Arbitrarily Shaped Periods in Multidimensional Discrete Time Periodicity
Traditionally, most of the analysis of discrete time multidimensional periodicity in DSP is based on defining the period as a parallelepiped. In this work, we study whether this framework can incorporate signals that are repetitions of more general shapes than parallelepipeds. For example, the famous Dutch artist M. C. Escher constructed many interesting shapes such as fishes, birds and animals, which can tile the continuous 2-D plane. Inspired from Escher's tilings, we construct discrete time signals that are repetitions of various kinds of shapes. We look at periodicity in the following way - a given shape repeating itself along fixed directions to tile the entire space. By transcribing this idea into a mathematical framework, we explore its relationship with the traditional analysis of periodicity based on parallelepipeds. Our main result is that given any such signal with an arbitrarily shaped period, we can always find an equivalent parallelepiped shaped period that has the same number of points as the original period.
© 2015 IEEE. Manuscript received February 05, 2015; revised April 09, 2015; accepted May 08, 2015. Date of publication May 12, 2015; date of current version May 18, 2015. This work was supported in part by the Office of Naval Research under Grants N00014-11-1-0676 and N00014-15-1-2118, and by the Information Science and Technology (IST) initiative of Caltech. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Chandra Sekhar Seelamantula.