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In addition, a sixth parameter, a6 , is used to additively alter the nominal 17-cm vocal tract length. The articulatory vector a is (a1 , a2 , . . , a6 ). The vocal tract model has three components: an oral cavity, a glottal source, and an acoustic impedance at the lips. We shall consider them singly ﬁrst and then in combination. As is commonly done, we assume that the behavior of the oral cavity is that of a lossless acoustic tube of slowly varying (in time and space) cross-sectional area, A(x), in which plane waves propagate in one dimension (see Fig.

3b) − and − respectively. In order to account for the losses we deﬁne Z(x, ω) and Y (x, ω) to be the generalized acoustic impedance and admittance per unit length, respectively. 4) which is recognized as the “lossy” Webster equation for the volume velocity. 4) in space and obtaining approximate solutions to the resulting difference equation for a sequence of frequencies. Let us write Uik to signify U (i x, k ω) where the spatial discretization assumes x = L/n with i = 0 at the glottis and i = n at the lips, as is shown in Fig.

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. 64) 2 will give perfect resolution of the FM chirp, xc (t) = ej (ω0 t+ 2 mt ) , at the expense of a discontinuous component, δ(ω)2 cos(2ω0 t). In comparison, Sx (ω, t) will give continuous but poor resolution of xi (t).