The vowel-tube model

Recall that we described the overall process of vowel production as one whereby a signal from the glottis t/[n] travels through and is modified by the vocal tract and lips to produce the speech waveform y n. In the z-domain, this is represented by Equation (11.2)  

After this, we can determine the final speech signal 7(z) by applying the radiation characteristic R(z) to Ul(z)- Rather than attempt to find F(z) from the tube properties directly, we will use the tube model to find U z) in terms of Ug z), and divide the former by the latter to find F(z). 

---

A special case of the two tube model is the single tube or uniform tube model where the cross sectional area is constant along the entire length of the tube. This is a reasonable approximation of the vocal tract when producing the schwa vowel. If.4i =.<42, then the refection coefficient ri = 0 and hence Equation 11.19 simplifies to... 

In Section 12.12, we showed that the lossless tube model was a reasonable approximation for the vocal tract during the production of a vowel. If we assume for now that H z) can therefore be represented by an all-pole filter, we can write... 

We will now develop a general expression for the transfer function of a tube model for vowels, in which we have an arbitrary number of tubes coimected together. 

All pole modelling Only vowel and approximant soimds can be modelled with complete accuracy by all-pole transfer functions. We will see in Chapter 12 that the decision on whether to include zeros in the model really depends on the application to which the model is put, and mainly concerns tradeoffs between accuracy and computational tractability. Zeros in transfer functions can in many cases be modelled by the addition of extra poles. The poles can provide a basic model of anti-resonances, but can not model zero effects exactly. The use of poles to model zeros is often justified because the ear is most sensitive to the peak regions in the spectrum (naturally modelled by poles) and less sensitive to the anti-resonance regions. Hence using just poles can often generate the required spectral envelope. One problem however is that as poles are used for purposes other than their natural one (to model resonances) they become harder to interpret physically, and have knock on effects in say determining the number of tubes required, as explained above. 

Further refinement can be achieved by the use of zeros. These can be used to create antiresonances, corresponding to a notch in the frequency response. Here the format synthesis model again deviates from the all-pole tube model, but recall that we only adopted the all-pole model to make the derivation of the tube model easier. While the all-pole model has been shown to be perfectly adequate for vowel sounds, the quality of nasal and fricative sounds can be improved by the use of some additional zeros. In particular, it has be shown [254] that the use of a single zero anti-resonator in series with a the normal resonators can produce realistic nasal sounds. 

The effect of a sound source in the middle of the vocal tract is to split the source such that some sound travels backwards towards the glottis while the remainder travels forwards towards the lips. The vocal tract is thus effectively split into a backward and forward cavity. The forward cavity acts a tube resonator, similar to the case of vowels but with fewer poles as the cavity is considerably shorter. The backwards cavity also acts as a further resonator. The backwards travelling source will be reflected by the changes in cross sectional area in the back cavity and at the glottis, creating a forward travelling wave which will pass through the constriction. Hence the back cavity has an important role in the determination of the eventual sound. This back cavity acts as a side resonator, just as with the oral cavity in the case of nasals. The effect is to trap sound and create antiresonances. Hence the back cavity should be modelled with zeros as well as poles in its transfer function. 

---