Due to the high levels of attenuation considered in this work, retrieving of the resonant frequencies requires some processing of the measured spectrum. Indeed, the resonant peaks are broad and systematically overlap each other, except for the very first few modes (Fig. 3.3). In that context, the resonant frequencies do not generally correspond to local maxima in the spectrum.

Despite damping, and assuming linear response of the solid sample, its frequency responseF Ris a sum of individual responses of single degree-of-freedom resonators, described by Lorentzian line shapes, i.e.

F R(f) =

M

X

k=1

a_{k}

(f_{k}^{2}−f^{2}) +i(f_{k}f /Q_{k}) (3.8)
whereakare the complex amplitudes,fkthe resonant frequencies andQkthe quality
factors. Therefore, a fit of the spectrum with a large number of Lorentzian line
shapes could be used to retrieve the parameters. However, it is difficult in such
situation to obtain convergence with conventional non-linear fitting algorithm (such
as the Levenberg-Marquardt algorithm), because this kind of algorithm needs a good
starting point to converge. Moreover, the correct numberM of resonant peaks in a
given frequency band is unknown a priori.

Lebedev (2002) proposed an approach that does not try to fit the spectrum with a sum of Lorentzian in the frequency domain but works mainly in the time domain, using the inverse Fourier transform of the spectrum to recover the reso-nant frequencies. While this method produces good estimates of the parameters, it was observed to lack accuracy in fitting the spectrum by Bernard et al. (2013). In this last study the estimates from the time domain method were used as the ini-tial point of a non-linear fitting algorithm in the frequency domain, which slightly modified the estimated parameters to improve the fit. The combination of the two methods was observed to reduce the variability of the estimated parameters. The same combined approach was used here. We firstly describe the time domain part of the method, following the work ofLebedevand then the non-linear fitting in the frequency domain.

3.4.1 Estimation of the resonant frequencies in time domain The inverse Fourier transform of the frequency response (5.1) is the impulse response of the system, which is a sum of damped exponentials involving the same parameters ak,fk andQk. Methods to estimates these parameters from discrete samples of the impulse response have been developed in the signal processing literature. A method based on a linear predictive filter originally proposed byKumaresan and Tufts(1982) has been used for RUS by Lebedev(2002).

3.4. Processing of the measured spectra 47

The N discrete samples of the measured spectrum in the range [fmin, fmax]are
transformed into a time series y[n] via inverse discrete Fourier transform. From
these samples, the matrix
sam-ples would be sufficient to exactly predict the remaining samsam-ples via the
predic-tion equapredic-tion Ag = b, with b a column vector containing the N −L last samples
[y[L+ 1] y[L+ 2] · · · y[N]]^{t} andg the column vector of the linear prediction

can be determined by solving the prediction equation and finding the zeros of H.

The correct system order M (the number of resonant peaks) has to be
deter-mined. To this purpose, the autocorrelation matrix R = A^{∗} ×A is constructed
(with (.)^{∗} denoting the hermitian conjugate) and an eigenvalue decomposition of
R is performed. In a noise free case, only M non-zero eigenvalues would be found
(Kumaresan and Tufts, 1982) but, in the presence of noise, all the L eigenvalues
are above zero. However, theL−M eigenvalues corresponding to random noise are
weaker than the M signal values. Thus, by looking for abrupt change of slope in
the eigenvalue spectrum ofR,M can be determined (Lebedev,2002).

The coefficients of the linear predictive filter g could be obtained as g = ˆAb, whereˆ.denotes the pseudo-inverse of A. However, due to the presence of noise this system of equation is numerically instable. The solution is to built a “noise-free"

approximation ofA. Because the number of resonant peaksM has been determined
it is possible to reconstruct an approximationR^{0}ofRusing theM largest eigenvalues
λof Rand the corresponding eigenvectors V

R^{0} =

M

X

i=1

V_{i}λ_{i}V_{i}^{∗}. (3.10)

This approximation ofR is noise-free in the sense that is has been constructed using only the eigenvalues and eigenvectors that correspond to resonant peaks present in the signal. Coefficients of g are then determined from the system of equations

g = R^{0}r, with r = A^{∗}b (Kumaresan and Tufts, 1982). The zeros of the filter
transfer function are localized, and the resonant frequencies and damping factors
are obtained.

The last step estimates the complex amplitudes ak. They are estimated in the
frequency domain. Indeed, the dependence of the Lorentzian toa_{k}is linear (Eq.5.1),
and the a_{k} can be obtain from a linear system of equations (Lebedev,2002).

The process requires us to compute the matrixAwith a fixed value ofL before M has been determined. We usedL= 128, this is much larger than the number of resonant frequencies present in the signal, and much smaller than the total number of samples. In the selection of the number M, there is generally a small ambiguity.

Indeed, the break in the spectrum of the eigenvalues is not always very clear and
2 or 3 distinct values of M can be good candidates. In this case, one should select
the highest candidate value, as selecting a value that is superior to the real number
M will only introduce a little noise in R^{0}, while selecting a smaller number could
discard an actual resonant peak.

The method presented here was originally described in the context of RUS by A.V. Lebedev. It was first applied to a simulated frequency response (Lebedev,2002) and subsequently to experimental signals (Lebedev et al., 2003). Since we applied the method in a similar way, these two last references contain relevant details and discussions about the processing and particularly about the determination of the number of resonances M.

3.4.2 Non-linear fitting in frequency domain

In a second step, the parameters determined in time domain were used as initial pa-rameters of a non-linear optimization algorithm (Levenberg-Marquardt), searching for the set of parameters minimizing the difference between Eq. (5.1) and the ex-perimental frequency response in a least-square sense. The number of independent parameters increases rapidly with the number of peaks (two real and one complex parameter for each peak), making the convergence of the algorithm hard to obtain.

Therefore, the processing was performed on small parts of the spectrum, each con-taining about 8-10 peaks (Fig. 3.4). The spectrum was cut in several overlapping sub-parts. Overlapping allows one to check for the good convergence (same frequen-cies should be obtained in the overlapping parts). Convergence was also checked visually by comparing the reconstructed and the experimental spectra.

This additional step improves the repeatability of the parameters estimation and the agreement between experimental and reconstructed spectra compared to the time-domain method only (Bernard et al., 2013) (Fig. 3.4). It also makes the estimated parameters less sensitive to the choice of the model order M. If M is slightly over evaluated from the time domain processing, the least-square fit runs