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This paper considers the implementation of a fast Fourier transform (FFT) structure using arrays of read-only memories. The arithmetic operations are based entirely on the residue number system. The most important aspect of the structure relates to the scaling arrays, which are required to prevent overflow. Because of the limitations of the number system, scaling factors have to be chosen on an a priori basis. This paper develops optimum procedures for choosing both scaling factors and the position of scaling arrays in the structure. Some examples are presented relating to the filtering of speech via a convolutional filter structure.