Statistical Errors in Remote Passive Wireless SAW Sensing Employing Phase Differences

Passive remote wireless sensing employing properties of the surface acoustic wave (SAW) has gained currency during a couple of decades to measure different physical quantities such as temperature, force (pressure, torque, and stress), velocity, direction of motion, etc. with a resolution of about 1% [1]. The basic principle utilized in such a technique combines advantages of the precise piezoelectric sensors [2, 3, 4], high SAW sensitivity to the environment, passive (without a power supply) operation, and wireless communication between the sensor element and the reader (interrogator). Several passive wireless SAW devices have been manufactured to measure temperature [1], identify the railway vehicle at high speed [5], and pressure and torque [6]. The information bearer in such sensors is primarily the time delay of the SAW or the central frequency of the SAW device. Most passive SAW sensors are designed as reflective delay lines with M reflectors1 and operate as sketched in Fig. 1. At some time instant t0 = 0, the reader transmits the electromagnetic wave as an interrogating radio frequency (RF) pulse (K = 1), pulse burst (K > 1), pulse train, or periodic pulse burst train. The interdigital transducer (IDT) converts the electric signal to SAW, and about half of its energy distributes to the reflector. The SAW propagates on the piezoelectric crystal surface with a velocity v through double distances (2L1 and 2L2), attenuates (6 dB per μs delay time [5]), reflects partly from the reflectors (R1 and R2), and returns back to the IDT. Inherently, the SAW undergoes phase delays on the piezoelectric surface. The returned SAW is reconverted by the IDT to the electric signal, and retransmitted to the interrogator. While propagating, the RF pulse decays that can be accompanied with effects of fading. At last, K pairs of RF pulses (Fig. 1b) appear at the coherent receiver, where they are contaminated by noise. In these pulses, each inter distance time delay Δ┬(2k)(2k–1) = 2(L2 – L1)/v, k ∈ [1,K], bears information about the measured quantity, i.e., temperature [1], pressure and torque [5], vehicle at high speed [6], etc. To measure Δ┬(2k)(2k–1), a coherent receiver is commonly used [7], implementing the maximum likelihood function approach. Here, the estimate of the RF pulse phase relative to the reference is formed to range either from –π/2 to π/2 or from –π to π by, respectively,