Mathematics Colloquia and Seminars

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I will show how recent methods from applied harmonic analysis can play a key role in modern wireless communications. I will first briefly describe Orthogonal Frequency Division Multiplexing (OFDM), which is one of the most promising transmission schemes for wireless communications. An important problem in OFDM is the design of transmission pulses that are robust against interference caused by time-varying channels. By exploiting the connection between Weyl-Heisenberg systems (a family of functions that consists of translations and modulations of some ``nice'' function), Banach algebras and OFDM I will construct a theoretical framework for the construction of orthogonal transmission functions that are optimally localized in the phase space. This localization property is crucial in order to mitigate the distortions caused by time-frequency dispersive channels. Some nice properties of Weyl-Heisenberg systems enable us to compute the aforementioned optimal pulses by a fast numerical algorithm, which is suitable for real-time applications. I will derive convergence rates for this algorithm and discuss some open problems. The proposed algorithm, which takes into account various practical constraints, has been recently used in the design of a modem for short-radio wave communications.