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Four-step FFT
The Bailey's FFT (also known as a 4-step FFT) is a high-performance algorithm for computing the fast Fourier transform (FFT). This variation of the Cooley–Tukey FFT algorithm was originally designed for systems with hierarchical memory common in modern computers (and was the first FFT algorithm in this so called "out of core" class). The algorithm treats the samples as a two dimensional matrix (thus yet another name, a matrix FFT algorithm) and executes short FFT operations on the columns and rows of the matrix, with a correction multiplication by "twiddle factors" in between. The algorithm got its name after an article by David H. Bailey (mathematician), David H. Bailey, ''FFTs in external or hierarchical memory'', published in 1989. In this article Bailey credits the algorithm to W. M. Gentleman and G. Sande who published their paper, ''Fast Fourier Transforms: for fun and profit'', some twenty years earlier in 1966. The algorithm can be considered a radix-\sqrt n FFT decompo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bailey 4-step FFT
(The) Bailey(s) may refer to: Arts and entertainment * Bailey, a type of robot in the television series ''Cleopatra 2525'' * Bailey, New Hampshire, a fictional town depicted in the comic book ''Mister Miracle'' * The Baileys, an Australian band comprising Charlie Collins (musician), Charlie Collins (née Bailey) and her siblings Constructions * Bailey (castle), or ward, a courtyard of a castle or fortification, enclosed by a curtain wall * Bailey bridge, a portable prefabricated truss bridge People and fictional characters * Bailey (surname) * Bailey (given name) Places United States * Bailey, Colorado, an unincorporated community * Bailey, Minnesota, an unincorporated community * Bailey, Mississippi, an unincorporated community * Bailey, Missouri, an unincorporated community * Bailey, North Carolina, a town * Bailey, Oklahoma, a ghost town * Bailey, Texas, a city * Bailey Brook (West Branch French Creek tributary), Pennsylvania * Bailey Park, Austin, Texas * Bailey Peninsula, Wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fast Fourier Transform
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by Matrix decomposition, factorizing the DFT matrix into a product of Sparse matrix, sparse (mostly zero) factors. As a result, it manages to reduce the Computational complexity theory, complexity of computing the DFT from O(n^2), which arises if one simply applies the definition of DFT, to O(n \log n), where is the data size. The difference in speed can be enormous, especially for long data sets where may be in the thousands or millions. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cooley–Tukey FFT Algorithm
The Cooley–Tukey algorithm, named after James Cooley, J. W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite number, composite size N = N_1N_2 in terms of ''N''1 smaller DFTs of sizes ''N''2, recursion, recursively, to reduce the computation time to O(''N'' log ''N'') for highly composite ''N'' (smooth numbers). Because of the algorithm's importance, specific variants and implementation styles have become known by their own names, as described below. Because the Cooley–Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT. For example, Rader's FFT algorithm, Rader's or Bluestein's FFT algorithm, Bluestein's algorithm can be used to handle large prime factors that cannot be decomposed by Cooley–Tukey, or the prime-factor FFT algorithm, prime-factor algorithm can be exploited for greater efficiency in s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twiddle Factor
A twiddle factor, in fast Fourier transform (FFT) algorithms, is any of the trigonometric constant coefficients that are multiplied by the data in the course of the algorithm. This term was apparently coined by Gentleman & Sande in 1966, and has since become widespread in thousands of papers of the FFT literature. More specifically, "twiddle factors" originally referred to the root-of-unity complex multiplicative constants in the butterfly operations of the Cooley–Tukey FFT algorithm, used to recursively combine smaller discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discre ...s. This remains the term's most common meaning, but it may also be used for any data-independent multiplicative constant in an FFT. The prime-factor FFT algorithm is one unusual ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David H
