Speech production can be modeled by a linear system representing the vocal cavity, which...

Speech production can be modeled by a linear system representing the vocal cavity, which is excited by puffs of air released through the vibrating vocal cords. One approach to synthesizing speech involves representing the vocal cavity as a connection of cylindrical acoustic tubes of equal length, but with varying cross-sectional areas, as depicted in Figure P6.53. Let us assume that we want to simulate this system in terms of the volume velocity representing airflow. The input is coupled into the vocal tract through a small constriction, the vocal cords. We will assume that the input is represented by a change in volume velocity at the left end, but that the boundary condition for traveling waves at the left end is that the net volume velocity must be zero. This is analogous to an electrical transmission line driven by a current source at one end and with an open circuit at the far end. Current in the transmission line is then analogous to volume velocity in the acoustic tube, whereas voltage is analogous to acoustic pressure. The output of the acoustic tube is the volume velocity at the right end. We assume that each section is a lossless acoustic transmission line.

At each interface between sections, a forward-traveling wave f⁺ is transmitted to the next section with one coefficient and reflected as a backward-traveling wave f⁻ with a different coefficient. Similarly, a backward-traveling wave f⁻ arriving at an interface is transmitted with one coefficient and reflected with a different coefficient. Specifically, if we consider a forward traveling wave f⁺ in a tube with cross-sectional area A1 arriving at the interface with a tube of cross-sectional area A2, then the forward-traveling wave transmitted is (1 + r)f⁺ and the reflected wave is rf⁺, where

Consider the length of each section to be 3.4 cm, with the velocity of sound in air equal to 34,000 cm/s. Draw a flow graph that will implement the four-section model in Figure P6.53, with the output sampled at 20,000 samples/s.

In spite of the lengthy introduction, this a reasonably straightforward problem. If you find it difficult to think in terms of acoustic tubes, think in terms of transmission-line sections with different characteristic impedances. Just as with transmission lines, it is difficult to express the impulse response in closed form. Therefore, draw the flow graph directly from physical considerations, in terms of forward- and backward-traveling pulses in each section.