Shape invariance

In the case of quasi-periodic sinusoidal signals, the buzziness can often be linked to the fact that the phase coherence between sinusoidal components is not preserved. Shape invariant modification techniques for quasi-periodic signals are an attempt to tackle this problem. As explained in 9.4.2, quasi-periodic signals such as speech voiced segments or sounds of musical instruments can be thought of as sinusoidal signals whose frequencies are multiples of a common fundamental COo(x), but with additional, slowly varying phases 0 (/) ... 

Although common knowledge has it that fixed phase relations do not influence the perception of timbre [Zwicker and Fasti, 1990, Zwicker, 1982], this is not true for time-varying phases disturbing phase relations is known to introduce buzziness or reverberation in the modified signal [McAulay and Quatieri, 1986b], In Eq. (7.26), phase relations are controlled through the terms 0,(/). Therefore, an ideal shape-invariant time-scale modification would be... 

Figure 7.10 Time-domain representation of an original speech signal (top), and of its time-stretched version (bottom), showing the loss of shape-invariance. Phase-vocoder...

Time-domain representation of a speech signal showing shape invariance 305... 

Reverberation and shape invariance. One problem often associated with the use of time-scale or pitch-scale modifications, pointed out in [Portnoff, 1981] is commonly called the reverberation, chorusing or phasiness effect (chorusing refers to the subjective sensation that several persons are speaking/playing at the same time, as in a chorus). For moderate to large modification factors (say, above 1.5 or under 0.7),... 

By contrast, Fig. 7.10 shows the modified signal obtained by use of the standard phase-vocoder time-scaling technique. Clearly, the shapes of the signals are quite different, illustrating the lack of shape invariance. The standard phase-vocoder technique described in section 7.3 cannot ensure shape invariance because the signal is not assumed to be quasi-periodic and the time-scale modification is at best that described by... 

Notice that while the preceding paragraph mentioned phase relations between harmonics in connection with shape invariance, this paragraph addresses the problem of phase coherence between successive... 

Quatieri and McAulay, 1992] Quatieri, T. and McAulay, R. (1992). Shape-invariant time-scale and pitch modification of speech. IEEE Trans, on Acoustics, Speech, and Signal Processing, ASSP-40(3) 497-510. 

The radius of string-like nanodomains observed under the fdb conditions is much smaller than its length, rm -C lm. Consequently, for the fdb phenomenon the shape invariant value should also be small. Equation (10.18) shows that the shape invariant has the lowest value if a ferroelectric material satisfies the following conditions ... 

In Table 10.2 below we summarize the calculated values of the shape invariant r 2//m and aspect ratio rm/lm of string-like domains in different fe crystals formed under the fdb conditions. The experimental parameters used for these calculations are U = 3kV (applied voltage), R = 50 nm (typical radius of curvature of the tip apex), d = 0.5 nm (distance between the tip apex and the sample surface). The calculated aspect ratios rm/lm are quite different for the chosen fe. 

Table 10.2 Shape invariant and aspect ratio of domains grown under FDB effect in different FE materials...

Figure 5.16 A direction-independent SGIM characterization of a space curve C, regarded as a molecular backbone. On the left-hand side the shape globe S of radius R is shown enclosing the space curve C. The centre of the sphere is chosen as the centre of mass of chain molecule C. On the right-hand side the shape invariance domains of the sphere are shown, as defined by the knot types derived from the projections. There are only two knot types in this example unknots and trefoil knots.

A standard set of solvable potentials with critical behavior can be found in many text books on quantum mechanics [49,50], like the usual square-well potentials and other piecewise constant potentials. Also there are many potentials that are solvable only at d - 1 or for three-dimensional, v waves like the Hulthen potential, the Eckart potential, and the Posch-Teller potential. These potentials belong to a class of potentials, called shape-invariant potentials, that are exactly solvable using supersymmetric quantum mechanics [51,52], There are also many approaches to make isospectral deformation of these potentials [51,53] therefore it is possible to construct nonsymmetrical potentials with the same critical behavior as that of the original symmetric problem. 

Local Shape Invariance of Density Domains and the Transfer... 

Ranges DD Shape Invariance Domains of the Configuration Space... 

These domains represent a partitioning of M, hence the union of these DD shape invariance domains is the entire nuclear configuration space M,... 

Having determined the necessary pulse duration, the transmitter power must be calibrated so that the pulse delivers the appropriate tip angle. This procedure differs from that for hard pulses where one uses a fixed pulse amplitude but varies its duration. For practical convenience, amplitude calibration is usually based on previously recorded calibrations for a soft rectangular pulse (as described below), from which an estimate of the required power change is calculated. Table 9.3 also summarises the necessary changes in transmitter attenuation for various envelopes of equivalent duration, with the more elaborate pulse shapes invariably requiring increased rf peak amplitudes (decreased attenuation of transmitter output). 

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