Variational Inference of Structured Line Spectra Exploiting Group-Sparsity

Abstract

This paper introduces a method of decomposing a signal into several groups of related spectral lines. The frequencies of the spectral lines in each group are related to a parameter common to all spectral lines within the same group, such as the fundamental frequency of a harmonic series of spectral lines. The parameters of each group are estimated on a continuum by the proposed variational expectation-maximization (EM) algorithm. Additionally, the number of groups and the number of spectral lines within each group are inferred through a group-sparse solution, obtained by latent variables in a hierarchical Bernoulli-Gamma-Gaussian prior model inspired by sparse Bayesian learning (SBL). The performance of the proposed algorithm is demonstrated on three tasks: multi-pitch estimation, extended object detection using radar signals, and variational mode decomposition (VMD). On the Bach 10 dataset, which contains recordings of ten musical pieces, the proposed algorithm outperforms state-of-the-art model-based and machine-learning-based multi-pitch estimation algorithms in terms of fundamental frequency, i.e. pitch, detection accuracy. In addition, the extended object detection task demonstrates how incorporating knowledge of the structural relationships between spectral lines into the estimation procedure can lead to performance gains compared to assuming independent spectral lines, especially under low signal-to-noise ratio (SNR) conditions. Finally, the VMD task is included to further demonstrate the versatility of the proposed algorithm.