Generalizing the Butterfly Structure of the FFT

Abstract

The structure of the various forms of the fast Fourier transform (FFT) is well described by patterns of “butterfly” operations, each involving only an individual pair of inputs or intermediate results, but ultimately yielding one of the most elegant, efficient, and ubiquitous computational algorithms known to mathematics. Here, the structure underlying the FFT is developed in a more general context of a decomposition applicable to any arbitrary complex unitary matrix. By developing this multi-layer decomposition (MLD), it becomes clear that:

• Any unitary matrix may be decomposed into a butterfly flow graph.

• Each individual butterfly may be interpreted as an equivalent 2 × 2 standard complex Givens rotation.

• Identification of the butterfly flow graph for any particular unitary matrix is intimately related to the singular value decomposition (SVD) of the different block sub-matrices of the original unitary matrix.

• The resulting butterfly flow graph may be interpreted as a prescription of the individual planar coordinate rotations necessary to precisely implement the general multi-dimensional coordinate rotation defined by the unitary matrix. It may also be interpreted as the equivalent polar representation for the matrix.

• The general form of the butterfly flow graph is not “fast.” The FFT achieves its computational efficiency due to certain specific mathematical symmetries, permitting removal of extraneous butterflies.

• The form of the butterfly flow graph is sensitive to the order in which the inputs and outputs are specified.

• A distinct advantage of the MLD is that any approximations or modifications made within this structure continue to result in a full unitary transformation.

An initial MATLAB code set implementing MLD is provided. The author believes the basic MLD concept to be a novel procedure which opens the possibility for subsequent exploitation in a number of different directions.