## 6 Local Oscillator Stability
This 10 \(\upmu\)s maximum returned echo delay is appropriate for probing SAW delay lines with delays up to a few microseconds. Thus, collecting 1024 samples requires a stable environment between the GPR and the target during \(1024\times 10\)\(\upmu\)s \(\simeq 10\) ms if no additional stacking is used during acquisition. The slow ramp is generated by a digital to analog converter, a digital component whose voltage range can be defined by a stable voltage reference, and clocked by a microprocessor whose frequency is determined by a quartz oscillator exhibiting better stability than the sensor sensitivity: little drift is to be expected from this part of the circuit. The fast ramp on the other hand is generated by the integration of a constant voltage defined by a resistor bridge. The integration is performed by an analog operational amplifier circuit with a capacitor in the feedback loop, which is short-circuited by a digitally-controlled analog switch for resetting the integrator at a PRR interval (Figure 8). Capacitors are well known to be dependent on environmental conditions, whether temperature or moisture: the poor sampling rate stability of Mala's ProEx is attributed to the fast ramp generation strategy. In the scheme demonstrated in Figure 8, the equivalent sampling rate \(f_{s}\) is given by:  
\[f_{s}=\frac{1}{dt}=\frac{\Delta T_{slow}}{\Delta V_{slow}}\cdot\frac{\Delta V_{ fast}}{\Delta T_{fast}}\cdot\frac{1}{\delta t} \tag{7}\]  
where \(\Delta T_{slow}\) and \(\Delta T_{fast}\) are respectively the slow and fast ramp period, \(\Delta V_{slow}\) and \(\Delta V_{fast}\) respectively the slow and fast ramp voltage amplitudes and \(\delta t\) the pulse repetition interval (10 \(\upmu\)s in the case of the ProEx). As shown on Figure 8, \(dt\) is the time increment between the emitted pulse and the sampling trigger between successive pulses and is therefore the inverse of the equivalent sampling  
Figure 8: **(a)** Schematic view of the voltage to time converter for generating the stroboscopic timing signals (top), and (bottom) implementation of the fast ramp generator in the ProEx GPR. **(b)** Comparison of the analog ramp generator (blue) and our proposed digital implementation of the ramp generator (red). The slightly different slopes mean that the nominal sampling rate is no longer valid. Bottom right: zoom on the linear parts of the ramps generated by analog (blue) and digital (red) means, the latter voltage being generated by an Analog Devices AD9871 DAC: both methods show similar noise levels, as seen here after removal of the linear trend (the two curves were offset for clarity). rate. This equation is useful in an error budget analysis, emphasizing the importance of the voltage ranges and most importantly the \(\Delta T_{fast}\), which is the core issue in this discussion, all the other signal timing being controlled by a stable quartz-clocked oscillator. Targeted time delay accuracy is defined by the acoustic velocity variation induced by the physical environment under investigation of the sensor: with a 70-ppm/K temperature sensitivity, a 300-ns echo delay difference and a targeted 1 K measurement resolution, the delay measurement accuracy should be in the 20-ps range. Similarly, a 1-\(\upmu\)g/cm\({}^{2}\) adsorbed film on the surface of an acoustic transducer exhibiting a gravimetric sensitivity of 200 cm\({}^{2}\)/g in a chemical sensing application will exhibit a 60-ps delay upon film formation: again, a time delay accuracy of a few tens of ps should be targeted. To demonstrate this point, we replace the analog ramp generation scheme with a quartz-disciplined oscillator clocked digital ramp synthesizer. As a first demonstration, a laboratory-based experiment is performed using an off-the-shelf arbitrary waveform generator (Tektronix AFG3102) (Figure 9). To demonstrate the ability to transfer the concept to an embedded implementation, a FPGA (Field Programmable Gate Array) controlling a fast digital to analog converter is used.