The single-tube model

This is a reasonable approximation of the vocal tract when producing the schwa vowel. If Ai = A2, then the refection coefficient ri = 0 and hence Equation (11.19) simplifies to 

Recall that D is the length of the vocal tract in units of the normalised distance that we used to simplify the equation for discrete time analysis. By use of Equation (11.9), we can determine a value for D from empirically measured values. For instance, if we set the length of the vocal tract as 0.17 m (the average for a man) and the speed of soimd in air as 340 m s , and choose a sampling rate of 10 000 Hz, Equation (11.9) gives the value of D as 5. Hence the transfer fimction becomes 

The numerator of the transfer function tells us that the output is delayed by a factor that is a function of the length of the tube. All these zeros lie at 0, and can be ignored for purposes of determining the frequency response of the transfer function. The poles are evenly spaced on the unit circle at intervals of n/5. The first is at tt/10 = 0.314, which, when converted to a real frequency value with Equation (10.29), gives 500 Hz. Subsequent resonances occur every 1000 Hz after that, i.e. at 1500 Hz, 2500 Hz and so on. 

Recall that the losses in the system come from the wave exiting the vocal tract at the glottis and lips, and the degree to which this occurs is controlled by the special reflection coefficients and vq. Let us first note that, since these are multiplied, the model is not affected by where the losses actually occur, and in fact in some formulations the losses are taken to occur only at the glottis or only at the lips. Secondly, recall from Section 11.3.3 that the special case of complete closme at the glottis would produce a reflection coefficient of 1, whereas the special condition of complete openness at the lips would produce a reflection coefficient of -1. This is a situation in which there are no losses at all in the system (which is unrealistic of comse, for the simple reason that no sound would escape and hence a listener would never hear the speech). For these special cases file product nj-Q is 1, which means fiiat the denominator term is 

It can be shown that, when this is factorised, all the poles are spaced as before, but lie on the unit circle. This is of course exactly what we would expect from a lossless case. 

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