Symmetric Hadamard Research Articles

Implementation and analysis of symmetrical signals in the sequency domain

Every transform has unique attributes and traits that are crucial to reducing computing costs and offering simple solutions. Many different frequency domain transformations, for instance, have properties that can be used in a variety of signal processing applications and analyses. Some of the Complex Hadamard Transform's variants' sequencies can be compared to those of the Discrete Fourier Transforms. It is proven the characteristics of the Conjugate Symmetric Sequency-Ordered Complex Hadamard Transform symmetry. These qualities are crucial for signal analysis and image processing. Due to duplicate spectra across the origin, it reduces computational complexity, makes an analytical analysis for symmetric signals simpler, and needs less storage. Its analysis shows that the Discrete Fourier Transform and this Complex Hadamard Transform version exhibit similar symmetry tendencies. By using elementary signals in the time domain to connect the positive and negative sequencies with their associated phasor conceptions, sequency domain spectra are used to highlight these properties. As a result of image representation, relative spectra are represented in related domains. Transform can be used to extract and analyze various aspects from a wide range of medical images.

Interrelation of Symmetry and Antisymmetry of Quasi-Orthogonal Cyclic Matrices with Prime Numbers

Quasi-orthogonal Hadamard matrices and Mersenne matrices with two and three values of the elements, used in digital data processing, are considered, as well as the basis of error-correcting codes and algorithms for transforming orthogonal images. Attention is paid to the structures of cyclic matrices with symmetries and antisymmetries. The connection between symmetry and antisymmetry of structures of cyclic Hadamard and Mersenne matrices on a orders equal to prime numbers, products of close primes, composite numbers, powers of a prime number is shown. Separately, orders equal to the degrees of the prime number 2 are distinguished, both the orders of Hadamard matrices and the basis of the composite orders of Mersenne matrices of block structures with two element values. It is shown that symmetric Hadamard matrices of cyclic and bicyclic structures, according to the extended Riser boundary, do not exist on orders above 32. Mersenne matrices of composite orders belonging to the sequence of Mersenne numbers 2k ‒ 1 nested in the sequence of orders of the main family of Mersenne matrices 4t ‒ 1 exist in a symmetric and antisymmetric form. For orders equal to the powers of a prime number, Mersenne matrices exist in the form of block-diagonal constructions with three element values. The value of prime power determines the number of blocks along the diagonal of the matrix on which the elements with the third value are located. The cyclic blocks are symmetrical and antisymmetric.

Some new symmetric Hadamard matrices

Introduction: It is conjectured that the symmetric Hadamard matrices of order 4v exist for all odd integers v>0. In recent years, their existence has been proven for many new orders by using a special method known as the propus construction. This construction uses difference families Xk (k=1, 2, 3, 4) over the cyclic group Zv (integers mod v) with parameters (v; k1, k2, k3, k4; λ) where X1 is symmetric, X2=X3, and k1+2k2+k4=v+λ. It is also conjectured that such difference families (known as propus families) exist for all parameter sets mentioned above excluding the case when all the ki are equal. This new conjecture has been verified for all odd v≤53. Purpose: To construct many new symmetric Hadamard matrices by using the propus construction and to provide further support for the above-mentioned conjecture. Results: The first examples of symmetric Hadamard matrices of orders 4v are presented for v=127 and v=191. The systematic computer search for symmetric Hadamard matrices based on the propus construction has been extended to cover the cases v=55, 57, 59, 61, 63. Practical relevance: Hadamard matrices are used extensively in the problems of error-free coding, and compression and masking of video information.

An alternative method for constructing Hadamard matrices

<p class="Default">Symmetric Hadamard matrices are investigated in this research and an alternative method of construction is introduced. Using the proposed method, we can construct Hadamard matrices of order 2ⁿ⁺¹(<em>q</em>+1) where <em>q ≡</em> 1 (mod 4) and <em>n ≥</em> 1. <p class="Default"> This construction can be used to construct an infinite number of Hadamard matrices. For the present study, we use quadratic non-residues over a finite field.

