## 3 Communication Results by Experiments
### Results from Experiment I  
We conducted the test on a carbon steel pipe to investigate the reliability of our technique by sending a relatively long stream of bits at different data rates, \(R=10,20,50,100\) kbps, and the feasibility of automatic decoding of the bit streams. A total of \(40\) independent test runs were performed. Under each test run, \(M=1,000\) binary bits were modulated, transmitted and decoded. Out of the \(40,000\) recovered information bits, we counted the corrected decoded bits and calculated the bit error rate. Figure 3 depicts snapshots of various signals collected in the experiment. Figure 3(a) depicts a short \(8\mu\)s Gaussian modulated pulse, \(p(t)\), with amplitude, \(A=10\) volts. The signal, \(p(t)\), is excited through the PZT sensor to generate guided acoustic waves that propagate along the carbon steel pipe. The received signal, \(y(t)\), is recorded by the receive PZT transducer at the other end of the pipe specimen (see Figure 3(b)). We can see from Figure 3(b) that the received signal has a very long series of arrivals as a result of multi-modal and elastic wave dispersion, which typically occurs in a cylindrical shell. Moreover, the attenuation level of the dynamic range of the received signal is on the order of \(10^{-2}\), which translates to about \(26\ \mathrm{dB}/\mathrm{m}\) peak power attenuation from the initial excitation signal to the received signal over a \(1.5\)-meter propagation distance. The severe dispersive channel makes it very difficult to achieve data communications using traditional techniques. By the time reversal principle, a time-reversed response waveform, \(y(-t)\) (see Figure 3(c)), is transmitted through the same dispersive channel and results in a much simpler waveform with a strong focused peak at the center (see Figure 3(d)). Time reversal signal processing effectively focuses the multiple dispersive wave modes and enhances the signal-to-noise ratio. Figure 3(e) depicts a snapshot of TR-PPM modulated information carrying waveforms, which is the superposition of \(1,000\) time reversed waveforms (Figure 3(c)) under the data rate of \(10\) kbps with the total time span of \(100\) ms. The TR-PPM modulated waveforms were excited through the PZT sensor, transmitted and recorded. We note that the TR-PPM transmission waveform was scaled to avoid saturation of the waveform generator of the PXI system. The received information carrying waveform consists of a series of focused peaks, whose positions within a frame indicate bit stream (see Figure 3(f)). Note that a \(5\) kHz - \(600\) kHz bandpass filter was applied during the data acquisition process at the receiving PZT sensor to remove low frequency vibrations and high frequency electrical noise. The filtering also prevented saturation of the PPM waveform, due to DC drift. Figure 3(g) is a zoomed-in plot of Figure 3(f) in the \(40\) ms - \(42\) ms time window, which includes the decoded \(10\)-bit stream \([0\ 0\ 0\ 0\ 1\ 1\ 1\ 0\ 1]\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Data** & **Frame** & **\# of bits** & **Sampling** & **Center** & **Bandwidth** & **Time** & **\# of** \\
**Rate** & **Time** & **Transmitted** & **Frequency** & **Frequency** & & **Shift** & **Trials** \\
**(kbps)** & **(\(\mu s\))** & & & **(MHz)** & **(kHz)** & **(kHz)** & **(\(\mu s\))** & \\ \hline \(10\) & \(100\) & \(1,000\) & \(2\) & \(250\) & \(500\) & \(25\) & \(40\) \\ \(20\) & \(50\) & \(1,000\) & \(2\) & \(250\) & \(500\) & \(12.5\) & \(40\) \\ \(50\) & \(20\) & \(1,000\Figure 5 depicts snapshots of the zoomed-in view of the waveforms in the \(4.0\) ms - \(4.2\) ms time window under different data rates. The peaks are more clearly distinguishable at the lower data rates. The results are what we expected in that a higher data rate, \(R\), means a shorter frame time, \(T_{f}\), which gives rise to higher sideband levels (_i.e._, interference), due to leakage from adjacent coded waveforms. Thus, the focused peaks become less distinguishable, which results in a higher bit error rate (BER).