Chaotic Dynamics Event Signal Processing

• The Chaotic Dynamics Event Signal Processing Framework is a suite of advanced time-frequency transforms that decompose complex, chaotic signals using adaptive chirp projections.
• It employs an efficient FFT-based implementation by reducing computations to phase deformation and dilation steps for rapid signal analysis.
• The framework effectively separates and analyzes non-stationary signals, including plasma reflectometry echoes and stochastic self-similar processes.

A Chaotic Dynamics Event Signal Processing Framework provides a suite of methodological and mathematical tools for the analysis, separation, and characterization of signals exhibiting chaotic behavior. Such frameworks rely on advanced time-frequency and time-scale transformations that are sensitive to the underlying structure of chaotic signals, allowing practitioners to disentangle overlapping components, quantify complexity, and adaptively represent the intricate, often non-stationary, evolution of the signals. In the context of (Ricaud et al., 2010), this approach is realized via two novel transformations—the Linear Chirp Transformation (LCT) and the Hyperbolic Chirp Transformation (HCT)—each tailored for specific classes of chaotic signals and each mathematically connected to foundational tools such as Fourier, fractional Fourier, Mellin, and Lamperti transforms.

1. Linear Chirp Transformation (LCT): Definition and Operator Structure

The LCT generalizes spectral representations for signals in $L^2(\mathbb R)$ (or over finite intervals) by projecting the signal $f(t)$ onto an orthonormal basis of linear chirps: $$\psi_x^\theta(t) = \frac{1}{\sqrt{2\pi\,|\sin\theta|}}\,\exp\left(-i\frac{t^2}{2\tan\theta} + i\,\frac{x}{\sin\theta}t\right).$$ Here, the parameter $\theta\notin \{n\pi\,|\, n\in\mathbb Z\}$ effectively “rotates” the time-frequency representation: $\theta=0$ corresponds to the time domain, $\theta=\pi/2$ is the frequency domain, and intermediate $\theta$ implement a continuum of oblique representations. Each basis function is a linear chirp with instantaneous frequency

$$\phi'(t) = -\frac{1}{\tan\theta}\,t + \frac{x}{\sin\theta}.$$

The associated selfadjoint operator is $$A_L(\theta) = \cos\theta\, t - i\,\sin\theta\, (d/dt)$$, whose eigenfunctions are the linear chirps themselves.

Importantly, the LCT computation can be efficiently realized by reducing the projection to a deformation by a phase factor and a dilation (via the operator $$D_\theta f(t) = \sqrt{|\sin\theta|} f(\sin\theta\, t)$$), followed by a Fast Fourier Transform (FFT)—thus making the framework compatible with large-scale data processing.

2. Hyperbolic Chirp Transformation (HCT): Scaling and Self-Similarity

Whereas the LCT is suited to general $L^2$ signals, the HCT is designed for signals on $\mathbb{R}^+$, particularly self-similar processes such as fractional Brownian motion. For a real $c(t)$, basis functions are defined as

$$\varphi_x^\theta(t) = \frac{1}{\sqrt{2\pi|\sin\theta|\, t}}\,\exp\left(-i\frac{1}{\tan\theta}\,c(t)+ i\,\frac{x}{\sin\theta}\,\ln t\right).$$

When $\theta = \pi/2$, the transformation reduces to the Mellin transform (modulo normalization), thus recasting the problem in a scale-invariant, logarithmic time domain. With the specific choice $c(t) = \ln t$, the HCT and LCT become related through the Lamperti transform, an isometry connecting stationary and self-similar signal representations: $\left(L_{-1/2} f\right)(t) = t^{-1/2}\, f(\ln t).$ The HCT thus facilitates the analysis of signals where self-similarity rather than stationarity dominates, broadening the framework’s applicability.

3. Separation and Extraction in Non-Stationary and Self-Similar Chaotic Signals

The framework’s practical strength is demonstrated in the analysis of reflectometry signals from turbulent plasma environments, where electromagnetic pulses exhibit both desired reflected signals (varying due to plasma turbulence) and undesired static echoes (e.g., from chamber walls). By tuning the LCT parameter $\theta$, one can perform a “slice” of the time-frequency plane along any desired slope $1/\tan\theta$, isolating distinct signal ridges corresponding to different propagation paths. After thresholding the energy localized by the LCT in the $(x,\theta)$ plane and reconstructing via inverse LCT, each individual component (e.g., wall echoes vs. turbulent reflections) can be separated and analyzed independently.

