The fast generalized discrete Fourier transforms: A unified approach to the discrete sinusoidal transforms computation

Abstract

Special forms of the generalized discrete Fourier transform (GDFT) matrices are investigated and their sparse matrix factorizations are presented to complete Wang's set of real sparse matrix factorizations for the family of discrete sinusoidal transforms. Different versions of the GDFT, different versions of the generalized discrete Hartley transform (GDHT) or equivalently of the discrete W transform (DWT), various versions of the discrete cosine transform (DCT) and discrete sine transform (DST) are members of the discrete sinusoidal transform family. There are intrinsic relationships among corresponding versions of the GDFT, GDHT (DWT), DCT and DST for real data sequences. A real sparse matrix factorization of GDFT matrices leads to simple fast algorithms for their computation, where only real arithmetic is involved. The resulting generalized signal flow graphs for the computation of different versions of the GDFT represent simple and compact unified approach to the fast discrete sinusoidal transforms computation. It is also shown that all algorithms are based on the universal DCT-II/DST-II (DCT-III/DST-III) computational structure which is used as the basic processing component.