## 3 Using a Ground-Penetrating RADAR to Probe Acoustic Transducers
All measurements are performed using the commercially available ProEx GPR control unit, provided by the same supplier. The custom control software available at [38]. Initial SAW delay line characterization is performed in the frequency domain using a network analyzer (Figure 2a,b top), assessing whether the SAW transducer response matches the GPR emitted pulse spectra. Frequency to time domain processing is needed to assess the expected GPR response, with echos delayed by durations defined by the IDT (InterDigitalated Transducer) to reflector distance \(d\) and the acoustic velocity \(v\). Since the distance is fixed and the velocity is dependent on the SAW transducer environment, the delay \(\tau=d/v\) is representative of the physical quantity under investigation. Two echoes induced by two reflectors patterned on both sides of the IDT are visible in Figure 2a (middle) and 2b (bottom), with delays of about 1.25 and 1.55 \(\upmu\)s, as defined by distances \(d\) of 2480 and 3100 \(\upmu\)m, respectively, consistent with the Rayleigh wave tabulated velocity of 3979 m/s [24] on lithium niobate YXI/128\({}^{\circ}\). The electromechanical coefficient:  
\[K^{2}=2\frac{\Delta v}{v}=5.4\% \tag{1}\]  
with \(\Delta v\) the metalized to free surface acoustic wave velocity difference, hints at insertion losses of \(10\log_{10}(K^{2})=-13\) dB: the higher losses (\(-40\) dB) seen in Figure 2a are due to the excessive bandwidth of the measurement on the network analyzer whose frequency span was widened to clearly show the acoustic bandwidth of the device. The measurement bandwidth in Figure 2b was adjusted to optimize returning power by matching the probe signal bandwidth with the SAW reflective delay line bandwidth. The time delay of the echoes is accurately measured as a phase:  
\[\varphi=2\pi d/\lambda \tag{2}\]  
with \(\lambda\) the acoustic wavelength, as seen in Figure 2a (bottom). Keep in mind that when converting from the frequency domain to the time domain (Figure 2a, middle and bottom) using numerical processing  
Figure 2: (**a**) One hundred megahertz lithium niobate reflective SAW (Surface Acoustic Wave) delay line transfer function, in the frequency domain (top: raw data in blue, windowed data in red) and the time domain (middle and bottom, magnitude and phase, respectively) as found by inverse Fourier transform of the former. Windowing (red: Blackman window) applied on the top and middle graphs (**a**) helps lower the baseline level with no effect on the insertion losses or the phase observed in the time domain. Inset: geometry of the delay line, defining the schematic position of the IDT (InterDigitalated Transducer) between the two reflectors patterned on the piezoelectric substrate. (**b**) One hundred ninety megahertz lithium niobate reflective SAW delay line characterized with various bandwidths (top, frequency domain), yielding different insertion losses and time resolution (bottom), emphasizing the need to tune the probe pulse spectral characteristics to match the SAW delay line transfer function. The thick black line is the result of the inverse Fourier transform of the blue (20 MHz bandwidth) measurement, with no windowing. software such as MATLAB or GNU/Octave, their convention of positioning the zero-frequency differs from the one expected from radiofrequency signal processing: in the former software, the zero-frequency is located on the left part of the chart and the sampling frequency on the right part of the chart. Radiofrequency signal processing and demodulation expect the zero-frequency to lie in the middle of the chart, with minus half the sampling frequency and half the sampling frequency lying on the left and right part of the chart, respectively. Converting the frequency measurement obtained on a network analyzer to the time domain requires converting from one convention to the other, as conveniently provided by the fftshift() function of these processing software packages.