Data Communications Using Guided Elastic Waves by Time Reversal Pulse Position Modulation: Experimental Study

2.1. General Signal Models for Pulse Position Modulation

For wideband guided elastic signals, we adopt a signaling method that is similar to the ultra-wide-band (UWB) signaling in radio communications [19,22,29]. A general signaling model can be described as follows. Every transmitted symbol is composed of N_f repeated pulses, called frame. Let T_s and T_f denote the symbol duration and frame duration, respectively. Then, T_s= N_fT_f, which means there is one pulse per frame. Each guided elastic pulse, g(t), is of a short duration, T₀ < T_f, at the microsecond scale, occupying a wide bandwidth of a few hundred kilo-hertz. Each frame contains N_c chips, each of duration, T_c. A general wideband symbol signature waveform transmitted during the acquisition process for a single user can be expressed as a series of pulses:

Next, we discuss the elastic wave propagation channel for wideband signaling. Under the linear elastic wave theory, the overall channel response can be modeled by a stochastic tapped delay line:

The symbol, τ₀, is the direct propagation delay between the transmitter and the receiver and can be expressed as τ₀ = n_sT_s + n_fT_f + ε, based on different time scales at the symbol-level, frame-level and pulse-level [19]. Here, n_s= ⌊τ₀/T_s⌋ stands for the symbol level offset, n_f= ⌊ (τ₀− n_sT_s)/ T_f⌋ is the frame-level offset, where n_f ∈ [0, N_f− 1], and the pulse level offset is ε ∈ [0, T_f). ⌊x⌋ represents the largest integer not greater than x. Typically, a symbol-by-symbol sliding correlation filer is applied to generate samples, x[i x], at the symbol rate by [19,22,32]:

2.2. 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 [34-37]. Under the invariance principle, if u(r,t) is the solution to the wave equation, the time reversed version of the original solution, i.e., u(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→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 [34-37].

Step (3)

Preambles transmission for timing acquisition (A→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:

$$s_p(t)=\sum _{n=0}^{N-1}k\cdot y(-t+n{T}_f),0\le N{T}_f$$

(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→B) Following the preamble sequences, the data symbols are transmitted. Here, we use a pulse position modulation scheme, which takes the form of:

$$s_d(t)=\sum _{j=0}^{M-1}k\cdot y(-t+j{T}_f+a_j\mathrm{\Delta }),t>N{T}_f+T_g$$

(13)

where Δ is the shifting time for distinguishing the pulses carrying bit ‘0′ or bit ‘1′. 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. All binary data, a_j ∈ {0,1}, are modulated by the pulse position modulation (PPM; scheme, where M is the total number of binary bits for each data package to be transmitted.

2.3. 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:

2.4. Signal Detection and Demodulation

The estimate of the timing, t₀, is then utilized for signal demodulation using either non-coherent detectors, such as energy detector (ED) and peak detector (PD) or a coherent detector (CD). For low system complexity and power consumption, we focus on the noncoherent reception of PPM signals [32], which is a generalized likelihood detection. The symbol decision is based on finding the pulse position that contains the maximum energy. The symbol-by-symbol detection process does not require the estimation of the channel parameters. The energy of the multipath components is collected to increase the detection probability of the actual transmitted pulse. For binary PPM, we first calculate two energy quantities:

3.1. Experimental Setup

Figure 1 shows the steel pipe specimens and the acoustic sensors that were used in experiments. Two sets of experimental configurations were employed. The first set of experiments included a carbon steel pipe specimen of 1.833 meters in length, 70 mm in outside diameter and 4 mm in wall thickness (see Figure 1(a)) and two PZT wafers (PSI-5A4E, Piezo Systems, Inc., Cambridge, MA) that were mounted on the surface of the carbon pipe using cyanoacrylate adhesive, located 1.5 meters apart. Each wafer was 12 mm long and 6 mm wide. The second set of experiments included a stainless steel pipe of 1.5 meters in length, 115 mm in outside diameter and 6 mm in wall thickness (see Figure 1(b)). On the stainless steel pipe, two PZT wafers (PSI-5A4E) were mounted on the surface of the pipe, also using cyanoacrylate adhesive, located 0.69 meters apart. Each wafer was 10 mm long and 10 mm wide.

