Ramanujan filter banks have been used for identifying periodicity structure in streaming data. This paper studies the locations of zeros of Ramanujan filters. All the zeros of Ramanujan filters are shown to lie on or inside the unit circle in the z-plane. A convenient factorization appears as a corollary of this result, which is useful to identify common factors between different Ramanujan filters in a filter bank. For certain families of Ramanujan filters, further structure is identified in the locations of zeros of those filters. It is shown that increasing the number of periods of Ramanujan sums in the filter definition only increases zeros on the unit circle in z-plane. A potential application of these results is that by identifying common factors between Ramanujan filters, one can obtain efficient implementations of Ramanujan filter banks (RFB) as demonstrated here.

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Ramanujan filter banks (RFB) have in the past been used to identify periodicities in data. These are analysis filter banks with no synthesis counterpart for perfect reconstruction of the original signal, so they have not been useful for denoising periodic signals. This paper proposes to use a hybrid analysissynthesis framework for denoising discrete-time periodic signals. The synthesis occurs via a pruned dictionary designed based on the output energies of the RFB analysis filters. A unique property of the framework is that the denoised output signal is guaranteed to be periodic unlike any of the other methods. For a large range of input noise levels, the proposed approach achieves a stable and high SNR gain outperforming many traditional denoising techniques.

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It has recently been shown that periodicity in discrete-time data can be analyzed using Ramanujan sums and associated dictionaries. This paper explores the role of dictionary learning methods in the context of period estimation and periodic signal representation using dictionaries. It is shown that a wellknown dictionary learning algorithm, namely K-SVD, is able to learn Ramanujan and Farey periodicity dictionaries from the noisy, sparse coefficient data generated from them without imposing any periodicity structure in the learning stage. This similarity between the learned dictionary and the underlying original periodicity dictionary reaffirms the power of the K-SVD in predicting the right dictionary from data without explicit application-specific constraints. The paper also examines how the choice of different parameter values affect the similarity of the learned dictionary to the underlying dictionary. Two versions of K-SVD along with different initializations are analyzed for their effect on representation and denoising error for the data.