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1

Discrete Fourier transform and permutations

Hui S., Żak S. H.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2019

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Vol. 67, nr 6

995--1005

EN

It is well known that the magnitudes of the coefficients of the discrete Fourier transform (DFT) are invariant under certain operations on the input data. In this paper, the effects of rearranging the elements of an input data on its DFT are studied. In the one-dimensional case, the effects of permuting the elements of a finite sequence of length N on its Discrete Fourier transform (DFT) coefficients are investigated. The permutations that leave the unordered collection of Fourier coefficients and their magnitudes invariant are completely characterized. Conditions under which two different permutations give the same DFT coefficient magnitudes are given. The characterizations are based on the automorphism group of the additive group ZN of integers modulo N and the group of translations of ZN. As an application of the results presented, a generalization of the theorem characterizing all permutations that commute with the discrete Fourier transform is given. Numerical examples illustrate the obtained results. Possible generalizations and open problems are discussed. In higher dimensions, results on the effects of certain geometric transformations of an input data array on its DFT are given and illustrated with an example.

2

Circulant matrices: norm, powers, and positivity

Lindner M.

Opuscula Mathematica

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2018

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Vol. 38, no. 6

849--857

EN

In their recent paper "The spectral norm of a Horadam circulant matrix", Merikoski, Haukkanen, Mattila and Tossavainen study under which conditions the spectral norm of a general real circulant matrix C equals the modulus of its row/column sum. We improve on their sufficient condition until we have a necessary one. Our results connect the above problem to positivity of sufficiently high powers of the matrix CTC. We then generalize the result to complex circulant matrices.

3

A note on circulant matrices of degree 9

Dubovský V.

Systemy Wspomagania w Inżynierii Produkcji

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2017

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Vol. 6, iss. 4

243--251

EN

Circulant matrices provide quite a wide range of applications in many different branches of mathematics, such as data and time-series analysis, signal processing or Fourier transformation. Huge number of results concerning circulant matrices could be found in algebraic number theory. This is because we could construct factor ring isomorphic to the p-th cyclotomic field Q(ζp) from the ring of circulant matrices degree p, where p is a prime. In this paper the connection between ring of circulant matrices of degree 9, C9, and the cyclotomic field Q(ζ9) is shown.

CS

Cirkulantní matice nabízí širokou škálu aplikací v mnoha ruzných odvetvích matematiky, jako jsou analýza dat a casový rad, zpracování signálu ci Fourierova transformace. Další výsledky využívající vlastností cirkulantních matic mužeme nalézt v algebraické teorii císel, což je dáno tím, že z okruhu cirkulantních matic prvocíselného stupne p, lze vytvorit faktorový okruh isomorfní s p-tým cyklotomickým telesem, Q(ζp). V clánku je ukázán vztah mezi okruhem cirkulantních matic stupne 9, C9, a devátým cyklotomickým telesem.

4

Construction of Efficient MDS Matrices Based on Block Circulant Matrices for Lightweight Application

Han H., Tang C., Lou Y., Xu M.

Fundamenta Informaticae

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2016

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Vol. 145, nr 2

111--124

EN

Maximum distance separable (MDS) codes introduce MDS matrices which not only have applications in coding theory but also are of great importance in the design of block ciphers. It has received a great amount of attention. In this paper, we first introduce a special generalization of circulant matrices called block circulants with circulant blocks, which can be used to construct MDS matrices. Then we investigate some interesting and useful properties of this class of matrices and prove that their inversematrices can be implemented efficiently. Furthermore, we present some 4×4 and 8×8 efficient MDS matrices of this class which are suitable for MDS diffusion layer. Compared with previous results, our construction provides better efficiency for the implementation of both the matrix and the its inverse matrix.