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Efficient Time–Frequency Representations via the Quadratic-Phase Hilbert Transform

This paper introduces the short-time quadratic-phase Fourier transform (ST-QPFT) and its multi-resolution variant to overcome the global limitations of the quadratic-phase Hilbert transform by localizing the kernel, while establishing their fundamental mathematical properties, clarifying their relationship to classical transforms via a chirp-reduction lemma, and validating the theory through numerical simulations.

Original authors: MUSADIQ SHAHEEN, FIRDOUS A. SHAH

Published 2026-09-03

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CC BY 4.0

Original authors: MUSADIQ SHAHEEN, FIRDOUS A. SHAH

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 are the language of the modern world, carrying everything from the voice of a caller on a phone to the seismic tremors of an earthquake. For signals that remain steady, like a pure musical note held forever, scientists have long had a reliable tool to decode them: a mathematical process that breaks the sound down into its constituent frequencies. However, the real world is rarely so static. Most signals we encounter are non-stationary, meaning their pitch or frequency changes as time passes. A bird's song, a radar pulse, or the rising whine of a passing siren are all examples of sounds that evolve. To understand these, researchers use a technique called time-frequency analysis, which attempts to map out not just what frequencies are present, but exactly when they occur. The standard method for this, known as the short-time Fourier transform, works by sliding a fixed window across the signal, taking a snapshot of the sound at each moment. While useful, this approach has a fundamental flaw when dealing with signals that change frequency rapidly, such as a chirp—a sound that sweeps smoothly from a low pitch to a high one. Because the window is fixed in size, the resulting map of the signal appears as a smeared, diagonal blur rather than a sharp line, obscuring the precise details of how the sound is changing.

A team of researchers at the University of Kashmir, Musadiq Shaheen and Firdous A. Shah, has developed a new mathematical framework designed to cut through this blur. Their work builds upon a more flexible mathematical tool called the quadratic-phase Fourier transform, which uses a curved, or "chirped," kernel to analyze signals. Think of this kernel as a lens that can be tuned to match the specific curvature of a signal's frequency change. While this tool was already known to be powerful for analyzing signals in a global sense, it lacked the ability to show how those signals evolve over time. The researchers' primary achievement was to combine this advanced, curved lens with the sliding-window technique, creating a new method they call the short-time quadratic-phase Fourier transform. By doing so, they created a way to view signals that change frequency with a level of clarity that the standard method simply cannot achieve.

The core of their discovery lies in a clever mathematical trick that simplifies the entire problem. They proved that their new, complex transform is mathematically equivalent to taking a standard time-frequency analysis of a signal that has first been "demodulated," or stripped of its specific frequency curve. In practice, this means that if a signal is a linear chirp—a sound that rises in pitch at a constant rate—and the researcher tunes their analysis tool to match that exact rate, the signal transforms from a messy diagonal streak into a perfectly sharp, horizontal line. This concentration of energy is not just a visual improvement; it represents a fundamental increase in efficiency. In their computer simulations, the researchers tested this method on signals that mimicked real-world chirps. When they used the standard method, the energy of the signal was spread out, making it harder to pinpoint exactly when a specific frequency occurred. When they used their new matched method, the energy collapsed into a tight, concentrated line. For one specific test signal, this new approach reduced the measure of this spread by nearly 25 percent, a statistically significant improvement that indicates a much sharper, more precise representation of the data.

Beyond simply sharpening the image of the signal, the researchers also clarified how to construct what is known as an "analytic signal" within this new mathematical domain. In signal processing, an analytic signal is a special version of a real-world sound that contains only positive frequencies, effectively removing the redundant mirror image that exists in standard analysis. The team demonstrated that their new method produces a clean, one-sided spectrum, meaning all the signal's energy is concentrated on the positive side of the frequency scale, just as it should be for a perfect analytic signal. They further defined a way to extract the "instantaneous frequency"—the exact pitch of the signal at any given split second—using this new framework. In their tests, this method recovered the starting frequency of a chirp with near-perfect accuracy, showing that the signal's pitch could be tracked without the smearing or error that plagues older techniques.

The researchers also introduced a multi-resolution version of their tool, which adjusts the size of the analysis window based on the frequency being observed. This allows for a finer look at high-frequency changes while maintaining a broader view of low-frequency trends, a feature that is crucial for signals that span a wide range of pitches. While their simulations showed that this method is highly effective for signals that match the specific curve of the analysis tool, they acknowledged a limitation: if a signal contains multiple components with different rates of change, a single setting cannot perfectly focus on all of them simultaneously. However, the ability to scan through different settings to find the best match offers a practical solution. The work suggests that by tailoring the mathematical lens to the specific shape of the signal, scientists can achieve a level of clarity in time-frequency analysis that was previously out of reach, turning blurred diagonals into sharp, informative lines.

Technical Summary: Efficient Time–Frequency Representations via the Quadratic-Phase Hilbert Transform

Problem Statement
Time-frequency analysis aims to represent signals that exhibit both temporal and spectral evolution, a characteristic common in communications, radar, and biomedicine. While the Short-Time Fourier Transform (STFT) is the standard tool for non-stationary signals, it relies on a fixed window and a basis of pure harmonics. This limits its resolution for signals with rapidly varying instantaneous frequencies, such as linear frequency-modulated (chirp) signals, which appear as smeared diagonal ridges rather than sharp curves in the time-frequency plane.

