Designing garbage-free reversible implementations of the integer cosine transform

Discrete linear transformations are important tools in information processing. Many such transforms are injective and therefore prime candidates for a physically reversible implementation into hardware. We present here reversible integer cosine transformations on n input integers. The resulting reversible circuit is able to perform both the forward transform and the inverse transform.

The detailed structure of such a reversible design strongly depends on the odd prime factors of the determinant of the transform: whether those are of the form 2k ± 1 or of the form 2k ± 2l ± 1 or neither of these forms.