## 3 Using a Ground-Penetrating RADAR to Probe Acoustic Transducers
The magnitude of the time domain response is not affected by this convention issue, but the phase is erroneous if the center frequency is not brought to the center of the chart during the conversion from the frequency to time domain. Throughout this document, time delay through phase measurement is performed in a differential approach to eliminate the additional delay depending on the RADAR to target distance: subtracting the phase of two echoes eliminates the common time of flight term and is representative solely of the acoustic velocity. As is classically known in RADAR signal processing, the matched filter is the cross-correlation: in our case, the incoming electromagnetic pulse, converted into an acoustic wave through the inverse piezoelectric effect of the substrate, is reflected twice by two reflectors located at different distances from the IDT. These two echoes are converted back into an electromagnetic wave by the direct piezoelectric effect and detected by the RADAR. Cross-correlating the signals within the two time intervals in which the echoes are known to lie provides optimal noise rejection and signal identification. Furthermore, the cross-correlation maximum position provides a fine estimate of the time delay between the two echoes: in all our processing steps, the cross-correlation maximum is detected and fitted by a polynomial function for oversampling, assuming the cross-correlation maximum is locally symmetric and approximated at the second order by a parabola. The fine phase estimate is given by the parabola fit maximum, with a time-resolution improvement with respect to the sampling period equal to the signal to noise ratio. A core aspect of the passive cooperative target with respect to RFID measurement is that the former is an analog measurement, with a measurement range that is not defined by reaching a threshold voltage to power a digital circuit, but by a signal to noise ratio deteriorating the measurement quality until it becomes no longer usable. The link budget is determined by the ability of the electromagnetic wave to propagate in the various media between the surface emitter and the sub-surface cooperative target, the electromechanical conversion efficiency, antenna impedance and transfer function matching to the incoming electromagnetic pulse and returning signal losses through the same medium in which the electromagnetic wave propagates. The link budget due to free space propagation of the electromagnetic power and conductivity losses in the medium is well documented and classically investigated when assessing GPR measurement range [39]. In the case of RADAR systems, with a target acting as a point like-source, the FSPL (Free Space Propagation Loss) rises with the distance to the fourth power. Indeed, power spreads first on a sphere centered on the emitter and decays with the square of the distance to the source. The target itself acts as a point-like source from which a new sphere defines the surface over which the minute returning power spreads again. The product of two quadratic power laws yields a global loss law with distance appearing as the fourth power, an unfavorable condition with respect to powered sensors whose power law only decays as the square of the distance. Considering such a link budget balance, using cooperative targets are only justified in unique operating conditions where active sensors are not applicable. The electromechanical conversion, which replaces the RADAR cross-section of a target, is addressed by selecting strongly-coupled piezoelectric substrates and acoustic wave modes meeting both sensing capability and strong conversion efficiency: from the user perspective, an acoustic wave transducer is an electrical dipole returning a fraction of the incoming power, through a linear process (i.e., whatever the incoming power, some power will be returned). We will here discuss the last design issues, which lie in impedance matching and spectrum matching.