A New Simple Algorithm for Deriving the Winograd 9-Point FFT by Using New Identical Equations for 3 × 3 Circulant and Quasi-Circulant Matrices

The Winograd small fast Fourier transform (FFT) is a method of efficiently computing the discrete Fourier transform (DFT) for data of small block length. The equations of post-additions, constant multiplication factors, and pre-additions for the Winograd 9-point FFT are given in references [3], [5], [6]. A 6 × 6 block matrix is obtained from 9-point DFT matrix by matrix manipulation. By using the 6 × 6 block matrix, 3 × 3 circular and quasi-circular matrices can be derived. New identical equations for 3 × 3 circular and quasi-circular matrices have been derived by the authors. A new simple algorithm is given for the Winograd 9-point FFT correctly by using new identical equations for 3 × 3 circular and quasi-circular matrices.