Signal periodic decomposition with conjugate subspaces

TL;DR

This paper introduces a novel conjugate subspace matching pursuit algorithm for signal periodic decomposition, capable of identifying all hidden periods, including time-varying ones, with low computational cost and broad applications.

Contribution

The paper proposes the CSMP algorithm based on conjugate subspaces, improving hidden period detection and signal decomposition over existing methods.

Findings

Accurately identifies all hidden periods in signals of any length.

Detects time-varying hidden periods effectively.

Demonstrates applications in speech pitch detection and signal approximation.

Abstract

In this paper, we focus on hidden period identification and the periodic decomposition of signals. Based on recent results on the Ramanujan subspace, we reveal the conjugate symmetry of the Ramanujan subspace with a set of complex exponential basis functions and represent the subspace as the union of a series of conjugate subspaces. With these conjugate subspaces, the signal periodic model is introduced to characterize the periodic structure of a signal. To achieve the decomposition of the proposed model, the conjugate subspace matching pursuit (CSMP) algorithm is proposed based on two different greedy strategies. The CSMP is performed iteratively in two stages. In the first stage, the dominant hidden period is chosen with the periodicity strategy. Then, the dominant conjugate subspace is chosen with the energy strategy in the second stage. Compared with the current state-of-the-art methods for hidden period identification, the main advantages provided by the CSMP are the following: (i) the capability of identifying all the hidden periods in the range from $1$ to the maximum hidden period $Q$ of a signal of any length, without truncating the signal; (ii) the ability to identify the time-varying hidden period with its shifted version; and (iii) the low computational cost, without generating and using a large over-complete dictionary. Moreover, we provide examples and applications to demonstrate the abilities of the proposed two-stage CSMP algorithm, which include hidden period identification, signal approximation, time-varying period detection, and pitch detection of speech.