Walsh-Hadamard Coefficients Research Articles

Analytic Representation vs. Angle Modulation of Hilbert Transform of Fast Walsh-Hadamard Coefficients (HTFWHC) in Epileptic EEG Classification

Analytic Representation vs. Angle Modulation of Hilbert Transform of Fast Walsh-Hadamard Coefficients (HTFWHC) in Epileptic EEG Classification

Boolean Functions with Affine Annihilators

In the article we study boolean functions with affine annihilators. We have obtained results in both, estimating the number of functions under study and defining the relationship between Walsh-Hadamard coefficients of an arbitrary boolean function and its affine annihilator available. The second section of this article focuses on estimating the number of boolean functions with affine annihilators. The value has top and bottom bound. Besides, we have obtained the asymptotic estimate of the number of boolean functions with affine annihilators. The third section studies the Walsh-Hadamard coefficients of boolean functions with affine annihilators. First, we have derived the dependence of the Walsh-Hadamard coefficient on the distance between an arbitrary boolean function and a vector space of the affine function’s annihilators. Based on this result, we have obtained the dependence of distance between an arbitrary boolean function and a set of functions with affine annihilators on the spectrum of given function. Also we have defined the necessary and sufficient condition for the arbitrary boolean function to be with an affine annihilator available. Using the results obtained we bounded an absolute value of Walsh-Hadamard coefficients.Also we suggested a method for boolean equations analysis, which is based on two known methods. Namely, we used an analysis using annihilators and an analysis using linear analogs. We have obtained an estimate of the success probability of the suggested method for an arbitrary boolean function. Also we proved that bent functions are the most resistant to this analysis.The results obtained can be used in analysis of boolean equations. Also obtained dependences can be used, for instance, to study bent functions and algebraic immunity of boolean functions.

Landscape Boolean functions

In this paper we define a class of generalized Boolean functions defined on $ {\mathbb F}_2^n $ with values in $ {\mathbb Z}_q $ (we consider $ q = 2^k $, $ k\geq 1 $, here), which we call landscape functions (whose class contains generalized bent, semibent, and plateaued) and find their complete characterization in terms of their Boolean components. In particular, we show that the previously published characterizations of generalized plateaued Boolean functions (which includes generalized bent and semibent) are in fact particular cases of this more general setting. Furthermore, we provide an inductive construction of landscape functions, having any number of nonzero Walsh-Hadamard coefficients. We also completely characterize generalized plateaued functions in terms of the second derivatives and fourth moments.

Decomposing Generalized Bent and Hyperbent Functions

In this paper, we introduce generalized hyperbent functions from $ {\mathbb F}_{2^{n}}$ to $ {\mathbb Z}_{2^{k}}$ , and investigate decompositions of generalized (hyper)bent functions. We show that generalized (hyper)bent functions $f$ from $ {\mathbb F}_{2^{n}}$ to $ {\mathbb Z}_{2^{k}}$ consist of components which are generalized (hyper)bent functions from $ {\mathbb F}_{2^{n}}$ to $ {\mathbb Z}_{2^{k^\prime }}$ for some $k^\prime . For even $n$ , most notably we show that the g-hyperbentness of $f$ is equivalent to the hyperbentness of the components of $f$ with some conditions on the Walsh–Hadamard coefficients. For odd $n$ , we show that the Boolean functions associated to a generalized bent function form an affine space of semibent functions. This complements a recent result for even $n$ , where the associated Boolean functions are bent.

Towards holographic “brain” memory based on randomization and Walsh–Hadamard transformation

The holographic conceptual approach to cognitive processes in the human brain suggests that, in some parts of the brain, each part of the memory (a neuron or a group of neurons) contains some information regarding the entire data.In Dolev and Frenkel (2010, 2012) we demonstrated how to encode data in a holographic manner using the Walsh–Hadamard transform. The encoding is performed on randomized information, that is then represented by a set of Walsh–Hadamard coefficients. These coefficients turn out to have holographic properties. Namely, any portion of the set of coefficients defines a “blurry image” of the original data.In this work, we describe a built-in error correction technique—enlarging the width of the matrix used in the Walsh–Hadamard transform to produce a rectangular Hadamard matrix. By adding this redundancy, the data can bear more errors, resulting in a system that is not affected by missing coefficients up to a certain threshold. Above this threshold, the loss of data is reflected by getting a “blurry image” rather than a concentrated damage. We provide a heuristic analysis of the ability of the technique to correct errors, as well as an example of an image saved using the system. Finally, we give an example of a simple implementation of our approach using neural networks as a proof of concept.

Walsh-Hadamard Optimization of Fixed Polarity Reed-Muller Transform

By investigating links between Reed-Muller transform and Walsh-Hadamard spectra an exact and non-exhaustive algorithm for the generation of optimal Reed-Muller expansions directly from just few Walsh-Hadamard spectral coefficients has been developed. The algorithm makes use of the properties of Walsh-Hadamard spectra and by using only few Walsh-Hadamard coefficients the optimal Reed-Muller expansion is obtained for all Boolean functions through the provided equations in the new algorithm.

On inherited properties of restricted Boolean functions

AbstractFor a property P of Boolean functions, a Boolean function f(x), x ∈ VIn this paper, this approach is applied to the following property of Boolean functions: the value f̂(α)/2We give convenient criteria for (H, α)-stability in terms of zeros of the Walsh-Hadamard coefficients, and establish relations between the (H, α)-stability, correlation immunity, and m-resiliency.