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

## Abstract

**:**

## 1. Introduction

#### 1.1. Lorenz Waveforms for Time Series Prediction

#### 1.2. Lorenz Waveforms for Communication Systems

#### 1.3. Lorenz Waveforms for Ranging Sensors

#### 1.4. Impact on Engineering Landscapes

## 2. Materials and Methods

#### 2.1. The Chaotic Lorenz System: Sensitivity, Information Creation & Symbolic Dynamics

#### 2.2. Extracting Approximate Basis Functions from Time-Series Simulations

#### 2.3. Regularizing Timing for the Lorenz Waveforms

- (i)
- measure and store the number of samples & indices between two peaks
- (ii)
- extend each data segment to convenient a sample interval (i.e., 50 samples or 100 samples)
- (iii)
- pad new x(t) values and increment new indices
- (iv)
- define a regeneration sample rate & corresponding period
- (v)
- time stretch/compress segment proportionally by original number of samples
- (vi)
- repeat for each pair of peaks

Algorithm 1 Algorithm that regularizes timing between Poincarè returns |

for each k in peaks do |

▹ Regularize timing between two peaks |

$r\leftarrow x\left(peaks\right(k):peaks(k+1\left)\right)$ |

$iROrig\leftarrow \left[peakI\right(k):1:peakI(k+1\left)\right]$ |

$numSamples\leftarrow length\left(r\right)$ |

$roundingPlace\leftarrow 50$ |

$numSamples\leftarrow ceil(numSamples/roundingPlace)*roundingPlace$ |

$rLong\leftarrow r$ |

$iRNew\leftarrow iROrig$ |

for i=1:numSamples-length(iROrig) do |

▹ Pad x(t) to a convenient length i.e., 6023 samples would map to 6050 |

$rLong(end+1)\leftarrow rLong\left(end\right)$ |

$iRNew(end+1)\leftarrow iRNew\left(end\right)+1$ |

end for |

▹ Define regeneration sample rate & period |

$shortLength\leftarrow roundingPlace$ |

$rShort\leftarrow zeros(1,shortLength)$ |

$iShort\leftarrow zeros(1,shortLength)$ |

$Fs\leftarrow shortLength/length\left(rLong\right)$ |

$Ts\leftarrow 1/Fs$ |

▹ Regenerate x(t) with sample-proportionate time compression |

$j\leftarrow 1$ |

$i\leftarrow 1$ |

while $i<length\left(rLong\right)-Ts$ do |

▹ Populate samples using regeneration period |

$rShort\left(j\right)\leftarrow rLong\left(i\right)$ |

$iShort\left(j\right)\leftarrow iRNew\left(i\right)$ |

$j\leftarrow j+1$ |

$i\leftarrow i+Ts$ |

end while |

$rShort\left(end\right)\leftarrow r(end-floor(Ts/2\left)\right);$ |

$iShort\left(end\right)\leftarrow iROrig(end-floor(Ts/2\left)\right);$ |

end for |

#### 2.4. Extracting a Single Basis Function as an Information Primitive

#### 2.5. Extracting Multiple Basis Functions to Form an Average Basis Function

- (i)
- set a global scanning index q to iterate template storage over the long waveform sample $x\left(\tau \right)$,
- (ii)
- store the ${q}^{\mathrm{th}}$ template segment of $x\left(\tau \right)$ to form the ${q}^{\mathrm{th}}$ instance of ${x}_{\mathrm{template}}\left(\tau \right)$,
- (iii)
- extract the symbolic content of the ${q}^{\mathrm{th}}$ instance of ${x}_{\mathrm{template}}\left(\tau \right)$.

- (i)
- set a scanning index k to iterate over the long waveform sample $x\left(\tau \right)$
- (ii)
- store the ${k}^{\mathrm{th}}$ template segment of $x\left(\tau \right)$ to form the ${k}^{\mathrm{th}}$ instance of ${x}_{\mathrm{compare}}\left(\tau \right)$
- (iii)
- extract the symbolic content of the ${k}^{\mathrm{th}}$ instance of ${x}_{\mathrm{compare}}\left(\tau \right)$
- (iv)
- perform a test of the Hamming distance criterion between the symbolic content of the ${k}^{\mathrm{th}}$ instance of ${x}_{\mathrm{compare}}\left(\tau \right)$ and the symbolic content of the ${q}^{\mathrm{th}}$ instance of ${x}_{\mathrm{template}}\left(\tau \right)$.

