Minimum number of possibly non-contiguous samples to distinguish two periods

Given that a sequence x(n) is periodic with period P belonging to a known integer set {P<inf>1</inf>, P<inf>2</inf>, … P<inf>L</inf>}, what is the minimum number of samples of x(n) required to find the period? For the special case where the samples of x(n) are constrained to be contiguous in time, this problem has recently been solved. More generally, when the samples are allowed to be non-contiguous, the problem is quite difficult. This paper provides the answer for the restricted situation where P ∈ {P<inf>1</inf>, P<inf>2</inf>}. With P<inf>1</inf> < P<inf>2</inf>, the necessary and sufficient number of (possibly noncontiguous) samples for period estimation turns out to be (a) P<inf>1</inf>, if P<inf>1</inf> is not a divisor of P<inf>2</inf>, and (b) P<inf>2</inf> otherwise. While the proof is quite involved even in this restricted case, it is likely to form the basis for addressing the more general situation where P ∈ {P<inf>1</inf>, P<inf>2</inf>, … P<inf>L</inf>}.

[1]  M. Nakashizuka.  A Sparse Decomposition for Periodic Signal Mixtures , 2007, 2007 15th International Conference on Digital Signal Processing.

[2]  William A. Sethares,et al.  Periodicity transforms , 1999, IEEE Trans. Signal Process..

[3]  Atul Grover,et al.  Searching microsatellites in DNA sequences: approaches used and tools developed , 2011, Physiology and Molecular Biology of Plants.

[4]  P. P. Vaidyanathan,et al.  A Unified Theory of Union of Subspaces Representations for Period Estimation , 2016, IEEE Transactions on Signal Processing.

[5]  P. P. Vaidyanathan.  Ramanujan Sums in the Context of Signal Processing—Part II: FIR Representations and Applications , 2014, IEEE Transactions on Signal Processing.

[6]  T. Parks,et al.  Maximum likelihood pitch estimation , 1976 .

[7]  M. Bertinelli,et al.  Analysis of T-wave alternans using the Ramanujan transform , 2008, 2008 Computers in Cardiology.

[8]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.

[9]  P. P. Vaidyanathan,et al.  Properties of Ramanujan filter banks , 2015, 2015 23rd European Signal Processing Conference (EUSIPCO).

[10]  Rolf Unbehauen,et al.  Separation of periodic signals by using an algebraic method , 1991, 1991., IEEE International Sympoisum on Circuits and Systems.

[11]  P. P. Vaidyanathan,et al.  Ramanujan filter banks for estimation and tracking of periodicities , 2015, 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[12]  A. Hewish,et al.  Observation of a Rapidly Pulsating Radio Source , 1968, Nature.

[13]  C. Ponting,et al.  Protein repeats: structures, functions, and evolution. , 2001, Journal of structural biology.

[14]  Petros Maragos,et al.  Harmonic analysis and restoration of separation methods for periodic signal mixtures: Algebraic separation versus comb filtering , 1998, Signal Process..

[15]  Thomas W. Parks,et al.  Orthogonal, exactly periodic subspace decomposition , 2003, IEEE Trans. Signal Process..

[16]  P. P. Vaidyanathan,et al.  Nested Periodic Matrices and Dictionaries: New Signal Representations for Period Estimation , 2015, IEEE Transactions on Signal Processing.

[17]  P. P. Vaidyanathan,et al.  Ramanujan Sums in the Context of Signal Processing—Part I: Fundamentals , 2014, IEEE Transactions on Signal Processing.