In this dissertation, we investigate periodicity estimation with nested periodic dictionaries (NPDs), a newly introduced family of matrices that can capture periodicity in the data. We address two main problems. First, we study the detection of periodic signals using their representations in NPDs. To this end, the detection problem is posed as a composite hypothesis testing problem, where the signals are assumed to admit sparse representations in a Ramanujan Periodicity Transform (RPT) dictionary as an instance of NPDs. For the binary case, we develop a generalized likelihood ratio detector and obtain exact distributions of the test statistics in terms of confluent hypergeometric functions, along with flexible approximate distributions. Subsequently, we extend our approach to multi-hypothesis and multi-channel settings, where we account for spatial correlations between the different channels. We study the application of the proposed method in the detection of periodic brain responses to external visual stimuli, known as steady-state visually evoked potentials (SSVEPs), which is fundamental to the development of Brain Computer Interfaces. Results based on experiments with synthesized and real-data demonstrate that the RPT detector outperforms conventional spectral-based methods. Second, we address the problem of period estimation. Periodic signals composed of periodic mixtures admit sparse representations in NPDs. Therefore, their underlying hidden periods can be estimated by recovering the exact support of said representations. In this dissertation, we investigate support recovery guarantees of such signals in noise-free and noisy settings. While sufficient recovery conditions of sparse signals have been studied in the literature on compressive sensing, these conditions are of limited use for NPDs, because their analysis does not capture their intrinsic structures. Therefore, we establish new conditions based on a newly introduced notion of nested periodic coherence. Our results show significant improvement over generic recovery bounds as the conditions hold over a larger range of sparsity levels.
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Doctor of Philosophy (Ph.D.)
College of Engineering and Computer Science
Electrical and Computer Engineering
Length of Campus-only Access
Doctoral Dissertation (Open Access)
Saidi, Pouria, "Periodicity Estimation Using Representations in Nested Periodic Dictionaries with Applications in Neural Engineering" (2021). Electronic Theses and Dissertations, 2020-. 1343.