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Taking into account the definition of the generalized Dirac function [mathematical expression not reproducible], the factor [mathematical expression not reproducible] in the singular component of the second derivative of Green's tensor leads to a transition from an integral to a constant value; that is, the integral operator [mathematical expression not reproducible] becomes a constant tensor:
Figure 3 shows a peak at 275.4 eV, also associated with [p.sub.[pi]] states, that has already been reported by the method of the second derivative [6,7,10].
We have found that for small values of the normalized time offset [epsilon] = 0.05, [epsilon] = 0.1, and [epsilon] = 0.2, the pulses reported in ascending order of performance with respect to error probability are also placed in ascending order of fractional energy [E.sub.s], descending order of the absolute value of the slope [s.sub.k] of the impulse response evaluated at t/T = 1, and ascending order of the second derivative of the impulse response [a.sub.k] evaluated at t/T = 1.
To construct methods with high orders and satisfactory stability properties, methods were introduced where use second derivatives of the solution [16,18, 21,27].
MFCC along with its first and second derivatives were not the only suitable features for the pronunciation training systems.
[D.sup.2][TE.sub.L] - Second derivative of the Transmission Error at Low load
For example scenario 12 is a steady state situation: all first and second derivatives are zeros.
The FCN amplitude and phase variations with their first and second derivatives are presented in Figure 2.
In order to better identify the differences between spectra, they were vector-normalized followed by a second derivative transformation (Figure 3(b)).
If a point belongs to the feature point of ridge/valley, the first order derivative at this point along n(t) equals to 0, meanwhile, the second derivative at this point reaches to the extreme.
Figure 2 illustrates the spectrum without resampling (a) and with two (b) and four (c) instances of resampling digital audio signal's second derivative. The original sampling characteristics of the spectrum are smooth, and the spectrum peak of the resampling audio signal appears more than once.