Design and Implementation of a Digital Dual Orthogonal Outputs Chaotic Oscillator

Abstract

Discrete time dynamical chaotic systems obey a set of recurrence equations involving one or more variables. Many chaotic maps have been proposed. None that simultaneously provides two sine–cosine outputs has stationary mean and standard deviation, or is quite robust with respect to the data format used in the hardware implementation. Here, we propose a chaotic oscillator based on a complex phasor whose angular argument evolves according to a geometric progression that is independent of the instantaneous amplitude. In order to maintain the oscillations, the phasor magnitude is normalized at each iteration using an approximation factor. The statistical characteristics of this oscillator are stationary in the short term, and do not depend on the initial conditions. The mean and standard deviation of the two orthogonal sequences quickly approach 0 and 1 / 2 , respectively. The resulting distribution is similar to that of a digital sine with a constant angular step. We also present an FPGA architecture and its implementation results. This oscillator can be used in modulation schemes, such as the chaotic shift keying one or for data and image encryption. Finally, we show an original application that exploits the orthogonality of the two chaotic signals for the simultaneous encryption of two digital images.