  
\[R_{p}(\tau) = \int_{0}^{NT_{f}}r_{p}(t)\cdot u(t-\tau)dt \tag{15}\]1We emphasize that the approximation given in Equation (23) has been verified using experimentally measured elastic signals. A detailed theoretical performance analysis regarding the impact of the approximation is out of the scope of this paper and will be discussed elsewhere. As a result of the approximation, Equation (17) can be simplified as:  
\[R_{p}(\tau) = \int_{0}^{NT_{f}}r_{p}(t)\cdot u(t-\tau)dt \tag{15}\]2  
Finally, the start time of the preamble symbols is determined by searching the peak of the output of the sliding correlation detector in Equation (24) as follows:  
\[R_{p}(\tau) = \int_{0}^{NT_{f}}r_{p}(t)\cdot u(t-\tau)dt \tag{15}\]3  
Once the starting time of the preamble symbols, \(\widehat{\tau}_{0}\), is determined, the receiver performs the demodulation by choosing:  
\[R_{p}(\tau) = \int_{0}^{NT_{f}}r_{p}(t)\cdot u(t-\tau)dt \tag{15}\]4  
as the start of the data symbols.