  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.
