Generic Feasibility of Perfect Reconstruction with Short FIR Filters in Multi-channel Systems

We study the feasibility of short finite impulse response (FIR) synthesis for perfect reconstruction (PR) in generic FIR filter banks. Among all PR synthesis banks, we focus on the one with the minimum filter length. For filter banks with oversampling factors of at least two, we provide prescriptions for the shortest filter length of the synthesis bank that would guarantee PR almost surely. The prescribed length is as short or shorter than the analysis filters and has an approximate inverse relationship with the oversampling factor. Our results are in form of necessary and sufficient statements that hold generically, hence only fail for elaborately-designed nongeneric examples. We provide extensive numerical verification of the theoretical results and demonstrate that the gap between the derived filter length prescriptions and the true minimum is small. The results have potential applications in synthesis FB design problems, where the analysis bank is given, and for analysis of fundamental limitations in blind signals reconstruction from data collected by unknown subsampled multi-channel systems.Comment: Manuscript submitted to IEEE Transactions on Signal Processin

Studying the Intervention of an Unusual Term in f(T)f(T) Gravity via Noether Symmetry

As usual, we observe an unknown coupling function, i.e. $F(\varphi)$, with a function of torsion and also curvature, i.e. $f(T)$ and $f(R)$, generally depending on a scalar field. In $f(R)$ case, it comes from quantum correlations and other reasons. Now, what if beside this term in $f(T)$-gravity context, we enhance the action through another term which depends on both scalar field and its derivatives? In this paper, we have added such an unprecedented term in the generic common action of $f(T)$-gravity such that in this new term, an unknown function of torsion has coupled with an unknown function of both scalar field and its derivatives. We have explained why we can append such term, in the introduction in details. By the use of Noether symmetry approach, we have considered its behavior and effect. We have shown that it does not produce an anomaly, but rather it works successfully and numerical analysis of the exact solutions of field equations coincide with all most important observational data particularly late-time-accelerated expansion. So, this new term may be added to the gravitational actions of $f(T)$-gravity