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 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 norm minimization techniques, we can incorporate a number of new and computationally simple dictionaries.Index Terms-Dictionary representations for periodic signals, nested periodic matrices, period estimation, periodicity, Ramanujan periodicity Transform, Ramanujan sums.
We propose a new filter-bank structure for the estimation and tracking of periodicities in time series data. These filter-banks are inspired from recent techniques on period estimation using high-dimensional dictionary representations for periodic signals. Apart from inheriting the numerous advantages of the dictionary based techniques over conventional periodestimation methods such as those using the DFT, the filterbanks proposed here expand the domain of problems that can be addressed to a much richer set. For instance, we can now characterize the behavior of signals whose periodic nature changes with time. This includes signals that are periodic only for a short duration and signals such as chirps. For such signals, we use a time vs period plane analogous to the traditional time vs frequency plane. We will show that such filter banks have a fundamental connection to Ramanujan Sums and the Ramanujan Periodicity Transform.
This paper studies a class of filter banks called the Ramanujan filter banks which are based on Ramanujan-sums. It is shown that these filter banks have some important mathematical properties which allow them to reveal localized hidden periodicities in real-time data. These are also compared with traditional comb filters which are sometimes used to identify periodicities. It is shown that non-adaptive comb filters cannot in general reveal periodic components in signals unless they are restricted to be Ramanujan filters. The paper also shows how Ramanujan filter banks can be used to generate time-period plane plots which track the presence of time varying, localized, periodic components.
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