Usually, the envelope of a SAWR signal is modelled as a decaying exponential . However, this model does not reflect the behaviour of all the SAWR devices. The envelope of some SAWR signals is sometimes closer to a Gaussian shape as shown in Fig. . We propose a model suitable for all envelope shapes. Its expression fλ,μ,α(t) is the probability density function of an inverse-Gaussian law defined as 1fλ,μ,α(t)=λ2π(αt)3⋅e-λ(αt)-μ22μ2αtfort>0,0otherwise, where μ>0 is its expectation, and λ>0 and α>0 are shape and scale parameters, respectively.

Both parameters λ and μ allow covering of all envelopes forms as shown in Fig. . The more λ increases, the closer the function becomes to a Gaussian shape. The scale parameter α simply adjusts the timescale. By applying an amplitude modulation of carrier frequency F we obtain the synthetic SAWR signal s(t) as Fig. a shows an example: 2s(t)=sin⁡2πFt⋅fλ,μ,α(t).

This modelling allows the identification of an experimental SAWR signal and roughly estimates its signal-to-noise ratio (SNR). Since the frequency range of the SAWR sensor is always known and since the signal contains a single main frequency, we can apply a low-pass filter with a cut-off frequency close to the resonance frequency F. Thus, by a local maximum detection on this filtered signal sf(t) and a cubic spline interpolation, we obtain the experimental signal envelope e(t). Finally, by using a least-squares algorithm, the different parameters λ, μ, and α of fλ,μ,α(t) can be calculated, which leads to the model of the SAWR signal s(t) and an estimation of the SNR. This is depicted in Fig. .

This modelling also the allows performance of a study concerning the best wavelet choice. If the EMD method offers no choice for the analysis functions, this is not the case for the wavelet transform. Indeed, there are many wavelet families, and it is necessary to select the best fit for the intended application. We do not neglect this rule, and hence we built a synthetic SAWR signal with the previous modelling to which we added a Gaussian white noise.

By varying the standard deviation of the noise, we generated 21 noisy signals with 4096 samples covering several reference signal-to-noise ratios (SNRRef) ranging from 0 to 20 dB (Fig. b). In order to obtain the best wavelet, we denoised each signal with 24 wavelets and computed the new SNR thereafter. The results are depicted in Fig. . Some wavelets stand out, i.e. Beylkin, Coiflet-5, Daubechies-20, Symmlet-10, and Vaidyanathan, for which a focus is given in Table . By minimizing the standard deviation, the one with the best average SNR is the Daubechies-20 wavelet, which was used for the wavelet-based method. The good results of this wavelet are directly related to its morphology, which agrees with that of the SAWR signal as shown in Fig. .

We implement spectral and temporal approaches which are particularly well suited to this work because the coherent signal contains only one main frequency. For the first one, an FFT is performed on the whole signal without any prior processing because this method is very robust to noise. The resonance frequency is obtained by finding the maximum modulus frequency. Figure  depicts this way.

Unlike Fourier, which is robust to noise, a very important step of denoising must be performed for the time-based methods. To accomplish this, two common techniques are implemented and compared: wavelets and EMD. In particular, we implement the Antoniadis and Kopsinis methods, respectively. In both cases, the default authors' settings were chosen. These two denoising techniques are very efficient, as depicted in Fig.  (the SNR gets better from 10 to 23 dB). The denoised signal allows a maximum detection procedure to be applied as described in . Finally, the researched frequency is obtained by computing the inverse of the average of all periods computed between each maximum.

The time domain offers the advantage of choosing the portion of the signal for computation (a variable number of periods) without having a major impact on the precision. The resonance frequency will thereafter be the average of the different periods considered. The signal portion used for this calculation depends directly on the denoising quality and therefore on the SNR. As the SNR reduces, the more necessary it will be for a higher number of periods to be taken. On the other hand, if the SNR is high enough (greater than 5 dB), fewer than 10 periods will be sufficient to obtain a similar precision to the entire signal. In the case of Fourier, the search for minimum uncertainty imposes the consideration of FFT on the totality of the signal, as described in the following section.

SAWR signals s(t) were generated as described above with the parameters of Fig.  and for M=100 frequencies close to a usual intermediate frequency at F=10.7 MHz and for Q SNR values (Q=21 and -10⩽SNR⩽30 dB) with N=8192 (213) samples, a sampling rate of Fs=1.6 GHz and a time duration T0=5.12 µs. These frequencies are represented in red in Fig. . The colour was chosen in order to avoid the masking of some represented figures. Also generated for each of the M frequencies are P=100 versions ηi(t) of the noised signal by an additive Gaussian white noise N(0,σh) with σh=Ps×10(-SNRh/10) and Ps: the signal power of s(t) is 3sj,i,h(t)=sin⁡2πFjt⋅fλ,μ,α(t)+σh⋅ηi(t), with j∈1,M, i∈1,P and h∈1,Q.

