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This survey paper is intended to integrate the subjects of digital signal processing and error control codes by studying their common dependence on the properties of the discrete Fourier transform. The two subjects are traditionally studied in different algebraic fields. Usually, the computations of digital signal processing are done using the complex number system, while the computations of error control codes are done using the arithmetic of Galois fields. We will argue that this dichotomy may be partly a historical accident. By viewing the two problems in the opposite number system, we shall find that there are parallels and that many techniques can be shared by the two subjects. The new material included within the paper is introduced in order to extend known techniques used in one algebraic field into another algebraic field where those techniques are not yet used.