Recent developments have introduced the Quadratic-Phase Fourier Transform (QPFT) and the Quadratic-Phase Hilbert Transform (QPHT) to address signals with quadratic phase structures. The QPFT unifies various integral transforms (Fourier, fractional, linear-canonical, and special-affine) under a single five-parameter quadratic kernel. However, both QPFT and QPHT are global operators; they provide a single spectral representation and fail to capture the temporal evolution of a signal's spectrum. The central problem addressed in this paper is the lack of a localized, efficient time-frequency representation within the quadratic-phase domain that can resolve the temporal evolution of non-stationary spectra, particularly for chirp-type signals.

Methodology
The authors propose a localization strategy by introducing a window function into the quadratic-phase framework. The core methodology relies on a "chirp-reduction lemma," which establishes a structural identity between the proposed transforms and classical transforms of demodulated signals.

Short-Time Quadratic-Phase Fourier Transform (ST-QPFT):
The authors define the ST-QPFT by sliding a window function $g(t)$ across the signal $f(t)$ before applying the QPFT kernel.
- Chirp-Reduction Lemma: The paper proves that the ST-QPFT of a signal $f$ is equivalent to an ordinary STFT of a chirp-demodulated signal $\tilde{f}(t) = f(t)e^{-i(At^2+Dt)}$ , evaluated on a rescaled frequency axis ( $B\omega$ ), multiplied by a quadratic phase factor.
- Derivation of Properties: Leveraging this identity, the authors derive boundedness, an energy (Parseval) relation with an explicit factor $B^{-1}$ , an inversion formula, and covariance laws. These laws reveal a "time-frequency shear," where a time shift induces a frequency shift proportional to the time displacement, a characteristic induced by the quadratic phase.
Quadratic-Phase Analytic Signal and Spectrum:
The paper clarifies the relationship between the QPHT and the classical Hilbert transform.
- It is proven that the QPHT is exactly a chirp-conjugated version of the classical Hilbert transform.
- Consequently, the "quadratic-phase analytic signal" is shown to possess a strictly one-sided QPFT spectrum (vanishing for negative frequencies).
- Based on this, the authors define a quadratic-phase Hilbert spectrum using instantaneous amplitude and frequency, derived from the analytic signal.
Multi-Resolution Variant (MR-ST-QPFT):
To address signals spanning wide frequency bands, the authors adapt the "window-size-fixed-in-frequency" (STFT-FD) construction by Mateo and Talavera. In this variant, the window length is fixed in terms of the number of cycles ( $N_c$ ) rather than time. This allows for shorter windows at high frequencies and longer windows at low frequencies, providing multi-resolution analysis without requiring adaptive filters or band-pass decomposition.

Key Contributions
The paper makes four primary contributions:

Introduction of ST-QPFT: A genuine joint time-frequency representation in the quadratic-phase domain, with rigorous proofs of its fundamental properties (boundedness, energy conservation, inversion, and covariance) derived directly from the chirp-reduction lemma.
Clarification of Analytic Signal Theory: A proof that the QPHT acts as a chirp-conjugated classical Hilbert transform, establishing that the resulting quadratic-phase analytic signal has a strictly one-sided spectrum.
Definition of Quadratic-Phase Hilbert Spectrum: The formulation of an instantaneous frequency representation based on the quadratic-phase analytic signal.
Multi-Resolution Extension: The adaptation of the STFT-FD concept to the quadratic-phase domain to create the MR-ST-QPFT, offering variable time-frequency resolution.

Results
The theoretical claims are validated through numerical simulations using Python/NumPy/SciPy on linear chirp signals (both single and superimposed).

Exact Reconstruction: The ST-QPFT allows for perfect signal reconstruction with machine-precision errors ( $\sim 10^{-16}$ ).
One-Sided Spectrum: The quadratic-phase analytic signal successfully eliminates negative frequency components, with residual energy in the negative domain reduced to $\sim 10^{-7}$ .
Energy Concentration: For signals matched to the quadratic-phase kernel (where the signal's chirp rate equals the kernel's parameter $A$ $A$ ), the ST-QPFT collapses the diagonal chirp ridge into a sharp horizontal line.
- In terms of Rényi entropy (a measure of time-frequency concentration), the matched ST-QPFT reduced entropy by 24.3% for a single chirp and 16.6% for a superposition of two parallel chirps compared to the classical STFT.
- The concentration is maximized precisely when the kernel parameter $A$ matches the signal's chirp rate.
Instantaneous Frequency: The quadratic-phase Hilbert spectrum recovers the instantaneous frequency of a linear chirp as a constant value with negligible error, effectively reducing the chirp to a single horizontal line in the time-frequency plane.

Significance and Claims
The paper claims that by localizing the quadratic-phase machinery, it bridges the gap between global quadratic-phase transforms and practical time-frequency analysis. The primary significance lies in the efficiency gain for chirp-type signals: when the quadratic phase of the transform matches the signal's chirp rate, the representation achieves a level of energy concentration unattainable by the classical STFT.

The authors posit that the "chirp-reduction principle" suggests that the entire apparatus of modern Gabor and time-frequency analysis (including frames, modulation spaces, and ridge extraction) can be ported to the quadratic-phase domain via demodulation. They conclude that these localized transforms offer a principled route to adaptive, signal-matched time-frequency analysis, potentially serving non-stationary signals as effectively as global transforms served stationary ones. The paper acknowledges limitations, noting that a single kernel cannot fully concentrate signals with multiple, differing chirp rates, and that the multi-resolution variant incurs higher computational costs due to denser sampling.

Technical Summary: Efficient Time–Frequency Representations via the Quadratic-Phase Hilbert Transform

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