The signal transmission and data acquisition unit was a National Instruments (NI) PXI system, which consists of an arbitrary waveform generator (AWG), a digitizer/oscilloscope and a controller. The waveform generator had a sampling rate 100 MS/s and a 16-bit resolution. The AWG can generate arbitrary waveforms in response to the dispersive medium, thus implementing time reversal. The digitizer had a sampling rate of 100 MS/s and a 14-bit resolution. The waveform generator drives the transmission of the PZT sensor. The digitizer was connected to the receiving PZT sensor and records waveforms. The recorded waveforms were processed online using LabView programming for symbol demodulation and detection. For simplicity, a peak detector was used. The controller synchronizes the operations of the waveform generator and the digitizer. The experimental setup is illustrated in Figure 2. Figure 2(a) depicts the cable connection between the PXI system with a carbon steel pipe specimen. Figure 2(b) depicts the connection for a stainless steel pipe specimen. Water can be filled into the pipe for testing.

3.2. Description of Experiments

We follow the four steps given in sub-section 2.2 to conduct experiments.

Step 1:

Channel sounding. In this step, we choose a Gaussian modulated pulse:

$$p(t)=\text{Aexp}\left\{-{\left(\frac{t-t_s}{{\tau }_p}\right)}^2\right\}cos(2\pi f_{c}t)$$

(32)

as the probe signal, where f_c is the center frequency, τ_p = 2 μs, 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 $E_p={\int }_{-\infty }^{\infty }{\left|p(t)\right|}^{2}dt$. 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.

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 ofy(-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. The length of the received signal after time-reversal transmission at the digitizer is twice as much as the length of the received signal attained from Step 1.

Step 3 & 4:

Data transmission via TR-PPM. Timing acquisition is not conducted in our experiment, because the PXI system synchronizes the transmission and reception process between the arbitrary waveform generator and the digitizer. A fixed time offset can be determined. Here, we focus on data transmission. The time reversed signal, y(-t), obtained in Step 2 is employed as a basic waveform for information data transmission. The information data are a stream of binary digits (i.e., zero's and one's) and are coded by the pulse position modulation scheme, which takes the form of:

$$s(t)=\sum _{j=0}^{M-1}ky(-t)\delta (t-{jT}_f-c_j\mathrm{\Delta })$$

(33)

where c_j ∈ {0,1} is the binary data to be transmitted, T_f is the frame time to send one bit data, Δ is the additional time shift to distinguish between the pulses carrying the bit, zero or one. δ(·) is the Dirac delta function. The energy-normalization coefficient is given by $k=\sqrt{E_p/E_y}$, where the total energy for signal y(-t) is given by $E_y={\int }_{-\infty }^{\infty }{\left|y(-t)\right|}^{2}dt$. In our experiments, the information data was a series of M = 1, 000 binary numbers, zero's or one's, randomly generated by Lab View programming. The frame time is determined by the prescribed transmitting data rate, R, as follows:

$$T_f=\frac{1}{R}$$

(34)

The tests were carried out at the data rates of 10 kbps, 20 kbps, 50 kbps and 100 kbps, which correspond to the frame time of 100 μs, 50 μs, 20 μs and 10 μs, respectively. In order to restore the transmitted bit data, the time shift should be larger than the full width of the single Gaussian pulse. Here, we use one fourth of the frame time as the time shift, i.e., $\mathrm{\Delta }=\frac{1}{4}{T}_f$. The parameters for the TR-PPM test are summarized in Table 1. The recorded length is mainly determined by the length of the modulated time reversal signal. Here, we use twice the signal length for signal recording at the digitizer.

3.3. 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 8μs 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 dB/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 00011110 1].

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).

The obtained bit error rates under the four data rates are summarized in Table 2. Each bit error rate is averaged over 40 independent trials. For each trial, the information stream consists of 1, 000 randomly generated binary bits. Both the mean and the standard deviation of BER is shown in the table and is plotted in Figure 6(a).

We study the effect of noise level in the data acquisition system on the communication BER. We average the received responses 10 times to reduce the white noise and compare the results from data acquisition without averaging. Figure 6(b) shows that the mean and the standard deviation of BER over multiple trials reduces to 4.33 × 10⁻² and 5.55 × 10⁻³, respectively, as compared to 5.01 × 10⁻² and 1.17 × 10⁻², when the white noise is not suppressed by averaging. However, the improvement is not significant. This implies that the improvement of the signal-to-noise ratio by averaging is insignificant compared to that by applying the time reversal focusing technique.

