Discrete Sampling: A Graph Theoretic Approach to Orthogonal Interpolation

Abstract

We study the problem of finding unitary submatrices of the N × 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 Z N and tiling 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.

Computing the Discrete Fourier Transform of signals with spectral frequency support

“…Lemma 1: when N is a power of 2, the submatrix F −1 (I, J ) is unitary iff for some set of pivots L 1) J corresponds to a conforming digit table with pivots L, 2) I corresponds to a conforming digit table with pivots N/2L. Lemma 1 is a direct consequence of earlier works [11], [12]; we discuss more about this in Section V.…”

“…Also, note that given a spectral set J , there could be many possible time domain samples I that result in a unitary submatrix [11]. For one specific choice of I, we prove in Theorem 1 that the resulting submatrix F(J , I) has a block structure (similar to F k ) that enables O(k log k) computation (down from O(k 2 )) in (2).…”

Fuglede’s conjecture fails in 4 dimensions over odd prime fields

“…Instead, we use an algorithm based on clique finding, a famous problem in computer science. This clique approach is also taken by Siripuram et al [5] in studying spectrality in groups Z N .…”

Convolution Idempotents with a given Zero-set

“…A third problem is that of finding unitary submatrices of the Fourier matrix F : Find all possible rows I ⊆ Z N and columns J ⊆ Z N with |I| = |J | such that the corresponding Fourier submatrix M satisfies M * M = kI, with k a scalar. This is related to the problem of finding all orthogonal interpolating bases for spaces of bandlimited signals, as treated in [4].…”

“…This work deals with recovering h when some of its elements are known to be zero. Our motivation for considering this comes from applications to several, apparently distant areas: multicoset sampling of analog signals ( [1], Section II-A), Fuglede's conjecture on spectral and tiling sets of integers [2]- [4], and finding unitary submatrices of the discrete Fourier transform matrix [4].…”