Piecewise Conical Acoustic Tube Modeling

Simple Example: Cylinder with Conical Cap

where

$$\begin{eqnarray*} R(z) &=& \left(\frac{1}{99}\right)\left(\frac{1+z^{-1}}{1-\frac{101}{99}z^{-1}}\right) \\ T(z) &=& \left(\frac{100}{99}\right)\left(\frac{1-z^{-1}}{1-\frac{101}{99}z^{-1}}\right) = 1+R(z) \end{eqnarray*}$$

• Problem: Reflection filter $R(z)$ and
  transmission filter $T(z)$ are unstable (pole at $z=101/99$ )
• Overall system is passive $\Rightarrow$ unstable pole is canceled