Topic
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.
Papers published on a yearly basis
Papers
More filters
17 Jan 2012
TL;DR: This work considers the sparse Fourier transform problem, and proposes a new algorithm, which leverages techniques from digital signal processing, notably Gaussian and Dolph-Chebyshev filters, and is faster than FFT, both in theory and practice.
Abstract: We consider the sparse Fourier transform problem: given a complex vector x of length n, and a parameter k, estimate the k largest (in magnitude) coefficients of the Fourier transform of x. The problem is of key interest in several areas, including signal processing, audio/image/video compression, and learning theory.We propose a new algorithm for this problem. The algorithm leverages techniques from digital signal processing, notably Gaussian and Dolph-Chebyshev filters. Unlike the typical approach to this problem, our algorithm is not iterative. That is, instead of estimating "large" coefficients, subtracting them and recursing on the reminder, it identifies and estimates the k largest coefficients in "one shot", in a manner akin to sketching/streaming algorithms. The resulting algorithm is structurally simpler than its predecessors. As a consequence, we are able to extend considerably the range of sparsity, k, for which the algorithm is faster than FFT, both in theory and practice.
342 citations
TL;DR: The discrete cosine transform (DCT) of an N-point real signal is derived by taking the discrete Fourier transform (DFT) of a 2N-point even extension of the signal and the method is extended to two dimensions, with a saving of 1/4 over the traditional method that uses the DFT.
Abstract: The discrete cosine transform (DCT) of an N-point real signal is derived by taking the discrete Fourier transform (DFT) of a 2N-point even extension of the signal. It is shown that the same result may be obtained using only an N-point DFT of a reordered version of the original signal, with a resulting saving of 1/2. If the fast Fourier transform (FFT) is used to compute the DFT, the result is a fast cosine transform (FCT) that can be computed using on the order of N \log_{2} N real multiplications. The method is then extended to two dimensions, with a saving of 1/4 over the traditional method that uses the DFT.
334 citations
TL;DR: Two recently developed ideas, the conversion of a discrete Fourier transform to convolution and the implementation of short convolutions with a minimum of multiplications, are combined to give efficient algorithms for long transforms.
Abstract: Two recently developed ideas, the conversion of a discrete Fourier transform (DFT) to convolution and the implementation of short convolutions with a minimum of multiplications, are combined to give efficient algorithms for long transforms Three transform algorithms are compared in terms of the number of multiplications and additions Timing for a prime factor fast Fourier transform (FFT) algorithm using high-speed convolution, which was programmed for an IBM 370 and an 8080 microprocessor, is presented
331 citations
TL;DR: The CORDIC iteration is applied to several Fourier transform algorithms and a new, especially attractive FFT computer architecture is presented as an example of the utility of this technique.
Abstract: The CORDIC iteration is applied to several Fourier transform algorithms. The number of operations is found as a function of transform method and radix representation. Using these representations, several hardware configurations are examined for cost, speed, and complexity tradeoffs. A new, especially attractive FFT computer architecture is presented as an example of the utility of this technique. Compensated and modified CORDIC algorithms are also developed.
304 citations
TL;DR: A complete set of fast algorithms for computing the discrete Hartley transform is developed, including decimation-in-frequency, radix-4, split radix, prime factor, and Winograd transform algorithms.
Abstract: The discrete Hartley transform (DHT) is a real-valued transform closely related to the DFT of a real-valued sequence. Bracewell has recently demonstrated a radix-2 decimation-in-time fast Hartley transform (FHT) algorithm. In this paper a complete set of fast algorithms for computing the DHT is developed, including decimation-in-frequency, radix-4, split radix, prime factor, and Winograd transform algorithms. The philosophies of all common FFT algorithms are shown to be equally applicable to the computation of the DHT, and the FHT algorithms closely resemble their FFT counterparts. The operation counts for the FHT algorithms are determined and compared to the counts for corresponding real-valued FFT algorithms. The FHT algorithms are shown to always require the same number of multiplications, the same storage, and a few more additions than the real-valued FFT algorithms. Even though computation of the FHT takes more operations, in some situations the inherently real-valued nature of the discrete Hartley transform may justify this extra cost.
275 citations