Interpolation Filter Model For Ramanujan Subspace Signals

Abstract

Ramanujan sums have been shown to have interesting applications in signal processing. Ramanujan subspaces, Ramanujan dictionaries, and Ramanujan filter banks are useful in representing and denoising discrete-time periodic signals. In this paper, we theoretically investigate an ideal interpolation filter model for Ramanujan subspace signals wherein an expander ↑ M is followed by the ideal q-th Ramanujan filter C<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</inf> (e<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">jω</sup>). The output space of this interpolation filter is, in general, only a proper subspace of the q-th Ramanujan subspace ${{\mathcal{S}}_q}$. For the special case when M and q are coprime, we prove that the output space is the entire Ramanujan subspace. We also discuss a more general form of this model for the representation of periodic signals, which may have a potential application in denoising periodic signals. When M and q are not coprime, we provide a bound on the dimension of the output space of the interpolation filter. For this general case, we also conjecture that the provided bound in fact equals the dimension of the output space.