Permutable Symmetric Hadamard Matrices in Quaternion Algebra and Engineering Applications

In this paper, aiming to develop the group and out-of-group formalization of the symmetry concept, the preservation of a matrix symmetry after row permutation is considered by the example of the maximally permutable \emph{normalized} Hadamard matrices which row and column elements are either plus or minus one. These matrices are used to extend the additive decomposition of a linear operator into symmetric and skew-symmetric parts using several commuting operations of the Hermitian conjugation type, for the quaternionic generalization of a vector cross product, as well as for creating educational puzzles and other applications.

Calculating symmetrical Hadamard matrices of Balonin-Seberry construction for coding and masking

Abstract We discuss the ways of calculating Hadamard matrices and estimating their possibilities when these matrices are found on orders 4t, symmetrical structures included. We demonstrate the shortcomings of Silvester, Paley and Scarpi classical methods regarding the orders of the matrices calculated. Based on Williamson array, Hadamard matrices of three-block construction are specified. We discuss two strategies of sequence search in order to form cyclic blocks of orthogonal matrices of Propus-like three-block construction, suggested by Balonin and Seberry. The results of our work can be a starting point for developing new approaches to search for symmetric Hadamard matrices used in noise-immune coding and filtering of radio signals, masking and compression of digital images, and other applied telecommunication problems.

Hadamard matrices from Goethals — Seidel difference families with a repeated block

Purpose: To construct Hadamard matrices by using Goethals — Seidel difference families having a repeated block, generalizingthe so called propus construction. In particular we construct the first examples of symmetric Hadamard matrices of order 236.Methods: The main ingredient of the propus construction is a difference family in a finite abelian group of order v consisting offour blocks (X1, X2, X3, X4) where X1 is symmetric and X2 X3. The parameters (v; k1, k2, k3, k4; λ) of such family must satisfythe additional condition ki  λ  v. We modify this construction by imposing different symmetry conditions on some of theblocks and construct many examples of Hadamard matrices of this kind. In this paper we work with the cyclic group Zv of order v.For larger values of v we build the blocks Xi by using the orbits of a suitable small cyclic subgroup of the automorphism groupof Zv. Results: We continue the systematic search for symmetric Hadamard matrices of order 4v by using the propus construction.Such searches were carried out previously for odd v  51. We extend it to cover the case v53. Moreover we construct thefirst examples of symmetric Hadamard matrices of order 236. A wide collection of symmetric and skew-symmetric Hadamardmatrices was obtained and the corresponding difference families tabulated by using the symmetry properties of their blocks.Practical relevance: Hadamard matrices are used extensively in the problems of error-free coding, compression and masking ofvideo information. Programs for search of symmetric Hadamard matrices and a library of constructed matrices are used in themathematical network Internet together with executable on line algorithms.

Helping Hadamard conjecture to become a theorem. Part 2

Introduction:Hadamard conjecture about the existence of specific square matrices was formulated not by Hadamard but by other mathematicians in the early 20th century. Later, this problem was revised by Ryser together with Bruck and Chowla, and also by Hall, one of the founders of discrete mathematics. This is a problem of the boundary mixed type, as it includes both the continuous and discrete components. The combinatorial approach used in the framework of the discrete component has run its course by the end of the century. The article discusses an alternative based on both concepts.Purpose:To analyze the reasons why the conjecture about the existence of Hadamard matrices of all ordersn =4tis considered unproven, and to propose possible ways to prove it.Methods:Transition, by lowering the ordern =4t— 2, to two-level quasiorthogonal matrices with elements 1 and –bwhose existence on all specified orders is not a difficult problem due to the possible irrationality of their entries. Subsequent construction of a chain of transformations to matrix ordersn= 4t–1,n= 4t,n= 4t+ 1.Results:It is proved that Gauss points on anx2 + 2y2 +z2 =nspheroid are in one-to-one correspondence with symmetric Hadamard matrices (constructed on the basis of the Balonin — Seberry arrays), covering up the gaps on the unsolvable orders 140, 112, etc. known in Williamson’s array theory. Solution tables are found and systematized, which include so-called «best» three-block matricesL(p,q), wherep³qis the number of non-conjugated symmetric matrices of the order in question, andqis the number of block-symmetric matrices which coincide with Williamson’s solutions. The iterative Procrustes algorithm which reduces the norm of the maximum entry in a matrix is proposed for obtaining Hadamard matrices by searching for local and global conditional extremes of the determinant.Practical relevance:The obtained Hadamard matrices and quasi-orthogonal matrices of ordersn =4t– 2,n =4t– 1,n =4t +1 are of immediate practical importance for the problems of noise-resistant coding, compression and masking of video information.

