A pde-Based Analysis of the Spectrogram Image for Instantaneous Frequency Estimation

Vittoria Bruni,Domenico Vitulano,Michela Tartaglione

doi:10.3390/math9030247

Vittoria Bruni, Domenico Vitulano + Show 1 more

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https://doi.org/10.3390/math9030247

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Abstract

Instantaneous frequency (IF) is a fundamental feature in multicomponent signals analysis and its estimation is required in many practical applications. This goal can be successfully reached for well separated components, while it still is an open problem in case of interfering modes. Most of the methods addressing this issue are parametric, that is, they apply to a specific IF class. Alternative approaches consist of non-parametric time filtering-based procedures, which do not show robustness to destructive interference—the most critical scenario in crossing modes. In this paper, a method for IF curves estimation is proposed. The case of amplitude and frequency modulated two-component signals is addressed by introducing a spectrogram time-frequency evolution law, whose coefficients depend on signal IFs time derivatives, that is, the chirp rates. The problem is then turned into the resolution of a two-dimensional linear system which provides signal chirp rates; IF curves are then obtained by a simple integration. The method is non-parametric and it results quite robust to destructive interference. An estimate of the estimation error, as well as a numerical study concerning method sensitivity and robustness to noise are also provided in the paper.

Highlights

Instantaneous frequency (IF) estimation is of great interest in many applications dealing with non-stationary signals, such as radar and Micro Doppler systems [1,2,3], seismic signals [4], some kinds of gravitational waves [5,6], audio [7] and human speech signals [8], animal sounds [9] and biomedical signals [10]
The case of amplitude and frequency modulated two-component signals is addressed by introducing a spectrogram time-frequency evolution law, whose coefficients depend on signal IFs time derivatives, that is, the chirp rates
This paper has proposed a non-linear time-frequency evolution law for the spectrogram of a frequency and amplitude modulated two-component signal, which is employed for instantaneous frequencies (IFs) estimation

Summary

Introduction

Instantaneous frequency (IF) estimation is of great interest in many applications dealing with non-stationary signals, such as radar and Micro Doppler systems [1,2,3], seismic signals [4], some kinds of gravitational waves [5,6], audio [7] and human speech signals [8], animal sounds [9] and biomedical signals [10]. Practical applications deal with multicomponent signals (MCS), that is, the superposition of individual waveforms, characterized by specific time-dependent frequency content, that is, IF, as well as timedependent amplitude, namely the instantaneous amplitude (IA) [11]. The correct analysis of MCS requires signal modes separation, that allows for IFs estimation. In many decomposition schemes, IFs estimation is a required step for the recovery of the individual modes by the observed mixture. The existing strategies addressing this issue can be grouped into two approaches: the former analyzes the target signal directly in the time domain, such as Empirical Mode Decomposition and its improvements [12,13,14], or in the frequency domain [15,16]. The second approach analyzes the signal in the joint time-frequency (TF) plane, that is, it is based on TF analysis

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