Frequency localisation

Each Fourier coefficient in a transform with apodisation represents a band of frequencies. The width of that band is controlled via the length of the signal that is transformed and the shape of the apodisation function. We can introduce the notion of frequency localisation as an extension of the previously introduced frequency resolution and in analogy to localisation in time. When the bands are wide, the frequency information returned by the transform is less localised than when the bands are narrow. In other words, when the time localisation is good, the frequency localisation is poor. 

A wavelet basis allows a time-frequency analysis similar to that of a short-time Fourier basis. It is different in that its time-localisation is better hence its frequency localisation is worse, for high frequencies than for low fre-... 

A small scale value permits us to perform a local analysis a large scale value is used for a global analysis. Combining local and global is a useful feature of the wavelet analysis. The wavelets having a compact support are used in local analysis. This is the case for Haar and Daubechies wavelets. As wavelet analysis constraint to Heisenberg uncertainty principle, it is impossible to reduce arbitrarily both time and frequency localisation. The resolution increases as the scale decreases. The greater the resolution, the smaller and finer are the details that can be accessed. 

Singularities can be thought of as either an abrupt change or impulse in a signal, or the sudden shift of the signal s mean value to a different level. The good time-frequency localisation property provides wavelet in singularity analysis (Mallat and... 

The ohmic case is the most complex. A particular result is that the system is localised in one of the wells at T = 0, for sufficiently strong friction, viz. rj > nhjlQo. At higher temperatures there is an exponential relaxation with the rate Ink oc (4riQllnh — l)ln T. Of special interest is the special case t] = nhl4Ql. It turns out that the system exhibits exponential decay with a rate constant which does not depend at all on temperature, and equals k = nAl/2co. Comparing this with (2.37), one sees that the collision frequency turns out to be precisely equal to the cutoff vibration frequency Vo = cojln. 

Fig. 7.9(b) Chemoinvestigation among Viverrids (c.f. Chap. 3, Heading Fig.) localised, site-specific investigation of skin-gland complexes in Genetta ano-genital (ag), perineal (p), scrotal (st), anal (al), and sub-caudal (sc), [nos. = sniffs (sec.), = % of all observations, frequency and duration.] (from Wemmer, 1977). 

Moreover, the increase is most marked for those frequencies that are localised in the CO groups trans to each other, rather than for these trans to halogen in the notation of Figure 2, k% is more affected than k. ... 

Localisation of photons to a specific area and of restricted frequencies... 

Due to the higher frequency involved in NMR, the broad distribution involved in the p transition observed by dynamic mechanical measurements will be reduced, the localised or cooperative processes tending to merge. 

In spite of the apparent sensitivity to the material properties, the direct assignment of the phase contrast to variation in the chemical composition or a specific property of the surface is hardly possible. Considerable difficulties for theoretical examination of the tapping mode result from several factors (i) the abrupt transition from an attractive force regime to strong repulsion which acts for a short moment of the oscillation period, (ii) localisation of the tip-sample interaction in a nanoscopic contact area, (iii) the non-linear variation of both attractive forces and mechanical compliance in the repulsive regime, and (iv) the interdependence of the material properties (viscoelasticity, adhesion, friction) and scanning parameters (amplitude, frequency, cantilever position). The interpretation of the phase and amplitude images becomes especially intricate for viscoelastic polymers. 

Fig. 1. The variation of the potential energy of an adatom moving parallel to the surface. Em is the barrier height measured from the zero point energy of the two degenerate modes of vibration parallel to the surface, i.e. Em = E — hvp, where is the frequency of vibration of a localised adatom parallel to the surface.

This is the customary relation [3] for the collision frequency in an ideal two-dimensional gas moving freely in a constant potential field provided by the surface, except for the absence of a /2 factor. The kinetic theory derivation of the collision frequency involves the relative velocity, which is the origin of they/2 factor. In the present model, there are two types of collisions, firstly with the localised adatoms, when the y/2 factor would not be appropriate, and secondly with mobile adatoms, when it would. However, the degree of approximation in the treatment is such that the refinement of a /2 factor is unwarranted it is mentioned merely to clarify its presence in expressions found in the literature. 

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