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Published July 15, 2015 | public
Journal Article

Nested Periodic Matrices and Dictionaries: New Signal Representations for Period Estimation

Abstract

In this paper, we propose a new class of techniques to identify periodicities in data. We target the period estimation directly rather than inferring the period from the signal's spectrum. By doing so, we obtain several advantages over the traditional spectrum estimation techniques such as DFT and MUSIC. Apart from estimating the unknown period of a signal, we search for finer periodic structure within the given signal. For instance, it might be possible that the given periodic signal was actually a sum of signals with much smaller periods. For example, adding signals with periods 3, 7, and 11 can give rise to a period 231 signal. We propose methods to identify these "hidden periods" 3, 7, and 11. We first propose a new family of square matrices called Nested Periodic Matrices (NPMs), having several useful properties in the context of periodicity. These include the DFT, Walsh–Hadamard, and Ramanujan periodicity transform matrices as examples. Based on these matrices, we develop high dimensional dictionary representations for periodic signals. Various optimization problems can be formulated to identify the periods of signals from such representations. We propose an approach based on finding the least l_{2} norm solution to an under-determined linear system. Alternatively, the period identification problem can also be formulated as a sparse vector recovery problem and we show that by a slight modification to the usual l_{1} norm minimization techniques, we can incorporate a number of new and computationally simple dictionaries.

Additional Information

© 2015 IEEE. Manuscript received August 21, 2014; revised February 26, 2015; accepted May 06, 2015. Date of publication May 18, 2015; date of current version June 09, 2015. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Akbar Sayeed. This work was supported in parts by the ONR grants N00014-11-1-0676 and N00014-15-1-2118, and the Information Science and Technology (IST) initiative of Caltech. The authors wish to thank the reviewers for insightful comments. One of the reviewers brought up an interesting question about shifting the signals which resulted in Section VII-C.

Additional details

Created:
August 20, 2023
Modified:
October 23, 2023