Winograd ' s Short DFT Algorithms ∗

In 1976, S. Winograd [20] presented a new DFT algorithm which had signi cantly fewer multiplications than the Cooley-Tukey FFT which had been published eleven years earlier. This new Winograd Fourier Transform Algorithm (WFTA) is based on the typeone index map from Multidimensional Index Mapping with each of the relatively prime length short DFT's calculated by very e cient special algorithms. It is these short algorithms that this section will develop. They use the index permutation of Rader described in the another module to convert the prime length short DFT's into cyclic convolutions. Winograd developed a method for calculating digital convolution with the minimum number of multiplications. These optimal algorithms are based on the polynomial residue reduction techniques of Polynomial Description of Signals: Equation 1 to break the convolution into multiple small ones [2], [12], [14], [23], [21], [9]. The operation of discrete convolution de ned by