By Dawson C.N., Martinez-Canales M.L.

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**Extra info for Acharacteristic-Galerkin Aproximation to a system of Shallow Water Equations**

**Example text**

Furthermore, the natural logarithm is monotonic in its argument so that distance and density are inversely monotonically related in the sense that p(x1 | ωi ) < p(x2 | ωi ) ⇐⇒ D(x1 , ωi ) > D(x2 , ωi ). 116) The inverse variation of the Mahalanobis distance and the normal density function extends to arbitrary probability densities and metrics. The general relationship is elucidated by the theory of non-parametric density estimation a central result of which is the following theorem due to Fix and Hodges [241], stated here without proof.

Since all feature sets of the signal are derived from time–frequency distributions of its energy, it follows that there is some ambiguity in any such representation of the speech signal. The following analysis serves to explain and quantify the effects of non-stationarity. 1 we deﬁned the spectrogram as a method for representing the time variation of energy in the speech signal. 39) in continuous time, the spectrogram Sx (ω, t), of the signal, x(t), observed through the window g(τ ), becomes ∞ Sx (ω, t) = −∞ 2 g(τ )x(t + τ )e−j ωτ dτ .

The following analysis serves to explain and quantify the effects of non-stationarity. 1 we deﬁned the spectrogram as a method for representing the time variation of energy in the speech signal. 39) in continuous time, the spectrogram Sx (ω, t), of the signal, x(t), observed through the window g(τ ), becomes ∞ Sx (ω, t) = −∞ 2 g(τ )x(t + τ )e−j ωτ dτ . 63) It is natural to ask whether or not we can ﬁnd a better time–frequency representation in some well-deﬁned sense. In particular, we seek another representation, Fx (ω, t), that will give better estimates in both time and frequency of the time variation of the spectrum due to non-stationarity.