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Necessary and sufficient conditions for a direct sum of local rings to support a generalized discrete Fourier transform are derived. In particular, these conditions can be applied to any finite ring. The function O(N) defined by Agarwal and Burrus for transforms over ZN is extended to any finite ring R as O(R) and it is shown that R supports a length m discrete Fourier transform if and only if m is a divisor of O(R) This result is applied to the homomorphic images of rings-of algebraic integers.
number theoretic transforms, Digital filtering, fast convolution, FFT, finite computation structures, generalized discrete Fourier transform, modular arithmetic
E. Dubois, A.N. Venetsanopoulos, "The Discrete Fourier Transform Over Finite Rings with Application to Fast Convolution", IEEE Transactions on Computers, vol. 27, no. , pp. 586-593, July 1978, doi:10.1109/TC.1978.1675158
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