The two-tube vocal-tract model

Here we will consider a model in which we have two connected tubes of different cross-sectional areas. In this formulation, the tubes have areas A and A2, and normalised lengths Di and Dx, such that D + Dx = D, the total normalised length of the vocal 

Our goal is to find the z-domain transfer function of the system, which we will do 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 (11.16a) and (11.16b), which relate the volume velocity in one tube to the volume velocity in the next. Since our end point is a z-domain expression, we will make repeated use of the z-transforms of Equations (11.16a) and (11.16b), which are 

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

We have now reached the glottis, which is modelled by a special tube of length 0 and reflection coefficient tq. We take the length of this tube to be 0, thus, since z = 1, 

---