###### 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.  
chaos; joint radar-communication; solvable chaos; time series prediction; chaotic test signal; reservoir computing +  
theory [24; 25]. [PERSON] identified oscillatory basis functions that decay in different directions (both causally and acausally), which can be linearly combined with random weighting factors to produce deterministic chaotic waveforms that approximate solutions to chaotic systems. This insight suggests that straight-forward signal processing techniques [26] that rely on linear analysis and synthesis methods (i.e., convolution sums) can be used to guide the application of chaos in engineered systems in a quantifiable and meaningful way. However, previous work [21] lacked details regarding the specific implementation of algorithms, the statistics of the bit sequences used, and the timing information that is discarded by the resulting algorithms. In this paper, we provide these essential details to enhance the accessibility and applicability of these techniques with the intent of quantifying advances in machine learning algorithms and waveform design for communication and sensing applications. Here, we report three new contributions: (1) a presentation of a detailed realization of [PERSON]'s algorithm, (2) the statistics of the bitstreams produced by the chaotic Lorenz system as associated with extracted basis functions, and (3) the statistics of the timing information that is removed from non-uniform chaotic Lorenz systems.  
This manuscript is structured as follows. In the introduction section, we describe the use and impact of chaotic Lorenz waveforms on various applications, including time series prediction algorithms, communication systems, and ranging sensors. In Section 2, we present the materials and methods used. We define the chaotic Lorenz system, evaluate the entropic content of its waveforms, and describe partitions to the phase space used to define symbol sequences. In Sections 2.2 and 2.3, we discuss the process of extracting approximate basis functions from time-regularized Lorenz waveforms. In this section, we provide timing and algorithmic details. In Sections 2.4 and 2.5, we detail the algorithm used for basis function extraction. The results of these algorithms for all three state variables of the Lorenz system are presented. Section 3 provides a discussion of these results. Finally, we conclude with our remarks in Section 4.