Winograd S. (1978) On Computing the Discrete Fourier Transform. Mathematics of Computation 32, pp. 175--199. 74  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Pass 0 Figure 4. 4K point FFT is accomplished using 3 passes through a 16 point kernel and phase rotator. A fourth pass is used to load and unload data from the memory. The 16 point FFT kernel is a very fast and compact fixed point hardware implementation of the Winograd 16 point FFT algorithm [9,10,11] with 16 bit inputs and 21 bit outputs. We selected this algorithm over a more traditional CooleyTukey approach in order to achieve maximum performance with minimum area. This IP core depends heavily upon the Xilinx SRL16 shift register primitives for intermediate storage, data reordering, and ....

Winograd, S., "On Computing the Discrete Fourier Transform," Mathematics of Computation, Vol. 32, No. 141, pp. 175-199 (1978).

....Both the linear algebra and the combinatorics communities have paid attention to this important class of matrices. The algebraic properties of circulant matrices have lead to efficient algorithms for several related computations, notably the Winograd Fourier Transform is based on circulants [Win78] and several preconditioners for solving Toeplitz or other related systems use circulant matrices [Cha88] Circulants also have several applications in combinatorics and counting [Min87] e.g. the solution of the well known rencontres and menage problems can be obtained as a permanent of a ....

Shmuel Winograd. On computing the discrete Fourier transform. Math. Comp., 32:175--199, 1978.

....if c is not a root of 1; this generalization was introduced by Rabiner, Schafer, and Rader in [80] as the chirp z transform. The fractional Fourier transform, announced many years later in [8] is the same as the chirp z transform. An alternative to Bluestein s trick is Winograd s method in [111]. Special cases of Winograd s method were published earlier by Rader in [81] and by Singleton in [94] See also [4] 22] 50] 52] 58] and [117] For results on the problem of decomposing non commutative group rings, see, e.g. 26] 8. Good s trick Let R be a ring. Let m;n be coprime ....

Shmuel Winograd, On computing the discrete Fourier transform, Mathematics of Computation 32 (1978), 175-199.

....a new structure of nested loop indexing. Later, the radix was increased to reduce the necessary arithmetic. Bergland [7] and Sande [69] developed algorithms for the calculation of FFTs in real sequences (RFFT) Winograd applied the theory of computational complexity to the calculation of the DFT [85] obtaining this way a lower limit for the number of multiplications required for the computation of a DFT of 2 n elements and designed a constructive method for the generation of these algorithms. Burrus [16] extended the application of the DFT to sequences whose length is the product of two ....

S. Winograd, \On computing the discrete Fourier transform". Math. Comput. , Vol. 32, No. 3, pp. 175-199, 1978.

....Both the linear algebra and the combinatorics communities have paid attention to this important class of matrices. The algebraic properties of circulant matrices have lead to efficient algorithms for several related computations, notably the Winograd Fourier Transform is based on circulants [Win78] and several preconditioners for solving Toeplitz or other related systems use circulant matrices [Cha88] Circulants also have several applications in combinatorics and counting [Min87] e.g. the solution of the well known rencontres and menage problems can be obtained as a permanent of a ....

Shmuel Winograd. On computing the discrete Fourier transform. Math. Comp., 32:175--199, 1978. 16

.... implementation of an FFT, but it is well understood and easily implemented using the mesh spectral archetype; in this paper, we use this algorithm primarily to illustrate the performance model, rather than attempt to achieve the fastest possible FFT (as is the focus of other work, such as [Win78] Duhamel and Vetterli provide an excellent survey of FFTs [DV90] A good comparison of the FFT algorithm we use with more efficient ones (such as a split radix algorithm) on a vanilla workstation is 49 given in [Arn96] although, with a multicomputer, a simpler butterfly structure might be ....

S. Winograd. On Computing the Discrete Fourier Transform. Mathematics of Computation, 32:175--199, January 1978. 84

.... fast method for computing the DFT is the prime factor algorithm (PFA) which 4 uses an index map developed by Thomas and Good, 14] Prime factorization is slow when n is large, but the DFT for small cases, such as n = 2; 3; 4; 5; 7; 8; 11; 13; 16, can be made fast using the Winograd algorithm, [8, 16, 17, 18]. H. W. Johnson and C. S. Burrus, 15] developed a method to use dynamic programming to design optimal FFT programs by reducing the number of ops as well as data transfers. This approach designs custom algorithms for particular computer architectures. Ecient programs have been developed to ....

