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Convolution and Correlation Theorems Associated with the Linear Canonical Empirical Wavelet Transform and Their Applications

This paper establishes a rigorous convolution-correlation calculus for a fixed-partition linear canonical empirical wavelet transform, proving exact representation theorems, frame bounds, and stability results while demonstrating the invariance of synthesized identities under dual choices and validating these findings through reproducible computations.

Original authors: Owais Ahmad, Aamir Ayoub Mir

Published 2026-10-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Owais Ahmad, Aamir Ayoub Mir

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Signals in the real world are rarely simple, steady tones. They are complex, shifting patterns—like the roar of a jet engine changing pitch as it passes, or the fluctuating rhythm of a heartbeat during exercise. To understand these signals, scientists break them down into their constituent frequencies, much like a prism splits white light into a rainbow. However, standard tools for this task often assume the signal's structure is fixed and predictable. When a signal changes its nature over time or follows a curved, accelerating path, these traditional methods struggle to keep up. To handle such complexity, researchers have developed more flexible mathematical frameworks that can adapt to the signal's specific geometry. One such framework is the linear canonical transform, a powerful tool that can analyze signals moving along curved trajectories in a way that standard frequency analysis cannot. Another is the empirical wavelet transform, a method that automatically detects where a signal's frequency content changes and builds a custom set of filters to isolate those parts.

The challenge arises when these two advanced tools are combined. While each works well on its own, merging them creates a mathematical puzzle: how do you perform operations like mixing two signals or comparing them when they have been analyzed by two different, independently generated sets of filters? If the filters do not align perfectly, the usual rules for combining them break down. This is the specific problem addressed by Owais Ahmad and Aamir Ayoub Mir in their recent work. They sought to establish a rigorous set of rules that allow two signals, analyzed through different adaptive filters, to be combined or compared with mathematical precision. Their goal was not just to find a workaround, but to prove that a stable, exact relationship exists between these differently analyzed signals, even when the filters used to detect them are distinct.

The researchers began by treating the detection of signal boundaries as a separate step from the actual analysis. Once the boundaries are detected and the filters are set, they are "frozen" into a fixed arrangement. This allows the team to treat the system as a stable mathematical structure rather than a moving target. They then proved that when you take two different sets of these filters and multiply them together, the result is a new, refined set of filters that covers the same ground but with greater precision. This new set acts as a common ground, or a "common refinement," where the information from both original systems can be safely combined. They demonstrated that this combined system has specific mathematical limits that guarantee the analysis remains stable and that the original signal can be perfectly reconstructed from the pieces.

A key finding of the study is the discovery of a unique, optimal way to reverse the process. When breaking a signal into pieces, there are often many ways to put it back together. The authors proved that there is one specific method—the "canonical dual"—that is mathematically the most efficient and stable way to reconstruct the signal. They showed that this method is unique because it minimizes the error at every single point. Furthermore, they established that the final result of combining two signals does not depend on which specific reconstruction method is chosen, as long as it is a valid one. This means the core information extracted from the signals is intrinsic and reliable, regardless of the mathematical "coordinate system" used to view it.

To ensure these findings are not just theoretical, the team tested their rules with computer simulations using compact, well-defined windows. They created two different sets of filters with different center points and tested whether the signals could be correctly combined and compared. The results were precise: the difference between the predicted outcome and the actual computer calculation was so small it was effectively zero, measuring around 1.97 × 10⁻¹⁶ for convolution and 1.98 × 10⁻¹⁶ for correlation. These numbers are so close to zero that they represent the limit of the computer's ability to calculate, confirming that the mathematical rules hold true in practice. They also tested how sensitive the system is to small errors. When they slightly shifted the position of the filters, the change in the output was directly proportional to the size of the shift, proving that the system is stable and does not amplify small mistakes into large errors.

The work also clarifies how these new rules connect to older, simpler methods. The researchers showed that if the complex, curved geometry of their system is simplified to a straight line, their new rules naturally reduce to the standard, well-known rules used in basic signal processing. This confirms that their advanced framework is a true generalization of existing knowledge, capable of handling the complex, curved paths of real-world signals while still agreeing with established theory in simpler cases. By separating the adaptive detection of signal boundaries from the rigid mathematical operations that follow, the authors have created a clear path for analyzing complex, non-stationary signals. Their work provides a solid foundation for future applications where signals change rapidly or follow unpredictable paths, ensuring that the tools used to understand them are both flexible and mathematically sound.