David (; , "beloved one") was a king of ancient Israel and Judah and the Kings of Israel and Judah, third king of the Kingdom of Israel (united monarchy), United Monarchy, according to the Hebrew Bible and Old Testament. The Tel Dan stele, an Canaanite and Aramaic inscriptions, Aramaic-inscribed stone erected by a king of Aram-Damascus in the late 9th/early 8th centuries BCE to commemorate a victory over two enemy kings, contains the phrase (), which is translated as "Davidic line, House of David" by most scholars. The Mesha Stele, erected by King Mesha of Moab in the 9th century BCE, may also refer to the "House of David", although this is disputed. According to Jewish works such as the ''Seder Olam Rabbah'', ''Seder Olam Zutta'', and ''Sefer ha-Qabbalah'' (all written over a thousand years later), David ascended the throne as the king of Judah in 885 BCE. Apart from this, all that is known of David comes from biblical literature, Historicity of the Bible, the historicit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. Transpose of a matrix Definition The transpose of a matrix , denoted by , , , A^, , , or , may be constructed by any one of the following methods: # Reflect over its main diagonal (which runs from top-left to bottom-right) to obtain #Write the rows of as the columns of #Write the columns of as the rows of Formally, the -th row, -th column element of is the -th row, -th column element of : :\left mathbf^\operatorname\right = \left mathbf\right. If is an matrix, then is an matrix. In the case of square matrices, may also denote the th power of the matrix . For avoiding a possible confusion, many authors use left upperscripts, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multidimensional Transform
In mathematical analysis and applications, multidimensional transforms are used to analyze the frequency content of signals in a domain of two or more dimensions. Multidimensional Fourier transform One of the more popular multidimensional transforms is the Fourier transform, which converts a signal from a time/space domain representation to a frequency domain representation.Smith, W. Handbook of Real-Time Fast Fourier Transforms:Algorithms to Product Testing, Wiley_IEEE Press, edition 1, pages 73–80, 1995 The discrete-domain multidimensional Fourier transform (FT) can be computed as follows: : F(w_1,w_2,\dots,w_m) = \sum_^\infty \sum_^\infty \cdots \sum_^\infty f(n_1,n_2,\dots,n_m) e^ where ''F'' stands for the multidimensional Fourier transform, ''m'' stands for multidimensional dimension. Define ''f'' as a multidimensional discrete-domain signal. The inverse multidimensional Fourier transform is given by : f(n_1,n_2,\dots,n_m) = \left(\frac\right ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Fourier Transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex number, complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex number, complex Sine wave, sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number-theoretic Transform
In mathematics, the discrete Fourier transform over a ring generalizes the discrete Fourier transform (DFT), of a function whose values are commonly complex numbers, over an arbitrary ring. Definition Let be any ring, let n\geq 1 be an integer, and let \alpha \in R be a principal ''n''th root of unity, defined by:Martin Fürer,Faster Integer Multiplication, STOC 2007 Proceedings, pp. 57–66. Section 2: The Discrete Fourier Transform. The discrete Fourier transform maps an ''n''-tuple (v_0,\ldots,v_) of elements of to another ''n''-tuple (f_0,\ldots,f_) of elements of according to the following formula: By convention, the tuple (v_0,\ldots,v_) is said to be in the ''time domain'' and the index is called ''time''. The tuple (f_0,\ldots,f_) is said to be in the ''frequency domain'' and the index is called ''frequency''. The tuple (f_0,\ldots,f_) is also called the ''spectrum'' of (v_0,\ldots,v_). This terminology derives from the applications of Fourier transforms i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Row-column FFT Algorithm
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors. As a result, it manages to reduce the complexity of computing the DFT from O(n^2), which arises if one simply applies the definition of DFT, to O(n \log n), where is the data size. The difference in speed can be enormous, especially for long data sets where may be in the thousands or millions. As the FFT is merely an algebraic refactoring of terms within the DF ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector-radix FFT Algorithm
The vector-radix FFT algorithm, is a multidimensional fast Fourier transform (FFT) algorithm, which is a generalization of the ordinary Cooley–Tukey FFT algorithm that divides the transform dimensions by arbitrary radices. It breaks a multidimensional (MD) discrete Fourier transform (DFT) down into successively smaller MD DFTs until, ultimately, only trivial MD DFTs need to be evaluated. The most common multidimensional FFT algorithm is the row-column algorithm, which means transforming the array first in one index and then in the other, see more in FFT. Then a radix-2 direct 2-D FFT has been developed, and it can eliminate 25% of the multiplies as compared to the conventional row-column approach. And this algorithm has been extended to rectangular arrays and arbitrary radices, which is the general vector-radix algorithm. Vector-radix FFT algorithm can reduce the number of complex multiplications significantly, compared to row-vector algorithm. For example, for a N^M element ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
