Complexity Analysis of Two Permutations Used by Fast Cosine Transform Algorithms

The fast cosine transform algorithms introduced in [ST91, Ste92] require fewer operations than any other known general algorithm. Similar to related fast transform algorithms (e.g., the FFT), these algorithms permute the data before, during, or after the computation of the transform. The choice of this permutation may be an important consideration in reducing the complexity of the permutation algorithm. In this paper, we derive the complexity to generate the permutation mappings used in [ST91, Ste92] for power-of-2 data sets by representing them as linear index transformations and translating them into combinational circuits. Moreover, we show that the permutation used in [Ste92] not only allows e cient implementation, but is also self-invertible, i.e., we can use the same circuit to generate the permutation mapping for both the fast cosine transform and its inverse, like the bit-reversal permutation used by FFT algorithms. These results may be useful to designers of low-level algorithms for implementing fast cosine transforms.