The formula for a general first-order (bilinear) conformal mapping of
functions of a complex variable is conveniently expressed by
[3, page 75]

Bilinear transformations map circles and lines into circles and lines
(lines being viewed as circles passing through the point at infinity).
In digital audio, where both domains are ``*z* planes,'' we normally
want to map the unit circle to itself, with dc mapping to dc
(
) and half the sampling rate mapping to half the
sampling rate (
). Making these substitutions in
Eq.(2) leaves us with transformations of the form

(3) |

(4) |

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