Statistical analysis of Bernoulli, logistic, and tent maps with applications to radar signal design

Abstract

The uniqueness of the Bernoulli frequency modulated signal, and other chaos-based FM signals, can be exploited to improve the performance of the Synthetic Aperture Radar systems. Recent work 1,2 suggests that the Bernoulli map has an unusual behavior compared to other one dimensional discrete maps, such as Logistic or Tent maps. Additional work3 indicates that the sum of consecutive Bernoulli samples is generally non-Gaussian, except when the map parameters A= 0.5 and B = 1.8. This motivates us to analyze the behavioral differences of the maps for various parameters using the Lyapunov exponent, pseudo-phase spatial trajectory and neighbor samples correlation. Specifically, the correlation of Bernoulli samples is analyzed in terms of the probability density function which is derived from experimental data.. Some of statistical tools used include the Forbenius-Perron Operator, and the correlation properties of chaotic sequences. In addition, other measurements of chaos derived from nonlinear dynamical modeling will be used such as: the Lyapunov exponent and the bifurcation diagram. Results show differences between the calculated features; for example, the Lyapunov exponent is bigger for Bernoulli FM than Logistic or Tent FM. In summary, we determined that Bernoulli FM is more chaotic than Logistic or Tent FM. We have also found another singularity in the correlation of sequence samples for the Bernoulli map.