Signal Periodic Decomposition With Conjugate Subspaces

One may often decompose the domain of a topologically transitive map into finitely many regular closed pieces with nowhere dense overlap in such a way that these pieces map into one another in a periodic fashion. We call decompositions of this kind regular periodic decompositions and refer to the number of pieces as the length of the decomposition. If $f$ is topologically transitive but $f^{n}$ is not, then $f$ has a regular periodic decomposition of some length dividing $n$. Although a decomposition of a given length is unique, a map may have many decompositions of different lengths. The set of lengths of decompositions of a given map is an ideal in the lattice of natural numbers ordered by divisibility, which we call the decomposition ideal of $f$. Every ideal in this lattice arises as a decomposition ideal of some map. Decomposition ideals of Cartesian products of transitive maps are discussed and used to develop various examples. Results are obtained concerning the implications of local connectedness for decompositions. We conclude with a comprehensive analysis of the possible decomposition ideals for maps on 1-manifolds.

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Signal periodicity detection using Ramanujan subspace projection

Journal of Electrical Engineering ◽

10.2478/jee-2020-0044 ◽

2020 ◽

Vol 71 (5) ◽

pp. 326-332

Author(s):

Deepa Abraham ◽

Manju Manuel

Keyword(s):

Signal Processing ◽

Detection Methods ◽

Subspace Projection ◽

Speech Signal Processing ◽

Periodicity Detection ◽

Periodic Decomposition ◽

Signal Period ◽

Ramanujan Sums ◽

Pitch Analysis ◽

Detection Of Signals

Abstract Signal periodic decomposition and periodic estimation are two crucial problems in the signal processing domain. Due to its significance, the applications have been extended to fields like periodic sequence analysis of biomolecules, stock market predictions, speech signal processing, and musical pitch analysis. The recently proposed Ramanujan sums (RS) based transforms are very useful in analysing the periodicity of signals. This paper proposes a method for periodicity detection of signals with multiple periods based on autocorrelation and Ramanujan subspace projection with low computational complexity. The proposed method is compared with other signal periodicity detection methods and the results show that the proposed method detects the signal period correctly in less time.

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