Most applications which measure physical quantities, especially in harsh environments, rely on surface acoustic wave resonators (SAWRs). Measuring the variation of the resonance frequency is a fundamental step in such cases. This article presents a comparison between three techniques for best determining the resonance frequency in one shot from the point of accuracy and uncertainty: fast Fourier transform (FFT), discrete wavelet transform (DWT) and empirical mode decomposition (EMD). After proposing a model for the generation of synthetic SAW signals, the question of wavelet choice is answered. The three techniques are applied to synthetic signals with different central frequencies and signal-to-noise ratios (SNRs). They are also tested on experimental signals with different sampling rates, number of samples and SNRs. Results are discussed in terms of the accuracy of the estimated frequency and measurement uncertainty. This study is successfully extended to SAWR temperature sensors.

For some years now, the industry has been experiencing a strong trend in the integration of preventive maintenance in its development strategy

A major advantage of this kind of sensor is its ability to act discreetly. It does not need any dedicated energy to operate because it picks up its energy from the wave that interrogates it. Thus, a wireless SAW device is completely passive and able to operate in harsh environments.

Figure

In most cases, the fast Fourier transform (FFT) method is applied with its possible variations (zero crossing, zero padding, peak detection, etc.)

The objective of the work described in this paper is to proceed, in one shot as precisely as possible, with the resonance frequency estimation of the SAWR by comparing three methods: on the one hand the FFT and on the other hand two time-based methods, wavelets

This article is an in-depth study of a conference contribution

Wireless SAW sensor in one-port resonator mode.

Usually, the envelope of a SAWR signal is modelled as a decaying exponential

Both parameters

Five examples of the probability density function

The synthetic SAWR signal used for finding the best wavelet, with

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

Experimental SAWR signal (

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 (SNR

SNR computation after denoising for 24 wavelets according to a reference SNR range (SNR

A focus on Fig.

Morphological behaviour of the

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

Resonance frequency estimation with an FFT performed on the whole signal (

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

Result of denoising techniques:

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

Randomly computed frequencies

Figures

First, by observing both figures, through the yellow plans, we can see the invariability of the measurement by Fourier, regardless of the signal-to-noise ratio and the averaging rank of the noisy versions. As we might expect, we also see a progressive regularity of both 3D curves for wavelets and EMD as the number of averaged signals increases. In contrast, this regularity sets in much more quickly for EMD than wavelets. Regarding the wavelet or Fourier comparison, a dividing line appears at 0 dB and ends at

Figure

Relative error for wavelets and Fourier between all the averaged estimated frequencies

Relative error for EMD and Fourier between all the averaged estimated frequencies

Positive values of the relative error between all the wavelet averaged estimated frequencies

Another way of observing the behaviour of the error according to the SNR is proposed in Fig.

For Fourier (Fig.

For wavelets (Fig.

For EMD (Fig.

Error between the estimated and reference frequencies versus SNR for

Figure

Mean of errors between estimated and reference frequencies versus SNR for Fourier, EMD, and wavelet approaches.

In addition to accuracy, it is important to consider the uncertainty of the measurement.

For the FFT approach, uncertainty is related to the spectral resolution

Therefore, FFT uncertainty depends only on duration and not on the number of samples or the signal frequency. This is the reason why the entire duration of the signal must be considered to obtain the smallest uncertainty.

Unlike Fourier, the uncertainty

Figure

Uncertainties

Figure

Tables

Frequency estimation on experimental signals.

Measurement uncertainty in experimental signals.

Signal no. 1 (Fig.

Signal no. 2 (Fig.

Signal no. 3 (Fig.

Block diagram of the RF module implemented for the SAWR signal acquisition.

Experimental SAWR signal no. 1 with low-pass filtering at 20 MHz (

Experimental SAWR signal no. 2 with low-pass filtering at 20 MHz and lag (yellow in the colour version) used for the time-based methods (

Experimental SAWR signal no.3 without filtering and lag (yellow in the colour version) used for the time-based methods (

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.

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:

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),

Sensor 2 (AlN-SAW),

Sensor 3,

Sensor 4 (SS2414BB2),

The considered elements characterize the TCF (Fig.

Equation (

The results are depicted in Table

Temperature precision according to the frequency measurement.

For a strongly noisy signal (SNR

Currently, the usual temperature accuracy is

Frequency behaviour according to the temperature ranges for different temperature SAWR sensors:

Measurement of resonance frequency variation is fundamental for all applications using a wireless SAWR sensor. We proposed a comparative study for the estimation of this frequency between the usually FFT spectral method and two time-based methods, the first using a wavelet approach and the second applying empirical mode decomposition (EMD). A model which describes the behaviour of SAWR signals and also allows the generation of synthetic signals was also proposed. This model was implemented to address the question of the wavelet choice as part of the first method, and it leads to the

Presented here is the theoretical background of the two main methods employed in this study: the continuous–discrete wavelet transform and the empirical mode decomposition.

The CWT is defined as

The mother wavelet

A number of vanishing moments

In this case the wavelet analysis is blind to any polynomial of degree lower than

The wavelet transform also has the ability to reconstruct the signal

The wavelet transform is a powerful tool for the detection of singularities in a signal. By calculating the Hölder exponent (or Lipschitz) and using wavelet transform modulus maxima, it is possible to emphasize the properties of these singularities. Figure

Continuous wavelet analysis on a test signal with 1024 samples and a wavelet

Let

For

Unlike the CWT, the DWT is defined by two functions: the scaling mother function

A link between this approach by the scale and wavelet functions and the filter theory was established in particular by Stéphane Mallat

The MRA is very well suited for denoising a signal since it naturally separates the signal into approximations and details at different scales.

This is depicted in Fig.

Multi-resolution analysis of a zoomed experimental SAWR signal using a wavelet

As wavelets, the EMD method analyses signals in scales. The difference between Fourier or wavelets and EMD is the analysing basis. The EMD is based on oscillating functions extracted from the signal itself

The EMD algorithm is detailed in Fig.

Thus, the signal

Like DWT, the EMD method analyses the signal into scales and is well suited for denoising. This is depicted in Fig.

Building of the upper, lower, and mean envelopes which are the key points of the iterative algorithm of an intrinsic mode function (IMF) calculation.

EMD algorithm that gradually extracts all the intrinsic mode functions

EMD analysis of the same zoomed experimental SAWR signal as Fig.

The data set for this publication can be accessed at

AnS came up with the idea for this study. The modelling of the SAWR signals, the investigation of the methods on simulated and experimental signals, and the production and analysis of results were done by PR in close discussion with AnS. The hardware setups and the experimental signal acquisition were provided by PR and AgS. PR and AnS wrote the paper. All the authors contributed to the review of the final paper.

The contact author has declared that none of the authors has any competing interests.

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This open-access publication was funded by the French Air and Space Force Academy.

This paper was edited by Alexander Bergmann and reviewed by one anonymous referee.