2.4 Digital-to-digital frequency transformations

Given a prototype digital filter design, transformations similar to the bilinear transform can also be developed.

Requirements on such a mapping $z^{-1}=g(z^{-1})$ :

• points inside the unit circle stay inside the unit circle (condition to preserve stability)
• unit circle is mapped to itself (preserves frequency response)

This condition implies that $e^{-(i{\omega }_1)}=g(e^{-(i\omega )})=\left|g(\omega )\right|e^{i\mathop{\arg }(g(\omega ))}$ requires that $\left|g(e^{-(i\omega )})\right|=1$ on the unit circle!

Thus we require an all-pass transformation: $$g(z^{-1})=\prod_{k=1}^p \frac{z^{-1}-{\alpha }_k}{1-{\alpha }_{k}z^{-1}}$$ where $\left|{\alpha }_K\right|< 1$ , which is required to satisfy this condition .

(Interesting and occasionally useful!)