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Instantaneous Frequency Estimation in Noisy Multicomponent Signals with Interfering Modes Based on Prony Method and Spline Approximation

Basile Dubois-Bonnaire, Sylvain Meignen, Kévin Polisano

TL;DR

A novel estimator of the instantaneous frequencies (IFs) of the modes making up multicomponent signals (MCSs) based on spline approximation, based on spline approximation is proposed.

Abstract

In this paper, we propose a novel estimator of the instantaneous frequencies (IFs) of the modes making up multicomponent signals (MCSs). We are particularly interested in dealing with noisy MCSs containing close modes in the time-frequency plane. Though it is possible to adapt Prony approach to estimate IFs in such situations, interference between the modes generates oscillations in the obtained estimations. After having investigated the nature of these oscillations, we propose an algorithm to remove these in IFs estimation, based on spline approximation. Numerical applications in various situations illustrate the benefit of mixing Prony technique with spline approximation for IF estimation in noisy MCSs containing close modes.

Instantaneous Frequency Estimation in Noisy Multicomponent Signals with Interfering Modes Based on Prony Method and Spline Approximation

TL;DR

A novel estimator of the instantaneous frequencies (IFs) of the modes making up multicomponent signals (MCSs) based on spline approximation, based on spline approximation is proposed.

Abstract

In this paper, we propose a novel estimator of the instantaneous frequencies (IFs) of the modes making up multicomponent signals (MCSs). We are particularly interested in dealing with noisy MCSs containing close modes in the time-frequency plane. Though it is possible to adapt Prony approach to estimate IFs in such situations, interference between the modes generates oscillations in the obtained estimations. After having investigated the nature of these oscillations, we propose an algorithm to remove these in IFs estimation, based on spline approximation. Numerical applications in various situations illustrate the benefit of mixing Prony technique with spline approximation for IF estimation in noisy MCSs containing close modes.
Paper Structure (8 sections, 13 equations, 4 figures)

This paper contains 8 sections, 13 equations, 4 figures.

Table of Contents

  1. Introduction
  2. IF Estimation Based on the Prony Method and Spline Approximation
  3. The Noiseless Case
  4. The Noisy Case
  5. Results
  6. The Case of a Two parallel Linear Chirps
  7. More General Signals
  8. Conclusion

Figures (4)

  • Figure 1: (a) Spectrogram of two pure tones (same amplitude $\omega_1= 220.5$, $\omega_2 = 240.5$, $\sigma = 0.02$), and IFs estimates computed with Prony method as explained in Sec. \ref{['sec:Prony']} (black curves) and estimations $\psi_1$ and $\psi_2$ computed with \ref{['def:B_spline']} (in white, $r = 1-10^{-4}$); (b) normalized $l_2$ error associated with the estimation of $\omega_1$ using either $\boldsymbol{\eta}_1$ or $\boldsymbol{\psi}_1$ defined in Eq. \ref{['def:B_spline']}.
  • Figure 2: (a) Normalized $l_2$ error of the mode $f_1$ of the signal of Fig. \ref{['Fig1']} (a), for varying $\sigma$ using different techniques for IF estimation: cad, cad-tlsa, cad-spline, and cad-tlsa-spline (input SNR is 5 dB, the results are averaged over 10 noise realizations); (b) same as (a) but for an input SNR of 10 dB.
  • Figure 3: (a) Spectrogram of two parallel linear chirp (SNR=10 dB, $\sigma=0.025$); (b) normalized $l_2$ error for mode $f_1$ of the signal in (a), associated with cad, cad-tlsa, cad-spline, and cad-tlsa-spline (results averaged over 10 noise realizations).
  • Figure 4: (a) spectrogram of two chirps (SNR =10 dB, $\sigma=0.025$); (b) normalized $l_2$ error for modes of the signal in (a) (top $f_2$, bottom $f_1$) associated with cad, cad-tlsa, cad-spline, and cad-tlsa-spline (results averaged over 10 noise realizations).