Abstract
Autocorrelation sidelobes are a form of self-noise that reduce the effectiveness of phase coding in radar and communication systems. The merit factor is a well-known measure related to the autocorrelation sidelobe energy of a sequence. In this paper an equation is derived that allows the change in the merit factor of a binary sequence to be computed for all single-element changes with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> log <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> ) operations, where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> is the sequence length. The approach is then extended to multiple-element changes, allowing the merit factor change to be calculated with an additional <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) operations, where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> is the number of changed elements, irrespective of sequence length. The multiple-element calculations can be used to update the single-element calculations so that in iterative use only <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> ) operations are required per element change to keep the single-element calculations current. A steep descent algorithm (a variation on the steepest descent method) employing these two techniques was developed and applied to quarter-rotated, periodically extended Legendre sequences, producing optimized sequences with an apparent asymptotic merit factor of approximately 6.3758, modestly higher than the best known prior result of approximately 6.3421. Modified Jacobi sequences improve after steep descent to an approximate asymptotic merit factor of 6.4382. Three-prime and four-prime Jacobi sequences converge very slowly making a determination difficult but appear to have a higher post-steep-descent asymptotic merit factor than Legendre or modified Jacobi sequences.
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