Why Won't the Field Wall Collapse in the Typhoon? : Mathematical Approach to Non-orthogonal Symmetric Weighted Hadamard Matrix I

Why Won't the Field Wall Collapse in the Typhoon? : Mathematical Approach to Non-orthogonal Symmetric Weighted Hadamard Matrix I

Helping Hadamard conjecture to become a theorem. Part 1

Introduction:Hadamard conjecture about the existence of specific square matrices was formulated not by Hadamard but by other mathematicians in the early 20th century. Later, this problem was revised by Ryser together with Bruck and Chowla, and also by Hall, one of the founders of discrete mathematics. This is a problem of the boundary mixed type, as it includes both the continuous and discrete components. The combinatorial approach used in the framework of the discrete component has run its course by the end of the century. The article discusses an alternative based on both concepts.Purpose:To analyze the reasons why the conjecture about the existence of Hadamard matrices of all orders n = 4t is considered unproven, and to propose possible ways to prove it.Methods:Transition, by lowering the order n = 4t– 2, to two-level quasiorthogonal matrices with elements 1 and –b whose existence on all specified orders is not a difficult problem due to the possible irrationality of their entries. Subsequent construction of a chain of transformations to matrix orders n = 4t – 1, n = 4t, n = 4t + 1.Results:It is proved that Gauss points on an x2+ 2y2+ z2= n spheroid are in one-to-one correspondence with symmetric Hadamard matrices (constructed on the basis of the Balonin — Seberry arrays), covering up the gaps on the unsolvable orders 140, 112, etc. known in Williamson’s array theory. Solution tables are found and systematized, which include so-called «best» three-block matrices L (p, q), where p ≥ q is the number of non-conjugated symmetric matrices of the order in question, and q is the number of block-symmetric matrices which coincide with Williamson’s solutions. The iterative Procrustes algorithm which reduces the norm of the maximum entry in a matrix is proposed for obtaining Hadamard matrices by searching for local and global conditional extremes of the determinant.Practical relevance:The obtained Hadamard matrices and quasi-orthogonal matrices of orders n = 4t – 2, n = 4t – 1, n = 4t + 1 are of immediate practical importance for the problems of noise-resistant coding, compression and masking of video information.

Symmetric Hadamard Matrices of Orders 268, 412, 436 and 604

Purpose.To investigate more fully, than what was done in the past, certain families of symmetric Hadamard matrices of small orders by using the so called propus construction.Methods.Orbit method for the search of three cyclic blocks to construct Hadamard matrices of propus type. This method speeds up the classical search of required sequences by distributing them into different bins using a hash-function.Results. Our main result is that we have constructed, for the first time, symmetric Hadamard matrices of order 268, 412, 436 and 604. The necessary difference families are constructed by restricting the search to the families which admit a nontrivial multiplier. A wide collection of new symmetric Hadamard matrices was obtained and tabulated, according to the feasible sets of parameters.Practical relevance.Hadamard matrices are used extensively in the problems of error-free coding, compression and masking of video information. Programs for search of symmetric Hadamard matrices and a library of constructed matrices are used in the mathematical network “Internet” together with executable on-line algorithms.

