André Barbé

Self-fractional Fourier functions and selection of modes

It is shown that any function from can be represented as a linear combination of M self-fractional Fourier functions of order M, which are orthogonal to each other. Each of them contains a selection of Hermite - Gauss modes of the generator function. The physical meaning of the synthesis of self-fractional Fourier functions is discussed.

Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2

Abstract A binary difference pattern (BDP) is a pattern obtained by covering an equilateral triangular grid by black and white circles in a dense hexagonal packing under a simple symmetric local matching rule. It is a subpattern in a specific graphical representation of the orbit of a cellular automaton that generates Pascals triangle modulo 2. Analytic conditions for certain types of geometric symmetry of these patterns are derived. These allow us to find all symmetric solutions and the cardinalities of the different symmetry classes. In the analysis, a central role is played by the so-called Pascal matrix — a square matrix that contains Pascals triangle modulo 2 (up to a certain size) — and by certain groups of geometric transformations of this matrix, featuring remarkable product properties for the Pascal matrix.

Self-fractional fourier images

Abstract Self-fractional Fourier images (SFFIs), which are invariant under the two-dimensional fractional Fourier transform, are considered. Necessary and sufficient conditions for an image to be a SFFI for given angles are formulated. An optical procedure for synthesizing SFFIs from a generator image is developed. The physical meaning of SFFI synthesis is discussed. An image decomposition on the set of orthogonal SFFIs, which simplifies image processing in fractional FT optical systems, is proposed.

Fractional Fourier and Radon-Wigner transforms of periodic signals

Abstract In this paper we consider specific features of the fractional Fourier transform (FT) for periodic signals. It is shown that the fractional FT of a periodic signal at some angles is the superposition of its scaled weighted and shifted replicas with additional quadratic phase factor. The modulus of the fractional FT (Radon–Wigner transform) at an angle α ≠π/2+π n of a periodic signal is also periodic. For a certain sequence of angles, the Radon–Wigner transforms are affine to each other. This feature simplifies the tomographic reconstruction procedure of periodic signals.

Self-imaging in fractional Fourier transform systems

The self-imaging phenomenon in a fractional Fourier transform optical system is described in the framework of self-fractional Fourier functions. The main properties of these functions are investigated. The possible application of fractional Fourier transform optical systems in optical signal processing is discussed.

Correlation and Spectral Properties of Multidimensional Thue-Morse Sequences

This paper considers higher-dimensional generalizations of the classical one-dimensional two-automatic Thue–Morse sequence on ℕ. This is done by taking the same automaton-structure as in the one-di...

Limit sets of automatic sequences

Abstract We show that for every k-automatic sequence there exists a natural number p>0 such that the sequences of the form (kpn+j)n⩾0 with j=0,…,p−1 are scaling sequences for f. Moreover, we demonstrate that every limit set is the union of certain basic limit sets.

Correlation functions of higher-dimensional automatic sequences

A procedure for calculating the (auto)correlation function , of an m-dimensional complex-valued automatic sequence , is presented. This is done by deriving a recursion for the vector correlation function Γker(f)(k) whose components are the (cross)correlation functions between all sequences in the finite set ker(f), the so-called kernel of f which contains all properly defined decimations of f. The existence of Γker(f)(k), which is defined as a limit, for all , is shown to depend only on the existence of Γker(f)(0). This is illustrated for the higher-dimensional Thue–Morse, paper folding and Rudin–Shapiro sequences.

Cellular automata, quasigroups and symmetries

Summary. We investigate cellular automata (CA) with a local rule

A measure for the mean level-crossing activity of stationary normal processes (Corresp.)

\phi : G^2 \rightarrow G