The multi-tube vocal-tract model

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 connected together. 

D] = 1/2. Importantly, for the given sampling rate, no extra accuracy is achieved by increasing the number of tubes beyond this value. 

The principle behind finding the iV-tube transfer function is the same as for the two-tube model - start at the lips and apply Equations (11.18a) and (11.18b) for each tube in turn. This is tedious to perform in the manner used for the two-tube system, but we can use matrix multiplication to speed up the process. 

We can use the special reflection coefficients at the lips and glottis to model the terminating boundary conditions as before. In the matrix formulation, this gives 

It is possible to simplify the calculation by making the further assiunption that the glottis is a completely closed end, with infinite impedance and reflection coefficient 1. If we take 

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