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Split-radix FFT algorithm
About: Split-radix FFT algorithm is a research topic. Over the lifetime, 1845 publications have been published within this topic receiving 41398 citations.
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TL;DR: This tutorial simply reviews the DFT and FFT, with a few characteristic examples.
Abstract: Frequency analysis is an important issue in the IEEE. Using a computer in a calculation means moving into a non-physical, synthetic environment. Numerically, discrete or fast Fourier transformations (DFTs or FFTs) are used to obtain the frequency content of a time signal, and these are totally different than the mathematical definition of the Fourier transform. This tutorial simply reviews the DFT and FFT, with a few characteristic examples.
44 citations
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TL;DR: A fast and quasi-optimal algorithm for computing the NUDFT based on the fast Fourier transform (FFT) is proposed, which is essentially the FFT, and is competitive with state-of-the-art algorithms.
Abstract: By viewing the nonuniform discrete Fourier transform (NUDFT) as a perturbed version of a uniform discrete Fourier transform, we propose a fast and quasi-optimal algorithm for computing the NUDFT based on the fast Fourier transform (FFT). Our key observation is that an NUDFT and DFT matrix divided entry by entry is often well approximated by a low rank matrix, allowing us to express a NUDFT matrix as a sum of diagonally scaled DFT matrices. Our algorithm is simple to implement, automatically adapts to any working precision, and is competitive with state-of-the-art algorithms. In the fully uniform case, our algorithm is essentially the FFT. We also describe quasi-optimal algorithms for the inverse NUDFT and two-dimensional NUDFTs.
44 citations
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TL;DR: The fast Fourier transform (FFT) algorithm of this transform is faster than the conventional radix-2 FFT and is used to filter a two-dimensional picture, and the results are presented with a comparison to the standard FFT.
Abstract: A transform analogous to the discrete Fourier transform is defined on the Galois field GF(p), where p is a prime of the form k X 2n + 1, where k and n are integers. Such transforms offer a substantial variety of possible transform lengths and dynamic ranges. The fast Fourier transform (FFT) algorithm of this transform is faster than the conventional radix-2 FFT. A transform of this type is used to filter a two-dimensional picture (e.g., 256 X 256 samples), and the results are presented with a comparison to the standard FFT. An absence of roundoff errors is an important feature of this technique.
44 citations
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TL;DR: A new algorithm for reducing an arbitrary unitary matrix U into a sequence of elementary operations that can be used to manipulate an array of quantum bits and shows that the Fast Fourier Transform (FFT) algorithm is a special case of this algorithm.
Abstract: We present a new algorithm for reducing an arbitrary unitary matrix U into a sequence of elementary operations (operations such as controlled-nots and qubit rotations). Such a sequence of operations can be used to manipulate an array of quantum bits (i.e., a quantum computer). Our algorithm applies recursively a mathematical technique called the CS Decomposition to build a binary tree of matrices whose product, in some order, equals the original matrix U. We show that the Fast Fourier Transform (FFT) algorithm is a special case of our algorithm. We report on a C++ program called “Qubiter” that implements the ideas of this paper. Qubiter(PATENT PENDING) source code is publicly available.
44 citations
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TL;DR: An algorithm that reduces the computational effort to two-thirds of the effort required by most radix-2 algorithms and its structure is particularly appealing when transforming pure real or imaginary sequences and/or symmetric or antisymmetric sequences.
Abstract: This paper develops and presents a radix-2 fast Fourier transform (FFT) algorithm that reduces the computational effort (as measured by the number of multiplications) to two-thirds of the effort required by most radix-2 algorithms. The resulting algorithm is similar to one obtained by applying a principle suggested by Rader and Brenner; however, its structure is particularly appealing when transforming pure real or imaginary sequences and/or symmetric or antisymmetric sequences; furthermore, memory requirements (other than those for storing the input data) do not grow with the size of the transform.
43 citations