2.2 Iir digital filter design via the bilinear transform

A bilinear transform maps an analog filter $H_a(s)$ to a discrete-time filter $H(z)$ of the same order.

If only we could somehow map these optimal analog filter designs to the digital world while preserving the magnitude response characteristics, we could make use of the already-existing body of knowledge concerning optimal analog filter design.

Bilinear transformation

The Bilinear Transform is a nonlinear $\to (\mathbb{C}, \mathbb{C})$ mapping that maps a function of the complex variable $s$ to a function of a complex variable $z$ . This map has the property that the LHP in $s$ ( $\Re (s)< 0$ ) maps to the interior of the unit circle in $z$ , and the $i\lambda =s$ axis maps to the unit circle $e^{i\omega }$ in $z$ .

Bilinear transform: $$s=\alpha \frac{z-1}{z+1}$$ $$H(z)=H_a(s=\alpha \frac{z-1}{z+1})$$

The magnitude response doesn't change in the mapping from $\lambda$ to $\omega$ , it is simply warped nonlinearly according to $H(\omega )=H_a(\alpha \tan \left(\frac{\omega }{2}\right))$ , .

Prewarping

Given specifications on the frequency response of an IIR filter to be designed, map these tospecifications in the analog frequency domain which are equivalent. Then a satisfactory analog prototype can bedesigned which, when transformed to discrete-time using the bilinear transformation, will meet the specifications.