This report presents a derivation of the modified discrete Fourier transform, which has the property that the origin in frequency space appears in the center of the plot rather than at the edges (one dimension) or at the four corners (two dimensions), as in conventional treatments. Also included is a listing of an unusual fast Fourier transform (FFT) program for calculating the twodimensional, modified discrete Fourier transform. The program makes use of the Eklundh fast matrix transposition

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... orithm and can transform arrays that are much too large to fit within the internal memory of the computer. By way of example, a complex array of size 2048 X 2048 can easily be transformed on a microcomputer with a 40-MB hard disk. in ACKNOWLEDGMENTS We thank Kent Edwards and Charles Meins for their helpful suggestions and comments and especially for their unfailing support during the course of this work. We also thank Marion Becker for her precise and careful preparation of this manuscript.