## 3 Communication Results by Experiments
### Description of Experiments  
We follow the four steps given in sub-section 2.2 to conduct experiments. 1. _Step 1: Channel sounding._ In this step, we choose a Gaussian modulated pulse: \[p(t)=A\mathrm{exp}\left\{-\left(\frac{t-t_{s}}{\tau_{p}}\right)^{2}\right\} \cos(2\pi f_{c}t)\] (32) as the probe signal, where \(f_{c}\) is the center frequency, \(\tau_{p}=2\ \mu s\), which translates to the nominal pulse width of \(T_{s}=\sqrt{2}\pi\tau_{p}=8\ \mu s\)[40], and \(t_{s}\) is the time shift from \(t=0\) when the pulse is generated. \(A\) is the amplitude (see the example in Figure 3(a)). The total transmission energy is defined as \(E_{p}=\int_{-\infty}^{\infty}|p(t)|^{2}dt\). We choose this modulated Gaussian signal as the probe signal for spectral control. The excitation signal has a center frequency, \(f_{c}=250\) kHz, with the signal bandwidth of \(500\) kHz. The signal is amplified by a factor of \(10\) using a built-in amplifier in the NI's data acquisition system. The maximum voltage of the Gaussian pulse is \(A=10\) volts. The sampling frequency is chosen to be at \(2\) MHz, four times as much as the maximum frequency of the modulated Gaussian pulse, in order to reconstruct the sampled digital signal into the analog waveform in high fidelity. We note that PZT wafers have been explored with continuous sinusoidal or pulsed excitation for defect detection in various applications. Optimal operations of PZT wafers have been studied by many researchers (e.g., [41]), including the procedure to determine the optimal frequency range [42] and analysis of its sensitivity to damage detection [43]. Our data communication experiment was performed by using a nominal frequency bandwidth of \(500\) kHz. Compared to narrowband excitation, wideband pulse excitation results in more modes and more dispersion [44]. For PZTs with different sizes and coupling conditions, the optimal excitation frequency that results in the highest intensity of the reflected elastic waves may vary. The excitation frequency range (_i.e._, \(0-500\) kHz) we chose would cover the optimal frequency for the PZT sensors observed in the experiment. Figure 4(a,b) depict the channel frequency response of the carbon steel pipe specimen and the stainless steel pipe, respectively. The channel frequency response includes the frequency response of the PZT wafers and the propagation channel. Figure 4(a) shows that the peak response of the spectrum falls in the \(500\) kHz range, while Figure 4(b) shows that the peak response is located at about \(200\) kHz, which could be caused by the resonance frequency of the PZT wafer. Figure 4: Frequency spectrum of channel responses that include PZT wafers. (**a**) Carbon steel pipe specimen; (**b**) stainless steel pipe specimen. Figure 3: Snapshots of waveforms obtained experimentally from time reversal-based pulse position modulation (TR-PPM) data communications at a data rate of \(10\) kbps. (**a**) Gaussian pilot signal of \(8\)\(\mu s\); (**b**) received response after channel sounding; (**c**) time reversal transmission; (**d**) received time reversal focused waveform; (**e**) modulated time reversal PPM waveform; (**f**) received information carrying waveforms; and (**g**) zoomed-in plot of (**f**). The red dashed lines indicate the starting positions of symbol frames. The recovered bit stream in the time window of \(40\) ms - \(41\) ms is \([0\)\(0\)\(0\)\(0\)\(1\)\(1\)\(1\)\(1\)\(0\)\(1]\). 2. _Step 2: Time reversal._ As expected, the channel response, \(h(t)\), that characterizes the elastic wave propagation between two positions on a steel pipe is highly dispersive. To realize time reversal, the received signal is time-reversed, energy-normalized and then fed back to the wave generator. We let \(y(t)\) denote the received signal; then, the time reversed signal is \(y(-t)\), as in Equation (11). Note that energy normalization of \(y(-t)\) is necessary to ensure that the newly constructed transmission waveform has the same energy as the Gaussian pulse transmission used in the channel sounding experiment.