S. Winograd, \On Computing the Discrete Fourier Transform," Mathematics of Computation, vol. 32, pp. 175-199, January 1978.

....d concurrent exchange communication steps. The proposed algorithms achieve these nice features by using a simple yet efficient mapping scheme and a restructuring for the FFT algorithm which enables overlapping communication with computation. Different strategies exist for the computation of FFT [5, 7, 8, 14]. The FFT scheme chosen for parallelization is radix 2, Cooley Tukey scheme using the decimation in time decomposition [5] Section 2 presents and discusses the computational structure of this FFT scheme. Parallelization of the chosen FFT scheme is discussed in Section 3. The mapping scheme ....

S. Winograd "On Computing the Discrete Fourier Transform", Math. Comp., 32, pp.175-199, 1978.

....(2.12) 2.4.1 FMA Optimized Radix 8 Butter y Computation The radix 8 butter y update x : B 8;L x, x 2 C L can be written as x : F 8 diag(1; j L ; 2j L ; 3j L ; 4j L ; 5j L ; 6j L ; 7j L )x (2.19) with j = 0 : L=8 1. Taking advantage of Winograd s cyclic convolution [16] and a factorization similar to (2.8) the radix 8 butter y update can be calculated using Algorithm 2.4. 2.4. RADIX 8 BUTTERFLY COMPUTATION 15 Algorithm 2.4 (FMA Optimized Radix 8 Butter y) do j = 0 : L=8 1 1 : x(j) 4 x(j 4L=8) 2 : 2x(j) 1 3 : x(j 2L=8) 4 x(j 6L=8) ....

S. Winograd, On Computing the Discrete Fourier Transform, Math. Comp. 32 (1978), pp. 175-199.

....RC algorithm has the happy property that 1 D mixed radix algorithms based on a selection of optimized small order transform components can directly be plugged into each row and column computation directly. Pseudo code for small order transforms can be found in an appendix to the seminal work of [ 23 ] where the routines are proven to be optimal in the number theoretic sense [ 24 ] The mixed radix algorithm results in computational savings over the single radix case. 6 Table 1 is a simple illustration of the relative savings, accurately predicted by earlier authors [ 25 ] which accrue by ....

S. Winograd. On computing the Discrete Fourier Transform. Mathematics of Computation, 32(141):175--199, 1978.

....Primos multitasking, virtual memory operating system (64K virtual memory) As a consequence, we were able to increase sequence lengths by a factor of four, and image dimensions by a factor of two. Key Concerns and Projects Winograd Fourier Transform algorithm (WFTA) McClellan and Rader, 1976, Winograd, 1978] number theoretic transforms (NTT) implemented using WFTA [Bailey and White, 1977] the Prime 300 was microprogrammable, so that key algorithm hot spots could be implemented in microcode, e.g. FFT butterfly . 3.5 IDP 3000 1975 80 First commercial digital image processing system in ....

Winograd, S. (1978). On Computing the Discrete Fourier Transform. Mathematics of Computation, 32(141). //---- dlj.Im.java ----------------------------------

....Both the linear algebra and the combinatorics communities have paid attention to this important class of matrices. The algebraic properties of circulant matrices have lead to efficient algorithms for several related computations, notably the Winograd Fourier Transform is based on circulants [Win78] and several preconditioners for solving Toeplitz or other related systems use circulant matrices [Cha88] Circulants also have several applications in combinatorics and counting [Min87] e.g. the solution of the well known rencontres and menage problems can be obtained as a permanent of a ....

Shmuel Winograd. On computing the discrete Fourier transform. Math. Comp., 32:175--199, 1978.

....for data storage is expensive and usually not large, it is more feasible to process small size problems one at a time. In addition, when this approach is combined with efficient short length algorithms, such as Rader s algorithm [20] or Winograd type minimum multiplication algorithms in DFT [31], or Heideman s small odd length DCT modules [8] etc. it would be of practical interest in reducing the scalar multiplication complexity. Although Chan and Ho [6] and Cho and Lee [7] derived prime factor DCT algorithms based on variant DFT algorithms before, their algorithms required additional ....

