Two tube vocal tract model

Our goal is to find the z-domain transfer function of the system, and we will find this by dividing the z-domain expressions for output at the lips by the input at the glottis. The best way to proceed is to define the volume velocity at the lips, use this to find the volume velocity at the junction between the two proper tubes, and use this to find the volume velocity at the glottis. To do this we use Equations 11.16b and 11.16b which relate the volume velocity in one tube to the 

We start by defining Ul z) as the output at the lips. We now feed this into Equations 11.17b and 11.17b and use the fact that there is no backwards wave (L(r = 0) to set the volume velocities at junction k = 2 as  

By feeding these values into Equation 11.17b again, we can calculate the volume velocities and 17f at the junction k=.  

We have now reached the glottis, modelled by a special tube of length 0 and reflection coefficient vq. We take the length of this tube to be 0, and as z = 1 then 

To find Uq we simply repeat the above procedure again. Note that we don t need to know Uq as we are assuming this gets fiilly absorbed into the lungs. 

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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... 

An alternative to using formants as the primary means of control is to use the parameters of the vocal tract transfer function directly. The key here is that if we assume the all-pole tube model, we can in fact determine these parameters automatically by means of linear prediction, performed by the covariance or autocorrelation technique described in Chapter 12. In the following section we will explain in detail the commonality between linear prediction and formant synthesis, where the two techniques diverge, and how linear prediction can be used to generate speech. 

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