For stochastic, self-similar signals (such as Brownian motion), where standard Fourier-type methods fail due to the absence of stationary spectral content, the HCT is adapted for detection and extraction of localized, scale-invariant chaotic features.

4. Mathematical Connections: Interrelations Amongst Canonical Transforms

The LCT and HCT embed and generalize several classical transforms:

Transform	Parameterization	Limiting Case
Fourier	$\theta = \pi/2$ (LCT)	Frequency domain
Fractional Fourier (FRFT)	LCT for general $\theta$	Varies smoothly from time to frequency domain
Mellin	$\theta = \pi/2$ (HCT)	Logarithmic scale analysis
Lamperti	$c(t) = \ln t$ (HCT)	Connects LCT and HCT

The FRFT is particularly noteworthy: the LCT is mathematically equivalent (modulo normalization and convention) to the FRFT, with $\theta$ playing the role of the rotation angle in time-frequency space. The equivalence with the Mellin transform (HCT at $\theta = \pi/2$) and the existence of the Lamperti operator (bridging stationary and self-similar process domains) provide a coherent theoretical framework, situating these new transformations within the landscape of integral transforms for signal analysis.

5. Computational Efficiency and Algorithmic Considerations

Both LCT and HCT are constructed so their action can be reduced, up to deformation and dilation, to an FFT (or, for HCT, Mellin-FFT hybrids). Specifically:

• For LCT:
  1. Multiply input $f(t)$ by a quadratic phase factor $\exp(-it^2/(2\tan\theta))$.
  2. Dilate argument: $t\mapsto \sin\theta\cdot t$.
  3. FFT of this result yields LCT coefficients.

• For HCT:

  1. Multiply by $\exp(-i c(t)/\tan\theta)$.
  2. Convert to logarithmic time through $t\mapsto \ln t$ if needed.
  3. FFT or Mellin transform to evaluate coefficients.

This FFT-based implementation is essential for scalability, ensuring applicability to high-dimensional or real-time event signal processing in physical experiments.

6. Scope, Limitations, and Real-World Impact

The framework addresses the need for flexible, adaptive decomposition of complex signals where chaotic dynamics dominate. Its particular strengths are:

• The ability to “rotate” the analysis domain to match the dominant structure of ridges or chirps in time-frequency (or time-scale) space, adapting to the local behavior of the chaotic features.

• Efficient separation of overlapping components in highly non-stationary environments, as in plasma reflectometry, where inverse transformation aids accurate component reconstruction without cross-talk.
• Applicability to both stationary (LCT) and self-similar (HCT) processes, offering a unified methodology for diverse physical, biological, and engineering contexts.

Limitations include the dependence of HCT on a well-chosen function $c(t)$ for optimal performance in self-similar cases, and the requirement for careful parameter selection ($\theta$) in practical deployments—potentially necessitating automated or adaptive parameter tuning in highly nonstationary environments.

7. Summary Table: Transform Features and Applications

Transformation	Signal Domain	Parameter(s)	Key Application	Computational Core
LCT	$L^2(\mathbb{R})$	$\theta$	Chaotic/ridge separation in non-stationary, square-integrable signals (e.g., plasma reflectometry)	FFT-based
HCT	$L^2(\mathbb{R}^+)$	$\theta$, $c(t)$	Detection of chaos in self-similar (scale-invariant) signals (e.g., stochastic motion)	Mellin/FFT-based

This encapsulates the distinguishing mathematical innovations and practical consequences of the Chaotic Dynamics Event Signal Processing Framework as established in (Ricaud et al., 2010). The interplay between the geometric perspective (rotational parameter $\theta$ and Lamperti transform) and algorithmic tractability (FFT reduction) is central, enabling precise, efficient, and interpretable analysis of chaotic event signals across a range of domains.