## Creating your own vowel resonators using tubing

Theory:

When modeling speech production for applications such as speech processing we have learned that we can approximate the vocal tract as a excitation and a filtering component. The excitation corresponds to the vocal cords being vibrated by air pushed by the diaphragm. This is analogous to the plucking of a guitar string or the vibration of a reed. The excitation is then filtered by the vocal cavity. The shape of this cavity corresponds to the phoneme created. The different shapes of the vocal tract increase certain harmonics in the original excitation and decrease others to produce different voiced vowel sounds such as "ah" and "ih". The resonant frequencies that are amplified by the vocal tract are called formants and specific formants correspond to the different phonemes. We can therefore model the vocal tract as a series of uniform tubes that filter the signal.

In order to find the transfer function of these tubes we can utilize the z transform and some matrix algebra. Each tube delays the airflow signal that passes through it. This is equivalent to a time delay and by taking the z transform and putting the equations in matrix form we can obtain the transfer function of the time delay. In matrix form the equation for time delay can be shown below:

$\begin{pmatrix}R_d(z) \\L_d(z)\end{pmatrix} = z\begin{pmatrix}1 & 0 \\0 & z^-2\end{pmatrix}\begin{pmatrix}\overline{R_d(z)}\\\overline{L_d(z)}\end{pmatrix}$

Where $R_d(z)$ is the z transform of the air moving right at the entrance to the tube and $L_d(z)$ is the z transform of the air moving left at the entrance to the tube. $\overline{R_d(z)}$ and $\overline{L_d(z)}$ are the z tranforms of the air moving right and left at the exit of the tube.

When air flows between tube junctions the transfer function can also be obtained. Using the principles of flow continuity and pressure continuity the equation of the transfer function of a tube junction is shown below:

$\begin{pmatrix}R_d(z) \\L_d(z)\end{pmatrix} = \dfrac{1}{1+r}\begin{pmatrix}1 & -r \\-r & 1\end{pmatrix}\begin{pmatrix}\overline{R_d(z)} \\\overline{L_d(z)}\end{pmatrix}$

Where $r<\math> is the reflection coefficient which is a ratio of the cross sectional area of the tubing given by the equation [itex]r=\dfrac{B-A}{B+A}<\math> Multiplying these equations for time delay and tube junctions together gives us an equation for the transfer function of an approximation of the vocal tract.$