## 3. Results

#### Discussion

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Detailed flowchart for algorithm used to extract basis functions from the chaotic Lorenz system.

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**Figure 1.**Runge-Kutta (MATLAB ODE45) integration of the Lorenz equations showing (

**a**) a partitioned phase space defining symbols from a Poincarè section (red dots) of the chaotic attractor (black) and (

**b**) the time series of the $x\left(t\right)$ variable (black) with peaks (red) defining symbols from a Poincarè section.

**Figure 2.**(

**a**) Histograms of timing intervals for the unmodified, irregular Lorenz waveform $x\left(t\right)$ (red) showing a broad distribution and for the timing-regularized Lorenz waveform $x\left(t\right)$ (blue) showing a narrow distribution. (

**b**) Time-series waveform for the unmodified, irregular Lorenz waveform $x\left(t\right)$ (black) with Poincarè sections indicated at irregular time intervals (red dots) and waveform for the timing-regularized Lorenz waveform $x\left(t\right)$ (dashed-blue) with Poincarè sections indicated at regular time intervals (blue dots).

**Figure 3.**Graphical representation of two time-regularized waveforms with symbolic content that meets the Hamming distance criterion. (

**Top**) Sample waveforms from state variable x after being time-regularized. (

**Bottom**) Resulting basis function $p\left(\tau \right)$.

**Figure 4.**(

**a**) Two long waveform segments from state variable x after being time-regularized. (

**b**) Two examples of waveform segments with symbolic content that meets the Hamming distance criterion and their resulting instances of basis function $p\left(\tau \right)$. (

**c**) Two examples of waveform segments similar to Figure 4b with the exception that the summed peak is not inverted. Cases for Figure 4b,c must be accounted for when designing basis function extraction algorithms.

**Figure 5.**Results of processing a 5000 [s], ≈5,000,000 timestep chaotic Lorenz waveform $x\left(t\right)$ via Runga-Kutta (MATLAB ODE45) integration with the parameters ${\sigma}^{\prime}=\sigma -\mathrm{\Delta}\approx 10$, ${\rho}^{\prime}=\rho -\mathrm{\Delta}\approx 28$ and ${\beta}^{\prime}=\beta -\mathrm{\Delta}\approx \frac{8}{3}$ with $\mathrm{\Delta}={10}^{-6}$, $dt=0.001$ and length 5000 [s]. (

**a**) Bit statistics for each 8-bit long segment used to establish that the Hamming distance criterion was met. The red histogram shows the natural density of symbolic content provided by the x state variable. This red histogram constitutes the symbol sequences that are used to average the basis function $<p\left(\tau \right)>$. The blue histogram shows uniformly distributed bit sequences (all bit sequences are represented evenly with ≈200 examples of each) and constitutes the average basis function $<\rho \left(\tau \right)>$. (

**b**) Averaged basis functions in various regions of the red histogram from (

**a**). (

**c**) The resulting averaged basis functions for both natural bit density (solid red) and uniformly distributed bit density (dashed blue). Below each of these averaged basis functions show their small difference.

**Figure 6.**(

**a**) Bit statistics for instances where the Hamming distance criterion was met resulting in a recorded instance of a basis function for the state variable x. (

**b**) Extracted, averaged basis function for state variable x. (

**c**) Bit statistics for state variable y. (

**d**) Extracted, averaged basis function for state variable y. (

**e**) Bit statistics for state variable z. (

**f**) Extracted, averaged basis function for state variable z.

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**MDPI and ACS Style**

Beal, A.N.
Extracting Communication, Ranging and Test Waveforms with Regularized Timing from the Chaotic Lorenz System. *Signals* **2023**, *4*, 507-523.
https://doi.org/10.3390/signals4030027

**AMA Style**

Beal AN.
Extracting Communication, Ranging and Test Waveforms with Regularized Timing from the Chaotic Lorenz System. *Signals*. 2023; 4(3):507-523.
https://doi.org/10.3390/signals4030027

**Chicago/Turabian Style**

Beal, Aubrey N.
2023. "Extracting Communication, Ranging and Test Waveforms with Regularized Timing from the Chaotic Lorenz System" *Signals* 4, no. 3: 507-523.
https://doi.org/10.3390/signals4030027