An Information Theoretic Analysis of Time-Limited Signaling

Abstract

Practical communication systems use finite duration transmissions and signaling. As the time-limited signals have infinite bandwidth, they lose some of their energy when passed through a band-limited channel. In 1948, Shannon derived the capacity of the band-limited Gaussian channel making use of band-limited and therefore infinite-duration pulses. In 1965, Wyner derived the channel capacity when using time-limited signals that are “approximately band-limited” in an asymptotic regime, as the time support of the signals tends to infinity. A major breakthrough in this theory was achieved by Polyanskiy who studied the channel coding rates in the finite block-length regime, however the derived results are only for discrete-time channels. A related theory is the rate distortion theory which was introduced by Shannon in 1948, where he derived the rate distortion function for band-limited white Gaussian sources. However his results are obtained by allowing compression at once an asymptotically infinite duration piece from the source, which is not possible for practical systems. Recently, Kostina studied the rate distortion theory in the finite block-length regime for discrete-time sources.

In this thesis I study the achievable rates when using signals with finite duration over a band-limited Gaussian channel, and the main goal is to quantify these rates when finite-duration codewords are employed over a band-limited channel. First I considered two system models where infinite duration codewords are transmitted over a band-limited Gaussian channel, however the pulses are restricted to be time-limited. Second, I considered transmitting finite duration codewords over a band-limited Gaussian channel and studied the achievable rates and the available degrees of freedom. Finally I extended this work to the “dual” problem of rate distortion theory and considered the coding of a time-limited piece of a band-limited Gaussian source.