## The Bilinear Transform

The formula for a general first-order (bilinear) conformal mapping of functions of a complex variable is conveniently expressed by [42, page 75]

It can be seen that choosing three specific points and their images determines the mapping for all 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  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 (E.2) leaves us with transformations of the form

 (E.1)

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. Note that this class of circle to circle'' bilinear transformations takes the form of the transfer function of an allpass filter. We therefore call it an allpass transformation''. The allpass coefficient'' can be written in terms of the frequencies and as

 (E.2)

In this form, it is clear that is real, and that the inverse of is . Also, since , and for an audio warping (where low frequencies must be stretched out'' relative to high frequencies), we have for audio-type mappings from the plane to the plane.

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Optimal Bilinear Bark Warping
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