3.4. Results from Experiment II

Next, we conducted experimental tests on a stainless steel pipe specimen of 1.5 meters in length, 115 mm in outside diameter and 6 mm in wall thickness. Our goals were to validate that our technique could be generalized to (1) pipes of different materials, dimensions and loading conditions; (2) sensors of different sizes and inter-sensor distances; and (3) different noise levels. Two PZT wafers (PSI-5A4E) were mounted on the surface of the pipe using cyanoacrylate adhesive, located 0.69 m apart. Each wafer was 10 mm long and 10 mm wide. The stainless steel pipe vertically stood on the floor, with the bottom end sealed with a rubber cap, which allows the pipe to hold water. Randomly generated bit streams at data rates of R = 10, 20, 50, 100 kbps were sent through the pipe specimen using the TR-PPM method under two testing scenarios: the pipe was empty (without water) and the same pipe was filled with water. Furthermore, we also tested the communication results when the time averaging (10 times) of received responses was employed when generating y(−t) for modulation.

Similar to the experimental results on the carbon steel pipe, we observed a complex channel sounding response on the stainless steel pipe (see Figure 7(b)) and a strong time reversal focused peak (see Figure 7(d)). However, we also noticed a higher level of sidebands around the time reversal focused peak compared to what we observed in the carbon steel pipe specimen in Experiment I (see Figure 3(d)). This phenomena are caused by the relatively small ratio of the transmitter/receiver distance (D = 0.69 meter) to the pipe diameter (r = 115 mm), i.e., D/r = 690/115 = 6 for the stainless steel pipe setup, whereas the ratio is about D/r = 1500/70 = 21 for the configuration of the carbon steel pipe experiment in Experiment I. We note that elastic waves that propagate in pipes typically consist of longitudinal waves, torsional waves and flexural modes [45,46]. After excitation, different wave modes would appear as the waves travel along the pipe and interrogate the pipe. Since only one PZT receiving sensor was used at a single position on the pipe, many wave modes that were excited by the transmitting PZT sensor could not be captured, which, in turn, gave rise to a smaller focusing peak and larger sidebands after time reversal. We anticipate that using an array of PZT receivers could achieve a better time reversal focusing, because the use of a sensor array could capture more transmitted wave energy caused by various wave modes. Due to space limitation, the test results using sensor arrays will be reported elsewhere. Figure 7(g) depicts a zoomed-in plot of Figure 7(f) in the 40 ms - 41 ms time window, which includes the decoded 10-bit stream as x[1 1 0 1 0 0 1 1 0 1 x]. Figure 8 depicts snapshots of the received information carrying waveforms in the 4.0 ms - 4.2 ms time window for the four different data rates.

We repeated the same experimental procedure on the stainless steel pipe specimen by filling the pipe with water. When filling the pipe with water, a few drops of detergent liquid were added to the water to release bubbles at the pipe-water interface inside of the pipe. The bubbles were observed accumulating at the water surface and released to air instead. Figure 9 depicts the snapshots of waveforms obtained during the TR-PPM data communication schemes similar to what was presented in Figure 7 for the test scenario where no water is in the pipe specimen. Figure 9(d) and (g) show that the waveforms have less sidebands compared with Figure 7(d) and (g), which implies less interference caused by the sidebands of the time reversal focused response. The bit stream turns out to be more distinguishable when the pipe is filled with water. Our hypothesis is that water suppresses certain guided wave modes in the pipe (such as flexural modes), due to the water damping effects [47]; thus, the adverse effect on time reversal focusing, due to a short inter-transducer distance, becomes less significant.

As a result, we achieved a lower BER for TR-PPM communications on the pipe that was filled with water at the data rates of R = 10, 20, 50, 100 kbps, compared to the scenario when the pipe was not filled with water. Next, Figure 10 depicts the snapshots of the received information carrying waveforms in the 4.0 ms - 4.2 ms time window under data rates of R = 10, 20, 50, 100 kbps, respectively. The BER results are summarized in Table 3 and plotted in Figure 11.

Furthermore, we also utilized time averaging (by a factor of 10) to reduce noise level in the received responses for both scenarios (water filled and no water) on the stainless steel pipe specimen. The improvement due to time averaging is insignificant, as shown in the Table 4, where the BER results for the case of non-averaging and that of averaging at a 100 kbps data rate are summarized. The results are what we have expected, because Equation (29) reveals that the bit error rate depends on the noise power and the inter-symbol interference. Averaging will increase the signal-to-noise ratio by reducing the noise power, but has a limited effect on the side-band level that characterizes the inter-symbol interference. However, it is expected that time averaging would be effective in the environment when the noise level is relatively high.