1978
DOI: 10.1147/rd.222.0134
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Computation of Convolutions and Discrete Fourier Transforms by Polynomial Transforms

Abstract: Discrete transforms are introduced and are defined in a ring of polynomials. These polynomial transforms. are shown to have the convolution property and can be computed in ordinary arithmetic, without multiplications. Polynomial transforms are particularly well suited for computing discrete two-dimensional convolutions with a minimum number of operations. Efficient algorithms for computing one-dimensional convolutions and Discrete Fourier Transforms are then derived from polynomial transforms.

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Cited by 52 publications
(11 citation statements)
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“…The two most interesting approaches are certainly the vector radix FFT (a direct approach to the multi-dimensional problem in a Cooley-Tukey mood) proposed in 1975 by Rivard [91] and the polynomial transform solution of Nussbaumer and Quandalle in 1978 [87,88].…”
Section: Multi-dimensional Dftsmentioning
confidence: 99%
See 1 more Smart Citation
“…The two most interesting approaches are certainly the vector radix FFT (a direct approach to the multi-dimensional problem in a Cooley-Tukey mood) proposed in 1975 by Rivard [91] and the polynomial transform solution of Nussbaumer and Quandalle in 1978 [87,88].…”
Section: Multi-dimensional Dftsmentioning
confidence: 99%
“…Working in the field of polynomials resulted in a simplification of the multiplications by the root of unity, which was changed from a complex multiplication to a vector reordering. This powerful tutorial on fast Fourier transforms 291 approach was applied in [87,88] to the computation of 2-D DFTs as follows.…”
Section: Polynomial Transformmentioning
confidence: 99%
“…The total computation time required for performing a given task i~ directly proportional to the number of operations, and hence for the efficient implementation of a task on such architectures the computations should be reduced. Fast algorithms that reduce the computations by orders of magnitude proved to be of great importance [3]- [15].…”
Section: Introductionmentioning
confidence: 99%
“…This method was shown to provide an efficient means of recursive filtering operation especially when used in conjunction with fast transform techniques. Later, Mitra, and Gnanasekaran [3] have developed several other 1-D block structures which possess different properties with regard to roundoff error, sensitivity, etc.…”
Section: Introductionmentioning
confidence: 99%