Power-Bandwidth Performance of Smoothed Phase Modulation Codes

Abstract

Constant envelope phase varying sinusoids of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sqrt{2E/T} \cos(\omega_{c}t + \phi(t))</tex> are studied, in which the phase function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\phi(t)</tex> follows some coded pattern in response to data. Power and bandwidth performance are studied for such patterns. The patterns depend on a phase shaping function, a modulation index ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</tex> ), and a sequence of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> -ary underlying changes in phase which are chosen at random. A cutoff rate-like parameter R <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> is computed, which guarantees existence of codes at all rates <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R &lt; R_{0}</tex> bits/ <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</tex> -interval whose error performance varies as exp <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[-N(R_{0} - R)]</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> is the code word length in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</tex> -intervals. Plots of R <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> are given as a function of interval energy <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</tex> , the shaping function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> . Extensive spectral calculations give the spectra of these phased sinusoids, and their performance is plotted in the power-bandwidth plane. The results give strong evidence that phase codes can approximate any power-bandwidth combination consistent with Shannon's Gaussian channel capacity, and that linear channels are not required for narrow-band transmission.