Discrete Sampling and Interpolation: Universal Sampling Sets for Discrete Bandlimited Spaces

We study the problem of interpolating all values of a discrete signal f of length N when d<N values are known, especially in the case when the Fourier transform of the signal is zero outside some prescribed index set J; these comprise the (generalized) bandlimited spaces B^J. The sampling pattern for f is specified by an index set I, and is said to be a universal sampling set if samples in the locations I can be used to interpolate signals from B^J for any J. When N is a prime power we give several characterizations of universal sampling sets, some structure theorems for such sets, an algorithm for their construction, and a formula that counts them. There are also natural applications to additive uncertainty principles.Comment: 24 pages, 5 figures, Accepted for publication in IEEE Transactions on Information Theor

Convolution Idempotents with a given Zero-set

We investigate the structure of N-length discrete signals h satisfying h*h=h that vanish on a given set of indices. We motivate this problem from examples in sampling, Fuglede's conjecture, and orthogonal interpolation of bandlimited signals. When N is a prime power, we characterize all such h with a prescribed zero set in terms of digit expansions of nonzero indices in the inverse DFT of h

Graph learning under spectral sparsity constraints

Graph inference plays an essential role in machine learning, pattern recognition, and classification. Signal processing based approaches in literature generally assume some variational property of the observed data on the graph. We make a case for inferring graphs on which the observed data has high variation. We propose a signal processing based inference model that allows for wideband frequency variation in the data and propose an algorithm for graph inference. The proposed inference algorithm consists of two steps: 1) learning orthogonal eigenvectors of a graph from the data; 2) recovering the adjacency matrix of the graph topology from the given graph eigenvectors. The first step is solved by an iterative algorithm with a closed-form solution. In the second step, the adjacency matrix is inferred from the eigenvectors by solving a convex optimization problem. Numerical results on synthetic data show the proposed inference algorithm can effectively capture the meaningful graph topology from observed data under the wideband assumption

Discrete Sampling: A graph theoretic approach to Orthogonal Interpolation

We study the problem of finding unitary submatrices of the $N \times N$ discrete Fourier transform matrix, in the context of interpolating a discrete bandlimited signal using an orthogonal basis. This problem is related to a diverse set of questions on idempotents on $\mathbb{Z}_N$ and tiling $\mathbb{Z}_N$. In this work, we establish a graph-theoretic approach and connections to the problem of finding maximum cliques. We identify the key properties of these graphs that make the interpolation problem tractable when $N$ is a prime power, and we identify the challenges in generalizing to arbitrary $N$. Finally, we investigate some connections between graph properties and the spectral-tile direction of the Fuglede conjecture.Comment: Submitted to IEEE Transactions on Information Theor

Fast DFT Computation for Signals with Structured Support

Suppose an $N-$length signal has known frequency support of size $k$. Given sample access to this signal, how fast can we compute the DFT? The answer to this question depends on the structure of the frequency support. We first identify some frequency supports for which (an ideal) $O(k \log k)$ complexity is achievable, referred to as homogeneous sets. We give a generalization of radix-2 that enables $O(k\log k)$ computation of signals with homogeneous frequency support. Using homogeneous sets as building blocks, we construct more complicated support structures for which the complexity of $O(k\log k)$ is achievable. We also investigate the relationship of DFT computation with additive structure in the support and provide partial converses.Comment: 45 pages, 16figure

Some results on convolution idempotents

We consider the problem of recovering N length vectors h that vanish on a given set of indices and satisfy h ∗ h = h. We give some results on the structure of such h when N is a product of two primes, and investigate some bounds and their connections to certain graphs defined on ZN. © 2020 IEEE

DUALITY IN GRAPH SIGNAL PROCESSING

In recent years ,there has been a lot of data being collected from diverse sources like temperature measurement of sensor networks, transportation networks, social networks, biological networks and brain connectivity. A common feature in these data is that it resides on a complex and irregular structures. Graphs offer the ability to model such complex and irregular structures and interactions between them, analyzing signals residing on this graph leads us to Graph signal processing framework. The important aspect of Graph signal processing is that not just analyzing the signals it also takes into account of the inherent structure on which data is evolving. Researchers in the past decade have tried to extend the concepts of classical signal processing like Fourier transform, filtering, frequency response, sampling to Graph signal processing. In classical signal processing, duality property says that,when we apply Fourier transform twice on a signal we get the reversal of the signal.In this thesis we are trying to extend the concept of duality in Graph signal processing. We define the notion of dual graph and propose an optimization based algorithm to find the dual of an arbitrary graph

LP relaxations and Fuglede's conjecture

Consider a unitary (up to scaling) submatrix of the Fourier matrix with rows indexed by I and columns indexed by J. From the column index set J we construct a graph G so that the row index set I determines a max-clique. Interpreting G as coming from an association scheme gives certain bounds on the clique number, which has possible applications to Fuglede's conjecture on spectral and tiling sets