Fast Gray Code Kernel Algorithm for the Sliding Conjugate Symmetric Sequency-Ordered Complex Hadamard Transform

A fast algorithm based on the gray code kernel (GCK) for computing the conjugate symmetric sequency-ordered complex Hadamard transform (CS-SCHT) in a sliding window is presented. The proposed algorithm computes the current projection value from the previously computed ones. In order to obtain the peculiar computation order of the projection values, we construct the CS-SCHT matrix tree and also introduce the $\alpha $ -related concept. The properties of the elements of the CS-SCHT matrix are also given for deriving the GCK sliding CS-SCHT algorithm. The proposed algorithm only needs N/2+log $_{2}N-2$ (or log $_{2}N-1$ ) multiplications with $j$ and $4N-2$ (or $2N-1$ ) real additions for complex (or real) input data, which is more efficient than the block-based CS-SCHT and other existing sliding complex transform algorithms, such as the radix-4 sliding CS-SCHT algorithm, sliding FFT algorithm, and sliding DFT algorithm. A comparison of the proposed algorithm with other sliding transforms in terms of computation time is also presented to validate the theoretical results.

Use of symmetric Hadamard and Mersenne matrices in digital image processing

Matrices with orthogonal columns (rows) are extensively used in data processing. One of the pressing challenges for developers of image processing systems is the ability to choose easily the optimal types of structured orthogonal matrices for their specific tasks. The purpose of this study is to systemize the main types of structured symmetric Hadamard matrices, such as the cyclic, negacyclic, bicyclic, four-block and Propus-form three-block ones, which can be used in image processing for compression, filtering, antinoise coding, and masking. The basis of Hadamard matrices is extended with quasi-orthogonal Mersenne matrices of odd orders with previously unknown symmetry properties. It was found that Mersenne matrices and Hadamard matrices built using Mersenne matrices are directly associated with numeric sequences, for some of which the cyclic, bicyclic matrices, and Propus-shaped matrices exist. The paper demonstrates that the use of Mersenne matrices as a "core" produces Haramard matrices with new symmetric structures. This fact expands the classification of orthogonal matrices. Images of previously unknown symmetric Hadamard matrices are provided. The practical value of the paper is that the obtained results provide wide opportunities to facilitate the selection of matrices, which best fit for processing of certain images, including images of non-standard sizes. The number of matrix fragments sufficient for reconstruction of the whole matrix is defined for all matrices under consideration. This way of representation of Hadamard symmetric matrices allows us to increase the efficiency of the image processing using lesser the computer memory.

Construction of symmetric Hadamard matrices of order 4v for v = 47, 73, 113

Abstract We continue our systematic search for symmetric Hadamard matrices based on the so called propus construction. In a previous paper this search covered the orders 4v with odd v ≤ 41. In this paper we cover the cases v = 43, 45, 47, 49, 51. The odd integers v &lt; 120 for which no symmetric Hadamard matrices of order 4v are known are the following: 47, 59, 65, 67, 73, 81, 89, 93, 101, 103, 107, 109, 113, 119. By using the propus construction, we found several symmetric Hadamard matrices of order 4v for v = 47, 73, 113.

CONSTRUCTION OF SYMMETRIC HADAMARD MATRICES

We systematically explore the new method of construction (known as the propus construction) of symmetric Hadamard matrices for small orders, $4v$. In particular we give the first examples of symmetric Hadamard matrices of order $156=4\cdot 39$. The smallest $v$ for which symmetric Hadamard matrices of order $4v$ are not known is now $v=47$.

Uniform Mixing and Association Schemes

We consider continuous-time quantum walks on distance-regular graphs. Using results about the existence of complex Hadamard matrices in association schemes, we determine which of these graphs have quantum walks that admit uniform mixing.First we apply a result due to Chan to show that the only strongly regular graphs that admit instantaneous uniform mixing are the Paley graph of order nine and certain graphs corresponding to regular symmetric Hadamard matrices with constant diagonal. Next we prove that if uniform mixing occurs on a bipartite graph $X$ with $n$ vertices, then $n$ is divisible by four. We also prove that if $X$ is bipartite and regular, then $n$ is the sum of two integer squares. Our work on bipartite graphs implies that uniform mixing does not occur on $C_{2m}$ for $m \geq 3$. Using a result of Haagerup, we show that uniform mixing does not occur on $C_p$ for any prime $p$ such that $p \geq 5$. In contrast to this result, we see that $\epsilon$-uniform mixing occurs on $C_p$ for all primes $p$.