Figure  shows the block diagram of the experimental setup while performing signal acquisition using a radiofrequency (RF) module. The signals are generated by a RF generator. They are then transmitted to the SAWR via an antenna and a RF switch. Their initial central frequency is Fn=868 MHz, and the responses of the SAWR are translated at F=10.7 MHz using a RF down-converter.

Tables  and show the results for the three signals with different numbers of samples, sampling rates, and a constant time duration T0=13.1 µs. The superiority of the time-based methods is confirmed in terms of uncertainty.

Signal no. 1 (Fig. ) is characterized by a Gaussian envelope, Fs=2.5 GHz, N=32768 (215), and an estimated SNR =16.2 dB. The estimated frequencies are close in all three cases (Table ), with an improved wavelet or EMD spectral resolution of 40 % with respect to Fourier (Table ). Taking into account the SNR and with regard to Fig. , the right value is probably given by the wavelet method with F^=10.716MHz±45.6kHz.

Signal no. 2 (Fig. ) is characterized by Fs=5 GHz, N=65536 (216) and an estimated SNR =13.7 dB. The frequency difference between the time-based and Fourier approaches is about 30 kHz, while between the two time-based methods this difference is only 4 kHz (Table ), which confirms the results of Fig. . Furthermore, the uncertainty of wavelets or EMD is clearly better since its improvement reaches a ratio of 3 in comparison with Fourier (Table ). For the same reasons as above, the right frequency seems to be given by the wavelet method with F^=10.592MHz±22.8kHz.

Signal no. 3 (Fig. ) is much noisier than the previous two and is characterized by Fs=20 GHz, N=262144 (218) and an estimated SNR =5.2 dB. The gap between Fourier and wavelets or EMD is about 40 kHz against 1.3 kHz between the wavelets and EMD (Table ). SNR =5.2 dB corresponds to a range for which the measurement accuracy by wavelets is better than EMD (Fig. ). The uncertainty value decreases to 5.7 kHz against 76.3 kHz for Fourier, which represents 0.05 % of the central frequency (Table ). The true value is probably F^=11.023MHz±5.7 kHz.

In this subsection, we describe the impact of previously established frequency results on the temperature accuracy measured for this kind of sensor.

The frequency translation shown in Fig.  shifts the response of the sensor centred around Fn=868 MHz to a lower intermediate frequency F=10.7 MHz but maintains its spectral characteristics (amplitude, frequency width, etc.). Therefore, the results obtained at F=10.7 MHz are applicable to the industrial, scientific, and medical (ISM) bands in the ranges 433 MHz, 868 MHz, and 2.4 GHz.

Most SAWR temperature sensors are based on the piezoelectricity phenomenon. Whatever the material category (single crystal, film, or ceramic), the central frequency of this kind of SAWR sensor can be connected to the temperature through the TCF (temperature coefficient of frequency) by a linear approximation: 7ΔFFn=βT+δ,withΔFFn(ppm),T(∘C),β(ppm∘C-1),δ(ppm), or by a quadratic approximation with this form: 8ΔFFn=αT2+βT+δ,withα(ppm∘C-2).

We chose four SAWR temperature sensors from one of the frequency bands mentioned above. Two of them come from scientific articles and the other two from a manufacturer's documentation. Their references and central frequencies are respectively as follows:

Sensor 1 (SS433FB2), F1n=434.56 MHz ,

The considered elements characterize the TCF (Fig. a). These data establish with the desired level of precision the temperature–frequency relationship (Fig. b). We apply our frequency measurement method to the different sensors mentioned above. To compare the results for these sensors, we choose to calculate the TCF as 9TCF=106×Fmax⁡-Fmin⁡FnTmax⁡-Tmin⁡.

Equation () leads to the temperature–frequency relationship: 10ΔT=106×1TCF×ΔFFn.

The results are depicted in Table . For a low-noise signal (SNR =30 dB, Fig. ), the best frequency accuracy obtained by the wavelets is 0.97 kHz, which corresponds to a temperature accuracy of ±0.06 ∘C for a 433 MHz sensor and ±0.01 ∘C if the sensor is in the 2.4 GHz band.

For a strongly noisy signal (SNR =5 dB, Fig. ), the best frequency accuracy is 5.7 kHz, which corresponds to a temperature accuracy between ±0.07 and ±0.38 ∘C according to the central frequency of the sensor.

Currently, the usual temperature accuracy is ±0.1 ∘C for a single measurement and reaches ±0.01 ∘C in the averaged multiple measurements. The results presented in this paper reach an accuracy of ±0.07 ∘C in one shot for the 2.4 GHz ISM band in a very noisy environment. The accuracy even reaches ±0.01 ∘C for a slightly noisy signal.