Pulse Shaping, Localization and the Approximate Eigenstructure of LTV Channels

TL;DR

This paper explores the relationship between pulse shaping, localization, and approximate eigenstructure in LTV channels, providing insights into optimal signaling schemes like OFDM and their fundamental limits.

Contribution

It establishes the connection between pulse shaping for WSSUS channels and approximate eigenstructure of LTV channels, offering new theoretical insights into localization and orthogonality trade-offs.

Findings

Eigenvalues of localization operators relate to pulse localization.

Fundamental limits on SINR are derived.

Balance between localization and orthogonality remains an open problem.

Abstract

In this article we show the relation between the theory of pulse shaping for WSSUS channels and the notion of approximate eigenstructure for time-varying channels. We consider pulse shaping for a general signaling scheme, called Weyl-Heisenberg signaling, which includes OFDM with cyclic prefix and OFDM/OQAM. The pulse design problem in the view of optimal WSSUS--averaged SINR is an interplay between localization and "orthogonality". The localization problem itself can be expressed in terms of eigenvalues of localization operators and is intimately connected to the concept of approximate eigenstructure of LTV channel operators. In fact, on the L_2-level both are equivalent as we will show. The concept of "orthogonality" in turn can be related to notion of tight frames. The right balance between these two sides is still an open problem. However, several statements on achievable values of certain localization measures and fundamental limits on SINR can already be made as will be shown in the paper.