S. Winograd. On computing the discrete Fourier transform. Mathematics of Computation, 32(141):175-- 199, January 1978.

.... d 2 j i log( j H j i j j K j ) where the O notation indicates a universal constant determined by the by the FFT on cyclic groups. Furthermore, I(G) is bounded by the same sum but with I(K) replacing T (K) In Theorem 1 ff denotes the exponent of matrix multiplication, currently 2. 38, see [W]) It is useful to point out that under the simplifying assumption of ff = 2 the bounds becomes T (G) j G j j K j Delta T (K) O(jG j Delta log( j G j j K j ) and I(G) j G j j K j Delta I(K) O(jG j Delta log( j G j j K j ) with the O notation as in Theorem 1. In brief, the ....

Winograd, S. (1978). On Computing the discrete Fourier transform. Math. Comp. 32, 175-199.

....= W (i) x (i Gamma1) i = 1; m. Thus both the initial and final permutation steps could be skipped. As each elementary transform is usually a DFT of a short length, they apply an optimal DFT Winograd module where appropriate to further improve the efficiency of their algorithm. In [53] Winograd combined PFA algorithm with the optimal short length DFT modules in such a way that he could obtain for the entire transform an algorithm structured the same way as in (5.2) Theorem 5.5 (Winograd Fourier Transform Algorithm (WFTA) Let N = N 1 N 2 : Nm where gcd(N i ; N j ) 1 ....

S. Winograd, On computing the discrete Fourier transform, Math. of Comput. 32 (1978), 175--199.

....If n is prime, genfft uses either Equation (1) directly, or Rader s algorithm [Rad68] which converts the transform into a circular convolution of size n Gamma 1. The circular convolution can be computed recursively using two Fourier transforms, or by means of a clever technique due to Winograd [Win78] genfft does not employ this technique yet, however) Other algorithms are known for prime sizes, and this is still the subject of active research. See [TAL97] for a recent compendiumon the topic. Any algorithm for the forward DFT can be readily adapted to compute the backward DFT, the ....

S. Winograd. On computing the discrete Fourier transform. Mathematics of Computation, 32(1):175--199, January 1978.

....Computation The radix 3 butterfly update x : B 3;L x can be written as 0 x(j) x(j L=3) x(j 2L=3) 1 A : F 3 diag(1; j L ; 2j L ) 0 x(j) x(j L=3) x(j 2L=3) 1 A (10.1) 28 CHAPTER 10. RADIX 3 BUTTERFLY COMPUTATION with j = 0 : L=3 Gamma 1: Utilizing Winograd s convolution[22] (10.1) can be evaluated as follows. x(j) 0 3 [x(j) j L x(j L=3) 2j L x(j 2L=3) x(j L=3) 0 3 x(j) w 1 x(j 2L=3) 0 3 x(j) w 2 ; 10.2) where w 1 w 2 : 1 3 2 3 2 3 1 3 diag( j L ; 2j L ) x(j L=3) x(j 2L=3) ....

....x(j 3L=5) x(j 4L=5) 1 C C C C A : F 5 diag(1; j L ; 2j L ; 3j L ; 4j L ) 0 B B B B x(j) x(j L=5) x(j 2L=5) x(j 3L=5) x(j 4L=5) 1 C C C C A 11.1. FMA OPTIMIZED RADIX 5 BUTTERFLY COMPUTATION 31 with j = 0 : L=5 Gamma 1: Using Winograd s cyclic convolution[22] and a factorization similar to (7.2) the FMA optimized radix 5 butterfly update can be calculated using the following algorithm. A detailed derivation can be found in Karner et al. 11] Algorithm 11.1 (FMA Optimized Radix 5 Butterfly) do j = 0 : L=5 Gamma 1 z 0 : x(j) z 1 : L x(j L=5) ....

S. Winograd, On Computing the Discrete Fourier Transform, Math. Comp. 32 (1978), pp. 175--199.

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Winograd S. (1978) On Computing the Discrete Fourier Transform. Mathematics of Computation 32, pp. 175--199. 74

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S. Winograd, On Computing the Discrete Fourier Transform, Math. Comp. 32 (1978), pp. 175-199.

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S. Winograd, On Computing the Discrete Fourier Transform, Math. Comp. 32 (1978), pp. 175--199.

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S. Winograd, `On computing the discrete Fourier transform', Mathematics of Computation, 32, 175-- 199 (1978).

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S. Winograd, On computing the discrete Fourier transform, Math. Comp, 32, 175-199 (1978). 21

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