A Group-Theoretic Approach to the WSSUS Pulse Design Problem

TL;DR

This paper introduces a novel mathematical framework for designing pulses in multicarrier transmission over WSSUS channels, linking the problem to quantum channel fidelity and eigenvalue problems, leading to explicit solutions.

Contribution

It establishes a new group-theoretic and operator-algebraic approach to pulse design, connecting it to quantum information theory and differential equations.

Findings

Optimal pulses are eigenstates of the harmonic oscillator Hamiltonian.

The approach provides exact solutions for various scattering environments.

A local approximation simplifies the design for underspread channels.

Abstract

We consider the pulse design problem in multicarrier transmission where the pulse shapes are adapted to the second order statistics of the WSSUS channel. Even though the problem has been addressed by many authors analytical insights are rather limited. First we show that the problem is equivalent to the pure state channel fidelity in quantum information theory. Next we present a new approach where the original optimization functional is related to an eigenvalue problem for a pseudo differential operator by utilizing unitary representations of the Weyl--Heisenberg group.A local approximation of the operator for underspread channels is derived which implicitly covers the concepts of pulse scaling and optimal phase space displacement. The problem is reformulated as a differential equation and the optimal pulses occur as eigenstates of the harmonic oscillator Hamiltonian. Furthermore this operator--algebraic approach is extended to provide exact solutions for different classes of scattering environments.