Technical Summary: Convolution and Correlation Theorems Associated with the Linear Canonical Empirical Wavelet Transform

Problem Statement
The paper addresses a fundamental gap in the mathematical theory of adaptive signal analysis: the interaction between convolution and correlation operations and distinct, data-derived empirical window systems within the Linear Canonical Transform (LCT) domain. While the LCT provides a quadratic-phase geometry suitable for oscillatory signals (chirps), and the Empirical Wavelet Transform (EWT) offers adaptive spectral partitioning, their combination presents significant algebraic difficulties. Specifically, when two signals are analyzed through independently detected empirical partitions (resulting in different index sets and overlap regions), standard coefficient-domain convolution or correlation theorems cannot be derived merely by modifying classical Fourier identities. The challenge lies in defining a common spectral geometry, establishing a rigorous frame-theoretic reconstruction, and proving that the resulting algebraic identities are invariant under the choice of frame duals.

Methodology
The authors develop a rigorous "fixed-partition" framework where empirical windows are detected and then frozen, separating the adaptive stage from the linear operator-theoretic stage. The methodology proceeds through the following steps:

Unitary LCT Foundation: The construction begins with a consistently normalized unitary Linear Canonical Transform (LCT), denoted $L_M$ , defined by a symplectic matrix $M = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$ with $B > 0$ . The transform is factorized into chirp multiplications and a scaled Fourier transform.
Empirical Window Systems: The authors define "admissible empirical canonical window systems" as locally finite families of bounded functions with compact essential support in the canonical frequency domain. These systems satisfy a frame condition (bounded overlap).
Common Refinement: For two distinct input window systems, $\mathcal{M}$ and $\mathcal{N}$ , the authors construct a "common refinement" $\mathcal{R}$ by taking the pointwise product of the input windows ( $r_{jk} = m_j n_k$ ). They prove that this product system forms a valid frame with an overlap function equal to the product of the input overlaps.
Dual Characterization: The canonical dual of the refinement is characterized as the unique pointwise minimum- $\ell^2$ norm dual.
Operator Derivation: Convolution and correlation operators are defined via "chirp conjugation" of their ordinary counterparts. This ensures that the phase factors are fixed by the LCT convention rather than chosen arbitrarily.
Stability Analysis: The paper establishes Lipschitz stability estimates for the analysis map, the product refinement, and the canonical duals under perturbations of the empirical window boundaries.

Key Contributions and Results

Exact Representation Theorem: The paper proves a bridge between the multiplier formulation (filter-bank) and scale–centre canonical empirical wavelet atoms. It establishes that the analyzing atoms are related to the wavelet atoms by a specific quadratic phase and a rescaling of the translation variable.
Cross-Partition Convolution and Correlation Theorems:
- The authors derive exact factorization formulas for the LCT of the convolution ( $f \star_M g$ ) and correlation ( $f \diamond_M g$ ) of two signals.
- Crucially, these theorems handle the case where $f$ and $g$ are analyzed by different empirical window systems. The resulting coefficient-domain identity involves a sum over the common refinement, weighted by the canonical dual.
- Dual Invariance: A significant result is that while individual coefficient terms depend on the choice of the dual frame, the synthesized convolution and correlation identities are invariant under any admissible choice of the dual. This separates intrinsic structural statements from coordinate-dependent representations.
Stability Estimates: The paper provides quantitative bounds showing that small perturbations in the empirical window boundaries lead to controlled (Lipschitz continuous) changes in the coefficient maps and the refinement geometry. This makes the dependence on boundary detection mathematically explicit.
Reduction Principles: The framework is shown to reduce correctly to standard empirical Fourier filter-bank identities when the LCT matrix corresponds to the standard Fourier transform, and to fractional Fourier transforms for specific rotation matrices.

Numerical Validation
The theoretical results are verified through reproducible computations using compactly supported cosine windows. The numerical experiments:

Instantiate the theory on a finite canonical-frequency interval with two distinct, non-aligned input window systems.
Verify the cross-partition convolution and correlation identities with residuals near machine precision ( $\approx 10^{-16}$ ).
Confirm dual invariance by testing with an alternative dual (perturbed by a null-space component), showing the synthesized result remains unchanged.
Demonstrate Lipschitz stability by perturbing window centers and observing that the coefficient changes remain within the theoretical bounds.

Significance and Claims
The paper claims to provide the first rigorous convolution–correlation calculus for a fixed-partition realization of the Linear Canonical Empirical Wavelet Transform (LCEWT). Its primary significance lies in resolving the structural incompatibility between independent empirical partitions and algebraic signal operations. By establishing the "common refinement" and proving the invariance of the synthesized identities under dual choice, the authors separate the intrinsic geometry of the signal decomposition from the coordinate representation.

The work does not claim to solve specific application benchmarks (e.g., denoising performance comparisons) but rather establishes the necessary mathematical infrastructure (frame bounds, reconstruction, stability, and algebraic identities) to support such applications. The authors note that the stability estimates address boundary sensitivity directly, a feature absent in purely algebraic same-partition expansions. The paper concludes by identifying open questions regarding sharper perturbation bounds based on boundary locations, unconditional synthesis for infinite interactions, and regularity classes for integrable translation-domain kernels.

Technical Summary: Convolution and Correlation Theorems Associated with the Linear Canonical Empirical Wavelet Transform

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