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## The Bilinear Transform

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

 (2)

It can be seen that choosing three specific points and their images determines the mapping for all z and .

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)

The constant provides one remaining degree of freedom which can be used to map any particular frequency (corresponding to the point on the unit circle) to a new location . All other frequencies will be warped accordingly. The allpass coefficient can be written in terms of these frequencies as
 (4)

In this form, it is clear that is real and that the inverse of is . Also, since , and for a Bark map, we have for a Bark map from the z plane to the plane.

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