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Hadamard transform
About: Hadamard transform is a research topic. Over the lifetime, 7262 publications have been published within this topic receiving 94328 citations.
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TL;DR: In this paper, the authors investigate (ω 1,ω 2 ) -convex functions and obtain characterization theorems and Hadamard-type inequalities for them, and show that these inequalities are equivalent to the same properties for (ω 2,ω 3 ) functions.
Abstract: In this paper we investigate (ω1,ω2) -convex functions and obtain characterization theorems and Hadamard-type inequalities for them. Mathematics subject classification (2000): 26A51, 26B25.
71 citations
TL;DR: Considering paired transforms, this work analyzes simultaneously the splitting of the multidimensional Fourier transform as well as the presentation of the processed multiddimensional signal in the form of the short one-dimensional "signals", that determine such splitting.
Abstract: A concept of multipaired unitary transforms is introduced. These kinds of transforms reveal the mathematical structure of Fourier transforms and can be considered intermediate unitary transforms when transferring processed data from the original real space of signals to the complex or frequency space of their images. Considering paired transforms, we analyze simultaneously the splitting of the multidimensional Fourier transform as well as the presentation of the processed multidimensional signal in the form of the short one-dimensional (1-D) "signals", that determine such splitting. The main properties of the orthogonal system of paired functions are described, and the matrix decompositions of the Fourier and Hadamard transforms via the paired transforms are given. The multiplicative complexity of the two-dimensional (2-D) 2/sup r//spl times/2/sup r/-point discrete Fourier transform by the paired transforms is 4/sup r//2(r-7/3)+8/3-12 (r>3), which shows the maximum splitting of the 5-D Fourier transform into the number of the short 1-D Fourier transforms. The 2-D paired transforms are not separable and represent themselves as frequency-time type wavelets for which two parameters are united: frequency and time. The decomposition of the signal is performed in a way that is different from the traditional Haar system of functions.
71 citations
TL;DR: In this paper, the Hermite-Hadamard inequalities for convex functions were established and applied to construct inequalities involving special means of real numbers, some error estimates for the formula midpoint were given, and new inequalities for some special and q -special functions were also pointed out.
70 citations
TL;DR: The Data Encryption Standard (DES) can be regarded as a nonlinear feedback shift register (NLFSR) with input and the properties of the S-boxes of DES under the Fourier transform, Hadamard transform, extended Hadamards transform, and the Avalanche transform are investigated.
Abstract: The Data Encryption Standard (DES) can be regarded as a nonlinear feedback shift register (NLFSR) with input. From this point of view, the tools for pseudo-random sequence analysis are applied to the S-boxes in DES. The properties of the S-boxes of DES under the Fourier transform, Hadamard transform, extended Hadamard transform, and the Avalanche transform are investigated. Two important results about the S-boxes of DES are found. The first result is that nearly two-thirds of the total 32 functions from GF (2/sup 6/) to GF(2) which are associated with the eight S-boxes of DES have the maximal linear span G3, and the other one-third have linear span greater than or equal to 57. The second result is that for all S-boxes, the distances of the S-boxes approximated by monomial functions has the same distribution as for the S-boxes approximated by linear functions. Some new criteria for the design of permutation functions for use in block cipher algorithms are discussed.
70 citations
TL;DR: A method is described that yields the fast Walsh transform (FWT) in sequency order based on the Cooley-Tukey-type fast Hadamard transform (FHT) algorithm, whose computational effort is identical to the conventional FHT.
Abstract: A method is described that yields the fast Walsh transform (FWT) in sequency order. The advantages of this method over others are: 1) it is based on the Cooley-Tukey-type fast Hadamard transform (FHT) algorithm, 2) the computational effort is identical to the conventional FHT, and 3) the transform remains its own inverse.
70 citations