## 2 Background Theory
### Description of Time Reversal PPM  
The principle of time reversal for acoustic applications originates from the invariance of the governing wave equations in underwater acoustics [33] and in guided acoustic waves [34, 35, 36, 37]. Under the invariance principle, if \(u(\mathbf{r},t)\) is the solution to the wave equation, the time reversed version of the original solution, _i.e._, \(u(\mathbf{r},-t)\), also satisfies the wave equations. This observation implies that if we record the waves radiated from a certain source and retransmit their reversed waves back to the same medium, the retransmitted waves will retrace their incoming paths back towards the original source. Time reversal has been successfully applied to ultra-wideband electromagnetic wireless communication systems of various configurations [13, 15, 38, 39]. In this paper, we consider a single-transmitter and single-receiver configuration in a pitch-catch mode for elastic wave communication. The time reversal method for timing acquisition can be summarized in the following four steps:  
* Step (1) **Channel probing (B \(\rightarrow\) A). ** In this step, a pilot elastic wave pulse, \(p(t)\), is initiated from the intended receiver, B, for probing the multipath propagation channel between B and the intended transmitter, A. The received signals at A can be expressed by: \[y(t)=p(t)*h(t)\] (10) where \(h(t)\) is the impulse response of the elastic wave channel coupled with the transmit and receiver piezoelectric transducers, which is typically very dispersive, due to the intrinsic properties of elastic waves. Note that the additive noise is omitted for simplicity purposes. * Step (2) **Time reversal. ** The received signal is time reversed in the time domain. The time reversing operation is equivalent to reading the data by first-in-last-out, which yields: \[y(-t)=p(-t)*h(-t)\] (11)Here, \(y(-t)\) is called the time reversed waveform. Note that by the temporal and spatial focusing characteristics of time reversal, if we retransmit \(y(-t)\) back to the channel and assume that the channel reciprocity holds, the retransmitted waveform will focus the scattered multipath energies on the initial source receiver temporally and spatially. Thus, the intensity of the signal received by the receiver is not only enhanced greatly, but the time spread is also compressed effectively. The validity of the focusing effects have been studied in elastic wave theory in [34; 35; 36; 37]. Step (3) **Preambles transmission for timing acquisition (A \(\rightarrow\) B)** In time-reversal-based timing acquisition, we use the time reversed waveform, \(y(-t)\), in Equation (11) as the basic signal waveform for carrying the timing preambles. The main advantage of using the time reversed waveform as the information carrier is that complicated channel estimation algorithms are not necessary at the transmitter, because the channel information is inherently included in the time reversed waveform. Thus, the transmitting waveforms take the form of:  
\[s_{p}(t)=\sum_{n=0}^{N-1}k\cdot y(-t+nT_{f}),\ 0\leq t\leq NT_{f} \tag{12}\]  
where \(T_{f}\) is the frame duration for one bit transmission, \(N\) is the number of preamble training bits and \(k\) is the energy normalization factor to ensure the transmission energy remains unchanged for each symbol. Step (4) **Data symbol transmission (A \(\rightarrow\) B)** Following the preamble sequences, the data symbols are transmitted. Here, we use a pulse position modulation scheme, which takes the form of:  
\[s_{d}(t)=\sum_{j=0}^{M-1}k\cdot y(-t+jT_{f}+a_{j}\Delta),\ t>NT_{f}+T_{g} \tag{13}\]  
where \(\Delta\) is the shifting time for distinguishing the pulses carrying bit \({}^{\prime}0^{\prime}\) or bit \(`1^{\prime}\). The guard time, \(T_{g}\), is pre-determined and is inserted between the training preamble symbols and the data symbols for the purpose of preventing the data waveforms from overlapping the timing preambles.