APPENDIX S1. DETAILLED EXPLANATION OF THE MATHEMATICAL ASPECTS AND ALGORITHMS OF THE EMPLOYED SIGNAL ANALYSIS METHODS

i SUPPORTING INFORMATION

APPENDIX S1. DETAILLED EXPLANATION OF THE MATHEMATICAL ASPECTS AND ALGORITHMS OF THE EMPLOYED SIGNAL ANALYSIS METHODS

STFT. The short-time Fourier transform (STFT) provides a magnitude-frequency-time spectrogram of a time-series signal:

ℱ𝑚,𝑛𝑤𝑖𝑛(𝑓)(𝜔0, 𝑡0) =

( 1 ) 1

√2 ∙ 𝜋∙ ∫^{𝑡𝑡𝑜𝑡}𝑒^{−𝑖𝑚𝜔0𝑡}∙ 𝑔(𝑡 − 𝑛 ∙ 𝑡0) ∙ 𝑓(𝑡) ∙ 𝑑𝑡

0

The absolute values of the STFT coefficients ℱ𝑚,𝑛𝑤𝑖𝑛 give the amplitude distribution of the signal sinusoid components 𝑒^{−𝑖𝑚𝜔0𝑡} found in each time window. The discrete windows (function 𝑔), acting like a filter, have a fixed duration of width 𝑙. These regularly spaced time frames are translated by the shifting parameter 𝑛 on the whole time-series. Finally, STFT decomposes the signal into 𝑚 regularly spaced frequency steps 𝜔0 in 𝑛 regularly spaced windows, which can overlap using the sliding step 𝑡0 (Sturmel and Daudet 2011).

The windows begin at 𝑛 ∙ 𝑡₀ (convention used for the graphical explanation in FIG. 10), with a sliding step of 1 ≤ 𝑡0≤ 𝑙 for consecutive windows, to permit overlapping of the windows by 𝑙 − 𝑡₀ without missing any sampling points (Sturmel and Daudet 2011).

A quick and efficient discrete computation of STFT can be conducted with the fast Fourier transform (FFT) algorithm (Isermann and Münchhof 2011; Franklin 2013). The windows’ width is chosen to be a power of two because FFT is optimized to work with such input vectors (Franklin 2013).

FIG. 10. SCHEMATIC REPRESENTATION OF THE STFT SIGNAL DECOMPOSITION METHOD

Transient signal as upper curve, and the components S_m,n in each time window n, without overlapping (t₀= l).

The sum of the sinusoid components 𝑆_𝑚,𝑛(𝜔₀, 𝑡) = 𝑒^{−𝑖𝑚𝜔0𝑡} of each window, weighed with their respective amplitude factors 𝐴𝑚,𝑛, reconstructs the time signal 𝑓𝑛(𝑡) in this window (FIG. 10):

𝑓𝑛(𝑡) = ∑ 𝐴𝑚,𝑛∙ 𝑆𝑗,𝑛(𝜔0, 𝑡)

𝑚

𝑗=1

( 2 ) The energy content in the raw signal segments and their Fourier transform is the same, as can be verified by the energy conservation theorem from Parseval (Franklin 2013). Thus, the discrete ℱ𝑚,𝑛𝑤𝑖𝑛 characterize 𝑓 and can reconstruct it through their inverse transform.

Three very important tuning parameters must be selected for the windowing procedure: the time window width, its overlapping domain, and its shape. Large windows generate narrowband spectrograms, which permit a higher frequency resolution but a coarser time resolution, and short windows generate wideband spectrograms, which have contrary resolution rules (Isermann and Münchhof 2011). A change in frequency occurring inside one window cannot be localized within the time window.

Consequently, in the two STFT examples of FIG. 3, one could not determine the exact location of an abrupt peak.

