RIFT: Entropy-Optimised Fractional Wavelet Constellations for Ideal Time-Frequency Estimation

TL;DR

RIFT presents a probabilistic, CFWT-based framework to estimate a cross-term–free ITFR with WVD-like resolution by weighting a constellation of CFWTs with a local entropic measure and deconvolving via a positivity-constrained LR-TV process. It further yields an IPD field for local curvature visualization and enables Kalman-tracked extraction of component trajectories, leading to a Spline-RIFT variant. Derivations of the Cohen's class kernels and the CFWT-to-WVD relationships underpin a rigorous forward model and efficient computation. Empirical results on synthetic and real signals demonstrate superior cross-term suppression and high TF precision, with promising applications in audio, biomedical, and related time-varying signal analysis.

Abstract

We introduce a new method for estimating the Ideal Time-Frequency Representation (ITFR) of complex nonstationary signals. The Reconstructive Ideal Fractional Transform (RIFT) computes a constellation of Continuous Fractional Wavelet Transforms (CFWTs) aligned to different local time-frequency curvatures. This constellation is combined into a single optimised time-frequency energy representation via a localised entropy-based sparsity measure, designed to resolve auto-terms and attenuate cross-terms. Finally, a positivity-constrained Lucy-Richardson deconvolution with total-variation regularisation is applied to estimate the ITFR, achieving auto-term resolution comparable to that of the Wigner-Ville Distribution (WVD), yielding the high-resolution RIFT representation. The required Cohen's class convolutional kernels are fully derived in the paper for the chosen CFWT constellations. Additionally, the optimisation yields an Instantaneous Phase Direction (IPD) field, which allows the localised curvature in speech or music extracts to be visualised and utilised within a Kalman tracking scheme, enabling the extraction of signal component trajectories and the construction of the Spline-RIFT variant. Evaluation on synthetic and real-world signals demonstrates the algorithm's ability to effectively suppress cross-terms and achieve superior time-frequency precision relative to competing methods. This advance holds significant potential for a wide range of applications requiring high-resolution cross-term-free time-frequency analysis.

Figures (11)

• Figure 1: Overview of the proposed process, graphically illustrated for a parallel pair of sinusoidal frequency-modulated signals, $x_1(t)$ (Eq. \ref{['x_1']}). Each block corresponds to a stage in Algorithm \ref{['alg:rift']}, where the workflow is formalised step-by-step.
• Figure 2: Visualisation of the log-normal constellation kernel selection for $\theta=0$ and $N=7$ (and $\sigma_{\text{iso}}=1$), with (a) $\sigma_l=1$, (b) $\sigma_l=1.5$. The ellipses correspond to the isodensity contours for each T--F kernel, given by $\frac{1}{2}\mathbf{x}^T\mathbf{\Sigma}^{-1}\mathbf{x}=1$, for $\sigma\left(1 \le n \le N\right)$.
• Figure 3: RIFT results across four examples (columns). Rows (top to bottom): (1) Reference CWT evaluated with a kernel using the isotropic standard deviation $\sigma_{\text{iso}}$; (2) RIFT result; (3) time–frequency streamlines, determined via RK4 integration on the IPD field; (4) Kalman-tracked trajectories. (5) smoothed Instantaneous Phase Direction (IPD) field; Columns: (a) $x_1(t)$, Eq. \ref{['x_1']}; (b) $x_4(t)$, Eq. \ref{['x_4']}; (c) speech extract from the "Harvard sentences" database speech with the phrase "The stale smell of old beer lingers"; (d) bat echolocation signal consisting of a 2.5 µ s echolocation pulse emitted by the Large Brown Bat, Eptesicus fuscus. The authors wish to thank Curtis Condon, Ken White, and Al Feng of the Beckman Institute of the University of Illinois for the bat data and for permission to use it in this paper.
• Figure 4: T--F representations for (a) $x_6(t)$ and (b) $x_1(t)$. Two method blocks (columns = methods; rows = SNR). Shown SNRs: $\infty$ and $-5$ dB. Axes (Time in seconds; Frequency in Hz).
• Figure 5: Comparison between the (a) WVD (red +, blue -), the standard CWT applied for (b) small and (c) large standard deviations, and the (d) generalised CFWT presented in Eq. \ref{['ex:Phi']} aligned with the signal trajectory, for signal $z_3(t)$.