Augmented Slepians: Bandlimited Functions that Counterbalance Energy in Selected Intervals

TL;DR

This paper introduces augmented Slepian functions, which optimize energy concentration in selected intervals while penalizing others, extending traditional time-frequency localization methods.

Contribution

It presents a new generalized optimization criterion for Slepian functions that balances energy concentration and penalization in different intervals, with mathematical foundations and applications.

Findings

New class of augmented Slepian functions introduced

Mathematical properties and examples demonstrated

Potential applications discussed

Abstract

Slepian functions provide a solution to the optimization problem of joint time-frequency localization. Here, this concept is extended by using a generalized optimization criterion that favors energy concentration in one interval while penalizing energy in another interval, leading to the "augmented" Slepian functions. Mathematical foundations together with examples are presented in order to illustrate the most interesting properties that these generalized Slepian functions show. Also the relevance of this novel energy-concentration criterion is discussed along with some of its applications.