Theory of False Lock in Costas Loops

Abstract

This paper is concerned with new results on data related false lock of Costas loops caused by arm filter distortion of the data. The false lock phenomena is demonstrated to occur for random as well as periodic data and the relative levels of lock are predicted. It is shown that, for the case of random data with Manchester symbols, strong lock points exist when the incoming carrier frequency and the loop VCO frequency differ by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nR_{s}/2 (n = 1, 2, 3, ...)</tex> Hz where R <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</inf> is the data symbol rate. In addition to the loop error signal the lock detection signal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(I^{2}Q^{2})</tex> is also analyzed for false lock effects. The theory also applies to the case of NRZ data symbols with the main result being strong false lock points also occur at VCO frequency offsets of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nR_{s}/2 (n = l, 2, 3, \ldot)</tex> but at different levels than the Manchester data. The results of these analyses lead to schemes capable of detecting false lock.