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Resolving Two-Dimensional Ambiguity in Subspace-Based Frequency Estimation for Harmonic Signals

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Abstract

This paper studies how to solve a matrix ambiguity to find the fundamental frequency in harmonic signals based on subspace methods. Using the eigenvectors obtained from the autocorrelation matrix of the measured signal, we obtain a model for the estimation of fundamental frequency. By a connection of an invertible matrix ambiguity, the model establishes a link between two matrices whose columns both span the same signal subspace of interest. One of the two matrices comprises the obtained eigenvectors, and the other one is an unknown Vandermonde matrix with a harmonic structure which contains the information of fundamental frequency. The special structure of the estimation model will enable us to construct a linear system so that the matrix ambiguity can be solved to construct the Vandermonde matrix. Thus, the estimate of fundamental frequency can be achieved from the entries of the Vandermonde matrix. In addition, we also discuss an LU-like factorization of the Vandermonde matrix for estimation of the fundamental frequency. Simulation results illustrate the performance of the proposed methods.

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Data Availability

There are two kinds of data used for simulations in this research. The first kind is the synthetic data which can be generated by MATLAB code according to Eq. (1). The second kind is a segment of a song, “The Moon Represents My Heart,” played by violin, which is available from the following hyperlink. https://www.youtube.com/watch?v=sq38zHAfeaA

Notes

  1. R-MUSIC [34] is the abbreviation of Root MUSIC, which is similar to MUSIC in many respects, except that the parameters are determined from the roots of a polynomial formed from the noise subspace.

  2. PHD [20] is the abbreviation of Pisarenko harmonic decomposition. This method is usually regarded as a special case of MUSIC.

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Authors and Affiliations

  1. Department of Communications Engineering, Feng Chia University, No. 100, Wenhwa Road, Taichung, 40724, Taiwan

    Yi-Sheng Chen & Ho-En Liao

  2. Department of Automatic Control Engineering, Feng Chia University, No. 100, Wenhwa Road, Taichung, 40724, Taiwan

    Yue-Der Lin

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  1. Yi-Sheng Chen
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Research sponsored by the Ministry of Science and Technology of Taiwan under Grant MOST-107-2221-E-035-046-.

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Chen, YS., Liao, HE. & Lin, YD. Resolving Two-Dimensional Ambiguity in Subspace-Based Frequency Estimation for Harmonic Signals. Circuits Syst Signal Process 40, 5616–5631 (2021). https://doi.org/10.1007/s00034-021-01736-3

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