Fast Fourier transform algorithms rely upon the choice of certain bijective mappings between the indices of the data arrays. The two basic mappings used in the literature lead to Cooley-Tukey algorithms or to prime fa...
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Fast Fourier transform algorithms rely upon the choice of certain bijective mappings between the indices of the data arrays. The two basic mappings used in the literature lead to Cooley-Tukey algorithms or to prime factor algorithms. But many other bijections also lead to FFT algorithms, and a complete classification of these mappings is provided. One particular choice leads to a new FFT algorithm that generalizes the prime factor algorithm. It has the advantage of reducing the floating point operation count by reducing the number of trigonometric function evaluations. A certain equivalence relation is defined on the set of bijections that lead to FFT algorithms, and its connection with isomorphism classes of group extensions is studied. Under this equivalence relation every equivalence class contains bijections leading to an FFT algorithm of the new type.
In this brief contribution, an efficient pipeline architecture is proposed for the realization of the prime factor algorithm (PFA) for digital signal processing. By using the extended diagonal feature of the Chinese R...
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In this brief contribution, an efficient pipeline architecture is proposed for the realization of the prime factor algorithm (PFA) for digital signal processing. By using the extended diagonal feature of the Chinese Remainder Theorem (CRT) mapping, we show that the input data sequence tan be directly loaded into a multidimensional array for the PFA computation without any permutation. Short length modules are modified such that an in-place and in-order computation is allowed. The computed results can then he directly restored back to the memory array without the need for further reordering. More importantly, the CRT mapping ran also be used to represent the output data, hence we can utilize the extended diagonal feature of the CRT mapping to directly send the computed results to the outside world. As compared to the previous approaches, the present approach requires no shifting or rotation during the data loading and retrieval processes. In the case of multidimensional PFA computation, it does not require the computation to be split up into a number of two-dimensional computations. Hence, the overhead required for data loading and retrieval in each two-dimensional stage can be saved.
A prime factor algorithm for computing the discrete Hartley transform (DHT) is presented. It is shown that the short length DHTs used by the prime factor algorithm can be nested to lead to the Winograd Hartley transfo...
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A prime factor algorithm for computing the discrete Hartley transform (DHT) is presented. It is shown that the short length DHTs used by the prime factor algorithm can be nested to lead to the Winograd Hartley transform algorithm.
An efficient algorithm for computing the discrete cosine transform (DCT) is presented. It is based on an index mapping which converts an odd-length DCT to a real-valued DFT of the same length using permutations and si...
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An efficient algorithm for computing the discrete cosine transform (DCT) is presented. It is based on an index mapping which converts an odd-length DCT to a real-valued DFT of the same length using permutations and sign changes only. The real-valued DFT can then be computed by efficient real-valued FFT algorithms such as the prime factor algorithm. The algorithm is more efficient than an earlier one because no postmultiplications are required.
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