## 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.

**Next Section:**

Optimal Bilinear Bark Warping

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The Bark Frequency Scale