The Eigenvalue Distribution of Hankel Matrix: A Tool for Spectral Estimation From Noisy Data

Abstract

One of the key challenges of digital signal processing is to estimate sinusoidal components of an unknown signal. Researchers and engineers have been adopting various methods to analyze noisy signals and extract essential features of a given signal. Singular spectrum analysis (SSA) has been a popular and effective tool for extracting sinusoidal components of an unknown noisy signal. The process of singular spectrum analysis includes embedding time series into a Hankel matrix. The eigenvalue distribution of the Hankel matrix exhibits significant properties that can be used to estimate an unknown signal’s rhythmic components and frequency response. This paper proposes a method that utilizes the Hankel matrix’s eigenvalue distribution to estimate sinusoidal components from the frequency spectrum of a noisy signal. Firstly, an autoregressive (AR) model has been utilized for simulating time series employed to observe eigenvalue distributions and frequency spectrum. Nevertheless, the approach has been tested on real-life speech data to prove the applicability of the proposed mechanism on spectral estimation. Overall, results on both simulated and real data confirm the acceptability of the proposed method. This study suggests that eigenvalue distribution can be a helpful tool for estimating the frequency response of an unknown time series. Since the autoregressive model can be used to model various real-life data analyses, this study on eigenvalue distribution and frequency spectrum can be utilized in those real-life data. This approach will help estimate frequency response and identify rhythmic components of an unknown time series based on eigenvalue distribution.