Development of Low-Complexity Video Watermarking With Conjugate Symmetric Sequency–Complex Hadamard Transform

In this letter, a blind video watermarking in phase domain is proposed with conjugate symmetric sequency-complex hadamard transform (CS-SCHT). The watermark is made imperceptible and robust using methods, such as low-amplitude block selection and amplitude boost. Simulations are conducted to perform objective and subjective evaluation on watermark imperceptibility and robustness against attacks, such as high efficiency video coding compression, rescaling, and cropping. Performance metrics, such as peak signal to noise ratio (PSNR), mean opinion score (MOS), and bit error rate (BER), are used, and a comparison is carried out with discrete Fourier transform (DFT). It has been shown that CS-SCHT offers comparable performance with DFT in terms of PSNR, MOS, and BER. As this transform require only few multiplications for computation of the transform kernel, this scheme has 37.5% significant saving on hardware for high definition videos compared with DFT.

Beyond graph energy: Norms of graphs and matrices

In 1978 Gutman introduced the energy of a graph as the sum of the absolute values of graph eigenvalues, and ever since then graph energy has been intensively studied.Since graph energy is the trace norm of the adjacency matrix, matrix norms provide a natural background for its study. Thus, this paper surveys research on matrix norms that aims to expand and advance the study of graph energy.The focus is exclusively on the Ky Fan and the Schatten norms, both generalizing and enriching the trace norm. As it turns out, the study of extremal properties of these norms leads to numerous analytic problems with deep roots in combinatorics.The survey brings to the fore the exceptional role of Hadamard matrices, conference matrices, and conference graphs in matrix norms. In addition, a vast new matrix class is studied, a relaxation of symmetric Hadamard matrices.The survey presents solutions to just a fraction of a larger body of similar problems bonding analysis to combinatorics. Thus, open problems and questions are raised to outline topics for further investigation.

A Parallel Douglas–Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds

We are interested in restoring images having values in a symmetric Hadamard manifold by minimizing a functional with a quadratic data term and a total variation--like regularizing term. To solve the convex minimization problem, we extend the Douglas--Rachford algorithm and its parallel version to symmetric Hadamard manifolds. The core of the Douglas--Rachford algorithm is reflections of the functions involved in the functional to be minimized. In the Euclidean setting the reflections of convex lower semicontinuous functions are nonexpansive. As a consequence, convergence results for Krasnoselski--Mann iterations imply the convergence of the Douglas--Rachford algorithm. Unfortunately, these general results do not carry over to Hadamard manifolds, where proper convex lower semicontinuous functions can have expansive reflections. However, splitting our restoration functional in an appropriate way, we have only to deal with special functions---namely, several distance-like functions and an indicator function of a special convex set. We prove that the reflections of certain distance-like functions on Hadamard manifolds are nonexpansive, which is an interesting result on its own. Furthermore, the reflection of the involved indicator function is nonexpansive on Hadamard manifolds with constant curvature so that the Douglas--Rachford algorithm converges here. Several numerical examples demonstrate the advantageous performance of the suggested algorithm compared to other existing methods such as the cyclic proximal point algorithm and half-quadratic minimization. Numerical convergence is also observed in our experiments on the Hadamard manifold of symmetric positive definite matrices with the affine invariant metric, which does not have a constant curvature.

Symmetric Hadamard matrices of order 116 and 172 exist

Abstract We construct new symmetric Hadamard matrices of orders 92, 116, and 172. While the existence of those of order 92 was known since 1978, the orders 116 and 172 are new. Our construction is based on a recent new combinatorial array (GP array) discovered by N. A. Balonin and J. Seberry. For order 116 we used an adaptation of an algorithm for parallel collision search. The adaptation pertains to the modification of some aspects of the algorithm to make it suitable to solve a 3-way matching problem. We also point out that a new infinite series of symmetric Hadamard matrices arises by plugging into the GP array the matrices constructed by Xia, Xia, Seberry, and Wu in 2005.