Rapid acquisition sequences

Abstract

A recurring communications systems problem involves the determination, with relatively high precision, of the phase of a periodic, noise-corrupted signal. Perhaps the most commonly proposed signals for such purposes are pseudorandom sequences. Indeed, such signals are known to be optimum, at least when the noise is white and Gaussian, in the sense of minimizing the time needed to find the correct phase with a specified reliability. Equipment limitations, however, often preclude the efficient use of these sequences. In particular, suppose the received signal is known to be of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y(t) = x(t - \nu \Delta T) + n(t)</tex> , with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> a signal periodic with period <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n(t)</tex> additive white Gaussian noise. The problem is to determine as reliably and rapidly as possible the "phase" <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\nu</tex> of the received signal. The optimum detector involves the formation of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N = \Delta T/T</tex> correlations <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I_{\nu} = \int y(t)x(t - \nu \Delta T) dt</tex> and the selection of the largest of these. Frequently, however, equipment constraints force these correlations to be made serially. In this event, the required search time can be as much as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> times as great as that needed in the absence of such constraints. Even the most sophisticated sequential search algorithms require search times directly proportional to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> , the number of contending phases. In this paper a class of binary sequences is demonstrated which require a search time on the order of only (log2 N)2, a substantial improvement when N is large. These sequences are shown to be essentially optimum under the conditions outlined.