Algorithm to retrieve real coefficients of a two-dimensional Fourier series using complex one-dimensional FFTs

Abstract

Complex notation is used almost exclusively when dealing with two-dimensional Fourier series. Fast Fourier transform algorithms efficiently compute the complex Fourier coefficients of these complex series. There are instances, however, where the real form of the series may be preferred or required. The coefficients of the real series are related in an implicit manner to the complex coefficients of the complex series. In this paper, an algorithm is developed to efficiently and accurately extract the Fourier coefficients of a real two-dimensional discrete Fourier series by utilizing the complex based one-dimensional fast Fourier transform. An illustrative example is presented for validation of the algorithm, and a FORTRAN program listing is provided.