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.