Depending on the sampling frequency (on the amount of points in each window), the amount of frequency segments (the frequency precision) is also limited. Thus, too short windows do not permit discrimination between close frequency levels. The advantage of allowing the windows to overlap a certain edge domain of their

FIG. 11. COMMONLY USED MOTHER WAVELET SHAPES (continuous curves) FOR CHILD FUNCTIONS OF THE WAVELET TRANSFORM AND THEIR CENTER FREQUENCY-BASED APPROXIMATION CURVES (dashed curves)

ii

FIG. 12. SCHEMATIC REPRESENTATION OF THE WAVELET SIGNAL DECOMPOSITION METHOD Transient signal as upper curve, and the components ψ_m,n, obtained from DWT for simplification.

respective neighbors is that the time resolution can be improved without losing accuracy in the frequency domain (Isermann and Münchhof 2011). Finally, many window shapes were developed, such as the simple rectangular window or the very popular Gaussian, Hamming, and Hanning windows, which influence the values of the Fourier transform in each window as it results from the convolution of the frequency spectra of the original signal and of the window function (Isermann and Münchhof 2011). Multiplying the time signal with an appropriate windowing function minimizes leakages resulting from sudden discontinuities in the finite time intervals (Isermann and Münchhof 2011).

CWT. The continuous wavelet transform (CWT) is an alternative to the STFT for dynamic extraction of magnitude-frequency-time information contained in a signal. A large varity of wavelet shapes have been developed (Marchant 2003; Isermann and Münchhof 2011), such as the simple Haar wavelet, a family of Daubechies wavelets, the Morlet wavelet, or the

“Mexican hat wavelet” (FIG. 11).

The mother wavelet 𝜓 constitutes a family of basis functions:

𝜓^𝑎,𝑏(𝑡) = 1

⁄√|𝑎|∙ 𝜓 (^𝑡−𝑏_𝑎 ) ( 3 ) Pseudo-frequencies 𝐹_𝑎 can be calculated from the scale level, the sampling interval ∆𝑡, and the center frequency 𝐹𝑐 depending on the wavelet shape (Abry 1997, cited by The MathWorks 2014):

𝐹_𝑎= 𝐹𝑐

𝑎 ∙ ∆𝑡 ( 4 )

The center frequency 𝐹_𝑐 is the leading dominant frequency of the wavelet associated to a purely periodic sinusoid oscillation (FIG. 11), thereby extracted from the Fourier transform of the mother wavelet. Therefore, the pseudo-frequencies in the wavelet-generated spectrogram are not exactly the inverse of the scale (represented in scalograms) but are approximations depending on the chosen wavelet shape.

The discrete wavelet transform (DWT) is the discrete form of the wavelet spectral analysis. It performs the wavelet transformation in fixed time frames of the set of scales, becoming discrete at the level m (with 𝑎 = 𝑎₀^𝑚).

The translation parameter is proportional to the wavelet width to avoid overlapping: narrow (high frequency) wavelets are translated by n steps covering the whole time span; wider (low frequency) wavelets are translated by larger steps. Therefore, 𝑏 = 𝑛 ∙ 𝑏₀∙ 𝑎₀^𝑚. The coefficients 𝑇_𝑚,𝑛^𝑤𝑎𝑣 can be calculated as follows (Daubechies 1992):

𝑇_𝑚,𝑛^𝑤𝑎𝑣(𝑓)(𝑎0, 𝑏₀) =

( 5 ) 1

√𝑎0𝑚∙ ∫ 𝜓 ( 𝑡

𝑎₀^𝑚− 𝑛 ∙ 𝑏0) ∙ 𝑓(𝑡) ∙ 𝑑𝑡

𝑡_𝑡𝑜𝑡 0

Analogous to the discrete inverse STFT transform, the raw signal can be reconstructed from the basis functions 𝜓 𝑛,𝑚(𝑎0, 𝑏0) weighed by their respective amplitudes 𝑇_𝑖,𝑗^𝑤𝑎𝑣, called wavelet coefficients (FIG. 12):

𝑓(𝑡) = ∑ ∑ 𝑇_𝑖,𝑗^𝑤𝑎𝑣(𝑎₀, 𝑏₀) ∙ 𝜓_𝑖,𝑗(𝑎₀, 𝑏₀)

𝑛

𝑗=1 𝑚

𝑖=1

( 6 )

