Derivation of Diagonalized Identical Equations for 3 by 3 Circulant and Quasi-Circulant Matrices and Their Application to Winograd 7-Point FFT

The derivation of the 7-point Winograd fast Fourier transform (FFT) requires complex steps such as using the Rader prime algorithm to turn an N-point discrete Fourier transform (DFT) into an (N − 1)-point convolution and then using the Chinese remainder theorem for polynomials to find the set of remainders. In this paper, we describe a simpler method for deriving the 7-point Winograd FFT using the diagonalized identical equations of 3 by 3 circulant and quasi-circulant matrices. These diagonalized identical equations of 3 by 3 matrices are not found in the literature and are newly derived.