Relation to functions positive real in the right-half plane

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Relation to functions positive real in the right-half plane

Theorem. re $\left\{H(z)\right\}\geq 0$ for $\left\vert z\right\vert\geq 1$ whenever

re $\displaystyle \left\{H\left(\frac{\alpha+s}{ \alpha-s}\right)\right\}\geq 0$

for re $\left\{s\right\}\geq 0$ , where $\alpha$ is any positive real number.

Proof. We shall show that the change of variable $z\leftarrow (\alpha+s)/(\alpha-s),\; \alpha>0$ , provides a conformal map from the z-plane to the s-plane that takes the region $\left\vert z\right\vert\geq 1$ to the region re $\left\{s\right\}\geq 0$ . The general formula for a bilinear conformal mapping of functions of a complex variable is given by

$\displaystyle \frac{(z-z_1)(z_2-z_3)}{ (z-z_3)(z_2-z_1)} = \frac{(s-s_1)(s_2-s_3)}{ (s-s_3)(s_2-s_1)}.$

(M.9)

In general, a bilinear transformation maps circles and lines into circles and lines [85]. We see that the choice of three specific points and their images determines the mapping for all and . We must have that the imaginary axis in the s-plane maps to the unit circle in the z-plane. That is, we may determine the mapping by three points of the form $z_i=e^{j\theta_i}$ and $s_i=j\omega_i,\;i=1,2,3$ . If we predispose one such mapping by choosing the pairs $(s_1=\pm\infty ) \leftrightarrow (z_1= -1)$ and $(s_3=0)\leftrightarrow(z_3=1)$ , then we are left with transformations of the form

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Physical Audio Signal Processing
    Passive Impedances
       Positive Real Functions
          Relation to functions positive real in the right-half plane