HHT. The Hilbert-Huang transform (HHT) is a data- driven alternative technique for dynamic spectral analysis. A sifting process iterates the extraction of an intrinsic mode function until it reaches acceptable quality. The most important steps of the procedure are illustrated in the empirical mode decomposition flow chart (FIG. 13): the maximum and minimum values are identified from the signal followed by cubic spline interpolations to constitute the upper and lower

FIG. 13. FLOW CHART FOR THE EMPIRICAL MODE DECOMPOSITION ALGORITHM

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envelopes. The mean envelope is calculated from the upper and lower envelopes and subtracted from the signal. The result is an IMF if the number of its extrema is equal to the number of zero crossings or differs at most by one and if the mean value of the envelopes defined by local maxima and local minima is zero at any point (Huang et al. 1998). If the result does not satisfy these conditions, it enters into a new sifting cycle. Each IMF (FIG. 14) thus obtained admits well-behaved Hilbert transforms. Then, the condition for stopping EMD is that no more IMFs can be extracted from the residue 𝑟 without IMF becoming monotonic. According to Wu and Huang (2009), the number of IMFs is fewer than the next power of two of the total number of data points.

FIG. 14. SCHEMATIC REPRESENTATION OF THE HHT SIGNAL DECOMPOSITION METHOD

Transient signal as upper curve, the IMF-modes, and the residue 𝑟 at time 𝑡.

Instantaneous frequencies (IFs) are identified by Hilbert spectral analysis (HSA) using the Hilbert transform 𝐻.

𝐼𝐹𝑗 of each 𝐼𝑀𝐹𝑗 is calculated from its phase 𝜃 (in radians) in Eq. 11.

𝐼𝐹𝑗(𝑡) = 1 2𝜋∙𝑑𝜃𝑗(𝑡)

𝑑𝑡 ( 7 )

Because 𝜃 is the argument of the complex expression (Eq. 12) of each IMF component, with 𝐼𝑀𝐹 as the real part and 𝐻 as the imaginary part, it can be evaluated by the arctangent of the ratio 𝐻 to 𝐼𝑀𝐹.

𝐴𝑗(𝑡) ∙ 𝑒^{𝑖∙𝜃𝑗(𝑡)}= 𝐼𝑀𝐹𝑗(𝑡) + 𝑖 ∙ 𝐻𝑗(𝑡) ( 8 ) The instantaneous amplitude 𝐴, the absolute value of the magnitude of each spectral component of the Hilbert spectrum, is the modulus of the previous complex expression (Eq. 12). It can thus be calculated from the square root of the sum of the squared components 𝐻 and 𝐼𝑀𝐹.

Finally, a complex analytic expression 𝑓_{𝑐𝑜𝑚𝑝𝑙} of the input signal 𝑓 is obtained so that 𝑓 can be recovered from the Hilbert spectrum by taking the real part of 𝑓𝑐𝑜𝑚𝑝𝑙: 𝑓_{𝑐𝑜𝑚𝑝𝑙}(𝑡) = ∑^𝑁 𝐴_𝑗(𝑡) ∙

𝑗=1 𝑒^{𝑖2𝜋∙∫0𝑡𝑡𝑜𝑡𝐼𝐹𝑗(𝑡)∙𝑑𝑡} ( 9 )

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APPENDIX S2. SPECTRAL ANALYSES OF M2 CURLIES AT ALL TESTED RELATIVE HUMIDITIES (11, 23, 33, 44, 53, and 76% RH)

From top to bottom: raw mechanical signal in the time domain; one-sided Fourier spectrum; two short-time Fourier spectra (STFT) spectrograms with small rectangular windows of 2⁷ points or 0.256 s without overlap and large Hanning windows of 2¹⁰ points or 2.048 s with 50% overlap; continuous wavelet transform (CWT) spectrogram with Morlet wavelet with 1012 scales, corresponding to pseudo-frequencies from 0.1 to 25 Hz; Hilbert-Huang transform (HHT) spectrogram with empirical mode decomposition algorithm.