Lifting-based Alternatives to the FFT for Computing the 4-, 8-, and 16-point Discrete Fourier Transforms

This paper presents a novel approach to efficiently compute the 4-, 8-, and 16-point variants of the Discrete Fourier Transform (DFT). Our solution requires as many (non-trivial) multiplications and additions of real numbers as fast algorithms of the $\mathbf{FFT}$ family. However, we saved operations in a completely different way. Instead of factorizing the complex matrix that describes the transform, we consider separately the real and imaginary parts of the DFT matrix. Both parts are rank-deficient matrices, which contain pairs of linearly dependent rows and columns, but are structured and symmetric. We factorize each of them into a product of sparse matrices by applying shear transforms to rows and columns, similarly as in two-side diagonalization. The obtainable factorizations describe data flow graphs in which non-trivial multiplications are well grouped, as they occur in the midst of sequences of lifting steps related to trivial multiplications by powers of 2, which can be implemented by bit-shifting. Such schemes could be used to implement the DFT in hardware, as they offer alternative possibilities of modularization, pipelining, and folding of digital circuits.