Extracting Communication, Ranging and Test Waveforms with Regularized Timing from the Chaotic Lorenz System

Abstract

We present an algorithm for extracting basis functions from the chaotic Lorenz system along with timing and bit-sequence statistics. Previous work focused on modifying Lorenz waveforms and extracting the basis function of a single state variable. Importantly, these efforts initiated the development of solvable chaotic systems with simple matched filters, which are suitable for many spread spectrum applications. However, few solvable chaotic systems are known, and they are highly dependent upon an engineered basis function. Non-solvable, Lorenz signals are often used to test time-series prediction schemes and are also central to efforts to maximize spectral efficiency by joining radar and communication waveforms. Here, we provide extracted basis functions for all three Lorenz state variables, their timing statistics, and their bit-sequence statistics. Further, we outline a detailed algorithm suitable for the extraction of basis functions from many chaotic systems such as the Lorenz system. These results promote the search for engineered basis functions in solvable chaotic systems, provide tools for joining radar and communication waveforms, and give an algorithmic process for modifying chaotic Lorenz waveforms to quantify the performance of chaotic time-series forecasting methods. The results presented here provide engineered test signals compatible with quantitative analysis of predicted amplitudes and regular timing.