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Phase-Based Signal Representations for Scattering

Daniel Haider, Peter Balazs, Nicki Holighaus

TL;DR

This work addresses the loss of phase information in traditional scattering representations by introducing phase scattering, which cascades derivatives of the time-frequency phase (CIF_f and LGD_t) as nonlinearities in a scattering framework built on the STFT. It establishes analytical and numerical properties of CIF_f and LGD_t under STFT-invariances, showing linear, affine behavior for key signals and localization within window supports. The authors define STFT-based phase scattering coefficients, demonstrate 2nd-order CIF_f scattering for frequency modulation (revealing the modulation frequency as the zero of an affine function), and illustrate 2nd-order mixed phase scattering that detects the fundamental frequency of Dirac combs. The results indicate that phase scattering can capture large-scale structure with precise time-frequency localization, motivating future theoretical development and real-world audio applications as a complementary or alternative feature set to magnitude-based scattering.

Abstract

The scattering transform is a non-linear signal representation method based on cascaded wavelet transform magnitudes. In this paper we introduce phase scattering, a novel approach where we use phase derivatives in a scattering procedure. We first revisit phase-related concepts for representing time-frequency information of audio signals, in particular, the partial derivatives of the phase in the time-frequency domain. By putting analytical and numerical results in a new light, we set the basis to extend the phase-based representations to higher orders by means of a scattering transform, which leads to well localized signal representations of large-scale structures. All the ideas are introduced in a general way and then applied using the STFT.

Phase-Based Signal Representations for Scattering

TL;DR

This work addresses the loss of phase information in traditional scattering representations by introducing phase scattering, which cascades derivatives of the time-frequency phase (CIF_f and LGD_t) as nonlinearities in a scattering framework built on the STFT. It establishes analytical and numerical properties of CIF_f and LGD_t under STFT-invariances, showing linear, affine behavior for key signals and localization within window supports. The authors define STFT-based phase scattering coefficients, demonstrate 2nd-order CIF_f scattering for frequency modulation (revealing the modulation frequency as the zero of an affine function), and illustrate 2nd-order mixed phase scattering that detects the fundamental frequency of Dirac combs. The results indicate that phase scattering can capture large-scale structure with precise time-frequency localization, motivating future theoretical development and real-world audio applications as a complementary or alternative feature set to magnitude-based scattering.

Abstract

The scattering transform is a non-linear signal representation method based on cascaded wavelet transform magnitudes. In this paper we introduce phase scattering, a novel approach where we use phase derivatives in a scattering procedure. We first revisit phase-related concepts for representing time-frequency information of audio signals, in particular, the partial derivatives of the phase in the time-frequency domain. By putting analytical and numerical results in a new light, we set the basis to extend the phase-based representations to higher orders by means of a scattering transform, which leads to well localized signal representations of large-scale structures. All the ideas are introduced in a general way and then applied using the STFT.
Paper Structure (13 sections, 5 theorems, 16 equations, 3 figures)

This paper contains 13 sections, 5 theorems, 16 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Instantaneous Frequency and Group Delay
  3. Analytical Shapes of CIF and LGD - Revisited
  4. CIFf Analytically
  5. LGDt Analytically
  6. Numerical Shapes of CIF and LGD - Revisited
  7. Phase Scattering
  8. Time-Frequency Scattering
  9. CIF and LGD Scattering
  10. CIFf Scattering Application - Frequency Modulation
  11. Mixed Phase Scattering
  12. Mixed Phase Scattering Application - Dirac Comb
  13. Conclusion and Outlook

Key Result

Proposition 1

Let $x \in L^2(\mathbb{R})$ and $g \in L^2(\mathbb{R}) \cap \mathcal{C}^1(\mathbb{R})$, such that $Tg \in L^2(\mathbb{R})$, where the time-weighted window $Tg$ is given by $Tg(\tau)=\tau g(\tau)$ for all $\tau\in\mathbb{R}$. Then $\mathcal{V}_gx(t,\omega)$ and $\mathcal{V}_g^tx(t,\omega)$ are infini

Figures (3)

  • Figure 1: (a) Full CIFf of sinusoid and one column for fixed $t$ (right). (b) full LGDt of impulse and one column for fixed $\omega$ (bottom). (c) full STFT magnitude of sinusoid and one column for fixed $t$ (right). (d) full STFT magnitude of impulse and one column for fixed $\omega$ (bottom).
  • Figure 2: (a),(b) $1$st and $2$nd order CIFf and STFT scattering layers of $x_1$ with propagation channel at $880$Hz. (c),(d) $1$st and $2$nd order CIFf and STFT scattering layers of $x_2$.
  • Figure 3: (a) $1$st and $2$nd order mixed phase scattering layers of the Dirac comb w.r.t. an arbitrary $p_1$. (b) $1$st and $2$nd order STFT scattering layers.

Theorems & Definitions (9)

  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Definition 1: Time-Frequency Scattering
  • Definition 2: Phase Scattering Coefficients
  • Definition 3: STFT-Phase Scattering Coefficients
  • Lemma 3
  • Definition 4: Mixed Phase Scattering
  • Lemma 4