Time-frequency

Therefore, it seems natural to turn to time-frequency techniques in order to perform an efficient defect detection through the signal provided by the sensors. 

Of course, it is well known that because of the duality of the time and frequency parameters, there is no universal time-frequency technique. Indeed the two parameters cannot be simultaneously known with arbitrarily high resolution. This is expressed by the Heisenberg - Gabor inequality ... 

However, several techniques have been developed according to the characteristics of the signal to be analyzed, and lead to interpretable time-frequency representation. 

However, it is easily shown that if the mother wavelet is located in the frequency domain "around"/o (fig 8), then the wavelet a.b(t) is located around f(/a. That is to say, by the mean of the formal identification f = fata it is possible to interpret a time-scale representation as a time-frequency representation [4]. 

From the two equalities given in (6), we can observe that the time-frequency compromise expressed by (1) evolves within the changes of scale performed by the CWT. 

Q f)(t) wavelets in the time frequency plane, with the help of the Heisenberg ellipses. The axes of the ellipses are sized with respect to the RKfS value of the time resolution and frequency resolution. 

Best time-frequency representations are performed when the Wavelet Transform is continuous, that is when both parameters a... 

The computed CWT leads to complex coefficients. Therefore total information provided by the transform needs a double representation (modulus and phase). However, as the representation in the time-frequency plane of the phase of the CWT is generally quite difficult to interpret, we shall focus on the modulus of the CWT. Furthermore, it is known that the square modulus of the transform, CWT(s(t)) I corresponds to a distribution of the energy of s(t) in the time frequency plane [4], This property enhances the interpretability of the analysis. Indeed, each pattern formed in the representation can be understood as a part of the signal s total energy. This representation is called "scalogram". 

Figure Al.6.30. (a) Two pulse sequence used in the Tannor-Rice pump-dump scheme, (b) The Husuni time-frequency distribution corresponding to the two pump sequence in (a), constmcted by taking the overlap of the pulse sequence with a two-parameter family of Gaussians, characterized by different centres in time and carrier frequency, and plotting the overlap as a fiinction of these two parameters. Note that the Husimi distribution allows one to visualize both the time delay and the frequency offset of pump and dump simultaneously (after [52a]).

RDOs is much smaller than that for the vibration itself, not to mention that for the near-IR FT-Raman teclmique already discussed. This is particularly striking for high energy modes such as the C-H vibrations [108]. Modem applications of I CRS now utilize a two-dimensional time-frequency detection scheme... 

Ulness D J, Stimson M J, Kirkwood J C and Albrecht A C 1997 Interferometric downconversion of high frequency molecular vibrations with time-frequency-resolved coherent Raman scattering using quasi-cw noisy laser light C-H stretching modes of chloroform and benzene J. Rhys. Chem. A 101 4587-91... 

Frequency (Section 12.5) The number of electromagnetic wave cycles that travel past a fixed point in a given unit of time. Frequencies are expressed in units of cycles per second. or hertz. 

Fig. 4-4 The age frequency function j/ x) and the residence time frequency function 4> x) and the corresponding average values and r, for the three cases described in the text (a) ta > t, (b) la = xy, (c) ta > t,. (Adapted from Bolin and Rodhe (1973) with permission from the Swedish Geophysical Society.)...

Pati, Y. C., Wavelets and time-frequency methods in linear systems and neural networks. 

Compression may be achieved if some regions of the time-frequency space in which the data are decomposed do not contain much information. The square of each wavelet coefficient is proportional to the least-squares error of approximation incurred by neglecting that coefficient in the reconstruction ... 

Multiscale process identification and control. Most of the insightful analytical results in systems identification and control have been derived in the frequency domain. The design and implementation, though, of identification and control algorithms occurs in the time domain, where little of the analytical results in truly operational. The time-frequency decomposition of process models would seem to offer a natural bridge, which would allow the use of analytical results in the time-domain deployment of multiscale, model-based estimation and control. 

The UPD and anodic oxidation of Pb monolayers on tellurium was investigated also in acidic aqueous solutions of Pb(II) cations and various concentrations of halides (iodide, bromide, and chloride) [103]. The Te substrate was a 0.5 xm film electrodeposited in a previous step on polycrystalline Au from an acidic Te02 solution. Particular information on the time-frequency-potential variance of the electrochemical process was obtained by potentiodynamic electrochemical impedance spectroscopy (PDEIS), as it was difficult to apply stationary techniques for accurate characterization, due to a tendency to chemical interaction between the Pb adatoms and the substrate on a time scale of minutes. The impedance... 

In single-scale filtering, basis functions are of a fixed resolution and all basis functions have the same localization in the time-frequency domain. For example, frequency domain filtering relies on basis functions localized in frequency but global in time, as shown in Fig. 7b. Other popular filters, such as those based on a windowed Fourier transform, mean filtering, and exponential smoothing, are localized in both time and frequency, but their resolution is fixed, as shown in Fig. 7c. Single-scale filters are linear because the measured data or basis function coefficients are transformed as their linear sum over a time horizon. A finite time horizon results infinite impulse response (FIR) and an infinite time horizon creates infinite impulse response (HR) filters. A linear filter can be represented as... 

In analogy to the time frequency, /, the spatial concentration behaviour may be characterized by the local frequency, focal = 1 /liocai- In Fig. 2.6, four types of spatial concentration functions are shown. These types and combinations of them can characterize all the variations of concentration in analytical practice both in one- and more-dimensional cases. 

Fourier pairs not only exist in time-/frequency domain but also in any other domain combined by a quantity q and the belonging dimension-inverted quantity l/q. 

Fig. 9 (a) Constant-time frequency-selective pulse sequence used for indirect measurement of 13C -13C distances in MB(i + 4)EK with uniformly 15N,13C-labeled L-alanine at Ala9 and AlalO. The 2D spectra in (b) and (c) and the extracted data points in (d) show that the dephasing is more rapid, with hence a shorter intemuclear distance, for 13C nuclei in the helical conformation (represented by squares) than for those in the non-helical conformation (represented by circles) (reproduced from [149] with permission)... 

Silicon is a model for the fundamental electronic and mechanical properties of Group IV crystals and the basic material for electronic device technology. Coherent optical phonons in Si revealed the ultrafast formation of renormalized quasiparticles in time-frequency space [47]. The anisotropic transient reflectivity of n-doped Si(001) featured the coherent optical phonon oscillation with a frequency of 15.3 THz, when the [110] crystalline axis was parallel to the pump polarization (Fig. 2.11). Rotation of the sample by 45° led to disappearance of the coherent oscillation, which confirmed the ISRS generation,... 

Fluctuations dominate for T > For typical values fiq (350-F500) MeV and for Tc > (50 A- 70) MeV in the weak coupling limit from (26), (22) we estimate Tq< (0.6 A- 0.8)TC. If we took into account the suppression factor / of the mean field term oc e A /T, a decrease of the mass m due to the fluctuation contribution (cf. (11)), and the pseudo-Goldstone contribution (25), we would get still smaller value of T < (< 0.5TC). We see that fluctuations start to contribute at temperatures when one can still use approximate expressions (22), (20) valid in the low temperature limit. Thus the time (frequency) dependence of the fluctuating fields is important in case of CSC. 

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