Schur Functions

A Schur function $S(z)$ is any stable discrete-time rational transfer function having a modulus not exceeding $1$ on the unit circle,

$$\left\vert S(e^{j\omega T})\right\vert\leq 1$$

The corresponding continuous-time transfer function $S_c(s)$ satisfies the same condition on the $j\omega$ axis, i.e.,

$$\left\vert S_c(j\omega)\right\vert\leq 1$$

The reflection transfer function (``reflectance'') corresponding to any positive-real immittance is a Schur function

• Reflectance for ``force'' waves (pressure, voltage) is
  $$S_f(z) = \frac{R(z) - R_0}{R(z) + R_0}$$
  where $R_0$ is the impedance of the adjacent wave-propagation medium
• Reflectance for ``velocity'' waves is
  $$S_v(z) = \frac{\Gamma(z) - \Gamma_0}{\Gamma(z) + \Gamma_0} = -S_f(z)$$

• Duality: $R\leftrightarrow \Gamma$, `$+$' $\leftrightarrow$ `$-$'
• Every passive reflectance is Schur
• Every lossless reflectance is an allpass filter
  (modulus 1 on the $j\omega$ axis)
• Every reactance has an allpass reflectance